*3.1. Calculation of the Contact Angle and Verification of Water Nanodroplet Size*

The contact angle is an essential parameter indicating the wettability of solid surfaces. The density contour of a water nanodroplet on the smooth copper surface illustrated in Figure 2 is based on the time-average density of every single chunk the water nanodroplet is being divided into. Considering the symmetrical projection of the water nanodroplet on *x–y* plane, only the right-sided water nanodroplet is investigated for contact angle calculation. The curve where the density is half that of bulk water is selected as the boundary of the liquid and gas phase (the black line in Figure 2) [52]. The density profile is considered as a part of a circle and is fitted by Equation (4). The contact angle can be obtained through Equation (5) [48]:

$$(x-a)^2 + (z-b)^2 = R^2 \tag{4}$$

$$\theta = \arcsin\left(\frac{b - z\_{\text{sub}}}{R}\right) + 90^{\circ} \tag{5}$$

where *a* and *b* are *x* and *z* coordinates of the centroid and *R* denotes the radius of the fitted circle. Only chunks that are more than 5 Å above the solid surface are considered to avoid the effects of density fluctuations at the liquid–solid interface. Therefore, *z*sub is the *z* coordinate of a virtual surface 5 Å above the solid surface. According to the calculation, the equilibrium contact angle *θ* of water droplets on the smooth copper surface is 108.7◦ , and the corresponding experimental result is 102◦ [53]. The slight difference between them is attributed to the surface used in the experiment being not as smooth as that used in the simulation. *Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 5 of 18

**Figure 2.** Density profile of the right-sided water nanodroplet on the smooth copper surface. **Figure 2.** Density profile of the right-sided water nanodroplet on the smooth copper surface.

To explore the effect of the number of molecules on the dynamical behaviors, nanodroplets with different radii impinging on the immobile surface are simulated. Radii and the corresponding number of water molecules are listed in Table 3. The dimensionless

**Figure 3.** Effects of droplets size on spreading time.

Å are adopted.

It can be seen that although the number of water molecules varies, the *R*/*R*0 of differ-

ent nanodroplets are similar to each other, and the maximum relative error is 6.9%. Therefore, the effect of the number of molecules arranged in the droplet can be neglected. Taking the computation accuracy and efficiency into consideration, nanodroplets with *R*0 = 35

*3.2. Effects of Translation Velocity When Water Nanodroplets Impinge on Translation Surfaces* 

Weber number (*We*) is chosen to represent the characteristics of water nanodroplets [54]. The effects of translation velocity of surfaces *Vs* on the dynamical behaviors of water

= ௭

where *R* is the radius of the droplet, *Vz* represents the velocity of droplets, and the surface tension *γ =* 72.75 mN/m [52]. The morphological evolution of water nanodroplets is illustrated in Figure 4. Due to the sufficient simulation period, water nanodroplets can exceed the simulation box boundary, and they re-enter the box from the opposite side because of the periodical boundary conditions. For the sake of clarity, the dimensions of boxes and

ଶ

(6)

nanodroplets with *We* = 7.41 are investigated. *We* is defined as

surfaces in *x* direction are assumed to be infinite.

Considering the inertia force plays an important role in the impinging process, the

number (*R*/*R*0, *R* is the spreading radius, and *R*<sup>0</sup> represents the initial radius) of those nanodroplets is shown in Figure 3.

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**Table 3.** Nanodroplets with different radii and the number of water molecules.


**Figure 3.** Effects of droplets size on spreading time. **Figure 3.** Effects of droplets size on spreading time.

It can be seen that although the number of water molecules varies, the *R*/*R*0 of different nanodroplets are similar to each other, and the maximum relative error is 6.9%. Therefore, the effect of the number of molecules arranged in the droplet can be neglected. Tak-It can be seen that although the number of water molecules varies, the *R*/*R*<sup>0</sup> of different nanodroplets are similar to each other, and the maximum relative error is 6.9%. Therefore, the effect of the number of molecules arranged in the droplet can be neglected. Taking the computation accuracy and efficiency into consideration, nanodroplets with *R*<sup>0</sup> = 35 Å are adopted.

