3.1. Evidence of the existence of scale effects

This section uses β_max as the key parameter to demonstrate the presence of scale-dependent viscous dissipation mechanisms. An established understanding is that spreading droplets on smooth surfaces is governed by dimensionless groups, including We, Oh and θ. According to this concept, if scale effects are absent and the values of all dominant dimensionless groups remain constant, β_max should be identical at all scales. Therefore, scale effects are tested by directly comparing β_max values at different scales while maintaining the same We, Oh and θ.

In MDPD simulations, the value of Oh is 0.21 for MDPD droplets with D ₀ = 20 (DPD units). To maintain a consistent Oh of 0.21, the high-viscosity liquid (glycerol–water mixtures with 85 % glycerol concentration) at the macroscale and the extremely low-viscosity liquid (mW liquid, with a viscosity of just one-third that of water) at the nanoscale are chosen. Additionally, the surface wettability is also fixed at approximately 164°. Figure 2 shows the β_max versus We curves at different scales, with the macroscale data from the experimental study by Abolghasemibizaki et al. (Reference Abolghasemibizaki, Dilmaghani, Mohammadi and Castano2019) and the nanoscale data from this work using MD simulations (detailed method refer to Wang et al. (Reference Wang, Wang, He, Zhang, Yang, Wang and Lee2022b)). As We < 30, the maximum spreading factors at different scales show negligible divergence. However, as We exceeds 30, deviation among the β_max values at different scales grows with increasing We, contradicting the traditional scale-independent behaviour. Consequently, this disparity confirms the presence of scale effects between different scales.

Figure 2.

Comparison of β_max for impacting droplets at the macroscale (Abolghasemibizaki et al. Reference Abolghasemibizaki, Dilmaghani, Mohammadi and Castano2019), the microscale and the nanoscale on superhydrophobic surfaces with θ = 163°, 164° and 165°, respectively.

3.2. Understanding viscous dissipation mechanisms at different scales

This section focuses on analysing the velocity contours of impacting droplets to decipher the viscous dissipation mechanism at the microscale. With an understanding at the microscale, the viscous dissipation mechanisms of the macroscale and the nanoscale could be revisited, and the whole picture of the viscous dissipation mechanism can be unveiled.

The velocity contours for an impacting droplet with a diameter of 230 nm on a surface with θ = 147° at We = 235.6 is shown in figure 3(a–c). The left-hand sides of the figures illustrate the contours of V_z, while the right-hand sides display those of V_r. The velocity profile near the solid wall at r = 7.3 in figure 3(a) is also highlighted in figure 3(d) to exhibit near-wall velocity gradients. A few observations are noticeable. First, the velocity gradients in the region highlighted by the blue square resemble those in the boundary layer seen in macroscale scenarios, i.e. dV_r/dz is dominant. The difference is that the velocity at the liquid–solid interface is non-zero at the microscale against the validity of the no-slip boundary condition. This occurrence results in a weaker boundary-layer viscous dissipation than at the macroscale. Second, the spreading rims still occur during spreading, and V_r peaks at the entrance of spreading rims, as shown in the region marked by the green square in figure 3(a–c). When entering the rims, V_r rapidly decreases, offering a substantial gradient of dV_r/dr, indicative of rim geometric head loss. Third, within the bulk droplet, the velocity gradients are also violent but differ markedly from those in the boundary layer. Unlike dV_r/dz observed in the boundary layer, dV_r/dr and dV_z/dz dominate the bulk droplet. Such a feature of velocity distribution is named extensional flow, whose viscous dissipation is mainly attributable to the deformation of droplets. Consequently, the viscous dissipation mechanism at the microscale is governed by three types of viscous dissipation: boundary-layer viscous dissipation due to shear flow near solid surfaces (E_{dis ,boundary}); rim head loss at the entrance of spreading rims (E_{dis ,rim}); and bulk viscous dissipation caused by extensional flow from droplet deformation (E_{dis ,bulk}). This mechanism has also been further corroborated by additional observations from MDPD simulations in broad ranges of We and θ at the microscale.

Figure 3.

(a–c) Velocity contours of microscale droplets impacting surfaces with θ = 147° at We = 235.6. (d–f) Velocity profiles near solid walls at the microscale, the macroscale and the nanoscale, respectively.

Previous studies have examined the viscous dissipation mechanisms at both the macroscale and nanoscale. However, due to the distinct characteristics of velocity gradients at these scales, they have traditionally been considered opposites (Li et al. Reference Li, Zhang and Chen2015; Wang et al. Reference Wang, Wang, Gao, Yang, Wang and Chen2020a). In this study, the viscous dissipation mechanism at the microscale has been uncovered, offering a valuable link that bridges the gap between the two extreme scales. This discovery presents a unique opportunity to explore how the viscous dissipation mechanism transitions from the macroscale to the nanoscale. To this end, the viscous dissipation mechanisms at the macroscale and the nanoscale are revisited below.

