^{1,a)}, Toshihiro Kaneko

^{1}, Kenji Yasuoka

^{1}and Xiao Cheng Zeng

^{2}

### Abstract

A novel route to the exponential trapping-time distribution within a solidlike state in water clusters is described. We propose a simple homogeneous network (SHN) model to investigate dynamics on the potential energy networks of water clusters. In this model, it is shown that the trapping-time distribution in a solidlike state follows the exponential distribution, whereas the trapping-time distribution in local potential minima within the solidlike state is not exponential. To confirm the exponential trapping-time distribution in a solidlike state, we investigate water clusters, (H2O)6 and (H2O)12, by molecular dynamics simulations. These clusters change dynamically from solidlike to liquidlike state and vice versa. We find that the probability density functions of trapping times in a solidlike state are described by the exponential distribution whereas those of interevent times of large fluctuations in potential energy within the solidlike state follow the Weibull distributions. The results provide a clear evidence that transition dynamics between solidlike and liquidlike states in water clusters are well described by the SHN model, suggesting that the exponential trapping-time distribution within a solidlike state originates from the homogeneous connectivity in the potential energy network.

I. INTRODUCTION

II. SIMPLE HOMOGENEOUS MODEL OF POTENTIAL ENERGY NETWORK

III. SIMULATION RESULTS

IV. DISCUSSION

A. Generalization of the simple homogeneous networkmodel

B. Origin of the Weibull distribution in the trapping-time distribution

V. CONCLUSION

### Key Topics

- Solid liquid phase transitions
- 16.0
- Cluster dynamics
- 10.0
- Liquid solid interfaces
- 10.0
- Networks
- 10.0
- Relaxation times
- 8.0

## Figures

Schematic picture of a simple homogeneous model of a coarse-grained PEN. Green and red spheres are potential minima in solidlike and liquidlike states, respectively. All potential minima in a solidlike state are connected to a potential minimum in a liquidlike state. Some configurations of solidlike and liquidlike states for (H2O)12 are shown for reference.

Schematic picture of a simple homogeneous model of a coarse-grained PEN. Green and red spheres are potential minima in solidlike and liquidlike states, respectively. All potential minima in a solidlike state are connected to a potential minimum in a liquidlike state. Some configurations of solidlike and liquidlike states for (H2O)12 are shown for reference.

Time series of potential energy E(t) with its coarse-grained one E MA(t) for (a) (H2O)6 at T = 60 K and (b) (H2O)12 at T = 135 K. Dashed line represents a boundary between the solidlike and liquidlike state (E s = −31.2 and −37 kJ/mol for (a) and (b), respectively).

Time series of potential energy E(t) with its coarse-grained one E MA(t) for (a) (H2O)6 at T = 60 K and (b) (H2O)12 at T = 135 K. Dashed line represents a boundary between the solidlike and liquidlike state (E s = −31.2 and −37 kJ/mol for (a) and (b), respectively).

Configurations of water molecules for local potential minima in the solidlike state. (a) Two different configurations of local potential minima in the solidlike state in (H2O)6. (b) Two different configurations of local potential minima in the solidlike state in (H2O)12, while there are many different local potential minima in it. These configurations are obtained by minimizing the potential energy of a water cluster in the solidlike state using the steepest descent method.

Configurations of water molecules for local potential minima in the solidlike state. (a) Two different configurations of local potential minima in the solidlike state in (H2O)6. (b) Two different configurations of local potential minima in the solidlike state in (H2O)12, while there are many different local potential minima in it. These configurations are obtained by minimizing the potential energy of a water cluster in the solidlike state using the steepest descent method.

Probability of the number of trials. (a) (H2O)6 at T = 60 K. (b) (H2O)12 at T = 149 K. Triangular symbols are the results of the MD simulations. Histogram represents the probability of p(1 − p) k−1 with p = 0.55 and 0.22 for (a) and (b), respectively.

Probability of the number of trials. (a) (H2O)6 at T = 60 K. (b) (H2O)12 at T = 149 K. Triangular symbols are the results of the MD simulations. Histogram represents the probability of p(1 − p) k−1 with p = 0.55 and 0.22 for (a) and (b), respectively.