### ing the computation accuracy and efficiency into consideration, nanodroplets with *R*0 = 35 *3.2. Effects of Translation Velocity When Water Nanodroplets Impinge on Translation Surfaces*

Å are adopted. *3.2. Effects of Translation Velocity When Water Nanodroplets Impinge on Translation Surfaces*  Considering the inertia force plays an important role in the impinging process, the Weber number (*We*) is chosen to represent the characteristics of water nanodroplets [54]. The effects of translation velocity of surfaces *V<sup>s</sup>* on the dynamical behaviors of water nanodroplets with *We* = 7.41 are investigated. *We* is defined as

$$\text{We} = \frac{\rho RV\_z^2}{\gamma} \tag{6}$$

tension *γ =* 72.75 mN/m [52]. The morphological evolution of water nanodroplets is illustrated in Figure 4. Due to the sufficient simulation period, water nanodroplets can exceed the simulation box boundary, and they re-enter the box from the opposite side because of the periodical boundary conditions. For the sake of clarity, the dimensions of boxes and

Weber number (*We*) is chosen to represent the characteristics of water nanodroplets [54]. The effects of translation velocity of surfaces *Vs* on the dynamical behaviors of water nanodroplets with *We* = 7.41 are investigated. *We* is defined as = ௭ ଶ (6) where *R* is the radius of the droplet, *V<sup>z</sup>* represents the velocity of droplets, and the surface tension *γ =* 72.75 mN/m [52]. The morphological evolution of water nanodroplets is illustrated in Figure 4. Due to the sufficient simulation period, water nanodroplets can exceed the simulation box boundary, and they re-enter the box from the opposite side because of the periodical boundary conditions. For the sake of clarity, the dimensions of boxes and surfaces in *x* direction are assumed to be infinite.

surfaces in *x* direction are assumed to be infinite.

**Figure 4.** Morphological evolution of droplets impinging on surfaces with different translation velocities: (**a**) *Vs* = 0.5 Å/ps; (**b**) *Vs* = 9 Å/ps. **Figure 4.** Morphological evolution of droplets impinging on surfaces with different translation velocities: (**a**) *Vs* = 0.5 Å/ps; (**b**) *Vs* = 9 Å/ps.

(**b**)

Although the surface is smooth, Figure 4 shows that both deformation and displacement of water nanodroplets can be observed, which is due to microscopic forces such as the Van der Waals force and capillary force taking a leading role in the interaction on a nanoscale. The component of *F*, interaction force between the water nanodroplet and surface, in the direction of relative sliding, is defined as friction force *Fx*, and the component in the direction perpendicular to the contact area is defined as normal pressure *Fz*. Therefore, the deformation and displacement of water nanodroplets result from the *Fx* applied by moving surfaces. Although the surface is smooth, Figure 4 shows that both deformation and displacement of water nanodroplets can be observed, which is due to microscopic forces such as the Van der Waals force and capillary force taking a leading role in the interaction on a nanoscale. The component of *F*, interaction force between the water nanodroplet and surface, in the direction of relative sliding, is defined as friction force *Fx*, and the component in the direction perpendicular to the contact area is defined as normal pressure *Fz*. Therefore, the deformation and displacement of water nanodroplets result from the *F<sup>x</sup>* applied by moving surfaces.