At the macroscale, only two types of dissipation, E_{dis ,boundary} and E_{dis ,rim}, have been reported (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016), while E_{dis ,bulk} is negligible, to our best knowledge. The no-slip condition is applicable to the boundary layer, as shown in figure 3(e). The model incorporating only E_{dis ,boundary} and E_{dis ,rim} accurately predicts β_max for low Oh droplets at the macroscale; however, the model would overestimate β_max for high Oh, with more pronounced overestimation on β_max with increasing Oh (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016). This observation suggests that deformation-induced E_{dis ,bulk} is significant at high Oh. In fact, the droplet-deformation-induced dissipation commonly participates in the natural and industrial processes. For instance, the freely decaying oscillations of levitated droplets serve as evidence of this mechanism. In such processes, both E_{dis ,boundary} and E_{dis ,rim} are absent due to the lack of solid surfaces and rim formation, leaving only deformation-induced E_{dis ,bulk} as the dominating dissipation mechanism. Furthermore, as Oh increases, the damping of oscillations in levitated droplets is significantly enhanced (Kremer, Kilzer & Petermann Reference Kremer, Kilzer and Petermann2018), supporting the strong positive correlation between E_{dis ,bulk} and Oh. Therefore, all three types of viscous dissipation coexist for the impact of macroscale droplets.

At the nanoscale, a consensus has emerged in recent studies acknowledging that velocity gradients are notably intense throughout entire nanodroplets, and that the viscous dissipation caused by spreading rims is relatively insignificant (Li et al. Reference Li, Zhang and Chen2015, Reference Li, Li and Chen2017). Given the minimal scale of nanodroplets, some studies posited that impacting nanodroplets entirely fall in the boundary layer during impact (Li et al. Reference Li, Li and Chen2017; Wang et al. Reference Wang, Wang, Gao, Yang, Wang and Chen2020a), suggesting that boundary-layer dissipation is the primary mechanism (Li et al. Reference Li, Li and Chen2017; Wang et al. Reference Wang, Wang, Gao, Yang, Wang and Chen2020a). However, this viewpoint is worth discussing. The dominant velocity gradient within the boundary layer is typically dV_r/dz. Contrary to this, velocity contours from Li et al. (Reference Li, Zhang and Chen2015) and Wang et al. (Reference Wang, Wang, Yang and Chen2019) indicate that gradients dV_r/dr and dV_z/dz are violent throughout nanodroplets, contradicting the expected velocity characteristics of the boundary layer. Secondly, as the droplet scale decreases, the slip effect continuously enhances, diminishing the strength of boundary-layer dissipation. The slip effect has been observed at the microscale, as illustrated in figure 3(a–d), and it is more pronounced at the nanoscale, as figure 3(f) shows, suggesting a further weak influence of boundary-layer dissipation. Therefore, it is improbable that the boundary layer alone governs viscous dissipation at the nanoscale. The observed velocity gradients dV_r/dr and dV_z/dz align more closely with deformation-induced gradients in the bulk droplet, indicating that deformation-induced dissipation might be the predominant mechanism at the nanoscale. This occurrence can be attributed to the naturally high Oh at the nanoscale. Moreover, although previous studies have not focused on viscous dissipation due to rims at the nanoscale, Wang et al. (Reference Wang, Wang, Zhang, He, Yang, Zheng, Lee and Wang2023) showed that the rims do form during nanoscale impacts, particularly at high We. The formation of these rims likely results in a sharp reduction in radial velocity (V_r) at the entrance of rims, generating E_{dis ,rim}. For the boundary-layer dissipation, although the slip effect is strongly enhanced and almost no boundary-layer gradients are present in velocity contours at the nanoscale, indicating an almost negligible intensity of E_{dis ,boundary} compared with other dissipation, it still exists owing to the presence of the solid walls. Therefore, the three types of viscous dissipation (E_{dis ,boundary}, E_{dis ,rim} and E_{dis ,bulk}) also contribute to the impact at the nanoscale as well.

According to the above discussion, a comprehensive understanding of the viscous dissipation mechanisms across different scales can be articulated. The decreased scale of droplets would increase Oh, highlighting the significance of E_{dis ,bulk} at both the nanoscale and microscale. Moreover, E_{dis ,bulk} would also become increasingly important as the viscosity of liquid increases. Therefore, to accommodate a broad range of Oh, E_{dis ,bulk} must be considered throughout the whole scale. Additionally, the presence of walls, which generate shear flow, ensures that boundary-layer dissipation (E_{dis ,boundary}) occurs at all scales. The critical difference is that the slip effect becomes progressively more pronounced as the droplet scale decreases, so the strength of E_{dis ,boundary} continuously weakens. Lastly, spreading rims are a consistent observation in impacts across all scales when We is sufficiently high, approximately 30, as Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) noted. Thus, the rim geometric head loss, E_{dis ,rim}, also plays a crucial role throughout the scale.

