3.1. Energy formulation and mobility of the CL

Our discussion about CL dissipation is framed around the mechanical energy balance for dynamic wetting (Dussan & Davis Reference Dussan and Davis1986, (2.7)), with modified notation,

(3.3)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t}\left(K+\mathcal{A}\right)={-}\int_V \operatorname{Tr}\left(\boldsymbol{\mathsf{T}}\boldsymbol{\cdot} \boldsymbol{\mathsf{D}}\right)\mathrm{d}V +2\mu \oint_{\varGamma} \frac{\dot{R}^{2} \sin{\alpha}^{2}}{\alpha - \sin{\alpha}\cos{\alpha}}\, \mathrm{d}l - \mathcal{D}_{CL}.\end{equation}

The first term on the right-hand side represents the bulk dissipation of the system, with $\boldsymbol{\mathsf{T}}$ being the Cauchy stress tensor and $\boldsymbol{\mathsf{D}}$ the deformation tensor. The second term represents dissipation in the viscous boundary layer integrated about the CL, where $\mu$ is the bulk viscosity, $\dot {R}$ the CL velocity and $\alpha = \bar {\alpha }+\Delta \alpha$. The third term represents dissipation from work done at the moving CL.

The kinetic energy associated with the velocity field $\boldsymbol{\mathsf{u}}$ is defined as

(3.4)\begin{equation} K=\tfrac{1}{2}\int_V \rho\boldsymbol{\mathsf{u}}^{2}\,\mathrm{d}V,\end{equation}

and the surface energy as

(3.5)\begin{equation} \mathcal{A}=\sigma A_{lg}+\sigma_{ls} A_{ls}+ \sigma_{sg} A_{sg}, \end{equation}

which is the sum of the interfacial areas $A$ weighted by the associated surface tensions $\sigma$. The subscripts $lg$, $ls$ and $sg$ denote liquid/gas, liquid/solid and solid/gas, respectively.

Applying the Young–Dupré equation $\sigma _{sg}-\sigma _{ls}=\sigma _{lg} \cos \bar {\alpha }$ and noting that ${\mathrm {d} }/{\mathrm {d} t}(A_{sg}+A_{ls})=0$, one can reduce (3.5) to

(3.6)\begin{equation} \mathcal{A}=\sigma ( A_{lg}- \cos{\bar{\alpha}} \,A_{ls}).\end{equation}

Note that the same result would be reached by assuming $\sigma _{sg}=0$, as per Horibe et al. (Reference Horibe, Fukusako and Yamada1996). Alternatively, the effective interfacial area can also be defined as

(3.7)\begin{equation} \mathcal{A} \equiv \sigma( A_{lg}-\mathcal{M} A_{ls}) \end{equation}

where the wettability parameter is defined as $\mathcal {M} \equiv (\sigma _{sg}-\sigma _{ls})/\sigma$. If material properties are such that $\mathcal {M}$ falls outside $(-1,1)$, the liquid completely dewets ($\mathcal {M} < -1$) or wets ($1 < \mathcal {M}$) the solid. Otherwise, the liquid is partially non-wetting ($-1 < \mathcal {M}<0$) or partially wetting ($0 < \mathcal {M}<1$). Equilibrium requires $\mathcal {M} = \cos {\bar {\alpha }}$ (Bangham & Razouk Reference Bangham and Razouk1937), known as the Young–Dupré equation, thus relating (3.5) to (3.6). Hence, partially wetting and partially non-wetting can be distinguished by the neutrally wetting case, $\bar {\alpha } = 90 ^{\circ }$ ($\mathcal {M}=0$). Our primary interest is in partially wetting and partially non-wetting systems with $\bar {\alpha }$ not too far from $90 ^{\circ }$.

3.2. Inertial spreading ‘fast time scale’

Dissipation during impact occurs across two different time scales. The first is the fast time scale associated with spreading in the interval $[t_o,t_p]$, which is approximately $O(10^{-2})$. For normal impact onto a homogeneous surface, we observe that CL spreading is azimuthally uniform such that $\mathrm {d}l=R\,\mathrm {d}\phi$. The liquids used in our experiments are relatively inviscid, $\mu \ll 1$, and we assume that liquid away from the CL does no work on the substrate, a situation which would be obtained if the no-slip condition were respected. These assumptions are rationalized by noting the small capillary numbers $ {\textit {Ca}}= {\textit {We}}/ {\textit {Re}}=0.01 - 0.04$ used in the experiments and validated by evaluating the dissipation in the viscous boundary layer using experimental data (a dominant balance argument). Together, these assumptions justify the neglect of the first two terms on the right-hand side of (3.3), which leaves after integration

(3.8)\begin{equation} \{K+\mathcal{A}\}_{o}-\{K+\mathcal{A}\}_{p}=\int_{t_o}^{t_p} 2{\rm \pi} R\dot{R}\mathcal{F_Y} \,\mathrm{d}t, \end{equation}

where $\{K+\mathcal {A}\}_{o}$ denotes the drop's pre-impact energy and $\{K+\mathcal {A}\}_{p}$ the drop's post-pinning energy. The right-hand side of (3.8) represents the energy dissipated at the CL during spreading. Implications from (3.8) are that the only form of dissipation in the system during spreading is from the work done by the CL as it moves along the surface, consistent with Carlson et al. (Reference Carlson, Bellani and Amberg2012) and Xia & Steen (Reference Xia and Steen2018).