As seen in Figure 4a, the morphology of the water nanodroplet remains nearly symmetrical in the spreading and retraction processes. With *Vs* increasing to 9 Å/ps, contact angle hysteresis is observed during the process, so that the water nanodroplet moves As seen in Figure 4a, the morphology of the water nanodroplet remains nearly symmetrical in the spreading and retraction processes. With *V<sup>s</sup>* increasing to 9 Å/ps, contact angle hysteresis is observed during the process, so that the water nanodroplet moves along

the *V<sup>s</sup>* direction and spreads. Contact angle hysteresis refers to the difference between the advancing contact angle *θ<sup>a</sup>* and the receding contact angle *θ<sup>r</sup>* . The extent of hysteresis experiences a gradual drop to zero as time passes, which means the contact angle hysteresis eventually disappears. The water nanodroplet spreads asymmetrically in *x* and *y* directions at time *t* = 40 ps, and the asymmetry becomes more significant as time elapses. In particular, the water nanodroplet evolves into an ellipsoid at *t* = 80 ps. Afterwards, the water nanodroplet gradually recovers the sphere shape under the action of surface tension and then moves alongside the surface. *y* directions at time *t* = 40 ps, and the asymmetry becomes more significant as time elapses. In particular, the water nanodroplet evolves into an ellipsoid at *t* = 80 ps. Afterwards, the water nanodroplet gradually recovers the sphere shape under the action of surface tension and then moves alongside the surface. The evolution of velocity of the water nanodroplet in the *x* direction, *Vx*, is shown in Figure 5. It can be seen that the variation of *Vx* can be divided into two stages: the relative sliding stage and the stable stage. At the relative sliding stage, the acceleration of *Vx* in-

along the *Vs* direction and spreads. Contact angle hysteresis refers to the difference between the advancing contact angle *θa* and the receding contact angle *θr*. The extent of hysteresis experiences a gradual drop to zero as time passes, which means the contact angle hysteresis eventually disappears. The water nanodroplet spreads asymmetrically in *x* and

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The evolution of velocity of the water nanodroplet in the *x* direction, *Vx*, is shown in Figure 5. It can be seen that the variation of *V<sup>x</sup>* can be divided into two stages: the relative sliding stage and the stable stage. At the relative sliding stage, the acceleration of *V<sup>x</sup>* increases with the surface velocity. Especially, during 0~50 ps, *V<sup>x</sup>* almost linearly increases with *t*. This is because the deformation of water nanodroplets becomes more severe with the increase in surface velocity, making the spreading area enlarged, and thus the interaction between droplets and surfaces is enhanced. Once the water nanodroplet reaches the maximum spreading state, it begins to retract, and the interaction becomes weakened, resulting in a slowdown in the acceleration of *Vx*. At a stable stage, the translation velocity of the water nanodroplet is roughly equal to that of the surface, and there is no relative sliding between them. creases with the surface velocity. Especially, during 0~50 ps, *Vx* almost linearly increases with *t*. This is because the deformation of water nanodroplets becomes more severe with the increase in surface velocity, making the spreading area enlarged, and thus the interaction between droplets and surfaces is enhanced. Once the water nanodroplet reaches the maximum spreading state, it begins to retract, and the interaction becomes weakened, resulting in a slowdown in the acceleration of *Vx*. At a stable stage, the translation velocity of the water nanodroplet is roughly equal to that of the surface, and there is no relative sliding between them.

**Figure 5.** Time evolution of velocity of the water nanodroplet in the *x* direction. **Figure 5.** Time evolution of velocity of the water nanodroplet in the *x* direction.

The mathematical expression, presented in Equation (7), quantitatively describes the change in *Vx* in the whole dynamical process. Water nanodroplets with *We* ranging from 7.41 to 66.67 impinging on the translation surface are also simulated, and this equation The mathematical expression, presented in Equation (7), quantitatively describes the change in *V<sup>x</sup>* in the whole dynamical process. Water nanodroplets with *We* ranging from 7.41 to 66.67 impinging on the translation surface are also simulated, and this equation can provide good approximations.

