Two mechanisms of nanoparticle generation in picosecond laser ablation in liquids: the origin of the bimodal size distribution† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c7nr08614h

Novel mechanisms of nanoparticle generation in laser ablation in liquids are revealed in atomistic simulations and verified in experiments.


Details of computational setup and Supplementary Figure S1
The simulations reported in this paper are performed with a hybrid computational model combining a coarse-grained representation of liquid, a fully atomistic description of laser interaction with metal targets, and acoustic impedance matching boundary conditions, designed to mimic the non-reflecting propagation of the laser-induced pressure waves through the boundaries of the computational domain. A schematic representation of the computational system is shown in Figure S1. The computational setup is designed and parametrized for a bulk Ag target covered by water and irradiated by a picosecond laser pulse. A brief description of the main components of the computational model as well as details of the computational setup are provided below.

TTM-MD model for laser interactions with metals:
Although the MD method is capable of providing detailed information on the microscopic mechanisms of laser ablation, several modifications have to be made in order to apply the classical MD for simulations of laser interactions with metals. In particular, a realistic description of the laser coupling to the target material, the kinetics of thermalization of the absorbed laser energy, and the fast electron heat conduction should be incorporated into the MD technique. These processes can be accounted for by incorporating the MD method into the general framework of the two-temperature model (TTM) 1, 2 commonly used in the simulations of short pulse lase interactions with metals. The idea of the combined TTM-MD model [3][4][5] is schematically illustrated in Figure   S1 and is briefly explained below.
In the original TTM, the time evolution of the lattice and electron temperatures, Tl and Te, is described by two coupled differential equations (Eqs. (1) and (2) in Figure S1) that account for the electron heat conduction in the metal target and the energy exchange between the electrons and atomic vibrations. In the combined TTM-MD method, MD substitutes the TTM equation for the lattice temperature in the surface region of the target, where laser-induced structural and phase transformations take place. The diffusion equation for the electron temperature, Te, is solved by a finite difference method simultaneously with MD integration of the equations of motion of atoms.
The cells in the finite difference discretization are related to the corresponding volumes of the MD system, and the local lattice temperature, , is defined for each cell from the average kinetic energy of thermal motion of atoms. The electron temperature enters a coupling term, , that is added to the MD equations of motion to account for the energy exchange between the electrons and atomic vibrations. In this coupling term, is a coefficient that depends on the instantaneous where the dynamic material decomposition may result in lateral density and temperature variations.
In the continuum part of the model, beyond the surface region represented by the MD method, the electron heat conduction and the energy exchange between the electrons and the lattice are described by the conventional TTM equations. A dynamic pressure-transmitting boundary condition 8,9 is applied at the bottom of the MD part of the system (marked as  in Figure S1) to ensure non-reflecting propagation of the laser-induced stress wave from the MD region of the computational system to the bulk of the target. The energy carried away by the stress wave is monitored, allowing for control over the total energy conservation in the combined model. 10 The electron temperature dependences of the thermophysical material properties included in the TTM equation for the electron temperature (electron-phonon coupling factor G and electron heat capacity Ce, see Figure S1) are taken in the forms that account for the thermal excitation from the electron states below the Fermi level. 11 The electron thermal conductivity is described by the Drude model relationship, , where is the electron heat capacity, v 2 is the mean square velocity of the electrons contributing to the electron heat conductivity, approximated in this work as the Fermi velocity squared, vF 2 , and is the total electron scattering time defined by the electron-electron and electron-phonon scattering rates, . The value of the coefficient A, 3.5710 6 s -1 K -2 , is estimated 12 within the free electron model, following the approach suggested in Ref. 13. The value of the coefficient B, 1.1210 11 s -1 K -1 is obtained from the experimental value of the thermal conductivity of solid Ag at the melting temperature, 363 Wm -1 K -1 . 14

