*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 *V<sup>s</sup>* = 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 *V<sup>s</sup>* , which suggests the dynamical behaviors of water nanodroplets are determined by the joint actions of *V<sup>s</sup>* and *We*.

The contact hysteresis can be seen for water nanodroplets with *We* = 0.82~20.58, as shown in Figure 9a,b. However, the degree of hysteresis decreases with increasing *We,* and the hysteresis cannot be observed for *We* = 66.67, as shown in Figure 9c. This can be explained by the fact that when *We* is relatively low, the spreading velocity after impinging is also low due to inertia, which makes the influence of friction force more remarkable, and thus the tensile deformation becomes more obvious. Therefore, the contact angle hysteresis is more remarkable. As *We* increases, the spreading velocity after impinging increases, and the friction force plays a less important role in the dynamical behaviors. As a consequence, the difference between *θ<sup>a</sup>* and *θ<sup>r</sup>* decreases gradually. It is important to note that a hole appears near the center of the water nanodroplet during 20~30 ps, as shown in Figure 9c, because the higher *We* facilitates the spreading of water nanodroplets, and the water molecules move towards the edge of the water nanodroplet constantly during the spreading process. However, the hole disappears under the action of surface tension during the retraction process.

Figure 10 illustrates the friction force *F<sup>x</sup>* applied to water nanodroplets with various *We*. *F<sup>x</sup>* increases with *We* at the relative sliding stage, and thus, the water nanodroplet comes into the stable stage early, where the water nanodroplet and surface have the same velocity. This is because the increased spreading area leads to the enhancement of the interaction between water and solid surfaces. *Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 10 of 18

**Figure 9.** Morphological evolution of water nanodroplets with various *We* impinging on the translation surfaces: (**a**) *We* = 0.82; (**b**) *We* = 20.58; (**c**) *We* = 66.67. **Figure 9.** Morphological evolution of water nanodroplets with various *We* impinging on the translation surfaces: (**a**) *We* = 0.82; (**b**) *We* = 20.58; (**c**) *We* = 66.67.

teresis is more remarkable. As *We* increases, the spreading velocity after impinging increases, and the friction force plays a less important role in the dynamical behaviors. As a

The contact hysteresis can be seen for water nanodroplets with *We* = 0.82~20.58, as shown in Figure 9a,b. However, the degree of hysteresis decreases with increasing *We,* and the hysteresis cannot be observed for *We* = 66.67, as shown in Figure 9c. This can be explained by the fact that when *We* is relatively low, the spreading velocity after imping-

**Figure 10.** Time evolution of the friction force applied to water nanodroplets with various *We*: (**a**) *We* = 0.82; (**b**) *We* = 20.58; (**c**) *We* = 40.33; (**d**) *We* = 66.67.

From the spreading factors shown in Figure 11, it can be seen that both *β<sup>x</sup>* and *β<sup>y</sup>* are approximately 1.0 after 150 ps despite the different *We*, which indicates that *We* has little influence on the steady state of water nanodroplets and mainly takes a role in the spreading and retraction processes. In addition, *βx*max and *βy*max increase with the *We*. It should be noted that in the dashed line region shown in Figure 11b, besides *We* = 0.82, *β<sup>y</sup>* decreases first and then gradually increases to a stable level. Therefore, the water nanodroplet experiences a secondary spreading in the *y* direction. *Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 12 of 18

*3.4. Effects of Vibration Amplitudes on Dynamical Behaviors When Water Nanodroplets* 

nanodroplets impinging on surfaces with vibration amplitudes ranging from 1 to 4 Å, and the vibration period *T* = 12 ps is simulated. Vibration is achieved by applying periodic displacement up and down along the *z* axis on the solid surface, as shown in Figure 12.

**Figure 12.** Morphological evolution of water nanodroplets impinging on vibration surfaces with

The spreading area reduces with the increasing *A*, as shown in Figure 12. In particular, when *A* = 3 and 4 Å, there are some gaps between the water nanodroplets and surfaces. Hence, the adhesion between the water nanodroplet and surface is weakened, and more water molecules escape from the bulk of water nanodroplets with the increase in *A*.