3.3. Viscous dissipation ‘slow time scale’

The second time scale is the slow time scale during post-pinning in the interval $[t_p,t_{\infty }]$, which is approximately $O(1)$. Here, viscous dissipation dominates and integrating the energy balance over $[t_p,t_{\infty }]$ yields

(3.9a,b)\begin{equation} \{K+\mathcal{A}\}_{\infty}-\{K+\mathcal{A}\}_{p}=\int_{t_p}^{t_{\infty}} \mathcal{D}_{bulk} \,\mathrm{d}t, \quad \mathcal{D}_{bulk} \equiv{-}\int_V \textrm{tr}\left(\boldsymbol{\mathsf{T}}\boldsymbol{\cdot}\boldsymbol{\mathsf{D}}\right)\mathrm{d}V, \end{equation}

showing that the viscous dissipation $\mathcal {D}_{bulk}$ is effectively decoupled from CL dissipation $\mathcal {D}_{CL}$ during post-pinning. Here, $\{K+\mathcal {A}\}_{p}$ denotes the post-pinning energy, $\{K+\mathcal {A}\}_{\infty }$ the final resting energy, $t_p$ the instant the CL comes to rest and $t_{\infty }$ the instant when the post-pinning $lg$ vibrations have come to rest.

3.4. ‘Pre-impact’ energy

Quantification of the drop's pre-impact energy $\{K+\mathcal {A}\}_{o}$ is the first step in discussing the energetics of impact. To quantify $\{K+\mathcal {A}\}_{o}$, we assume that the drop is a sphere with some impacting velocity inferred from the fall height. Then we can write

(3.10)\begin{equation} \{K+\mathcal{A}\}_{o}=\{\rho Vgh+4{\rm \pi}\sigma R_d^{2}\}, \end{equation}

where the height $h$ is measured in the laboratory frame. The volume $V$ and radius $R_d$ are controlled by the gauge of the syringe tip. Alternatively, the pre-impact energy can be written as

(3.11)\begin{equation} \{K+\mathcal{A}\}_{o}=\left\{\tfrac{1}{2}\rho Vu_o^{2}+4{\rm \pi}\sigma R_d^{2}\right\},\end{equation}

where the impact velocity, $u_o$, is measured directly from the experimental images using MATLAB to track the drop's centre of mass. The variation between these two methods of measurement was on average ${<}5\,\%$. Hereafter, all calculations were performed using (3.11).

We assume based on observation that the surface energy contribution of the capillary waves generated when the drop breaks off the needle tip can be ignored. At higher impacting velocities these waves are sufficiently damped out before impact, and at lower impacting velocities the contribution to the overall surface energy remained negligible. Similarly we ignored oblate–prolate vibration of the drop prior to impact, a phenomenon shown to be important for vorticity generation during impact (Peck & Sigurdson Reference Peck and Sigurdson1994).

3.5. ‘Post-pinning’ energy

The post-pinning energy $\{K+\mathcal {A}\}_{p}$ at the instant of pinning $t=t_p$ is defined as

(3.12)\begin{equation} \{K+\mathcal{A}\}_{p}=\tfrac{1}{2}\int_V \rho\boldsymbol{\mathsf{u}}^{2}\,\mathrm{d}V+\sigma\left( A_{lg} - A_{ls}\cos{\left(\alpha_p\right)}\right),\end{equation}

which is the sum of the drop's kinetic and interfacial energies. In order to calculate $\{K+\mathcal {A}\}_{p}$ from experimental images, it would be helpful to simplify (3.12) to avoid calculating the velocity field $\boldsymbol{\mathsf{u}}$ in the entire drop domain. We assume an incompressible $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol{\mathsf{u}}=0$ and irrotational potential flow $\boldsymbol{\mathsf{u}}=\boldsymbol {\nabla }\varPsi$ and use the identity $\boldsymbol {\nabla } \boldsymbol {\cdot } (\varPsi \boldsymbol{\mathsf{u}}) = \boldsymbol {\nabla }\varPsi \boldsymbol {\cdot }\boldsymbol{\mathsf{u}} + \varPsi \boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol{\mathsf{u}}$ to rewrite (3.12) as

(3.13)\begin{equation} \{K+\mathcal{A}\}_{p}=\frac{\rho}{2}\int_V(\boldsymbol{\nabla}\varPsi\boldsymbol{\cdot}\boldsymbol{\mathsf{u}}+\varPsi \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{\mathsf{u}})\,\mathrm{d}V +\sigma\left( A_{lg} - A_{ls}\cos{\left(\alpha_p\right)}\right).\end{equation}

Application of Green's identity $\int _V{(\boldsymbol {\nabla }\varPsi \boldsymbol {\cdot } \boldsymbol {\nabla } \varPsi + \varPsi \nabla ^{2}\varPsi )}\,\mathrm {d}V = \int _{\partial V}{\varPsi (\boldsymbol {\nabla }\varPsi \boldsymbol {\cdot }\boldsymbol {n})}\,\mathrm {d}S$ and noting that $\nabla ^{2}\varPsi =0$ for incompressible flows allows us to rewrite (3.13) as

(3.14)\begin{equation} \{K+\mathcal{A}\}_{p}=\frac{\rho}{2}\int_S \varPsi \frac{\mathrm{d}\varPsi}{\mathrm{d}\textrm{n}} \mathrm{d}S +\sigma\left( A_{lg} - A_{ls}\cos{\left(\alpha_p\right)}\right).\end{equation}

