^{1,a)}, Alfons van Blaaderen

^{1}and Marjolein Dijkstra

^{1,b)}

### Abstract

We investigated the effect of size polydispersity on the crystal-fluid transition in hard-core repulsive Yukawa systems by means of Monte Carlo simulations for several state points in the Yukawa parameter space. Size polydispersity was introduced in the system only with respect to the hard particle cores; particles with different diameters had the same surface potential ψ_{0}, but the charge per particle was not varied with packing fraction or distance. We observed a shift to higher packing fraction of the crystal-fluid transition of bulk crystals with a fixed log-normal size distribution upon increasing the polydispersity, which was more pronounced for weakly charged particles (ψ_{0} ≈ 23 mV) compared to more highly charged particles (ψ_{0} ≈ 46 mV), and also more pronounced for larger Debye screening length. At high polydispersities (⩾0.13) parts of the more highly charged systems that were initially crystalline became amorphous. The amorphous parts had a higher polydispersity than the crystalline parts, indicating the presence of a terminal polydispersity beyond which the homogeneous crystal phase was no longer stable.

The authors acknowledge financial support from the Netherlands Organisation for Scientific Research (NWO) through a Vici Grant (MD) and Toptalent Grant (MNvdL).

I. INTRODUCTION

II. METHOD

A. Simulation details

B. Order parameters

III. RESULTS AND DISCUSSION

A. Weakly charged particles

B. More highly charged particles

C. Terminal polydispersity

D. Slow dynamics

IV. CONCLUSION

### Key Topics

- Colloidal systems
- 18.0
- Monte Carlo methods
- 11.0
- Surface charge
- 8.0
- Glass transitions
- 7.0
- Free energy
- 6.0

## Figures

Pair potentials for two reference particles with diameter for four different combinations of Yukawa potential parameters and .

Pair potentials for two reference particles with diameter for four different combinations of Yukawa potential parameters and .

Crystalline fraction versus packing fraction η of a system of particles which interact with a hard-core repulsive Yukawa pair potential with reference contact value and (a) , (b) 4.0, (c) 6.7, and (d) 10 after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc ( and 4.0) or fcc ( and 10) crystal structure, for different polydispersities *s* in the range 0.00–0.10 as labeled.

Crystalline fraction versus packing fraction η of a system of particles which interact with a hard-core repulsive Yukawa pair potential with reference contact value and (a) , (b) 4.0, (c) 6.7, and (d) 10 after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc ( and 4.0) or fcc ( and 10) crystal structure, for different polydispersities *s* in the range 0.00–0.10 as labeled.

Shift in packing fraction of the crystal-fluid transition Δη(*s*) (as defined in Eq. (17) ) with size polydispersity *s* of a system of hard-core repulsive Yukawa particles with reference contact value and , 4.0, 6.7, and 10 as labeled.

Shift in packing fraction of the crystal-fluid transition Δη(*s*) (as defined in Eq. (17) ) with size polydispersity *s* of a system of hard-core repulsive Yukawa particles with reference contact value and , 4.0, 6.7, and 10 as labeled.

Crystalline fraction versus packing fraction η of a hard-core repulsive Yukawa system with reference contact value and (a) , (b) 3.3, (c) 6.7, and (d) 10 after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc ( and 3.3) or fcc ( and 10) crystal structure, for different polydispersities *s* in the range 0.00–0.15 as labeled. The arrows indicate the edge of the plateau for *s* = 0.13 and may be used for comparison with Figs. 7 and 8 .

Crystalline fraction versus packing fraction η of a hard-core repulsive Yukawa system with reference contact value and (a) , (b) 3.3, (c) 6.7, and (d) 10 after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc ( and 3.3) or fcc ( and 10) crystal structure, for different polydispersities *s* in the range 0.00–0.15 as labeled. The arrows indicate the edge of the plateau for *s* = 0.13 and may be used for comparison with Figs. 7 and 8 .

Snapshot after 2 × 10^{4} Monte Carlo cycles of a hard-core repulsive Yukawa system with reference contact value , , η = 0.20, and *s* = 0.15. The color of a particle indicates (a) the average crystallinity in a series of six configurations between 1.5 × 10^{4} and 2 × 10^{4} Monte Carlo cycles, (b) the average local bond-orientational order parameter (Eq. (14) ) after 2 × 10^{4} MC cycles, (c) local bond-orientational order parameter *q* _{6}(*i*) (Eq. (12) ) after 2 × 10^{4} MC cycles, and (d) the square displacement from the particle's ideal lattice position for τ = 2 × 10^{4} Monte Carlo cycles.

