2.1. Capillary flow along narrow scratches

A scratch much narrower than the impinging droplet diameter ($w\ll 1$) is expected to have little effect on the early stage inertial spreading of the droplet. However, once deposited, the drop becomes a source of liquid from which capillary flow can occur over a longer time scale along sufficiently narrow and deep scratches. Capillary flow has been extensively studied, including in cylindrical micro-channels, since the early twentieth century (Bell & Cameron Reference Bell and Cameron1906; Lucas Reference Lucas1918; Washburn Reference Washburn1921). With the advent of microfluidics, recent attention has been paid to other micro-channel geometries; most relevant to our geometry is the work on open rectangular micro-channels by Yang et al. (Reference Yang, Krasowska, Priest, Popescu and Ralston2011) who, based on experiments and a model balancing capillary and viscous forces, proposed a critical channel width below which capillary flow occurs

(2.1)\begin{equation} w_{cap} = 2d\cos\theta_A/(1-\cos\theta_A). \end{equation}

The line given by (2.1) is shown in figure 3, with $\theta _A$ having a representative value of $75^{\circ }$. Below this line, i.e. where $w< w_{cap}$, the final printed droplet shape will consist of a localised droplet (since the small receding contact angle prevents full contraction of the droplet) with liquid filaments extending in both directions along the scratch. An example numerical simulation result (discussed later) is shown in figure 3 as image I. As a result of the mechanism producing it, we label this the capillary morphology. The long filament within the scratch has been observed by Seemann et al. (Reference Seemann, Brinkmann, Kramer, Lange and Lipowsky2005), but not the liquid outside the scratch because they used vapour condensation rather than droplet deposition. Above the line (2.1), i.e. for shallow, narrow scratches for which $w>w_{cap}$, there will be only a small deviation from perfect sphericity, as indicated by image II in figure 3. This shape will be referred to as the quasi-spherical cap.

2.2. Predicting edge pinning and overspill

The impacting droplet will spread over the top of the side ridges while at the same time penetrating the scratch. Depending upon the width and depth of the scratch, overspill from the side ridges onto the original substrate may occur. To develop an estimate of conditions under which this will happen, we need to consider the expected maximum spreading diameter, $D_{max}$, of the droplet on a flat substrate with the same advancing and receding contact angles.

For a droplet impacting a planar surface, the maximum spreading diameter (normalised as $\beta _{max}=D_{max}/D_0$) has been extensively studied and found to depend on the impact velocity and fluid properties captured in the Reynolds and Weber numbers (1.1a,b). Pasandideh-Fard et al. (Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996) showed, using experiments and numerical simulations, that for low Weber and Reynolds numbers, surface wettability also becomes a significant variable. Clanet et al. (Reference Clanet, Béguin, Richard and Quéré2004) showed experimentally that there are two regimes for droplet deposition: one is the capillary regime with low $We$ and high $Re$, where viscous effects are negligible, and the other is the viscous regime with high $We$ and low $Re$, where capillary forces can be ignored. Early studies reported two conflicting $\beta _{max}$ scalings with $We$ in the capillary regime: studies that used an energy balance predicted a $We^{1/2}$ dependence (Madejski Reference Madejski1976; Chandra & Avedisian Reference Chandra and Avedisian1991; Bennett & Poulikakos Reference Bennett and Poulikakos1993), while those using a momentum balance predicted $We^{1/4}$ (Clanet et al. Reference Clanet, Béguin, Richard and Quéré2004).

In the viscous regime, most studies predict a $Re^{1/5}$ dependence (Madejski Reference Madejski1976; Chandra & Avedisian Reference Chandra and Avedisian1991; Bennett & Poulikakos Reference Bennett and Poulikakos1993). Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010) showed, using a dynamical model with a viscous boundary layer, that if the $We^{1/2}$ scaling holds, then $\beta _{max}= Re^{1/5}F(P)$, where $F$ is a function of $P=We Re^{-2/5}$. Laan et al. (Reference Laan, De Bruin, Bartolo, Josserand and Bonn2014) used experiments with three different viscosity fluids and showed that $We^{1/2}$ holds rather than $We^{1/4}$ and used Padé interpolation to approximate $f(P)$ for $We>10$. Lee et al. (Reference Lee, Laan, De Bruin, Skantzaris, Shahidzadeh, Derome, Carmeliet and Bonn2015) extended this to $We>1$ by incorporating surface wettability. In fact, some models developed earlier than Eggers et al. (Reference Eggers, Fontelos, Josserand and Zaleski2010), including Roisman (Reference Roisman2009) and Scheller & Bousfield (Reference Scheller and Bousfield1995), can be written in the form $\beta _{max}= Re^{1/5}F(P)$. Wildeman et al. (Reference Wildeman, Visser, Sun and Lohse2016) recently used extensive simulations and modelling to bridge the energy balance and the momentum balance approaches and developed a model for $\beta _{max}$. All these models (summarised in table 1) have shown agreement with experimental and numerical data for certain (different) parameter ranges (Lee et al. Reference Lee, Laan, De Bruin, Skantzaris, Shahidzadeh, Derome, Carmeliet and Bonn2015; Josserand & Thoroddsen Reference Josserand and Thoroddsen2016; Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016), see also the rightmost columns of table 1. We will compare the models with new numerical data for micro-droplets in § 3.4.

