2T-MD simulation details

MM Mianzhen Mo SM Samuel Murphy ZC Zhijiang Chen PF Paul Fossati RL Renkai Li YW Yongqiang Wang XW Xijie Wang SG Siegfried Glenzer

This protocol is extracted from research article:

Visualization of ultrafast melting initiated from radiation-driven defects in solids

**
Sci Adv**,
May 24, 2019;
DOI:
10.1126/sciadv.aaw0392

Visualization of ultrafast melting initiated from radiation-driven defects in solids

Procedure

The two temperature models used here were developed by Duffy and Rutherford and implemented in the DL_POLY code (*29*, *40*). The crystal was represented using a classical MD supercell. This supercell was coupled to a continuum cell representing the electronic subsystem with energy able to transfer between the two to mimic electron phonon coupling. The electronic subsystem was represented using the following heat diffusion equation$${C}_{\mathrm{e}}({T}_{\mathrm{e}})\frac{\partial {T}_{\mathrm{e}}}{\partial t}=\nabla \cdot ({\mathrm{\kappa}}_{\mathrm{e}}\nabla {T}_{\mathrm{e}})-{G}_{\text{ei}}({T}_{\mathrm{e}}-{T}_{\mathrm{i}})+S(z,t)$$(4)where κ_{e} is the electronic heat conductivity and *S*(*z*, *t*) is a source term that represents the energy deposited by the laser. The laser pulse is assumed to be Gaussian in time, with an exponentially decreasing amplitude as a function of depth in the film, *z*, that is$$S(z,t)=\left(\frac{2F}{{l}_{\mathrm{p}}{t}_{\mathrm{p}}}\frac{\sqrt{\text{ln}2}}{\mathrm{\pi}}\right){e}^{-4\text{ln}{2(t-{t}_{0})}^{2}/{t}_{\mathrm{p}}^{2}}{e}^{-z/{l}_{\mathrm{p}}}$$(5)where *F* is the fluence from the experiments (46 mJ/cm^{2}), *l*_{p} is the optical penetration depth of the sample at the wavelength of the pulse length (12.5 nm), *t*_{p} is the duration of the pulse (130 fs), and *t*_{0} is the time zero corresponding to the maximum of the laser pulse on the sample surface. For thin films, κ_{e} can be considered infinite and so the first term on the right-hand side of Eq. 4 disappears.

The traditional equations of motion governing the evolution of atoms in the MD supercell are modified, such that$${m}_{\mathrm{i}}\frac{\partial {\mathbf{v}}_{i}}{\partial t}={\mathbf{F}}_{\mathrm{i}}-\mathrm{\gamma}{\mathbf{v}}_{\mathrm{i}}+{\mathbf{f}}_{\mathrm{L}}({T}_{\mathrm{e}})$$(6)where *v*_{i} is the velocity of an atom with mass, *m*_{i}. **F**_{i} is the classical force acting on the atom calculated using an extended Finnis-Sinlair model (*28*), γ represents a frictional drag force, and **f**_{L} is the stochastic force. The extended Finnis-Sinclair potential was selected as it accurately reproduces the melting temperature of W. The friction and stochastic terms allow energy to be added or removed from the ions to represent energy transfer with electrons; hence, γ is related to the electron-phonon coupling strength according to$$\mathrm{\gamma}=\left(\frac{V}{{N}^{\text{'}}}\right)\frac{m}{3{k}_{\mathrm{B}}}{G}_{\text{ei}}$$(7)where *V* is the volume of a coarse grain ionic voxel, *N*^{'} is the number of atoms in the voxel, and *G*_{ei} is the electron-phonon coupling strength. In our 2T-MD simulations, we set *G*_{ei} as a constant of 2.0 × 10^{17} W/m^{3} K^{–1}, which was inferred from the Debye-Waller factor measurements performed at below the damage threshold of W (see the Supplementary Materials for additional details).

To represent the experimental thin films simulation, supercells were created by taking 62 × 62 × 94 repetitions of the bcc W unit cell, resulting in a film that is 30 nm thick with a cross-sectional area of 383.85 nm^{2}. To represent irradiation-induced defects, a 5% concentration of vacancy defects was randomly introduced into the sample. The total number of atoms was kept approximately constant by adding layers to the top of the sample. Simulation supercells were equilibrated for 500 ps, with a 1-fs time step at 300 K under constant volume and temperature (NVT) conditions using the Nose-Hoover thermostat with a relaxation time of 0.01 ps. After equilibration, atoms in the bottom 5 Å were tethered using a harmonic potential to their initial lattice site to represent bonding to the substrate. The simulation supercells were then subjected to a 130-fs laser pulse delivering 46 mJ/cm^{2}.

The connectivity was calculated using the core dynamics package. Here, we define the connectivity as the number of lattice atoms within the limit of the second nearest neighbor shell, which is determined dynamically. Note that this excludes interstitial-type atoms, ensuring that the maximum connectivity is then 14 for the ideal bcc-structured materials. Neglect of the interstitial atoms in the determination of the connectivity makes identification of amorphous regions easier.

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