$$V\_{\mathbf{x}} = V\_{\mathbf{s}} - V\_{\mathbf{s}} \left( 0.962 + 0.019 \frac{V\_{\mathbf{s}}}{We} \right)^{t} \quad \text{7.41} \le We \le 66.67 \tag{7}$$
 
$$\text{Taking the surface width } V = 0 \text{ \AA} \text{ (one one normalized the reaction mechanism of motion)}$$

ing to the snap shots, water molecules in the blue and red regions rotate counterclockwise

250 ps, as shown in Figure 6b. During the period of 250~500 ps, water molecules in blue

<sup>൰</sup> 7.41 ≤ ≤ 66.67 (7) Taking the surface with *Vs* = 9 Å/ps as an example, the motion mechanisms of water molecules at different stages are explored. According to Figure 5, the water nanodroplet is at a relative sliding stage during 0~250 ps and at a stable stage during 250~500 ps. As illustrated in Figure 6a, the top and bottom sections of the water nanodroplet are colored blue and red at *t* = 0 ps, respectively. The color property is fixed during 0~250 ps. Accord-Taking the surface with *V<sup>s</sup>* = 9 Å/ps as an example, the motion mechanisms of water molecules at different stages are explored. According to Figure 5, the water nanodroplet is at a relative sliding stage during 0~250 ps and at a stable stage during 250~500 ps. As illustrated in Figure 6a, the top and bottom sections of the water nanodroplet are colored blue and red at *t* = 0 ps, respectively. The color property is fixed during 0~250 ps. According to the snap shots, water molecules in the blue and red regions rotate counterclockwise around the center of the water nanodroplet, accompanying the diffusion of water molecules

into the entire water nanodroplet. Then, the water nanodroplet is colored again at 250 ps, as shown in Figure 6b. During the period of 250~500 ps, water molecules in blue and red regions only move downward or upward due to the diffusion, and no rotation is observed. Therefore, diffusion aside, water molecules in the water nanodroplets rotate around the centroid while they move along the surface in the relative sliding stage, and the rotation of water molecules disappears in the stable stage. and red regions only move downward or upward due to the diffusion, and no rotation is observed. Therefore, diffusion aside, water molecules in the water nanodroplets rotate around the centroid while they move along the surface in the relative sliding stage, and the rotation of water molecules disappears in the stable stage.

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**Figure 6.** Different motion mechanisms of water molecules inside the water nanodroplet at two stages: (**a**) relative sliding stage; (**b**) stable stage. **Figure 6.** Different motion mechanisms of water molecules inside the water nanodroplet at two stages: (**a**) relative sliding stage; (**b**) stable stage.

Friction force *Fx* applied to water nanodroplets is shown in Figure 7. When the surfaces translate at high speed (e.g., *Vs* = 9 Å/ps), there are continuous positive values of *Fx* in a long period, which is about 80 ps, as shown in Figure 7c. Additionally, that period shrinks as *Vs* decreases, as illustrated in Figure 7a,b. As *Vs* increases to 3 and 9 Å/ps from 0.5 Å/ps, the maximum friction force climbs to 27.5 and 42.3 KJ·mole−1·Å−1, which means that *Fx* increases with the vs. In the stable stage, *Fx* fluctuates around 0. The fluctuation results from the random thermal motion that leads to a slight change in the spreading Friction force *F<sup>x</sup>* applied to water nanodroplets is shown in Figure 7. When the surfaces translate at high speed (e.g., *V<sup>s</sup>* = 9 Å/ps), there are continuous positive values of *F<sup>x</sup>* in a long period, which is about 80 ps, as shown in Figure 7c. Additionally, that period shrinks as *V<sup>s</sup>* decreases, as illustrated in Figure 7a,b. As *V<sup>s</sup>* increases to 3 and 9 Å/ps from 0.5 Å/ps, the maximum friction force climbs to 20.5 and 36 Kcal·mole−<sup>1</sup> ·Å−<sup>1</sup> , which means that *F<sup>x</sup>* increases with the *V<sup>s</sup>* . In the stable stage, *F<sup>x</sup>* fluctuates around 0. The fluctuation results from the random thermal motion that leads to a slight change in the spreading area.