Table 1.

A selection of previously published models for predicting the maximum spreading diameter of a droplet impacting on a flat surface. Here, $P=We Re^{-2/5}$ and $\beta _{eq}$ is the equilibrium spread diameter. $\theta _d$ in the model by Ukiwe & Kwok (Reference Ukiwe and Kwok2005) is the dynamic contact angle (here assumed to be equal to the advancing contact angle $\theta _A$). The ranges of $Re$ and $We$ are of experiments used to validate these studies. The asterisks indicate the explicit models used in constructing figure 3.

Although originally developed for axisymmetric conditions, maximum spreading laws like the above can be modified by introducing a correction factor to account for the presence of the scratch, and hence used to estimate when overspill occurs. For ease of implementation we consider only models that can be written explicitly as $\beta _{max}=f(We,Re)$ or $\beta _{max}=f(We,Re,\theta _A)$. In particular, we use the models of Scheller & Bousfield (Reference Scheller and Bousfield1995), Pasandideh-Fard et al. (Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi1996), Roisman (Reference Roisman2009) and Laan et al. (Reference Laan, De Bruin, Bartolo, Josserand and Bonn2014); Lee et al. (Reference Lee, Laan, De Bruin, Skantzaris, Shahidzadeh, Derome, Carmeliet and Bonn2015) listed in table 1.

Assuming that the droplet spreads on the tops of the side ridges and the inner scratch walls at the same rate, a correction factor can be derived by subtracting from the initial droplet volume the volume of liquid that will go into the scratch before the droplet reaches the outer edges. The new volume available for spreading can then be used to derive a new equivalent initial droplet diameter to be used in the maximum spreading laws. Assuming that $d\leqslant w$, the dimensional volume to be subtracted is approximately $dD_0\times wD_0\times D_0 = wdD_0^{3}$, so the new volume of liquid available for spreading over the side ridges is $V_{new}=({\rm \pi} D_0^{3}-6dwD_0^{3})/{6}=(D_0^{3}({\rm \pi} -6dw))/{6}$. This is equivalent to a free droplet of dimensional diameter $D_{0,new}= {\rm \pi}^{-1/3}D_0({\rm \pi} -6dw)^{1/3}$. The maximum spreading laws are of the form $D_{max}/D_0=f(Re,We,\theta _A)$, so using the new diameter $D_{0,new}$ to account for the presence of the scratch gives $D_{max}/D_{0,new}=D_{max}/({\rm \pi} ^{-1/3}D_0({\rm \pi} -6dw)^{1/3}) = f(Re,We,\theta _A)$. Hence, the new dimensionless maximum spreading diameter allowing for liquid entering the scratch is given by

(2.2)\begin{equation} \beta_{max} = D_{max}/D_{0} ={\rm \pi}^{{-}1/3}D_0\left({\rm \pi}-6dw\right)^{1/3}f(Re,We,\theta_A). \end{equation}

This holds when $d\leqslant w$, otherwise the droplet reaches the outer edges of the side ridges before the entire depth is covered by the impacting droplet. If $d>w$, the dimensional volume to be subtracted will be $w^{2}D_0^{3}$ and hence $D_{0,new}$ will not depend on $d$, and the adjusted $\beta _{max}$ will be the same value as that when $d=w$ for the entire range $d>w$. The condition for the droplet contact line reaching the outer edges of the side ridges is $\beta _{max}=3w$, i.e. the dimensionless width of the scratch and the two side ridges. Hence, the outer edges of the side ridges will be reached only for scratches with widths $w_{pin}$ satisfying the condition

(2.3)\begin{equation} 3w_{pin} ={\rm \pi}^{{-}1/3} f(We,Re,\theta_A) \begin{cases} \left({\rm \pi}-6dw_{pin}\right)^{1/3} & \textrm{if} \ d\leqslant w \\ \left({\rm \pi}-6w_{pin}^{2}\right)^{1/3} & \textrm{if} \ d > w \end{cases}. \end{equation}

The five different explicit spreading models listed (with asterisks) in table 1 give different predictions for $f(We,Re,\theta _A)$. Therefore, in plotting the values of $w_{pin}$ in figure 3, the three solid black lines correspond to the mean and standard deviation of the five predictions.