This is a boundary integral approach. Here, ${\boldsymbol {n}}$ is the unit outward normal to the liquid/gas interface, which we can identify with the radial coordinate $r$ defined in the spherical coordinate system shown in figure 2(c). For small interface disturbances, $\varPsi \approx r\,\mathrm {d}\varPsi /\mathrm {d}r\equiv r u_r$ at the $lg$ boundary and we can write (3.14) as

(3.15)\begin{equation} \{K+\mathcal{A}\}_{p}=\frac{\rho}{2}\int_0^{2{\rm \pi}}\int_{\theta_0}^{\rm \pi} u_r^{2}r^{3}\sin{(\theta)} \,\mathrm{d}\theta\,\mathrm{d}\phi +\sigma\left( A_{lg} - A_{ls}\cos{\left(\alpha_p\right)}\right),\end{equation}

which can be evaluated directly from the side view experimental images, as shown in figure 1.

4.1. Perfectly slipping CL

This important special case will aid in our interpretation of the experimental results. Models of CL response relate $\Delta \alpha$ to CL speed, sometimes denoted $U$, where $U=\dot {R}$ in this study. One of the simplest of models of CL response is ‘perfect slip’, where the CL moves at constant CA independent of $\dot {R}$,

(4.1)\begin{equation} \alpha =\begin{cases} \alpha_A & \text{for} \ \dot{R} > 0 \\ \alpha_R & \text{for} \ \dot{R} < 0. \end{cases} \end{equation}

Here, $\alpha _A$ and $\alpha _R$ are the advancing and receding CAs. Along both the advancing and receding slip legs, the driving forces $\mathcal {F}_{\mathcal {Y}A}$ and $\mathcal {F}_{\mathcal {Y}R}$ are constant,

(4.2)\begin{gather} \mathcal{F}_{\mathcal{Y}A} \equiv{-}\sigma \left(\cos{\alpha_A}-\cos{\bar{\alpha}}\right)\quad \text{for}\ \dot{R} > 0, \end{gather}

(4.3)\begin{gather}\mathcal{F}_{\mathcal{Y}R} \equiv{-}\sigma \left(\cos{\alpha_R}-\cos{\bar{\alpha}}\right)\quad \text{for}\ \dot{R} < 0. \end{gather}

An immediate consequence is that the right-hand side of (3.8) can be integrated along either leg (advancing here),

(4.4)\begin{equation} \left.\{K+\mathcal{A}\}\right\vert_{t_0}^{t_p}= \left.({\rm \pi} R^{2} \mathcal{F}_{\mathcal{Y}A}) \right\vert_{t_0}^{t_p}. \end{equation}

Noting that $({\rm \pi} R^{2})\vert _{t_0}^{t_p} = A_{ls} \vert _{t_0}^{t_p}$ and that $\mathcal {F}_{\mathcal {Y}A}/\sigma = \mathcal {M} - \mathcal {M_A}$, one can rearrange (4.4) into the form,

(4.5)\begin{equation} \left.\{ K+(\mathcal{A}_{lg}- \mathcal{M}_{\mathcal{A}} \mathcal{A}_{ls})\}\right\vert_{t_0}^{t_p}=0. \end{equation}

This striking result says that the loss of kinetic energy along a slip leg precisely equals the change in interfacial energy that would occur as if on a surface with wettability $\mathcal {M}_{\mathcal {A}}$.

Graphically, we can realize the result of (4.5) by looking at the cyclic driving diagram for a P0 surface with a given wettability, $\mathcal {M_A}$, shown in figure 4. In figure 4(a), we see the results for a P0 surface driven from below at the $[2,0]$ resonance mode frequency, nomenclature from Bostwick & Steen (Reference Bostwick and Steen2014) and experimental procedure from Xia & Steen (Reference Xia and Steen2018). The resonant response of the drop causes CL motion. The blue data represent ‘slip’ for $\mathcal {F}_{\mathcal {A}}$ and $\mathcal {F}_{\mathcal {R}}$ when $\Delta \hat {\alpha }\approx 1$ and $\Delta \hat {\alpha }\approx -1$, respectively. The red data represent ‘stick’, when the CL is stationary. For a ‘perfect slip’ system, only the blue and red lines would be present, as shown in figure 4(b). However, our system only approximates this ‘perfect slip’ scenario, with the green data representing the time where the CL is moving while $\Delta \alpha$ is changing and $\mathcal {F}_{\mathcal {Y}}$ is no longer constant.

Figure 4.

Cyclic diagram showing deviation in CA $\Delta \hat {\alpha }$ against CL radius $\hat {R}$ for a drop on a P0 substrate driven from below. Blue corresponds to the region of perfect slip where the CA is constant while the CL advances or recedes. Red corresponds to an immobile CL. Green corresponds to CL advancing or receding with a changing CA. Time proceeds clockwise. The dissipation during one cycle of CL motion is proportional to the area inside the rectangle, $\varDelta (\Delta \hat {\alpha })\Delta \hat {R}$ (Xia & Steen Reference Xia and Steen2020). (a) Laboratory generated results for 30 oscillations of a $20\ \mathrm {\mu }\textrm {l}$ sessile drop on P0 driven at an acceleration, $a = 5.7\ \textrm {m}\ \textrm {s}^{-2}$. (b) Idealized diagram for the perfect slip case.