(**c**) **Figure 7.** Time evolution of the friction force applied to droplets: (**a**) *Vs* = 0.5 Å/ps; (**b**) *Vs* = 3 Å/ps; (**c**) **Figure 7.** Time evolution of the friction force applied to droplets: (**a**) *Vs* = 0.5 Å/ps; (**b**) *Vs* = 3 Å/ps; (**c**) *Vs* = 9 Å/ps.

*Vs* = 9 Å/ps.

Figure 8 shows the spreading factors *β<sup>x</sup>* and *βy*. *β<sup>x</sup>* (*βy*) denotes the ratio spreading length *D<sup>x</sup>* (*Dy*) of water nanodroplets in the *x*(*y*) direction to the initial diameter *D*0. *βx*max (*βy*max) is the maximum of *β<sup>x</sup>* (*βy*). accelerates the retraction in the *y* direction since the volume of the water nanodroplet is constant. Once the *βx*max is realized, the water nanodroplet begins to retract in the *x* direction, which results in the water nanodroplet spreading again in the *y* direction.

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(*βy*max) is the maximum of *βx* (*βy*).

Figure 8 shows the spreading factors *βx* and *βy*. *βx* (*βy*) denotes the ratio spreading length *Dx* (*Dy*) of water nanodroplets in the *x*(*y*) direction to the initial diameter *D*0. *βx*max

For the case *Vs* = 9 Å/ps, it can be observed from Figure 8b that *βy* reaches its maximum at 30 ps and then descends to 0.76 rapidly at 80 ps. After, it climbs again and finally obtains stability. This inflection point indicates the water nanodroplet spreads twice in the *y* direction, which is perpendicular to the direction of relative sliding. This is due to the fact that the water nanodroplet suffers reactive force when it impinges on solid surfaces, and that makes the water nanodroplet spread at first. The *βy*max is realized soon, and then the water nanodroplet begins to retract in the *y* direction. Meanwhile, the tensile deformation of water nanodroplets in the *x* direction becomes severe with the increase in *Vx* and thus

**Figure 8.** Time evolution of spreading factors in *x* and *y* directions: (**a**) *βx*; (**b**) *βy*. **Figure 8.** Time evolution of spreading factors in *x* and *y* directions: (**a**) *βx*; (**b**) *βy*.

*3.3. Effects of the Weber Number When Water Nanodroplets Impinge on Translation Surfaces*  To investigate the influence of *We* on the behavioral evolution, the water nanodroplet with *We* ranging from 0.82 to 66.67 hitting the translation surfaces is simulated. The solid surface translates at the speed of *Vs* = 7 Å/ps along the positive direction of *x* axis. The performance of water nanodroplets is shown in Figure 9. It can be observed that water nanodroplets with different *We* lie on various positions of the surfaces despite the identical *Vs*, which suggests the dynamical behaviors of water nanodroplets are determined by the joint actions of *Vs* and *We*. For the case *V<sup>s</sup>* = 9 Å/ps, it can be observed from Figure 8b that *β<sup>y</sup>* reaches its maximum at 30 ps and then descends to 0.76 rapidly at 80 ps. After, it climbs again and finally obtains stability. This inflection point indicates the water nanodroplet spreads twice in the *y* direction, which is perpendicular to the direction of relative sliding. This is due to the fact that the water nanodroplet suffers reactive force when it impinges on solid surfaces, and that makes the water nanodroplet spread at first. The *βy*max is realized soon, and then the water nanodroplet begins to retract in the *y* direction. Meanwhile, the tensile deformation of water nanodroplets in the *x* direction becomes severe with the increase in *V<sup>x</sup>* and thus accelerates the retraction in the *y* direction since the volume of the water nanodroplet is constant. Once the *βx*max is realized, the water nanodroplet begins to retract in the *x* direction, which results in the water nanodroplet spreading again in the *y* direction.