3.3. Modelling the maximum spreading factor throughout the whole scale

Revealing the viscous dissipation mechanism is crucial for comprehending the impact dynamics of droplets; however, directly testing it is exceptionally intractable. The maximum spreading factor serves as an extensively concerned feature parameter, and modelling this factor requires a profound understanding of viscous dissipation mechanism, making it a valuable bridge for inspecting this mechanism within the scientific community focused on droplet impact. This section will establish a model of β_max to thoroughly check the full-scale viscous dissipation mechanism thoroughly. Moreover, this model is also expected to predict a full spectrum of β_max, covering droplet scales from the macroscale to the nanoscale.

The model of β_max is established using the energy conservation equation, i.e. (1.1). Based on the spherical shape of a droplet before impacts, E_{k ,0} and E_{s ,0} can be expressed as

(3.1)\begin{gather}{E_{k,0}} = {\textstyle{1 \over {12}}}{\rm \pi} \rho D_0^3V_0^2,\end{gather}

(3.2)\begin{gather}{E_{s,0}} = \gamma {\rm \pi}D_0^2.\end{gather}

Assuming the cylindrical shape of the droplet at the maximum spreading state (Ukiwe & Kwok Reference Ukiwe and Kwok2005), the surface energy at this state could be obtained as

(3.3)\begin{equation}{E_{s,m}} = {\rm \pi}D_0^2\gamma \left[ {\frac{1}{4}\beta_{max}^2(1 - \textrm{cos}\,\theta ) + \frac{2}{{3{\beta_{max}}}}} \right],\end{equation}

where the first term, ${\rm \pi} D_0^2\gamma {\beta _{max}}^{2}(1 - \textrm{cos}\,\theta )/4$, represents the surface energy of the upper and lower surfaces of spreading films, and the second term, $2{\rm \pi} D_0^2\gamma /(3{\beta _{max}})$, stands for the surface energy of the periphery surface of spreading films. These terms are named E_{s ,m1} and E_{s ,m2}, respectively, for discussion later.

The viscous dissipation during spreading is contributed by three terms, i.e. E_{dis ,boundary}, E_{dis ,bulk} and E_{dis ,rim}. As a result, the viscous dissipation during spreading can be expressed as

(3.4)\begin{equation}{E_{dis}} = {E_{dis,boundary}} + {E_{dis,bulk}} + {E_{dis,rim}}.\end{equation}

The expressions of E_{dis ,boundary} and E_{dis ,bulk} can be obtained by the integration of the dissipation functions of the shear flow (ϕ_s) and the extensional flow (ϕ_e), respectively, expressed as

(3.5)\begin{gather}{E_{dis,boundary}} = \int_0^{{t_{sp}}} {\int_{{\varOmega _{boundary}}} {{\phi _s}\,\textrm{d}\varOmega \,\textrm{d}t} } ,\end{gather}

(3.6)\begin{gather}{E_{dis,bulk}} = \int_0^{{t_{sp}}} {\int_{{\varOmega _{bulk}}} {{\phi _e}\,\textrm{d}\varOmega \,\textrm{d}t} } ,\end{gather}

where $\varOmega_{bulk}$ is the volume occupied by the extensional flow, which approximates the volume of an entire droplet, and ${\varOmega _{boundary}}\sim D_{max}^2\delta$ is the volume of the boundary layer. The general dissipation function is

(3.7)\begin{equation}\phi = \mu \left( {\frac{{\partial {v_i}}}{{\partial {x_j}}} + \frac{{\partial {v_j}}}{{\partial {x_i}}}} \right)\frac{{\partial {v_i}}}{{\partial {x_j}}}.\end{equation}

Considering V_θ = 0, (3.7) can be simplified to

(3.8)\begin{equation}\phi = 2\mu \left[ {{{\left( {\frac{{\partial {V_r}}}{{\partial r}}} \right)}^2} + {{\left( {\frac{{{V_r}}}{r}} \right)}^2} + {{\left( {\frac{{\partial {V_z}}}{{\partial z}}} \right)}^2} + \frac{1}{2}{{\left( {\frac{{\partial {V_r}}}{{\partial z}} + \frac{{\partial {V_z}}}{{\partial r}}} \right)}^2}} \right].\end{equation}

