^{1,2,a)}, Marco Bernabei

^{3}and Angel J. Moreno

^{3,4}

### Abstract

We present molecular dynamics (MD) simulations results for dense fluids of ultrasoft, fully penetrable particles. These are a binary mixture and a polydisperse system of particles interacting via the generalized exponential model, which is known to yield cluster crystal phases for the corresponding monodisperse systems. Because of the dispersity in the particle size, the systems investigated in this work do not crystallize and form disordered cluster phases. The clusteringtransition appears as a smooth crossover to a regime in which particles are mostly located in clusters, isolated particles being infrequent. The analysis of the internal cluster structure reveals microsegregation of the big and small particles, with a strong homo-coordination in the binary mixture. Upon further lowering the temperature below the clusteringtransition, the motion of the clusters’ centers-of-mass slows down dramatically, giving way to a cluster glass transition. In the cluster glass, the diffusivities remain finite and display an activated temperature dependence, indicating that relaxation in the cluster glass occurs via particle hopping in a nearly arrested matrix of clusters. Finally we discuss the influence of the microscopic dynamics on the transport properties by comparing the MD results with Monte Carlo simulations.

We acknowledge financial support from the projects MAT2007-63681 (Spain) and IT-436-07 (GV, Spain). We acknowledge Centro de Supercomputación de Cataluña (Barcelona, Spain) and the HPC@LR Center of Competence in High-Performance Computing of Languedoc-Roussillon (France) for allocation of CPU time. We thank A. Ikeda, G. Kahl, and C.N. Likos for valuable discussions.

I. INTRODUCTION

II. METHODS

III. RESULTS AND DISCUSSION

A. Structure and thermodynamics

B. Dynamics

C. Dependence on thermal history and microscopic dynamics

IV. CONCLUSIONS

### Key Topics

- Cluster dynamics
- 43.0
- Cluster analysis
- 36.0
- Glass transitions
- 19.0
- Glasses
- 18.0
- Monte Carlo methods
- 15.0

## Figures

Radial distribution functions of the binary mixture along the isochore ρ = 4.0 for selected temperatures (see legend). (a) *g* _{11}(*r*), (b) *g* _{12}(*r*), and (c) *g* _{22}(*r*).

Radial distribution functions of the binary mixture along the isochore ρ = 4.0 for selected temperatures (see legend). (a) *g* _{11}(*r*), (b) *g* _{12}(*r*), and (c) *g* _{22}(*r*).

Radial distribution functions of the polydisperse model along the isochore ρ = 5.0 for selected temperatures (see legend). (a) *g* _{11}(*r*), (b) *g* _{13}(*r*), and (c) *g* _{33}(*r*). Correlation functions involving particles of intermediate size (α = 2) are not shown.

Radial distribution functions of the polydisperse model along the isochore ρ = 5.0 for selected temperatures (see legend). (a) *g* _{11}(*r*), (b) *g* _{13}(*r*), and (c) *g* _{33}(*r*). Correlation functions involving particles of intermediate size (α = 2) are not shown.

Distribution of cluster population numbers *n* _{cl} for various temperatures (see legend) in (a) the binary mixture and (b) the polydisperse model.

Distribution of cluster population numbers *n* _{cl} for various temperatures (see legend) in (a) the binary mixture and (b) the polydisperse model.

Distribution of chemical compositions of the clusters (a) for the binary mixture in the plane and (b) for the polydisperse model in the plane. The radii of the circles are proportional to the probability of finding clusters with a given chemical composition. The state points are (a) *T* = 0.35 for the binary mixture and (b) *T* = 0.45 for the polydisperse model.

Distribution of chemical compositions of the clusters (a) for the binary mixture in the plane and (b) for the polydisperse model in the plane. The radii of the circles are proportional to the probability of finding clusters with a given chemical composition. The state points are (a) *T* = 0.35 for the binary mixture and (b) *T* = 0.45 for the polydisperse model.

Thermodynamic and cluster properties of the binary mixture (black and white symbols) as a function of *T* : (a) total potential energy *U*(*T*), (b) specific heat *C* _{ V }(*T*), and (c) fraction *P*(*n* _{cl}) of selected cluster populations *n* _{cl}. The vertical dotted line in (b) marks the position of the peak of the specific heat. In (a) data for the monodisperse GEM-4 model at a density ρ = 4.097 are included for comparison (red symbols).

Thermodynamic and cluster properties of the binary mixture (black and white symbols) as a function of *T* : (a) total potential energy *U*(*T*), (b) specific heat *C* _{ V }(*T*), and (c) fraction *P*(*n* _{cl}) of selected cluster populations *n* _{cl}. The vertical dotted line in (b) marks the position of the peak of the specific heat. In (a) data for the monodisperse GEM-4 model at a density ρ = 4.097 are included for comparison (red symbols).