For near neutral wetting systems, $\bar {\alpha }\approx {\rm \pi}/2$ and $\Delta \alpha \ll {\rm \pi}/2$, Xia & Steen (Reference Xia and Steen2020) showed that the damping ratio of the harmonically driven drop is a measure of dissipation within the drop, and that the Young's force, (3.1), can be expanded in Taylor series resulting in the damping ratio being proportional to the area of the region enclosed in the $\Delta \hat {\alpha } - \hat {R}$ plane (cf. figure 4). For our system the only source of dissipation within the drop is from CL motion, implying that the area of the region enclosed in the $\Delta \hat {\alpha } - \hat {R}$ plane can be used to quantitatively interpret $\mathcal {D}_{CL}$.

4.2. Inertial ‘spreading’ and CL dissipation

Up until now we have assumed that CL dissipation, $\mathcal {D}_{CL}$, is the primary form of dissipation during spreading, accounting for the difference between a drop's pre-impact, $\{K+\mathcal {A}\}_o$, and post-pinning, $\{K+\mathcal {A}\}_p$, energy. To test the validity of this assumption, the right-hand side of (3.8) was quantified. Throughout spreading, $R$ and $\dot {R}$ were measured from experimental images in MATLAB using subpixel edge detection techniques (Trujillo-Pino Reference Trujillo-Pino2019), and $\Delta \alpha$ was measured using the software DROPImageAdvanced. Figure 5 shows the results of directly integrating the CL dissipation (right-hand side of (3.8)) compared with the difference between the pre-impact and post-pinning energy of the drop (left-hand side of (3.8)). For low $K_o/\mathcal {A}_o$ we find that (3.8) holds, implying that the primary form of dissipation in the system is CL dissipation. However, as $K_o/\mathcal {A}_o$ increases the accuracy of (3.8) decreases. This deviation most likely occurs because the inviscid assumption fails, consequently increasing the contribution to the dissipation from the viscous boundary layer (Pasandideh-Fard et al. Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996) and from bulk dissipation, i.e. the second and first right-hand side terms from (3.3), respectively.

Figure 5.

Scaled spreading energy against $K_o/\mathcal {A}_o$ for (a) a $8.4$ and $13.8\ \mathrm {\mu }\textrm {l}$ drop on an F0 surface and (b) a $6.3\ \mathrm {\mu }\textrm {l}$ drop on an A0 surface. Solid and empty symbols denote the right- and left-hand sides of (3.8), respectively. For small $K_o/\mathcal {A}_o$, CL dissipation ($\mathcal {D}_{CL}$) is the primary form of dissipation in the system. Increasing $K_o/\mathcal {A}_o$ increases bulk and viscous boundary layer dissipation.

Figure 6 shows the results of typical spreading events for drop impacts on F0 and A0 surfaces at $K_o/\mathcal {A}_o = 0.10$ and $K_o/\mathcal {A}_o = 0.33$, respectively. For hydrophobic systems such as P0, F0, F5 and F10 at moderate $K_o/\mathcal {A}_o$, as for example in figure 6(a), the motion of the CL is under-damped, oscillating about its pinning angle $\alpha _p$. Figure 6(b) shows that, as $K_o/\mathcal {A}_o$ decreases, the drop approaches critical damping, only needing one advancing leg before enough energy is dissipated to halt CL motion. Interestingly, for drop impacts onto hydrophilic systems such as F100 and A0 (cf. figure 6c,d) the motion of the CL followed the same qualitative behaviour of the low $K_o/\mathcal {A}_o$ hydrophobic impact, regardless of $K_o/\mathcal {A}_o$. A similar phenomenon was observed for F100, however, a long relaxation back to equilibrium $\bar {\alpha }$ did occur after the drop's $lg$ interface came to rest, $t=t_{\infty }$.

Figure 6.

CL dynamics dependence on substrate wettability for (a) F0 with $K_o/\mathcal {A}_o = 0.33$, (b) F0 with $K_o/\mathcal {A}_o = 0.1$, (c) A0 with $K_o/\mathcal {A}_o = 0.33$, (d) A0 with $K_o/\mathcal {A}_o = 0.1$. CL radius against scaled time (left), CA hysteresis $\Delta \alpha$ against scaled time (centre) and $\Delta \alpha$ against scaled $R$ (right). Green symbols represent an advancing CL, red symbols a receding CL and blue symbols a pinned CL.

The rightmost panels in figure 6 are analogous to the cyclic diagram shown in figure 4. In the case of the driven drop, the work inputted into the system equals the work dissipated by the system, causing the loop to close. For the freely impacting drop no work is inputted into the system, energy is irreversibly dissipated at the CL and the result is the spiral decay to the rest state shown in figure 6. Here the rest state of the CL is represented by $\hat {R} = 1$.

This observed behaviour is similar to the experiments done by Noblin et al. (Reference Noblin, Buguin and Brochard-Wyart2004), where a drop was subject to a forced vibration at resonant frequencies, and the transition from a pinned CL to a mobile CL was observed. When the drop was not forced at a high enough acceleration the authors observed that $\Delta \alpha$ oscillated within its hysteresis, $[\Delta \alpha _r,\Delta \alpha _a]$ with no CL motion. Above a certain threshold acceleration they observed the transition to a moving CL. In our experiments we observe the opposite. At the beginning of spreading the drop's CL is moving with $\Delta \alpha >\Delta \alpha _a$, and only moves when $\Delta \alpha$ is outside the CL hysteresis shown in the middle column of figure 6. Once enough energy is dissipated at the CL, the drop does not contain enough energy for the $lg$ interface to deform in such a way for $\Delta \alpha$ to exceed the CL hysteresis, causing $\Delta \alpha$ to oscillate only within the hysteresis range. Consequently, the CL comes to rest at a static contact angle $\alpha _p$ determined from its rest position.