Material properties predicted by EAM Ag potential:
The interatomic interactions in the MD part of the model are described by the embedded atom method (EAM) potential with the functional form and parameterization developed in Ref. 15. A cut-off function suggested in Ref. 16 is added to the potential to smoothly bring the interaction energies and forces to zero at interatomic distance of 5.5 Å. Although the potential is fitted to lowtemperature values of the equilibrium lattice constant, sublimation energy, elastic constants, and vacancy formation energy, it also provides a good description of high-temperature thermodynamic properties of Ag 17 relevant to the simulation of laser-induced processes. In particular, the equilibrium melting temperature, Tm, determined in liquid-crystal coexistence simulations, is 1139 12 about 8% below the experimental values of 1235 K. 18 The threshold temperature for the onset of the explosive phase separation into liquid and vapor, T*, determined in simulations of slow heating of a metastable liquid, is found to be ~3450 K at zero pressure and ~4850 K at 0.5 GPa. 19 The onset of the phase explosion can be expected at 10% below the critical temperature [20][21][22] and the values of T* calculated for the EAM Ag material are not in conflict with the range of experimental values of the critical temperature of Ag spanning from 4300 K to 7500 K. 23 Coarse-grained representation of liquid environment: The direct application of the conventional all-atom MD representation of liquids in large-scale simulations of laser processing or ablation is not feasible due to the high computational cost. Thus, a coarse-grained representation of the liquid environment, 24, 25 where each particle represents several molecules, is adapted in this work. The coarse-graining reduces the number of degrees of freedom that have to be treated in the MD simulations and significantly increases the time and length scales accessible for the simulations. At the same time, however, the smaller number of the dynamic degrees of freedom results in a severe underestimation of the heat capacity of the liquid.
To resolve this problem, the degrees of freedom that are missing in the coarse-grained model are accounted for through a heat bath approach that associates an internal energy variable with each coarse-grained particle. [26][27][28] The energy exchange between the internal (implicit) and dynamic (explicit) degrees of freedom are controlled by the dynamic coupling between the translational degrees of freedom and the vibrational (breathing) mode associated with each coarse-grained particle (the particles are allowed to change their radii, or to "breath" 25,28 ). The energy exchange is implemented through a damping or viscosity force applied to the breathing mode, which connects it to the energy bath with capacity chosen to reproduce the real heat capacity of the group of atoms represented by each coarse-grained particle. Effectively, the breathing mode serves as a "gate" for accessing the energy stored in the molecular heat bath.
The first implementation of the coarse-grained model with heat bath approach was recently developed for water and applied to simulations of laser interactions with water-lysozyme system. 28 Each coarse-grained particle in the model has a mass of 50 Da and represents about three real water molecules. The potential describing the inter-particle interactions is provided in Ref. 28   The electron temperature dependences of the electron-phonon coupling factor and electron heat capacity are taken in the forms that account for the thermal excitation from the electron states below the Fermi level. 11 The electron thermal conductivity of Au is approximated as suggested in Ref. 38. The lattice heat capacity of Au is taken to be 25.41 Jmol -1 K -1 , density is 19300 kg/m 3 , melting temperature is 1337 K, and the heat of fusion is 12.7 kJ/mol. 18 The reflectivity and the absorption depth are calculated based on complex permittivity coefficient according to:   Figure S2a for a broad range of electron temperatures and the lattice temperature fixed at 300 K.
The evolution of the surface temperature obtained in a TTM simulation is shown in Figure S2b.
At an incident laser fluence 3400 mJ/cm 2 , the absorbed fluence predicted in the simulation is 1940 mJ/cm 2 . Since reflectivity of Au rapidly drops with increase of the electron temperature, the absorbed laser fluence and, therefore, the increase of the electron and lattice surface temperatures is much larger than that predicted in a simulation performed with constant optical properties of Au, shown as dashed lines in Figure S2b. In the latter case, the reflectivity of water-gold interface is assumed to be 0.972, the optical absorption depth is 12 nm, 42 and the temperature dependent ballistic range, similar to the simulation discussed above, is used. Due to the constant high value of reflectivity, the absorbed laser fluence is found to be 93.8 mJ/cm 2 . On the other hand, if the ballistic range is fixed at 100 nm, 43

Formation of small clusters via vapor condensation and growth, Supplementary Figure S4
The analysis of the evolution of sizes of the metal clusters and nanoparticles in the Ag-water mixing region is performed with a cluster identification algorithm applied to atomic configurations generated in the simulation between 100 ps and 5500 ps after the laser pulse, with a 100 ps interval.
The three large nanoparticles separated from the liquid jet are not considered in this analysis. The evolution of the cumulative number of Ag atoms present above the liquid layer as individual atoms (vapor) and small atomic clusters with the diameter below 1 nm (less than 30 atoms) as well as the larger clusters that we denote as nanoparticles is shown in Figure S4a. While the total number of Ag atoms in the mixing region steadily increases due to the continuous evaporation from the hot molten metal layer, the number of atoms in the Ag vapor and atomic clusters stays at an approximately the same level starting from 3 ns, and the increase in the total number of Ag atoms in the mixing region is largely sustained by the growing populations of nanometer-scale particles.
Overall, the nanoparticle size distribution broadens and shifts to the larger sizes as time progresses, as shown in Figure S4c,d. As can be seen from the snapshots of the metal-water mixing region shown in Figure 2, the largest nanoparticles formed through the nucleation and growth are mostly found in the middle part of the mixing region, where the sufficiently low temperature of the water environment and the high Ag vapor concentration provide the optimum conditions for condensation into Ag nanoparticles.   (c) Double pulse ablation of a Au target. The first pulse has a fluence of 1.2 J/cm 2 , the second pulse, which is delayed by 10 µs, has a fluence of 4 J/cm². We speculate that due to the low fluence of the first pulse, less particles are produced and, hence, no satellite bubbles formed. Comparable to the situation in Figure S5a, the energy of the second pulse is deposited into the cavitation bubble leading to the expansion of a second cavitation bubble. However, unlike Figure S5a where the residual laser energy can propagate freely into the interior of the first cavitation bubble, the residual laser energy is attenuated by the nanoparticles present inside the cavitation bubble. Finally, the remaining laser energy reaches the target, which leads to a second ablation. This produces an ablation plume expanding in the gas phase environment of the interior of the first cavitation bubble