*n*-eicosane melts

^{1}and Gregory C. Rutledge

^{2,a)}

### Abstract

Homogeneous nucleation of *n*-eicosane crystals from the supercooled melt was studied by molecular simulation using a realistic, united-atom model for *n*-alkanes. Using molecular dynamics simulation, we observed nucleation events directly at constant pressure and temperature, corresponding to about 19% supercooling. Under these conditions, the induction time is found to be 80.6 ± 8.8 ns for a system of volume (1.882 ± 0.006) × 10^{−19} cm^{3}, corresponding to a nucleation rate of (6.59 ± 0.72) × 10^{25} cm^{−3} s^{−1}. The nucleationfree energy was calculated separately for three temperatures, ranging from 10% to 19% supercooling, by a Monte Carlo method with umbrella sampling. Values for the nucleationfree energy range from 7.3 to 13.2 (in units of *k* _{B} *T*). Detailed examination of the simulations reveals the critical nucleus to be a bundle of stretched segments about eight methylene groups long, organized into a cylindrical shape. The remaining methylene groups of the chains that participate in the nucleus form a disordered interfacial layer. By fitting the free energy curve to the cylindrical nucleus model, the solid-liquid interfacial free energies are calculated to be about 10 mJ/m^{2} for the side surface and 4 mJ/m^{2} for the end surface, both of which are relatively insensitive to temperature.

Financial support from Exxon-Mobil is gratefully acknowledged. The authors are also grateful to Dr. David Lohse and Dr. Rebecca Locker for helpful discussions.

I. INTRODUCTION

II. THEORY

III. METHOD

A. System and force field

B. Reaction coordinate

C. Molecular dynamics simulation

D. Monte Carlo umbrella sampling

IV. RESULTS AND DISCUSSION

A. Melting point and heat of fusion

B. MD simulation of the nucleation process

C. MC sampling of the nucleationfree energy barrier

V. CONCLUSION

### Key Topics

- Free energy
- 61.0
- Nucleation
- 57.0
- Molecular dynamics
- 18.0
- Crystallization
- 14.0
- Monte Carlo methods
- 13.0

## Figures

Probability distributions of local order parameter *p* _{2}(*i*), calculated using 5 different values for the cutoff radius *r* _{p2}.

Probability distributions of local order parameter *p* _{2}(*i*), calculated using 5 different values for the cutoff radius *r* _{p2}.

Evolution of characteristic variables for a system of 336 *n*-eicosane chains during a typical MD simulation, after quenching from 400 to 250 K at *t* = 0. (a) Potential energy and volume per chain. (b) Size of the largest nucleus, *n* _{max}, and the global orientation order parameter *P* _{2}.

Evolution of characteristic variables for a system of 336 *n*-eicosane chains during a typical MD simulation, after quenching from 400 to 250 K at *t* = 0. (a) Potential energy and volume per chain. (b) Size of the largest nucleus, *n* _{max}, and the global orientation order parameter *P* _{2}.

The first-passage time of the largest nucleus size, *n* _{max}, from one MD simulation of a system of 336 *n*-eicosane chains quenched from 400 to 250 K at *t* = 0. The open circles are simulation data, and the solid line is the formula of Eq. (10), with *n**, *τ**, and *Z* parameterized to fit the simulation data.

The first-passage time of the largest nucleus size, *n* _{max}, from one MD simulation of a system of 336 *n*-eicosane chains quenched from 400 to 250 K at *t* = 0. The open circles are simulation data, and the solid line is the formula of Eq. (10), with *n**, *τ**, and *Z* parameterized to fit the simulation data.

The probability distribution of *τ**, *n**, and *Z*, parameters obtained by fitting the first-passage time Eq. (10) to each of the 56 independent MD simulations: (a) *τ** and the exponential fitting; (b) *n** and the Gaussian fitting; (c) *Z* and the Gaussian fitting. A waiting time *τ* ^{ ′ } = 10 ns was subtracted from *τ** for the fitting and two systems that nucleate within *τ* ^{ ′ } were excluded from all fittings.