For low $K_o/\mathcal {A}_o$ it was shown that (3.8) holds and contact-line dissipation $\mathcal {D}_{CL}$ is the primary form of dissipation in the system. In figure 7 we see that the drop's pre-impact energy $\{K+\mathcal {A}\}_o$ must decay below a certain threshold for the CL to come to rest, where this threshold is the post-pinning energy, $\{K+\mathcal {A}\}_p$. Otherwise stated, for a drop with a pre-impact energy above the post-pinning threshold, CL motion ensues until enough energy is dissipated to cause the CL to come to rest. Accordingly, the motion of the CL during spreading acts as a natural filter for the energy of the impacting drop. The energy that remains is the post-pinning energy $\{K+\mathcal {A}\}_p$, which contributes to the post-pinning $lg$ vibrations, which we discuss in the next section.

Figure 7.

The pre-impact energy, $\{K+\mathcal {A}\}_{o}/\mathcal {A}_o$, and post-pinning energy, $\{K+\mathcal {A}\}_{p}/\mathcal {A}_o$, against $K_o/\mathcal {A}_o$ for substrate (a) F0; $V = 8.39\ \mathrm {\mu } \textrm {l}$, $\alpha _p = 98.9^{\circ }$ and (b) A0; $V = 6.3\ \mathrm {\mu } \textrm {l}$, $\alpha _p = 47 - 73^{\circ }$. The post-pinning energy is invariant to the pre-impact energy when $\alpha _p$ is invariant to $K_o/\mathcal {A}_o$ as for F0, and decreases with increasing pre-impact energy when $\alpha _p$ decreases with increasing $K_o/\mathcal {A}_o$ as for A0.

4.3. ‘Post-pinning’ energy

Figure 7 plots the post-pinning energy $\{K+\mathcal {A}\}_p$ from (3.15) against $K_o/\mathcal {A}_o$ for sets of experiments on the F0 and A0 surfaces. Interestingly, we find that regardless of $K_o/\mathcal {A}_o$, $\{K+\mathcal {A}\}_{p}/\mathcal {A}_o$ remains approximately constant for a given liquid/solid/gas combination when the rest angle $\alpha _p$ is approximately constant, as for F0 shown in figure 7(a). Note that the pinning angle $\alpha _p$ is not necessarily the equilibrium angle $\bar {\alpha }$, but can fall anywhere within the system's hysteresis interval $[\alpha _r,\alpha _a]$, graphically represented in figure 2(b). For systems such as F0, this hysteresis interval is relatively small, resulting in the approximately constant $\alpha _p$ and $\{K+\mathcal {A}\}_{p}/\mathcal {A}_o$, as shown in figure 7(a). For systems such as A0, however, a large hysteresis interval led to a large range of $\alpha _p$. Figure 7(b) shows how decreasing $\alpha _p$ with increasing $K_o/\mathcal {A}_o$ decreased the post-pinning energy $\{K+\mathcal {A}\}_{p}/\mathcal {A}_o$. This result establishes a direct connection between the post-pinning energy $\{K+\mathcal {A}\}_{p}/\mathcal {A}_o$ and the contact-angle hysteresis through $\alpha _p$. We explore this result further in the next sections.

In order to predict the post-pinning energy $\{K+\mathcal {A}\}_p$, it will be helpful to reframe (3.8) in terms of the energy contributions made by each advancing and receding segment of the spreading event. From the right-hand panels of figure 6 it would be reasonable to assume that each advancing and receding CL event occurs at a constant $\Delta \alpha$, as was rationalized in § 4.1, allowing us to rewrite (3.8) as a sum of ‘perfect slip’ legs

(4.6)\begin{equation} \{K+\mathcal{A}\}_{o}-\{K+\mathcal{A}\}_{p}=\sum_{n=1}^{m}\int_{R_n-1}^{R_{n}} 2{\rm \pi} R \mathcal{F_Y}_n \,\mathrm{d}R.\end{equation}

Then we can imagine that as $K_o/\mathcal {A}_o\rightarrow 0$ only one spreading event will occur (cf. figure 6). We can also assume that the only source of dissipation for this singular spreading event will be CL dissipation from (3.8), since as $K_o/\mathcal {A}_o\rightarrow 0$, (3.8) holds (cf. figure 5). Last, we assume that the advancing $\Delta \alpha$ for this singular spreading event will be the minimal $\Delta \alpha$ at which advancing contact-line spreading occurs, $\Delta \alpha _a$. With these assumptions, we can infer $\{K+\mathcal {A}\}_p$ by manipulating (4.6) and rewriting $R$ in terms of $\alpha _p$,

(4.7)\begin{equation} \frac{\{K+\mathcal{A}\}_p}{\mathcal{A}_o}=1-\frac{1}{4^{{1}/{3}}}\left[\frac{\sin^{3}{\alpha_p}}{2-3\cos{\alpha_p}+\cos^{3}{\alpha_p}}\right]\left(\cos{\bar{\alpha}}-\cos\left({\bar{\alpha}+ \Delta\alpha_a}\right)\right).\end{equation}

We will contrast this theoretical prediction with experimental measurements of the post-pinning energy in § 6.

5.1. Modal decomposition

During post-pinning the drop's $lg$ interface vibrates dissipating energy through viscous dissipation before coming to rest. In this section we aim to understand how these freely disturbed $lg$ interfacial vibrations can be understood using the work of Bostwick & Steen (Reference Bostwick and Steen2014) and Bostwick & Steen (Reference Bostwick and Steen2016).