At the macroscale, the no-slip condition holds so that ϕ_s can be simplified to ∼μ(V ₀/δ)², as recognised by previous studies (Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996; Ukiwe & Kwok Reference Ukiwe and Kwok2005; Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016). However, when the droplet diameter decreases from the macroscale to the nanoscale, the slip effect takes place and is continuously enhanced. To obtain the whole-scale expression of E_{dis ,boundary}, according to figure 3(d), the slip effect is incorporated to revise the ϕ_s to

(3.9) \begin{equation}{\phi _s}\sim \mu {\left( {\frac{{{V_0}}}{{\delta + \lambda }}} \right)^2} = {\left( {\frac{1}{{1 + \dfrac{\lambda }{\delta }}}} \right)^2} \times \mu {\left( {\frac{{{V_0}}}{\delta }} \right)^2} = {C_{slip}}\mu {\left( {\frac{{{V_0}}}{\delta }} \right)^2},\end{equation}

where λ is the average slip length, and C_slip is the slip factor to describe the slip effect with its value ranging from 0 (λ → ∞, free-slip) to 1 (λ = 0, no-slip). Integrating (3.5), the expression of boundary-layer dissipation is

(3.10)\begin{equation}{E_{dis,boundary}}\sim {C_{slip}}\mu {\left( {\frac{{{V_0}}}{\delta }} \right)^2}(\beta _{max}^2D_0^2\delta ){t_{sp}}.\end{equation}

According to Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016), t_sp = (D ₀/V ₀)(β_max − 1). This expression has been validated by impacts on both free-slip and no-slip surfaces, so it is expected to be universal throughout the scale. The thickness of the boundary layer is estimated by δ/D ₀ ∼ [t_sp/(Re D ₀/V ₀)]^1/2 (Eggers et al. Reference Eggers, Fontelos, Josserand and Zaleski2010; Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016). Substituting them in (3.10), the whole-scale expression of E_{dis ,boundary} is obtained as

(3.11)\begin{equation}{E_{dis,boundary}} = 0.6{C_{slip}}\frac{{\beta _{max}^2{{({\beta _{max}} - 1)}^{1/2}}}}{{R{e^{1/2}}}}{E_{k,0}},\end{equation}

where the prefactor of 0.6 is determined by Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) by data at the macroscale.

The velocity distribution in the bulk droplet meets the extensional flow, expressed as (Wang et al. Reference Wang, Wang, Xie, Liu, Wang, Yang, Gao and Wang2020b, Reference Wang, Wang, Cai, Ma, Yang, Zheng, Lee and Wang2024)

(3.12)\begin{equation}\left. {\begin{array}{*{20}{c@{}}} {{V_r} = \dfrac{r}{R}{V_{bulk}},}\\ {{V_z} =- \dfrac{{2z}}{R}{V_{bulk}},} \end{array}} \right\}\end{equation}

where V_bulk is an equivalent velocity characterising the strength of extensional flow. Substituting (3.12) in (3.8), the dissipation function for the extensional flow is obtained as

(3.13)\begin{equation}{\phi _e} = 48\mu \frac{{V_{bulk}^2}}{{{\beta ^2}D_0^2}}.\end{equation}

Subsequently, substituting (3.13) in (3.6) and integrating (3.6) with respect to space, the bulk dissipation is transformed into

(3.14)\begin{equation}{E_{dis,bulk}} = 8{\rm \pi} \mu {D_0}V_{bulk}^2\int_0^{{t_{sp}}} {\frac{1}{{{\beta ^2}(t)}}\,\textrm{d}t} .\end{equation}

To ensure consistent estimations of E_{dis ,boundary} and E_{dis ,bulk}, t_sp for calculating E_{dis ,bulk} is also t_sp = (β_max − 1)D ₀/V ₀. Here, the integration of time can be replaced by the integration of spreading factor, i.e. dt = (D ₀/V ₀) dβ, expressed as

(3.15)\begin{equation}{E_{dis,bulk}} = 8{\rm \pi} \mu D_0^2C_{bulk}^2{V_0}\int_1^{{\beta _{max}}} {\frac{1}{{{\beta ^2}}}\,\textrm{d}\beta } = 8{\rm \pi} \mu D_0^2C_{bulk}^2{V_0}\left( {1 - \frac{1}{{{\beta_{max}}}}} \right),\end{equation}

where C_bulk = V_bulk/V ₀.

For the rim dissipation, Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) found that it almost always consumes half of the initial kinetic energy when We > 30, expressed as

(3.16)\begin{equation}{E_{dis,rim}} = 0.5{E_{k,0}}.\end{equation}

This expression has been verified for impacting droplets on both free-slip and no-slip surfaces, so it is expected to be valid throughout the whole scale.