The probability distribution of *τ**, *n**, and *Z*, parameters obtained by fitting the first-passage time Eq. (10) to each of the 56 independent MD simulations: (a) *τ** and the exponential fitting; (b) *n** and the Gaussian fitting; (c) *Z* and the Gaussian fitting. A waiting time *τ* ^{ ′ } = 10 ns was subtracted from *τ** for the fitting and two systems that nucleate within *τ* ^{ ′ } were excluded from all fittings.

The first-passage time of the largest nucleus size, *n* _{max}, from one MD simulation of a system of 336 *n*-eicosane chains quenched from 400 to 250 K at *t* = 0, using three different parameter sets to define the crystal nucleus.

The first-passage time of the largest nucleus size, *n* _{max}, from one MD simulation of a system of 336 *n*-eicosane chains quenched from 400 to 250 K at *t* = 0, using three different parameter sets to define the crystal nucleus.

The free energy of formation for a crystal nucleus in a melt of 336 *n*-eicosane chains at 250, 265, and 280 K, respectively.

The free energy of formation for a crystal nucleus in a melt of 336 *n*-eicosane chains at 250, 265, and 280 K, respectively.

A snapshot of a crystal nucleus that consists of 130 united atoms from a MC simulation of a melt of 336 *n*-eicosane chains at 250 K: (a) end view, showing only those united atoms that are part of the nucleus; (b) side view, showing only those united atoms that are part of the nucleus; (c) side view, showing all united atoms belonging to chains that participate in the nucleus. United atoms in the crystal phase are black, while those in the melt phase are red.

A snapshot of a crystal nucleus that consists of 130 united atoms from a MC simulation of a melt of 336 *n*-eicosane chains at 250 K: (a) end view, showing only those united atoms that are part of the nucleus; (b) side view, showing only those united atoms that are part of the nucleus; (c) side view, showing all united atoms belonging to chains that participate in the nucleus. United atoms in the crystal phase are black, while those in the melt phase are red.

The free energy of formation, the thickness and the radius as a function of nucleus size, *n*, for a crystal nucleus observed in a melt of 336 *n*-eicosane chains by Monte Carlo simulation at three different temperatures, (a) 250 K, (b) 265 K, and (c) 280 K. (+) = nucleus free energy from simulation; (Δ) = thickness *l* of nucleus; (◻) = radius *r* of nucleus. Two fitting schemes to the cylinder model Eq. (1) were used: dashed curve = two-parameter (*σ* _{ e }, *σ* _{ s }) fitting and solid curve = three-parameter (*σ* _{ e }, *σ* _{ s }, Δ*G* _{ v }) fitting.

The free energy of formation, the thickness and the radius as a function of nucleus size, *n*, for a crystal nucleus observed in a melt of 336 *n*-eicosane chains by Monte Carlo simulation at three different temperatures, (a) 250 K, (b) 265 K, and (c) 280 K. (+) = nucleus free energy from simulation; (Δ) = thickness *l* of nucleus; (◻) = radius *r* of nucleus. Two fitting schemes to the cylinder model Eq. (1) were used: dashed curve = two-parameter (*σ* _{ e }, *σ* _{ s }) fitting and solid curve = three-parameter (*σ* _{ e }, *σ* _{ s }, Δ*G* _{ v }) fitting.

## Tables

Potential energy per chain and average density of *n*-eicosane systems at pressure *P* = 1 atm and several different temperatures.

Potential energy per chain and average density of *n*-eicosane systems at pressure *P* = 1 atm and several different temperatures.

The crystal nucleation free energy and critical nucleus size at several temperatures for an *n*-eicosane melt containing 336 chains.

The crystal nucleation free energy and critical nucleus size at several temperatures for an *n*-eicosane melt containing 336 chains.

Crystal-liquid interfacial free energy of *n*-eicosane molecules.

Crystal-liquid interfacial free energy of *n*-eicosane molecules.

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