Black and white symbols: as Fig. 5 for the polydisperse model. Red symbols in (a) are data for the monodisperse GEM-8 model at a density ρ = 5.0.

Black and white symbols: as Fig. 5 for the polydisperse model. Red symbols in (a) are data for the monodisperse GEM-8 model at a density ρ = 5.0.

Snapshots of the particles’ positions, above and below the clustering temperature *T* * of the binary mixture: (a) *T* = 0.75 and (b) *T* = 0.35. Particles of species 1 and 2 are depicted as small white spheres and big red spheres, respectively. For clarity, only particles contained within a vertical slab of thickness 4 are shown.

Snapshots of the particles’ positions, above and below the clustering temperature *T* * of the binary mixture: (a) *T* = 0.75 and (b) *T* = 0.35. Particles of species 1 and 2 are depicted as small white spheres and big red spheres, respectively. For clarity, only particles contained within a vertical slab of thickness 4 are shown.

Same as Fig. 7 but for the polydisperse model: (a) *T* = 2.44 and (b) *T* = 0.64. Particles of species 1, 2, and 3 are depicted as small white spheres, intermediate blue spheres, and big red spheres, respectively. For clarity, only particles contained within a vertical slab of thickness 4 are shown.

Same as Fig. 7 but for the polydisperse model: (a) *T* = 2.44 and (b) *T* = 0.64. Particles of species 1, 2, and 3 are depicted as small white spheres, intermediate blue spheres, and big red spheres, respectively. For clarity, only particles contained within a vertical slab of thickness 4 are shown.

Radial distribution function of the clusters’ centers of mass in the binary mixture at some selected temperatures (see legend): (a) , (b) , and (c) .

Radial distribution function of the clusters’ centers of mass in the binary mixture at some selected temperatures (see legend): (a) , (b) , and (c) .

As Fig. 9 for the polydisperse system: (a) , (b) , and (c) .

As Fig. 9 for the polydisperse system: (a) , (b) , and (c) .

Static structure factors of the clusters’ centers of mass in the binary mixture at some selected temperatures (see legend): (a) , (b) , and (c) .

Static structure factors of the clusters’ centers of mass in the binary mixture at some selected temperatures (see legend): (a) , (b) , and (c) .

Static structure factors of the clusters’ centers of mass in the polydisperse system at some selected temperatures (see legend): (a) , (b) , and (c) .

Static structure factors of the clusters’ centers of mass in the polydisperse system at some selected temperatures (see legend): (a) , (b) , and (c) .

Arrhenius plot of the diffusion coefficients (symbols). (a) *D* _{1} and *D* _{2} for the binary mixture. (b) Total diffusion coefficient *D* for the polydisperse model. The vertical dotted lines indicate the location of the clustering transition (*T* *) and cluster glass transition (*T* _{ g }). Dashed lines are fits to an VFT law (for *T* > *T* _{g}) and to an Arrhenius law (for *T* < *T* _{g}, activation energies are indicated).

Arrhenius plot of the diffusion coefficients (symbols). (a) *D* _{1} and *D* _{2} for the binary mixture. (b) Total diffusion coefficient *D* for the polydisperse model. The vertical dotted lines indicate the location of the clustering transition (*T* *) and cluster glass transition (*T* _{ g }). Dashed lines are fits to an VFT law (for *T* > *T* _{g}) and to an Arrhenius law (for *T* < *T* _{g}, activation energies are indicated).

Diffusivities of the GEM-8 model vs. ρ/*T*. Filled symbols are data for the monodisperse system.^{22} Empty symbols are data for the polydisperse system of this work. The thick dashed lines indicate Arrhenius-like behavior in the cluster crystal and cluster glass of the monodisperse and polydisperse systems, respectively.

Diffusivities of the GEM-8 model vs. ρ/*T*. Filled symbols are data for the monodisperse system.^{22} Empty symbols are data for the polydisperse system of this work. The thick dashed lines indicate Arrhenius-like behavior in the cluster crystal and cluster glass of the monodisperse and polydisperse systems, respectively.

Intermediate scattering functions for the binary mixture evaluated at various temperatures (see legend): (a) , (b) , (c) *F* _{11}(*k* = 6, *t*) , and (d) *F* _{22}(*k* = 5, *t*). The clustering and cluster glass transitions are highlighted with bold dashed and bold continuous lines, respectively.

Intermediate scattering functions for the binary mixture evaluated at various temperatures (see legend): (a) , (b) , (c) *F* _{11}(*k* = 6, *t*) , and (d) *F* _{22}(*k* = 5, *t*). The clustering and cluster glass transitions are highlighted with bold dashed and bold continuous lines, respectively.

As Fig. 15 for the polydisperse model: (a) *F* _{ s }(*k* = 5.8, *t*) and (b) *F*(*k* = 5.8, *t*).