We find that the drop frequency spectrum from impact is the same as a forced drop. Figure 8(a) shows a frequency response diagram computed by taking the FFT of the time trace of the $lg$ interface at the drop's apex during post-pinning. The resonance peaks correspond to the natural frequencies associated with the $l=0$ vibrational modes for the sessile drop predicted by Bostwick & Steen (Reference Bostwick and Steen2014). This procedure can be repeated for all of our substrates.

Figure 8.

FFT of the post-pinning $lg$ interface. Modes are categorized by their wavenumber, $[k,l]$ (Bostwick & Steen Reference Bostwick and Steen2014). (a) Drop amplitude against frequency shows a series of resonance peaks associated with axisymmetric modes $[k,0]$. (b) Scaled resonance frequency against contact angle $\alpha _p$, as it depends upon the liquid/solid/gas system.

Figure 8(b) plots the scaled frequencies $\hat {f}=f\sqrt {\rho a^{3}/\sigma }$, where $a$ is the footprint radius of the drop, during post-pinning over a large range of CAs $\alpha _p$. The agreement between experiment and theory is good. Experiments ranged from $K_o/\mathcal {A}_o=0$ to $1.3$, where we have scaled the drop's pre-impact kinetic energy $K_o$ by its pre-impact interfacial energy $\mathcal {A}_o$ in order to account for the variation in drop size. This range encompasses impacting energies from when the drop is just touching the surface to energies large enough to cause partial rebound. Interestingly, variation of $K_o/\mathcal {A}_o$ did not shift the measured $\hat {f}$ from that predicted by the linear theory (Bostwick & Steen Reference Bostwick and Steen2014) despite the increasingly violent behaviour of the drop's interface during spreading, as seen in figure 1. However, variation in $K_o/\mathcal {A}_o$ did affect $\alpha _p$ for both the hydrophilic A0 and F100 systems, with increases in $K_o/\mathcal {A}_o$ leading to decreases in $\alpha _p$. For $\alpha _p<60^{\circ }$, $\hat {f}$ was suppressed relative to the values predicted by Bostwick & Steen (Reference Bostwick and Steen2014), as was observed and discussed in Chang et al. (Reference Chang, Bostwick, Daniel and Steen2015).

We have shown that the frequencies at which the drop's $lg$ interface vibrates during post-pinning match the frequencies from theory, but knowing only that the frequencies at the apex of the drop match those of the $l=0$ mode shapes does not ensure that the vibration of the $lg$ interface as a whole can be decomposed into modal components or that these modal components are exclusively $l=0$ vibrational modes. In order to show that the vibrations of the drop's $lg$ interface excited during normal surface impact can be decomposed into exclusively $l=0$ modal components, it is important to perform this same analysis at multiple points along the drop's $lg$ interface, as illustrated in figure 9(a). In performing such an analysis, we found that regardless of the location $s$, the FFT only picks up frequencies corresponding to the $l=0$ shapes, allowing us to conclude that for normal impact $l=0$ modes are excited exclusively. We postulate that this exclusivity occurs because the motion of the centre of mass of the drop remains normal to the surface (Wesson & Steen Reference Wesson and Steen2020).

Figure 9.

Drop oscillation shapes in (a) definition sketch and (b) as experimentally determined by the normalized FFT taken at different locations $s$ along the drop's interface. Empty symbols are experimental values and solid lines are predictions from theory. Maximum deflection for each mode occurs at $s=0$. Each mode has a value of $s$ that maximizes its contribution to the $lg$ disturbance and values of $s$ that minimize it.

In order to capture the modal decomposition of the shape of the drop from the FFT at the start of post-pinning, we restrict the time interval of the FFT to the minimum allowed by the Nyquist sampling theorem, beginning at $t_p$. Figure 9(b) plots the amplitude of the resonance peak for the $[k,0]$ mode from the FFT at different locations $s$ along the $lg$ interface and overlays these points with the $lg$ interface shape predicted by theory for each mode. These locations are defined by the ray, $r=r(s,0,t)$ extending from the centre of the spherical cap to drop's $lg$ interface from $s=0$ to the CL $s=\alpha _p$. Notably we see that when theory predicts the drop shape has a node that the FFT has a corresponding minimum. For all the detected mode shapes, the prediction from theory for the shape of the drop matches well with the decomposed shapes from the experiment, especially for the two most dominant shapes, the $[2,0]$ and $[4,0]$, allowing us to conclude that during post-pinning the motion of the drop's $lg$ interface can be meaningfully decomposed into mode shapes predicted by Bostwick & Steen (Reference Bostwick and Steen2014). This result is important for quantification of the post-pinning energy.

5.2. Modal decay

Viscous dissipation causes amplitude decay of the vibrations during post-pinning. The study of the decay of vibrational modes dates back to Lamb (Reference Lamb1932), who developed approximate expressions for the damping rates of different vibrational modes of a free drop. Notable studies since include that of Valentine, Sather & Heideger (Reference Valentine, Sather and Heideger1965) who considered the case when both fluids, in our case liquid/gas, have a non-zero viscosity, and that of Prosperetti et al. (Reference Prosperetti1980) who provide a detailed comparison of theoretically predicted damping rates for similar viscous drops in immiscible solutions to the experimental results of Valentine et al. (Reference Valentine, Sather and Heideger1965). Miller & Scriven (Reference Miller and Scriven1968) have additionally shown how inviscid velocity profiles do not account for the boundary layer flow near the free interface, consequently suppressing the theoretical predicted damping rate for each individual mode. In this section we report on the experimentally determined damping rate, $\gamma _{[k,l]}$, for each of our observed mode shapes during normal impact and compare to those predicted by Miller & Scriven (Reference Miller and Scriven1968) for the free drop case.