Substituting (3.1)–(3.3), (3.11), (3.15) and (3.16) in (1.1) and normalised by the initial surface energy, the model of β_max is finally obtained, expressed as

(3.17) \begin{align} \dfrac{1}{{12}}We + 1 & = \left[ {\dfrac{1}{4}\beta_{max}^2(1 - \textrm{cos}\,\theta ) + \dfrac{2}{{3{\beta_{max}}}}} \right] + \dfrac{1}{{20}}{C_{slip}}W{e^{3/4}}O{h^{1/2}}\beta _{max}^2{({\beta _{max}} - 1)^{1/2}}\notag\\ & \quad + 8C_{bulk}^2W{e^{1/2}}Oh\left( {1 - \dfrac{1}{{{\beta_{max}}}}} \right) + \dfrac{1}{{24}}We. \end{align}

On the left-hand side of this equation, the first and second terms stand for the initial kinetic energy and initial surface energy, respectively; on the right-hand side, the four terms represent the surface energy at the maximum spreading state, boundary-layer dissipation, bulk dissipation and rim dissipation, respectively.

Here, an analysis is presented to demonstrate how the proposed model can effectively predict the dynamics of impacting droplets across various scales, taking water, a representative low-viscosity liquid, as an example. Water droplets exhibit an extremely low Ohnesorge number at the macroscale, approximately O(10⁻³). Consequently, E_{dis ,bulk}, which is proportional to Oh, becomes negligible compared with E_{dis ,boundary}, which is proportional to Oh ^1/2. However, as droplet diameter decreases, reaching the microscale or the nanoscale, the value of Oh for water droplets approaches O(1). This shift results in the values of Oh and Oh ^1/2 becoming comparable, making E_{dis ,bulk} and E_{dis ,boundary} similarly significant. This observation elucidates why, as the diameter of low-viscosity droplets reduces from the macroscale to the nanoscale, viscous dissipation becomes increasingly essential, extending from the boundary layer to the entire droplet. Furthermore, this model accounts for the change in boundary-layer dissipation across scales. As droplet diameters decrease from the macroscale to the nanoscale, C_slip decreases due to increased slip length. This change leads to a spontaneous reduction in boundary-layer dissipation. This trend is observable in figure 2, where β_max increases with decreasing scale when We, Oh and θ are held constant. Thus, the model provides a comprehensive perspective on how the dynamics of impacting droplets evolves with scales.

The proposed model has two independent parameters, C_bulk and C_slip, which govern the strength of E_{dis ,bulk} and E_{dis ,boundary}, respectively. Given their independence, these parameters cannot be determined simultaneously by data fitting. Here, C_bulk, associated with the dissipation caused by droplet deformation, is expected to remain constant across all scales. Conversely, C_slip is influenced by the slip effect and thus varies depending on the scale of droplets. Specifically, C_slip approaches zero at the nanoscale and equals one at the macroscale. To determine these parameters, C_bulk should be first established by fitting data from either the nanoscale (C_slip = 0) or macroscale (C_slip = 1). Once C_bulk is determined, C_slip can be treated as an independent parameter.

This study has collected 16 series of data, including a diverse range of scales: five series at the macroscale from experiments (Series 1–5) (Stow & Hadfield Reference Stow and Hadfield1981; Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004; Antonini, Amirfazli & Marengo Reference Antonini, Amirfazli and Marengo2012; Abolghasemibizaki et al. Reference Abolghasemibizaki, Dilmaghani, Mohammadi and Castano2019; Du, Zhang & Min Reference Du, Zhang and Min2021b); seven series at the microscale from our MDPD simulations (Series 6–12); and four series at the nanoscale from MD simulations (Series 13–16) (Wang et al. Reference Wang, Wang, Gao, Yang, Wang and Chen2020a; Reference Wang, Wang, He, Zhang, Yang, Wang and Lee2022b; Zhang et al. Reference Zhang, Farokhirad, Lee and Koplik2014). Detailed information about these data series (Series 1–16) is available in supplementary material (table S2). In parameter determination, only one series of data from each scale is utilised to obtain C_bulk and C_slip. In contrast, the remaining data series are employed to validate the robustness and applicability of the proposed model.