Figure 13 shows the centroid height *h* of water nanodroplets. In general, *h* increases, and water nanodroplets have a tendency to depart from surfaces as *A* increases. During 0~20 ps, *h* decreases rapidly. That is because in this period, inertia is the dominant factor determining the dynamical behaviors, and the water nanodroplet spreads at high speed, and as a result, *h* almost decreases linearly. Once *h* decreases to the minimum, indicating

**Figure 11.** Time evolution of spreading factors in *x* and *y* directions: (**a**) *βx*; (**b**) *βy*. the maximum spreading state being realized, the water nanodroplet starts to retract due **Figure 11.** Time evolution of spreading factors in *x* and *y* directions: (**a**) *βx*; (**b**) *βy*.

*Impinge on Vibration Surfaces* 

various amplitudes.

#### *3.4. Effects of Vibration Amplitudes on Dynamical Behaviors When Water Nanodroplets Impinge on Vibration Surfaces Impinge on Vibration Surfaces*  To explore the influence of vibration amplitudes *A* on the dynamical behaviors, water

*3.4. Effects of Vibration Amplitudes on Dynamical Behaviors When Water Nanodroplets* 

To explore the influence of vibration amplitudes *A* on the dynamical behaviors, water nanodroplets impinging on surfaces with vibration amplitudes ranging from 1 to 4 Å, and the vibration period *T* = 12 ps is simulated. Vibration is achieved by applying periodic displacement up and down along the *z* axis on the solid surface, as shown in Figure 12. nanodroplets impinging on surfaces with vibration amplitudes ranging from 1 to 4 Å, and the vibration period *T* = 12 ps is simulated. Vibration is achieved by applying periodic displacement up and down along the *z* axis on the solid surface, as shown in Figure 12.

*Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 12 of 18

(**a**) (**b**)

**Figure 11.** Time evolution of spreading factors in *x* and *y* directions: (**a**) *βx*; (**b**) *βy*.

**Figure 12.** Morphological evolution of water nanodroplets impinging on vibration surfaces with various amplitudes. **Figure 12.** Morphological evolution of water nanodroplets impinging on vibration surfaces with various amplitudes.

The spreading area reduces with the increasing *A*, as shown in Figure 12. In particular, when *A* = 3 and 4 Å, there are some gaps between the water nanodroplets and surfaces. Hence, the adhesion between the water nanodroplet and surface is weakened, and more water molecules escape from the bulk of water nanodroplets with the increase in *A*. The spreading area reduces with the increasing *A*, as shown in Figure 12. In particular, when *A* = 3 and 4 Å, there are some gaps between the water nanodroplets and surfaces. Hence, the adhesion between the water nanodroplet and surface is weakened, and more water molecules escape from the bulk of water nanodroplets with the increase in *A*.

Figure 13 shows the centroid height *h* of water nanodroplets. In general, *h* increases, and water nanodroplets have a tendency to depart from surfaces as *A* increases. During 0~20 ps, *h* decreases rapidly. That is because in this period, inertia is the dominant factor determining the dynamical behaviors, and the water nanodroplet spreads at high speed, and as a result, *h* almost decreases linearly. Once *h* decreases to the minimum, indicating the maximum spreading state being realized, the water nanodroplet starts to retract due Figure 13 shows the centroid height *h* of water nanodroplets. In general, *h* increases, and water nanodroplets have a tendency to depart from surfaces as *A* increases. During 0~20 ps, *h* decreases rapidly. That is because in this period, inertia is the dominant factor determining the dynamical behaviors, and the water nanodroplet spreads at high speed, and as a result, *h* almost decreases linearly. Once *h* decreases to the minimum, indicating the maximum spreading state being realized, the water nanodroplet starts to retract due to surface tension. However, energy dissipation in the spreading process leads to a low retraction velocity. Hence, the vibration of surfaces gradually comes into play, and the saw-tooth fluctuation appears in a single vibration period. After 70 ps, the retraction almost finishes, and the dynamical behaviors of water nanodroplets are solely affected by the vibration surfaces. The amplitudes of saw-tooth fluctuations decrease, and the shape of the saw tooth becomes uniform.