In order to calculate the energy contribution of the individual mode shapes during post-pinning, it is important to understand the amount of energy contained within the drop at the instant post-pinning begins. As we have shown in § 5.1, the motion of the drop's interface can be understood as a linear superposition of modal components, each vibrating independently and containing its own contribution to the drop's total energy during post-pinning. By breaking up the motion of the drop's interface into modes, we can then think of the energy of the interface as having a point in time when the kinetic energy $K$ is maximized and a point where $K$ is minimized. When $K$ is minimized, all of the drop's energy is contained as $lg$ interfacial energy $\mathcal {A}$ for each mode shape $\mathcal {A}_{[k,l]}$. At this point in time we can determine the magnitude of the disturbance $c$ associated with this maximal $\mathcal {A}$ at $s=0$. This value of $c$ corresponds to the instant of pinning $t=t_p$, before any energy has been lost due to viscous dissipation $\mathcal {D}_{bulk}$.

In order to quantify the disturbance $c$ at time $t_p$, we determine the rate of decay of the individual mode shapes $\gamma _{[k,l]}$ and use this rate to determine the disturbance at the instant of pinning $c_{[k,l]}$ from the disturbance $c$ at any point during post-pinning. In this work we choose to measure $\gamma _{[k,l]}$ directly from the experiment for each mode. In order to calculate $\gamma _{[k,l]}$, the post-pinning experimental images were broken into intervals, such that the Nyquist condition is satisfied, and a series of FFTs were performed. The magnitude of the disturbance $c$ during post-pinning is expected and experimentally confirmed to take the form

(5.1)\begin{equation} c(t) = c_{[k,l]}\exp({-\gamma_{[k,l]} (t-t_p)})\cos{(2{\rm \pi} f_{[k,l]}(t-t_p)+\beta)}, \end{equation}

where $\beta$ is the phase shift. The amplitude of the peak from the FFT corresponding to each mode was recorded. The natural logarithm of these amplitudes was taken and $\gamma _{[k,l]}$ extracted by measuring the slope over time. The decay rate $\gamma _{[k,l]}$ was found to be constant throughout post-pinning for each of the mode shapes. Alternatively, $\gamma _{[k,l]}$ can be determined by performing a wavelet transformation on the waveform traced by $r=r(s,t)$ at a fixed $s$ from the experiment.

Figure 10 shows a series of FFTs taken over different time intervals during post-pinning. In figure 10(a), just after the start of post-pinning, mode shapes up to $k=8$ are observed. As $t$ increases, bulk dissipation $\mathcal {D}_{bulk}$ damps the modal contributions to the post-pinning $lg$ interfacial vibrations. After approximately $0.1\ {\rm s}$ post pinning the higher-order modes are almost completely dissipated, as seen in figure 10(b). For longer times, all of the higher-order modes have decayed leaving only the $[2,0]$ mode, as shown in figure 10(c).

Figure 10.

FFTs of sequential 0.12 s time intervals of $r(s,t)$ measured at $s=0$ for a $13.8\ \mathrm {\mu }\textrm {l}$ drop for liquid/solid system F0. Higher-order mode shapes, larger $k$, decay faster than their lower-order counterparts.

Figure 11 summarizes the measured decay rates compared with Miller & Scriven (Reference Miller and Scriven1968), equation (28), for a low viscosity free drop oscillating in a gas of negligible density,

(5.2)\begin{equation} \hat{\gamma}_{[k,0]} = \frac{\mu (k^{2}+3k)}{2\rho}\frac{R}{R_d^{3}}\sqrt{\frac{\rho a^{3}}{\sigma}}.\end{equation}

Here, we have used the sessile drop radius $R$ as the length scale and multiplied the decay rate by the scaled volume $R^{3}/R_d^{3}$ to account for the difference in volume between the sessile drop and a free drop. Equation (5.2) lends itself to a good first order approximation of the decay rates of the individual modes, where $k=2$ corresponds to the $[2,0]$ mode, $k=4$ the $[4,0]$ mode, etc. In our case, the pinned CL constrains the sessile drop, causing its modes to decay faster than a similar free drop for 0 wt/wt%, 5 wt/wt% and 10 wt/wt% glycol/water solutions as observed in figure 11(a–c). However, for the 100 wt/wt% glycol/water case, the non-negligible viscosity of the solution caused the approximation to over predict the decay rate as seen in figure 11(d). In all cases, higher order mode shapes decayed faster than their lower-order counterparts, as expected.

Figure 11.

Experimentally determined (M) and theoretically approximated (T) scaled decay rates $\hat {\gamma }$ against rest angle $\alpha _p$ for (a) 0 wt/wt% glycol/water, (b) 5 wt/wt% glycol/water, (c) 10 wt/wt% glycol/water, (d) 100 wt/wt% glycol/water impacts. For low viscosity systems (a–c), a scaled equation (28) from Miller & Scriven (Reference Miller and Scriven1968) provides a good first-order approximation of the decay rates of individual modes. The value of $K_o/\mathcal {A}_o$ was not found to influence $\hat {\gamma }$.