Snapshot after 2 × 10^{4} Monte Carlo cycles of a hard-core repulsive Yukawa system with reference contact value , , η = 0.20, and *s* = 0.15. The color of a particle indicates (a) the average crystallinity in a series of six configurations between 1.5 × 10^{4} and 2 × 10^{4} Monte Carlo cycles, (b) the average local bond-orientational order parameter (Eq. (14) ) after 2 × 10^{4} MC cycles, (c) local bond-orientational order parameter *q* _{6}(*i*) (Eq. (12) ) after 2 × 10^{4} MC cycles, and (d) the square displacement from the particle's ideal lattice position for τ = 2 × 10^{4} Monte Carlo cycles.

Normalized probability distribution functions *p*(σ) of the particle diameter σ for , , *s* = 0.15, and η = 0.20 (55% most-disordered particles, 25% most-ordered particles). Filled gray curve: all particles, solid blue line: most-ordered particles (particles that are crystalline in at least five out of six configurations between 1.5 × 10^{4} and 2 × 10^{4} Monte Carlo cycles), dashed red line: most-disordered particles (particles that are crystalline in at most one out of six configurations between 1.5 × 10^{4} and 2 × 10^{4} Monte Carlo cycles).

Normalized probability distribution functions *p*(σ) of the particle diameter σ for , , *s* = 0.15, and η = 0.20 (55% most-disordered particles, 25% most-ordered particles). Filled gray curve: all particles, solid blue line: most-ordered particles (particles that are crystalline in at least five out of six configurations between 1.5 × 10^{4} and 2 × 10^{4} Monte Carlo cycles), dashed red line: most-disordered particles (particles that are crystalline in at most one out of six configurations between 1.5 × 10^{4} and 2 × 10^{4} Monte Carlo cycles).

Polydispersity *s* versus packing fraction η of the most-ordered and most-disordered parts of hard-core repulsive Yukawa systems with reference contact value and an overall polydispersity *s* = 0.13. For (a) , (b) 3.3, (c) 6.7, and (d) 10 after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc ( and 3.3) or fcc ( and 10) crystal structure. The arrows indicate the same state points as in Fig. 4 .

Polydispersity *s* versus packing fraction η of the most-ordered and most-disordered parts of hard-core repulsive Yukawa systems with reference contact value and an overall polydispersity *s* = 0.13. For (a) , (b) 3.3, (c) 6.7, and (d) 10 after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc ( and 3.3) or fcc ( and 10) crystal structure. The arrows indicate the same state points as in Fig. 4 .

Mean square displacement ⟨Δ*r*(τ)^{2}⟩ from the ideal lattice position (Eq. (16) ) versus packing fraction η for reference contact value and after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc crystal structure, for different polydispersities *s* in the range 0.00–0.15 as labeled. The arrow indicates the same state point as in Fig. 4(a) .

Mean square displacement ⟨Δ*r*(τ)^{2}⟩ from the ideal lattice position (Eq. (16) ) versus packing fraction η for reference contact value and after a simulation of 2 × 10^{4} Monte Carlo cycles, starting from a bcc crystal structure, for different polydispersities *s* in the range 0.00–0.15 as labeled. The arrow indicates the same state point as in Fig. 4(a) .

2D projections of 3D trajectories during 2 × 10^{5} Monte Carlo cycles of 25 particles from a configuration with , , *s* = 0.15, and η = 0.20. The initial configuration is a perfect bcc crystal; the initial positions of the particles are indicated by *solid* red symbols. Initially, the 25 particles are in two parallel {100} planes: 16 particles (solid red circles and solid red squares) in one plane occupy the corners of 3 × 3 unit cells, 9 particles (solid red triangles and solid red diamonds) in the second plane are in the centers of the unit cells. The end positions after 2 × 10^{5} MC cycles are indicated by *empty* red symbols (again circles, squares, triangles and diamonds; for each particle we used a symbol of the same shape to indicate the initial and end position). The trajectories are shown with four different colours (black, blue, green, and cyan, which correspond to the circles, squares, triangles, and diamonds, respectively).

2D projections of 3D trajectories during 2 × 10^{5} Monte Carlo cycles of 25 particles from a configuration with , , *s* = 0.15, and η = 0.20. The initial configuration is a perfect bcc crystal; the initial positions of the particles are indicated by *solid* red symbols. Initially, the 25 particles are in two parallel {100} planes: 16 particles (solid red circles and solid red squares) in one plane occupy the corners of 3 × 3 unit cells, 9 particles (solid red triangles and solid red diamonds) in the second plane are in the centers of the unit cells. The end positions after 2 × 10^{5} MC cycles are indicated by *empty* red symbols (again circles, squares, triangles and diamonds; for each particle we used a symbol of the same shape to indicate the initial and end position). The trajectories are shown with four different colours (black, blue, green, and cyan, which correspond to the circles, squares, triangles, and diamonds, respectively).

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