The variation of spreading factors *β* with time is recorded in Figure 14. It also shows apparent fluctuation due to the vibration surfaces. The maximum spreading factor *β*max and the maximum spreading area reduce with the increasing *A*, suggesting a larger vibration amplitude not conducive to the spreading of water nanodroplets.

of the saw tooth becomes uniform.

of the saw tooth becomes uniform.

**Figure 13.** Centroid height of water nanodroplets impinging on vibration surfaces with various am-**Figure 13.** Centroid height of water nanodroplets impinging on vibration surfaces with various amplitudes. plitudes.

tion amplitude not conducive to the spreading of water nanodroplets.

tion amplitude not conducive to the spreading of water nanodroplets.

*Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 13 of 18

to surface tension. However, energy dissipation in the spreading process leads to a low retraction velocity. Hence, the vibration of surfaces gradually comes into play, and the saw-tooth fluctuation appears in a single vibration period. After 70 ps, the retraction almost finishes, and the dynamical behaviors of water nanodroplets are solely affected by the vibration surfaces. The amplitudes of saw-tooth fluctuations decrease, and the shape

to surface tension. However, energy dissipation in the spreading process leads to a low retraction velocity. Hence, the vibration of surfaces gradually comes into play, and the saw-tooth fluctuation appears in a single vibration period. After 70 ps, the retraction almost finishes, and the dynamical behaviors of water nanodroplets are solely affected by the vibration surfaces. The amplitudes of saw-tooth fluctuations decrease, and the shape

The variation of spreading factors *β* with time is recorded in Figure 14. It also shows

The variation of spreading factors *β* with time is recorded in Figure 14. It also shows

apparent fluctuation due to the vibration surfaces. The maximum spreading factor *β*max and the maximum spreading area reduce with the increasing *A*, suggesting a larger vibra-

apparent fluctuation due to the vibration surfaces. The maximum spreading factor *β*max and the maximum spreading area reduce with the increasing *A*, suggesting a larger vibra-

**Figure 14.** Time evolution of spreading factors of droplets impinging on surfaces with different vibration amplitudes. **Figure 14.** Time evolution of spreading factors of droplets impinging on surfaces with different vibration amplitudes.

#### **Figure 14.** Time evolution of spreading factors of droplets impinging on surfaces with different vibration amplitudes. *3.5. Effects of Vibration Periods on Dynamical Behaviors When Water Nanodroplets Impinge on 3.5. Effects of Vibration Periods on Dynamical Behaviors When Water Nanodroplets Impinge on Vibration Surfaces*

*3.5. Effects of Vibration Periods on Dynamical Behaviors When Water Nanodroplets Impinge on Vibration Surfaces Vibration Surfaces*  A water nanodroplet with *We* = 7.41 impinging on vibration surfaces with periods *T* A water nanodroplet with *We* = 7.41 impinging on vibration surfaces with periods *T* ranging from 2 to 16 ps and *A* = 2 Å is simulated to analyze the influence of *T* on dynamical behaviors.

A water nanodroplet with *We* = 7.41 impinging on vibration surfaces with periods *T* ranging from 2 to 16 ps and *A* = 2 Å is simulated to analyze the influence of *T* on dynamical behaviors. Figure 15 presents the snapshots in that process. Water nanodroplets eventually deposit on the solid surfaces when *T* = 4~16 ps. It can be seen that a longer *T* makes the connection between water nanodroplets and solid surfaces tighter, e.g., *T* = 16 ps in Figure ranging from 2 to 16 ps and *A* = 2 Å is simulated to analyze the influence of *T* on dynamical behaviors. Figure 15 presents the snapshots in that process. Water nanodroplets eventually deposit on the solid surfaces when *T* = 4~16 ps. It can be seen that a longer *T* makes the connection between water nanodroplets and solid surfaces tighter, e.g., *T* = 16 ps in Figure 15c. It should be noted that the water nanodroplet finally bounces off the solid surface Figure 15 presents the snapshots in that process. Water nanodroplets eventually deposit on the solid surfaces when *T* = 4~16 ps. It can be seen that a longer *T* makes the connection between water nanodroplets and solid surfaces tighter, e.g., *T* = 16 ps in Figure 15c. It should be noted that the water nanodroplet finally bounces off the solid surface when *T* = 2 ps. This provides a method to facilitate water nanodroplets bouncing, except when designing special rough structures on the surfaces mentioned in the previous study [55]. Moreover, the water nanodroplet has a shorter contact time with the solid surface than the alternative.