5.3. Modal energy decomposition

In § 3.5 we derived an expression for the post-pinning energy, (3.15), that allowed us to calculate $\{K+\mathcal {A}\}_{p}$ directly from experimental images. In order to minimize the numerical error associated with calculating the $lg$ interfacial velocity, $u_r$, from discrete experimental images, it would be advantageous to recast (3.15) to remove $u_r$. Since we know the $lg$ interface is oscillating, we can consider the case when $u_r=0$, which corresponds to the maximum deformation of the drop's $lg$ interface. We know a point in time must exist where $u_r=0$ for each mode shape at all values of $s$ by looking at the time derivative of (1.1)

(5.3)\begin{equation} \dot{r}(s,t) = u_r ={-}\epsilon \psi(s) \varOmega \cos{(l\phi)} \sin{(\varOmega t)},\end{equation}

justifying a simplification of (3.15) to

(5.4)\begin{equation} \{K+\mathcal{A}\}_{p}=\sigma\left( A_{cap}+\sum_{k=2,4,6\ldots}^{\infty}\left(2{\rm \pi} S_{\left[k,0\right]}\overline{x}_{\left[k,0\right]}-A_{cap}\right)- A_{ls}\cos{\left(\alpha_p\right)}\right),\end{equation}

where $S_{[k,0]}$ represents the arc length of each modal shape and is quantified by fitting the solution from Bostwick & Steen (Reference Bostwick and Steen2014) to the disturbance, $c_{[k,0]}$, measured for each mode using the procedure outlined in § 5.2. The energy associated with each mode shape is then calculated using Pappus's first centroid theorem, $\mathcal {A}_{[k,0]}=2{\rm \pi} \bar {x}S_{[k,0]}$, since the mode shapes are azimuthally symmetric, where $\bar {x}$ represents the geometric centroid of $S$, as shown in figure 12.

Figure 12.

Schematic of an oscillating drop with CA $\alpha _p$ illustrating the arclength $S$, geometric centroid $\bar {x}$ and disturbance magnitude $c$. The post-pinning energy $\{K+\mathcal {A}\}_p$ can be determined by evaluating the $lg$ interfacial energy from Bostwick's solution and application of Pappus's theorem.

This simplification to (5.4) relies on the assumptions that the motion of the $lg$ interface can be broken into linearly superimposable components as described in § 5.1 and that the energies of each of these modal components can be calculated. In simplifying (3.15) to (5.4) we assume that the energy of the drop is completely contained within the surface energy of the drop $\mathcal {A}_p$. This assumption is justified because the motion of the drop during post-pinning can be described as a linear superposition of mode shapes, and for each mode a point in time exists where $K=0$, i.e. $u_r=0$ from (5.3). We can then express $\{K+\mathcal {A}\}_{p}$ as a sum of three contributions to $\mathcal {A}_p$ as in (5.4): (i) the energy of the $lg$ interface at time $t_{\infty }$, $\sigma A_{cap}$, (ii) the additional contribution from each mode shape to the drop's $lg$ rest state $\sigma 2{\rm \pi} S_{[k,0]}\overline {x}_{[k,0]}-A_{cap}$ and (iii) the contribution from the $ls$ interface $-\sigma A_{ls} \cos {(\alpha _p)}$.

Figure 13 shows the energy decomposition using (5.4) for a set of experiments from F0 and A0 systems with varying $K_o/\mathcal {A}_o$. Figure 13(a,c) shows the total energy of the drop during post-pinning $\{K+\mathcal {A}\}_p$ compared with the drop's pre-impact energy $\{K+\mathcal {A}\}_o$ both scaled by $\mathcal {A}_o$. Earlier, we saw that variation in $\{K+\mathcal {A}\}_p$ for any given system is a result of varying $\alpha _p$. Here, we show the results of calculating the modal components of $\{K+\mathcal {A}\}_p$, i.e. $\sigma 2{\rm \pi} S_{[k,0]}\overline {x}_{[k,0]}-A_{cap}$, in figure 13b,d). We see that when $\alpha _p$ and $\{K+\mathcal {A}\}_p$ are constant, the modal contributions to $\{K+\mathcal {A}\}_{p}/\mathcal {A}_o$ are relatively constant as in figure 13(b), and that variation in $\alpha _p$ leads to variation in the modal energy contributions in figure 13(d). For increasing $K_o/\mathcal {A}_o$, figure 13(d) shows increasing modal energy contributions, despite a decreasing $\{K+\mathcal {A}\}_p$. This seemingly contradictory result is a consequence of the $ls$ energy term $-\sigma A_{ls}\cos {(\alpha _p)}$ from (5.4), which increases in magnitude with increasing $K_o/\mathcal {A}_o$ as $\alpha _p$ decreases with increasing $K_o/\mathcal {A}_o$ for the A0 system.

Figure 13.

(a,c) Scaled energy and (b,d) modal decomposition of $lg$ interfacial energy against $K_o/\mathcal {A}_o$ for (a,b) F0 system $V = 8.39\ \mathrm {\mu } \textrm {l}$, $\alpha _p = 98.9^{\circ }$ and (c,d) A0 system $V = 6.3\ \mathrm {\mu } \textrm {l}$, $\alpha _p = 47 - 73^{\circ }$. The post-pinning energy is invariant to $K_o/\mathcal {A}_o$ when $\alpha _p$ is constant but decreases when $\alpha _p$ changes with $K_o/\mathcal {A}_o$. Note the $[4,0]$ and $[6,0]$ shapes were present during each impacting event but not always shown due to clarity of images.