Based on the above analysis, one series of nanoscale data (Series 14) is first employed to determine C_bulk by setting C_slip to 0. Subsequently, one series of macroscale data (Series 4) is used to validate the universality of the obtained C_bulk, fixing C_slip at 1. As figure 4(a) illustrates, with C_bulk = 0.4 and C_slip = 0, the proposed model successfully predicts β_max at the nanoscale. To convince us that C_bulk = 0.4 reasonably describes the velocity fields of impacting droplets, the velocity fields captured by simulations for the cases at We = 30.6, 45.7 and 73.9 in data Series 14 are used to validate it directly. Satisfactory agreement between the fitted C_bulk and the values of C_bulk by measuring the velocity fields is found (see supplementary material for details), directly proving this fitted result, C_bulk = 0.4. It is important to note that the significance of E_{dis ,bulk} is only pronounced for high Oh at the macroscale. Therefore, for the macroscale validation, β_max data of high-viscosity droplets (Series 4) from Du et al. (Reference Du, Zhang and Min2021b) are chosen. As depicted in figure 4(b), the proposed model, with C_bulk = 0.4 and C_slip = 1, also aligns satisfactorily with the β_max data at the macroscale. This occurrence effectively demonstrates that C_bulk remains constant across the scales. Furthermore, the model proposed by Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) is also included for comparison in figure 4(b). For droplets with high Oh at the macroscale, our model, which accounts for deformation-induced bulk dissipation, exhibits significantly improved accuracy. This comparison preliminarily validates the effectiveness and accuracy of our proposed model in capturing the dynamics of droplet impacts across various scales.

Figure 4.

(a,b) Fitting C_bulk by the data of impacting nanodroplets with Oh = 0.35 on superhydrophobic surfaces with θ = 148° (Wang et al. Reference Wang, Wang, He, Zhang, Yang, Wang and Lee2022b) and of impacting macroscale droplets with Oh = 0.56 on superhydrophobic surfaces with θ = 120° (Du, Zhang & Min Reference Du, Zhang and Min2021b). (c) Fitting C_slip by the data of impacting microscale droplets with Oh = 0.21 on hydrophobic surfaces with θ = 125°. (d) The relationship between C_slip and D ₀ for water at We = 300 and θ = 125°.

Since C_slip varies with droplet scale, it cannot be straightforwardly determined through simple data fitting. In addition to the traditional dimensionless group (We, Re and θ) dominating the impact dynamics, the slip length also becomes a key parameter when considering dynamics across scales, which can be non-dimensionalised by D ₀. Therefore, C_slip should be C_slip = f_slip(We, Re, λ/D ₀, θ). Considering the estimation of δ ∼ D ₀[(β_max − 1)/(Re)]^1/2 previously used in the derivation of (3.11), the key ratio, λ/δ, can be expressed as

(3.18)\begin{equation}\frac{\lambda }{\delta }\sim \frac{{\lambda \,R{e^{1/2}}}}{{{D_0}{{({\beta _{max}} - 1)}^{1/2}}}}.\end{equation}

Incorporating this scaling relationship into C_slip = (1 + λ/δ)⁻², with a dimensionless prefactor c, yields C_slip as

(3.19)\begin{equation}{C_{slip}} = {\left( {1 + \frac{{c\lambda \,R{e^{1/2}}}}{{{D_0}{{({\beta_{max}} - 1)}^{1/2}}}}} \right)^{ - 2}}.\end{equation}

The parameters, Re and λ/D ₀, are explicitly considered; while the dependence of We and θ is implicitly included in β_max. Here, λ is a material property, depending on the liquid involved and the substrate. In MDPD simulations, the liquid is fixed and only the contact angle is altered for the substrate. According to McBride & Law (Reference McBride and Law2009), λ is not sensitive to the wettability of molecularly smooth hydrophobic surfaces. Therefore, λ = 4.6 nm measured from MDPD simulations, as shown in figure 3(d), is available for all MDPD cases in this study. When other liquids are considered or the property of substrates changes significantly, λ should be remeasured to obtain the correct one corresponding to the impact condition.

The dimensionless prefactor, c, is determined by fitting MDPD data. As shown in figure 4(c), for MDPD data of 230 nm water droplets (Series 8), fixing λ = 4.6 nm and using c = 4, the proposed model can align well with the data. Figure 4(d) shows the relationship between C_slip and D ₀ at fixed We = 300, θ = 125° and λ = 4.6 nm for water. It is worth restating that the current literature has shown that C_slip = 0 for D ₀ < 10 nm (Li et al. Reference Li, Zhang and Chen2015; Wang et al. Reference Wang, Wang, Xie, Liu, Wang, Yang, Gao and Wang2020b; Xie et al. Reference Xie, Lv, Yang and Wang2020) and C_slip = 1 for D ₀ > 40 μm (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016). The predicted relationship of C_slip intriguingly approaches zero at the nanoscale and one at the macroscale, preliminarily validating the expression of C_slip.