Note that (2.3) simply provides an estimate of scratch widths for which the droplet can reach the outer edges of the side ridges. Since we assume that the receding contact angle is close to zero, the contact line will not recede from the edges if it is able to reach them. However, the droplet contact line may continue spreading beyond the outer edges of the ridges if it has sufficient momentum. If $\beta _{max}>3w$, the droplet will continue spreading horizontally while the contact line remains pinned on the edge. This will increase the dynamic contact angle $\theta _d$ (i.e. the angle of the free surface measured from the horizontal top of the ridges) until either the droplet stops spreading and recoils or $\theta _d>\theta _A+90^{\circ }$, i.e. $\theta _d$ exceeds the Gibbs criterion for pinning. In the latter case, the droplet will then advance down the vertical outer walls of the side ridges and spill onto the original substrate surface; the final droplet morphology will look like image III in figure 3. This will be referred to as the inertial morphology, since the droplet has sufficient inertia to spill over the side ridges. Hence (2.3) provides an estimate for conditions under which the droplet contact line will pin on the outer edge of the side ridges. This will be discussed again with the benefit of numerical simulations in § 4.1.

2.3. Droplet splitting and imbibition

For $w>w_{pin}$, the droplet does not cross the outer edges of the side ridges. For sufficiently shallow scratches, its final shape is expected to be one where the continuous liquid volume rests on top of side ridges while also filling the scratch. This morphology is labelled semi-imbibed, and an example is shown in image IV in figure 3. As the scratch depth increases, more of the liquid volume will occupy the scratch. Given that the low receding contact angle prevents significant contraction of the droplet contact line from the side ridges, for a deep enough scratch the droplet may split along the inner edges of the side ridges such that the liquid on the top of the side ridges and that in the scratch become separated. Splitting along the inner edges is expected when the cross-sectional area of the scratch multiplied by $D_{Along}$, the length of spreading along the scratch, results in a greater volume than the droplet. Thus the critical width corresponds to when $wD_0\times dD_0 \times D_{Along}={4}({\rm \pi} (D_0/2 )^{3})/{3}$. Assuming that $D_{Along}=\beta _{max}D_0$, where $\beta _{max}$ is again the maximum spreading on a flat surface, this gives the following estimate of the critical width for which splitting will occur

(2.4)\begin{equation} w_{split}= \frac{\rm \pi}{6d\beta_{max}}. \end{equation}

Again the different explicit models asterisked in table 1 have been used to give a spread of estimates of $\beta _{max}$ (for $\theta _A=75^{\circ }$) and these result in the blue dotted curves in figure 3. For depths below these curves and $w<1$, the final droplet shape will be a split semi-imbibed morphology where most of the liquid occupies the scratch, but separate small droplets remain on the upper surfaces of the side ridges. See the example image V in figure 3.

The droplet fully imbibes into the scratch without spreading on the side ridges when $w\geqslant 1$ and the scratch depth is sufficient to contain the entire liquid volume. The result is a fully imbibed morphology as indicated in image VI in figure 3. By a similar argument to that above for droplet splitting, the condition for full imbibition into the scratch is

(2.5a,b)\begin{equation} w \geqslant 1, \quad \textrm{and} \quad d > \frac{\rm \pi}{6\beta_{max}w_{split}}. \end{equation}

This corresponds to the region to the right of the vertical dash-dot line in figure 3.

3.1. The LBM solver

The LBMs solve the Boltzmann equation on a lattice that discretises the spatial and velocity domains to form the lattice Boltzmann equation so that the probability distribution density function $f$ at position $\boldsymbol {x}$ and time $t$ evolves according to

(3.1)\begin{align} f_\alpha(\boldsymbol{x}+\boldsymbol{e}_\alpha \delta t,t+\delta t) &=f_\alpha(\boldsymbol{x},t) - \varOmega_{\alpha\beta} (f_\beta(\boldsymbol{x},t) - f_\beta^{eq}(\boldsymbol{x},t))\nonumber\\ &\quad + \delta t (S_\alpha(\boldsymbol{x},t) -0.5\varOmega_{\alpha\beta} S_\alpha(\boldsymbol{x},t) ). \end{align}

Here, $\delta t$ is the time step, $\boldsymbol {e}_\alpha$ are velocities chosen according to the velocity discretisation used in the model such that $\boldsymbol {x}+\boldsymbol {e}_\alpha \delta t$ moves the particle distribution density function to a neighbouring lattice site and $\alpha$ and $\beta$ are indices for the velocity vector space. In this paper, we use the D3Q19 velocity discretisation (that is 3 space dimensions and 19 velocity vectors) detailed in Appendix A. Also, $\varOmega$ in (3.1) is the collision operator, which mimics the effect of molecular collisions and is how viscosity is captured; for more detail on the role of the collision operator, refer to Krüger et al. (Reference Krüger, Kuzmin, Kusumaatmaja, Shardt, Silva and Viggen2016). The multiple relaxation time (MRT) collision operator used here is written as