15c. It should be noted that the water nanodroplet finally bounces off the solid surface when *T* = 2 ps. This provides a method to facilitate water nanodroplets bouncing, except

when *T* = 2 ps. This provides a method to facilitate water nanodroplets bouncing, except

than the alternative.

than the alternative.

**Figure 15.** Dynamical behaviors of water nanodroplets impinging on surfaces with different vibration periods: (**a**) *T* = 2 ps; (**b**) *T* = 8 ps; (**c**) *T* = 16 ps. **Figure 15.** Dynamical behaviors of water nanodroplets impinging on surfaces with different vibration periods: (**a**) *T* = 2 ps; (**b**) *T* = 8 ps; (**c**) *T* = 16 ps. nearly remains stable after 25 ps, when *T* = 12 and 16 ps. As *T* decreases, *h* increases. Especially, the water nanodroplet bounces off the solid surfaces at 55 ps when *T* = 2 ps.

when designing special rough structures on the surfaces mentioned in the previous study [55]. Moreover, the water nanodroplet has a shorter contact time with the solid surface

(**a**)

*Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 14 of 18

when designing special rough structures on the surfaces mentioned in the previous study [55]. Moreover, the water nanodroplet has a shorter contact time with the solid surface

The variation in centroid height *h* for water nanodroplets is shown in Figure 16. *h* nearly remains stable after 25 ps, when *T* = 12 and 16 ps. As *T* decreases, *h* increases. Especially, the water nanodroplet bounces off the solid surfaces at 55 ps when *T* = 2 ps. The variation in centroid height *h* for water nanodroplets is shown in Figure 16. *h* nearly remains stable after 25 ps, when *T* = 12 and 16 ps. As *T* decreases, *h* increases. Especially, the water nanodroplet bounces off the solid surfaces at 55 ps when *T* = 2 ps. According to Figure 17, as a whole, *β* decreases with the decrese in the vibration period, and even it finally decreases to 0 for *T* = 2 ps. As a result, the increasing *T* induces the further spread of the water nanodroplet.

According to Figure 17, as a whole, *β* decreases with the decrese in the vibration pe-

**Figure 16.** Time evolution of water nanodroplets impinging on surfaces with different vibration **Figure 16.** Time evolution of water nanodroplets impinging on surfaces with different vibration **Figure 16.** Time evolution of water nanodroplets impinging on surfaces with different vibration periods.

periods.

periods.

According to Figure 17, as a whole, *β* decreases with the decrese in the vibration period, and even it finally decreases to 0 for *T* = 2 ps. As a result, the increasing *T* induces the further spread of the water nanodroplet. *Nanomaterials* **2022**, *12*, x FOR PEER REVIEW 15 of 18

**Figure 17.** Spreading factors of water nanodroplets impinging on solid surfaces with various vibration periods**. Figure 17.** Spreading factors of water nanodroplets impinging on solid surfaces with various vibration periods. To quantify the bounce of water nanodroplets, the bounce domain, determined by *T*

To quantify the bounce of water nanodroplets, the bounce domain, determined by *T* and *A*, of water nanodroplets impinging on vibration surfaces, is showcased in Figure 18. To quantify the bounce of water nanodroplets, the bounce domain, determined by *T* and *A*, of water nanodroplets impinging on vibration surfaces, is showcased in Figure 18. This provides the theoretical basis for practical applications. and *A*, of water nanodroplets impinging on vibration surfaces, is showcased in Figure 18. This provides the theoretical basis for practical applications.

**Figure 18.** The bounce domain as water nanodroplets impinge on vibration surfaces. **Figure 18.** The bounce domain as water nanodroplets impinge on vibration surfaces.