3.4. Validation of the β_max model

In this section, the established model of β_max will be validated by data from the macroscale (Series 1–3 and 5), microscale (Series 6, 7 and 9–12) and nanoscale (Series 13, 15 and 16). For microscale data, C_slip is calculated by λ = 4.6 nm as mentioned above. Nonetheless, the lack of λ for macroscale and nanoscale data is an obstacle to validating the model. Fortunately, the orders of magnitude for D ₀ are O(1 mm) at the macroscale and O(10 nm) at the nanoscale; therefore, C_slip = 1 and C_slip = 0 are reasonably used for respective scale data according to the literature (Li et al. Reference Li, Zhang and Chen2015; Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016; Wang et al. Reference Wang, Wang, Xie, Liu, Wang, Yang, Gao and Wang2020b; Xie et al. Reference Xie, Lv, Yang and Wang2020). In addition, the widely recognised model of β_max (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016) that considers two kinds of viscous dissipation (E_{dis ,boundary} and E_{dis ,rim}) has been examined at the macroscale and is also tested by these series of data to present how the added or modified energy terms in the proposed model affect the prediction results.

Here, the two models are tested first by four series of data at the macroscale, as shown in figure 5(a), with Series 1, millimetre-sized water droplets (Oh = 0.002) impacting superhydrophobic surfaces (Antonini et al. Reference Antonini, Amirfazli and Marengo2012); Series 2, millimetre-sized water droplets (Oh = 0.003) impacting weakly hydrophobic surfaces (Stow & Hadfield Reference Stow and Hadfield1981); Series 3, millimetre-sized glycerol−water droplets (Oh = 0.21) impacting superhydrophobic surfaces (Abolghasemibizaki et al. Reference Abolghasemibizaki, Dilmaghani, Mohammadi and Castano2019); Series 5, millimetre-sized silicone oil droplets (Oh = 1.05) (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004) impacting weakly hydrophobic surfaces. Based on these data series, extensive parametric ranges are covered, i.e. 8 ≤ We ≤ 6815, 0.002 ≤ Oh ≤ 1.05, 90° ≤ θ ≤ 163°.

Figure 5.

(a) Validating the proposed model and the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) by macroscale data of β_max in wide ranges of We from 8 to 6815, Oh from 0.002 to 1.05 and θ from 90° to 163°. The data are marked by blue (Antonini, Amirfazli & Marengo Reference Antonini, Amirfazli and Marengo2012), green (Stow & Hadfield Reference Stow and Hadfield1981), red (Abolghasemibizaki et al. Reference Abolghasemibizaki, Dilmaghani, Mohammadi and Castano2019) and orange (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004) points, respectively. (b) The proportion of the surface energy of the periphery surface (E_{s ,m2}) to the one of the upper and lower surfaces (E_{s ,m1}) and (c) the proportion of the bulk dissipation (E_{dis ,bulk}) to the boundary-layer one (E_{dis ,boundary}).

For low-viscosity droplets (Series 1–2), both the proposed model and the model by Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) demonstrate equally high prediction accuracy when We exceeds 100. This similarity in accuracy is attributed to the fact that the added energy terms in the current model (E_{dis ,bulk} and E_{s ,m2}) are relatively insignificant compared with the existing terms (E_{dis ,boundary} and E_{s ,m1}). This inference is supported by the energy proportions (E_{dis ,bulk}/E_{dis ,boundary} and E_{s ,m2}/E_{s ,m1}) extracted from the present model, as depicted in figure 5(b,c). However, as We drops below 100, the dynamics accordingly changes: the spreading film at the maximum spreading state becomes thicker, and the maximum spreading factor (β_max) is reduced. Consequently, E_{s ,m2} becomes a non-negligible factor compared with E_{s ,m1}. In this lower We range, the present model, which includes E_{s ,m2}, delivers a more accurate prediction of β_max than the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016). This enhanced accuracy in lower We scenarios underscores the importance of incorporating comprehensive energy terms to fully capture the dynamics of droplet spreading.

For high-viscosity droplets (Series 3), both the present model and the model by Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) struggle to align with the data when We < 30. The current model tends to underestimate β_max in this range, which can be attributed to the difficulty in forming spreading rims at these Weber numbers. This limitation is also evident in the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016); however, their model tends to overestimate β_max because it does not consider E_{dis ,bulk} and E_{dis ,m2}. As We exceeds 30, the present model, which considers all forms of viscous dissipation, aligns satisfactorily with the data in both Series 3 and 4. This comprehensive approach is particularly crucial for high-viscosity droplets at the macroscale, where the effect of E_{dis ,bulk} becomes significant. This aspect of viscous dissipation is not included in the model by Wildeman et al., leading to their model showing a substantial overestimation of β_max in these scenarios. Therefore, including E_{dis ,bulk} in the present model provides a more accurate representation of the spreading dynamics for high-viscosity droplets.