As Fig. 15 for the polydisperse model: (a) *F* _{ s }(*k* = 5.8, *t*) and (b) *F*(*k* = 5.8, *t*).

Structural relaxation times (a) τ_{1} and τ_{2} for the binary mixture and (b) τ for the polydisperse model as a function of 1/*T*. The dashed lines represents VFT fits.

Structural relaxation times (a) τ_{1} and τ_{2} for the binary mixture and (b) τ for the polydisperse model as a function of 1/*T*. The dashed lines represents VFT fits.

Typical displacements of selected small particles in the binary mixture at *T* = 0.3843 (panels (a) and (b)) and *T* = 0.4983 (panels (c) and (d)). As indicated in the legend, the color of the line indicates the population of the cluster to which the particle belongs at time *t*. Portions of the trajectory during which the particle is isolated are highlighted with a thick red line.

Typical displacements of selected small particles in the binary mixture at *T* = 0.3843 (panels (a) and (b)) and *T* = 0.4983 (panels (c) and (d)). As indicated in the legend, the color of the line indicates the population of the cluster to which the particle belongs at time *t*. Portions of the trajectory during which the particle is isolated are highlighted with a thick red line.

(Symbols) Coherent scattering functions evaluated for the clusters’ centers of mass in the binary mixture. The respective data for the coherent scattering functions *F* _{αβ}(*k*, *t*), calculated on a particle basis, are included as solid lines. Panels (a) and (b) show data at different temperatures (see legend) and fixed wave vector. The selected wave vectors are *k* * = 6 for 1-1 correlations (a) and *k* * = 5 for 2-2 correlations (b). Panels (c) and (d) show data at fixed temperature *T* = *T* _{ g } = 0.48 and different wave vectors (see legend) for 1-1 correlations (c) and 2-2 correlations (d).

(Symbols) Coherent scattering functions evaluated for the clusters’ centers of mass in the binary mixture. The respective data for the coherent scattering functions *F* _{αβ}(*k*, *t*), calculated on a particle basis, are included as solid lines. Panels (a) and (b) show data at different temperatures (see legend) and fixed wave vector. The selected wave vectors are *k* * = 6 for 1-1 correlations (a) and *k* * = 5 for 2-2 correlations (b). Panels (c) and (d) show data at fixed temperature *T* = *T* _{ g } = 0.48 and different wave vectors (see legend) for 1-1 correlations (c) and 2-2 correlations (d).

*T*-dependence of the diffusion coefficients *D* _{1} and *D* _{2} in the binary mixture for different equilibration criteria. The different data sets correspond to different values of the total RMSD (open symbols) and partial RMSD (full symbols) targeted during the equilibration run (see legend). Inset: Equilibration time *t* _{ eq } as a function of *T* for the different equilibration criteria.

*T*-dependence of the diffusion coefficients *D* _{1} and *D* _{2} in the binary mixture for different equilibration criteria. The different data sets correspond to different values of the total RMSD (open symbols) and partial RMSD (full symbols) targeted during the equilibration run (see legend). Inset: Equilibration time *t* _{ eq } as a function of *T* for the different equilibration criteria.

Dependence of the coherent intermediate scattering functions (a) *F* _{1}(*k* = 6, *t*) and (b) *F* _{2}(*k* = 6, *t*) on the equilibration criterion in the binary mixture at *T* = *T* _{ g } = 0.48. Colors are the same as in Fig. 20.

Dependence of the coherent intermediate scattering functions (a) *F* _{1}(*k* = 6, *t*) and (b) *F* _{2}(*k* = 6, *t*) on the equilibration criterion in the binary mixture at *T* = *T* _{ g } = 0.48. Colors are the same as in Fig. 20.

Dependence of the radial distribution functions (a) and (b) on the equilibration criterion in the binary mixture at *T* = 0.3843 < *T* _{ g }. Colors are the same as in Fig. 20.

Dependence of the radial distribution functions (a) and (b) on the equilibration criterion in the binary mixture at *T* = 0.3843 < *T* _{ g }. Colors are the same as in Fig. 20.

Comparison of the temperature dependence of the diffusion coefficients obtained from Newtonian (circles) and Monte Carlo (squares) dynamics in the polydisperse system. The diffusion coefficients of the MC data set have been multiplied by a factor 75. The vertical dotted lines indicate the clustering transition *T* * and glass transition *T* _{g}. The dashed lines indicate power-law and linear behavior, and are included for comparison with the simulation data.

Comparison of the temperature dependence of the diffusion coefficients obtained from Newtonian (circles) and Monte Carlo (squares) dynamics in the polydisperse system. The diffusion coefficients of the MC data set have been multiplied by a factor 75. The vertical dotted lines indicate the clustering transition *T* * and glass transition *T* _{g}. The dashed lines indicate power-law and linear behavior, and are included for comparison with the simulation data.

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