**2. Simulation Model and Methodology**

The initial configuration of water nanodroplets impinging on a moving surface is schematically shown in Figure 1. The flat substrate is composed of copper atoms, which are arranged in the face-centered-cube (FCC) structure with a lattice constant of 3.615 Å [34]. A TIP4P model of water is adopted because it can present more accurate dynamical characteristics [35,36]. The parameters corresponding to the TIP4P model are listed in Table 1 (*q*O: the charge of oxygen atom; *q*H: the charge of hydrogen atom; *e*: electron charge; *r*OH: the bond length between oxygen and hydrogen atoms; *θ*HOH: the bond angle of hydrogenoxygen-hydrogen). The water nanodroplet is constituted by 5991 water molecules, with 35 Å in radius. To prevent atoms in the initial configuration from overlapping, which leads to the infinite interaction force among atoms, water molecules are arranged in the body-centered-cube (BCC) structure, whose lattice constant is determined by the density of water *ρ* at 298 K. *Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 3 of 18

**Figure 1.** The initial configuration of a water nanodroplet impinging on a moving surface: (**a**) front view; (**b**) perspective view. **Figure 1.** The initial configuration of a water nanodroplet impinging on a moving surface: (**a**) front view; (**b**) perspective view.

The whole simulation is programmed and conducted with LAMMPS (large-scale atomic/molecular massively parallel simulator) [37]. Periodical boundary conditions are

tion during simulation [39–41]. The interaction between water molecules [42], containing a 12-6 Lennard-Jones (LJ) term and a long-range Coulomb term, is shown as follows:

> − ൬୭୭ ୭୭ ൰

where *r*OO denotes the distance between oxygen atoms in water molecules *i* and *j*. *ε*OO and *σ*OO represent the well depth and equilibrium distances, respectively. *ria,jb* stands for the distance between charge site *a* of molecule *i* and charge site *b* of molecule *j*. *qia* and *qjb* are charges of sites *a* and *b*, respectively. *ε*0 denotes the vacuum permittivity. Although Lennard-Jones potential was presented for noble gas at first, it essentially describes the interaction between electronic neutral atoms [43]. Hendrik et al. justified the reliability of this potential for copper and showed that 12-6 LJ potential is more suitable for large simulation systems (~106 atoms) compared with embedded atom model (EAM) and density functional method [44]. In this paper, a large-sized solid surface consisting of 201,344 copper atoms is adopted to avoid the influences of adjacent mirror images during the spreading of the droplet. Therefore, both the interactions between copper atoms and those between oxygen atoms and copper atoms are implemented by using the 12-6 LJ potential [45–47]. The interaction parameters for oxygen and copper are presented in Table 2. Since the solid surface is frozen in the simulation, the parameters of copper atoms are mainly used to calculate the interaction between copper and oxygen through Lorentz–Berthelot mix rules

**Atom Type O Cu**  *ε* (kcal/mol) *ε*O = 0.1628 *ε*Cu = 0.2379 *σ* (Å) *σ*O = 3.1644 *σ*Cu = 2.3400

<sup>=</sup> <sup>1</sup>

where *σm*, *εm*, *σn*, and *εn* are parameters of 12-6 LJ potential for atoms *m* and *n*; *σmn* and *εmn*

+

ୀଵ

ଷ

ଷ

ୀଵ

4,

<sup>2</sup> ( + ) (2)

= ඥ (3)

(1)

୭୭ ൰ ଵଶ

= 4 ቈ൬୭୭

that are shown in Equations (2) and (3).

**Table 2.** Interaction parameters for oxygen and copper atoms [48].

are parameters of 12-6 LJ potential between atoms *m* and *n*.

**Table 1.** Parameters of TIP4P model [35].


The whole simulation is programmed and conducted with LAMMPS (large-scale atomic/molecular massively parallel simulator) [37]. Periodical boundary conditions are applied in all three directions. A SHAKE algorithm is adopted to fix the bond length and angle of water molecules [38]. Copper atoms are constrained to initial equilibrium position during simulation [39–41]. The interaction between water molecules [42], containing a 12-6 Lennard-Jones (LJ) term and a long-range Coulomb term, is shown as follows:

$$\mathcal{U} = 4\varepsilon\_{\rm CO} \left[ \left( \frac{\sigma\_{\rm co}}{r\_{\rm co}} \right)^{12} - \left( \frac{\sigma\_{\rm co}}{r\_{\rm co}} \right)^{6} \right] + \sum\_{a=1}^{3} \sum\_{b=1}^{3} \frac{q\_{\rm ia} q\_{jb}}{4 \pi \varepsilon\_{0} r\_{\rm ia,jb}} \tag{1}$$

where *r*OO denotes the distance between oxygen atoms in water molecules *i* and *j*. *ε*OO and *σ*OO represent the well depth and equilibrium distances, respectively. *ria,jb* stands for the distance between charge site *a* of molecule *i* and charge site *b* of molecule *j*. *qia* and *qjb* are charges of sites *a* and *b*, respectively. *ε*<sup>0</sup> denotes the vacuum permittivity. Although Lennard-Jones potential was presented for noble gas at first, it essentially describes the interaction between electronic neutral atoms [43]. Hendrik et al. justified the reliability of this potential for copper and showed that 12-6 LJ potential is more suitable for large simulation systems (~10<sup>6</sup> atoms) compared with embedded atom model (EAM) and density functional method [44]. In this paper, a large-sized solid surface consisting of 201,344 copper atoms is adopted to avoid the influences of adjacent mirror images during the spreading of the droplet. Therefore, both the interactions between copper atoms and those between oxygen atoms and copper atoms are implemented by using the 12-6 LJ potential [45–47]. The interaction parameters for oxygen and copper are presented in Table 2. Since the solid surface is frozen in the simulation, the parameters of copper atoms are mainly used to calculate the interaction between copper and oxygen through Lorentz–Berthelot mix rules that are shown in Equations (2) and (3).

$$
\sigma\_{mn} = \frac{1}{2} (\sigma\_m + \sigma\_n) \tag{2}
$$

$$
\varepsilon\_{mn} = \sqrt{\varepsilon\_m \varepsilon\_n} \tag{3}
$$

where *σm*, *εm*, *σn*, and *ε<sup>n</sup>* are parameters of 12-6 LJ potential for atoms *m* and *n*; *σmn* and *εmn* are parameters of 12-6 LJ potential between atoms *m* and *n*.

**Table 2.** Interaction parameters for oxygen and copper atoms [48].


The cutoff distances of 12-6 LJ potential and Coulomb potential are 10 Å and 12 Å [34], respectively. Velocity–Verlet algorithm is used to integrate the Newton equation of motion and update the velocities and coordinates of atoms with a time step of 1.0 fs. The long-range Columbic interaction is calculated by PPPM (particle–particle–particle–mesh) method in *K* space [49].

Water nanodroplets experience minimization and relaxation lasting for 0.5 ns in NVT ensemble, leading them equilibrated and the potential energy being the local minimum. During the relaxation process, a Nose–Hoover thermostat is adopted to realize temperature control and the initial equilibrium velocity of water molecules is in accord with Maxwell–Boltzmann distribution [50,51]. After relaxation, the simulation system achieves

equilibrium state, and there are some water molecules in gas phase randomly distributed in simulation domain. Then, the NVT ensemble is removed, eliminating the temperature control, and the system is simulated under NVE ensemble. Meanwhile, a vertical initial velocity *V<sup>z</sup>* is applied to the water nanodroplet. *V<sup>z</sup>* is in the negative *z* direction. The moment water nanodroplets touch the solid surface is considered 0 ps.