At the microscale, six data series (Series 6, 7 and 9–12) are employed to evaluate both the present model and the model by Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016). The first five series encompass a wide range of We from 3 to 930 and θ from 86° to 174° for droplets with a diameter of D ₀ = 230 nm. The last series covers We from 50 to 1000, with θ = 125° and D ₀ = 1.15 μm. As depicted in figure 6(a,b), the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) fails to predict the results in the first five series. For We < 90, it overestimates β_max due to the non-negligible contribution of E_{s ,m2} compared with E_{s ,m1} in this low We range. More importantly, their model does not account for the increasing importance of E_{dis ,bulk} at the microscale, attributable to higher Oh due to reduced droplet scales. As We > 90, the model conversely underestimates β_max, indicating an overestimation of boundary-layer dissipation due to the slip effect as they fixed C_slip = 1. The present model, however, incorporates not only E _{dis, bulk} and E_{s ,m2} but also across-scale boundary-layer dissipation. Figure 6(a,b) shows that the present model aligns satisfactorily with MDPD simulation results. Further analysis of E_{dis ,bulk}/E_{dis ,boundary} and E_{s ,m2}/E_{s ,m1} extracted from the present model, as shown in figure 6(c,d), reveals that at low We, E_{s ,m2} and E_{dis ,bulk} are more significant than E_{s ,m1} and E_{dis ,boundary}. This observation supports the idea that the overestimation of the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) is the lack of these terms at low Weber numbers. At high Weber numbers, E_{s ,m2} and E_{dis ,bulk} become negligible, affirming the dominance of E_{s ,m1} and E_{dis ,boundary} and verifying the underestimation of β_max by the model Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) due to overestimated no-slip boundary-layer dissipation. To further validate the proposed expression for C_slip and the present model, Series 12 with larger D ₀ = 1.15 μm at θ = 125° is also included in the comparison, as shown in figure 6(e). As anticipated, the model shows satisfactory agreement with simulation data, underscoring the universality and reliability of the expression of C_slip.

Figure 6.

(a) Validating the proposed model and the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) by the MDPD data on (a) weak hydrophobic surfaces with θ from 100° to 109° and on (b) highly hydrophobic surfaces with θ from 138° to 174°. (c) The proportion of the surface energy of the periphery surface (E_{s ,m2}) to the one of the upper and lower surfaces (E_{s ,m1}) and (d) the proportion of the bulk dissipation (E_{dis ,bulk}) to the boundary-layer one (E_{dis ,boundary}). (e) Validating the models by a larger water droplet with D ₀ = 1.15 μm at θ = 125°.

Eventually, the two models are tested by three series of nanoscale data, with Series 13, mW nanodroplets (Oh = 0.35) impacting hydrophobic surfaces (Wang et al. Reference Wang, Wang, Gao, Yang, Wang and Chen2020a,Reference Wang, Wang, He, Zhang, Yang, Wang and Leeb); Series 15, mW nanodroplets (Oh = 0.35) impacting superhydrophobic surfaces; Series 16, Ar nanodroplets (Oh = 0.48) impacting superhydrophobic surfaces (Zhang et al. Reference Zhang, Farokhirad, Lee and Koplik2014). These data series span a wide range of We, Oh and θ. The enhanced slip effect at the nanoscale significantly reduces the boundary-layer viscous dissipation, rendering it almost negligible. Additionally, the increased Oh at this scale makes E_{dis ,bulk} more significant. Similar to observations at the microscale, the model Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) tends to overestimate β_max in the low We range but underestimates it in the high We range, as shown in figure 7(a). In contrast, the proposed model effectively accounts for these changes, accurately capturing the boundary-layer viscous dissipation at the nanoscale. As shown in figure 7(a), the proposed model successfully predicts the data across these nanoscale series. In addition, the energy proportion E_{s ,m2}/E_{s ,m1} is extracted from the proposed model (figure 7b). This ratio reinforces the reasoning behind the inaccuracy of the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) to predict β_max at the nanoscale, emphasising the importance of incorporating E_{dis ,bulk} and including slip effect for E_{dis ,boundary} to achieve accurate predictions in droplet impact dynamics.

Figure 7.

(a) Validating the proposed model and the model of Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) by the data of β_max for impacting droplets at the nanoscale in wide ranges of We, Oh and θ from 1 to 290, from 0.35 to 0.48 and from 125° to 180°, respectively (Wang et al. Reference Wang, Wang, Gao, Yang, Wang and Chen2020a, Reference Wang, Wang, He, Zhang, Yang, Wang and Lee2022b; Zhang et al. Reference Zhang, Farokhirad, Lee and Koplik2014). (b) The proportion of the surface energy of the periphery surface (E_{s ,m2}) to the one of the upper and lower surfaces (E_{s ,m1}).