Weibull plot, i.e., (a) ln |ln [1 − F(τα)]| vs. ln τα and (b) ln |ln [1 − F(τβ)]| vs. ln τβ. Symbols are the results of the MD simulations. Lines are the fitting lines. The slopes of the fitting lines indicate the exponent γ of the Weibull distribution (9) . The Weibull exponents for the trapping-time distribution of τα are almost γ = 1, which implies the exponential distribution.

Weibull plot, i.e., (a) ln |ln [1 − F(τα)]| vs. ln τα and (b) ln |ln [1 − F(τβ)]| vs. ln τβ. Symbols are the results of the MD simulations. Lines are the fitting lines. The slopes of the fitting lines indicate the exponent γ of the Weibull distribution (9) . The Weibull exponents for the trapping-time distribution of τα are almost γ = 1, which implies the exponential distribution.

Schematic picture of a generalized SHN model. Potential minima within solidlike states (green spheres) are not directly connected to those within solidlike states.

Schematic picture of a generalized SHN model. Potential minima within solidlike states (green spheres) are not directly connected to those within solidlike states.

Probability density functions for τα and τβ in a semi-log scale for T = 135 K. Histograms are the results of the MD simulations. The value of E s is set to be −36.5 kJ/mol. The dashed line is a fitting curve of the Weibull distribution (9) obtained by the Weibull plot. The dotted line is a fitting curve of a superposition of the exponential distribution (11) for n = 2. The fitting parameters are p 1 = 0.4 and p 2 = 0.6. The two relaxation times are obtained by the mean trapping times for E MA < −37 kJ/mol and −37 < E MA < −36.5 kJ/mol (τ1 = 5 ns and τ2 = 0.5 ns).

Probability density functions for τα and τβ in a semi-log scale for T = 135 K. Histograms are the results of the MD simulations. The value of E s is set to be −36.5 kJ/mol. The dashed line is a fitting curve of the Weibull distribution (9) obtained by the Weibull plot. The dotted line is a fitting curve of a superposition of the exponential distribution (11) for n = 2. The fitting parameters are p 1 = 0.4 and p 2 = 0.6. The two relaxation times are obtained by the mean trapping times for E MA < −37 kJ/mol and −37 < E MA < −36.5 kJ/mol (τ1 = 5 ns and τ2 = 0.5 ns).

The mean trapping times ⟨τα⟩ and ⟨τβ⟩ vs. 1000/T in (H2O)12 water cluster. The values of E s are set to be −37, −37, −36.9, −36.5, and −36.5 kJ/mol for T = 135, 138, 142, 149, and 155 K, respectively. Circles and triangles are the results for ⟨τα⟩ and ⟨τβ⟩, respectively. Solid lines are the linear fitting lines: ⟨τα⟩ = τ0 exp (1000A/T) and ⟨τβ⟩ = τ1 exp (1000B/T) (τ0 = 5.6 × 10−6, A = 1.86, τ1 = 2.9 × 10−6, and B = 1.74). Dashed line represents τ0 exp (1000B/T)/p.

The mean trapping times ⟨τα⟩ and ⟨τβ⟩ vs. 1000/T in (H2O)12 water cluster. The values of E s are set to be −37, −37, −36.9, −36.5, and −36.5 kJ/mol for T = 135, 138, 142, 149, and 155 K, respectively. Circles and triangles are the results for ⟨τα⟩ and ⟨τβ⟩, respectively. Solid lines are the linear fitting lines: ⟨τα⟩ = τ0 exp (1000A/T) and ⟨τβ⟩ = τ1 exp (1000B/T) (τ0 = 5.6 × 10−6, A = 1.86, τ1 = 2.9 × 10−6, and B = 1.74). Dashed line represents τ0 exp (1000B/T)/p.

## Tables

The probability p, the Weibull exponent γ, the mean trapping time ⟨τβ⟩, ⟨τβ⟩/p, the mean trapping time ⟨τα⟩, and the relaxation time a for (H2O)12.

The probability p, the Weibull exponent γ, the mean trapping time ⟨τβ⟩, ⟨τβ⟩/p, the mean trapping time ⟨τα⟩, and the relaxation time a for (H2O)12.

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