(3.2)\begin{equation} \varOmega_{\alpha\beta} = (\boldsymbol{\mathsf{M}}^{-\boldsymbol{\mathsf{1}}}\boldsymbol{\varLambda} \boldsymbol{\mathsf{M}})_{\alpha \beta}, \end{equation}

where $\boldsymbol{\mathsf{M}}$ is the collision matrix and $\boldsymbol{\varLambda}$ is a diagonal relaxation matrix used to relax the various moments of the distribution density function at various rates. The most commonly used collision matrix is that derived using the Gram–Schmidt procedure (D'Humières et al. Reference D'Humières, Ginzburg, Krafczyk, Lallemand and Luo2002); however, an equivalent but more efficient and easier to implement matrix has been derived by Li et al. (Reference Li, Du, Fei and Luo2019), which is used in this work. Instead of a single relaxation time for all moments, moments are relaxed at different rates in MRT simulations. The corresponding diagonal relaxation matrix is given by

(3.3)\begin{equation} \boldsymbol{\varLambda} ={\rm diag}(1,1,1,1,s_\zeta, s_\nu,s_\nu,s_\nu,s_\nu,s_\nu,s_q,s_q,s_q,s_q,s_q,s_q,s_{\rm \pi} s_{\rm \pi}, s_{\rm \pi},s_{\rm \pi}), \end{equation}

where $s_\zeta$ and $s_\nu$ determine the bulk and kinematic shear viscosities, respectively; $s_{\rm \pi}$ and $s_q$ are relaxation rates for non-hydrodynamic moments that can be tuned to ensure the stability of the simulation. The term $S_\alpha (\boldsymbol {x},t)$ in (3.1) is a source term used here to incorporate inter-molecular forces needed to produce coexistence of liquid and vapour phases. To simulate multiphase flow, Shan & Chen (Reference Shan and Chen1993) introduced local attractive interaction forces written in discrete form as

(3.4)\begin{equation} \boldsymbol{F}(\boldsymbol{x},t)={-}G\psi(\boldsymbol{x},t) \sum_\alpha w_\alpha \psi(\boldsymbol{x}+\boldsymbol{e}_\alpha\delta t, t+\delta t)\boldsymbol{e}_\alpha, \end{equation}

where $G$ is the strength of the interactive force, $\psi (\boldsymbol {x},t)$ is the pseudo-potential used to prevent the force from diverging at high densities and $w_\alpha$ are weights that depend on the velocity directions; they are $w_0=1/3$, $w_1,\ldots ,w_6=1/6$ and $w_7,\ldots ,w_{18}=1/12$. This interaction force is incorporated differently into $S_\alpha (\boldsymbol {x},t)$ in the original Shan & Chen (Reference Shan and Chen1993) model compared with Li et al. (Reference Li, Luo and Li2013), where the latter solves the thermodynamic inconsistency in the former. The source term used in this work is that proposed by Li et al. (Reference Li, Luo and Li2013), and is detailed in Appendix A. This model results in a pressure, $p=c_s^{2}\rho +\psi ^{2} c_s^{2}G/{2}$, supporting two phases, where $c_s=1/\sqrt {3}$ is the speed of sound of the lattice (Shan & Chen Reference Shan and Chen1993). Yuan & Schaefer (Reference Yuan and Schaefer2006) showed that this equation of state, with the choice of $\psi (\boldsymbol {x},t)=\rho _0(1- \exp {(-\rho /\rho _0))}$ originally proposed by Shan & Chen (Reference Shan and Chen1993), is limited in terms of achievable density ratio between the liquid and vapour phases and instead proposed using a different expression for $\psi (\boldsymbol {x},t)$,

(3.5)\begin{equation} \psi(\boldsymbol{x},t)=\sqrt{\frac{2(p-c_s^{2}\rho(\boldsymbol{x},t))}{c_s^{2}G}}. \end{equation}

This enables using different equations of state for pressure such as the Carnahan–Starling equation of state used in this work:

(3.6)\begin{equation} p = \rho RT \frac{1+b\rho/4+(b\rho/4)^{2}-(b\rho/4)^{3}}{(1-b\rho/4)^{3}} - a\rho^{2}, \end{equation}

where $T$ is temperature, $R$ is the universal gas constant (set to 1), $a=0.49963R^{2}T_c^{2}/p_c$ and $b=0.18727RT_c/p_c$, with $T_c$ and $p_c$ being the critical temperature and pressure, respectively. The parameter $a$ physically represents the strength of the molecular interaction in the equation of state and lowering it results in a thicker interface and a more stable simulation at higher density ratios (Li et al. Reference Li, Luo and Li2013). Reducing $T$ in (3.6) increases the density of the liquid and lowers that of the gas, hence increasing the density ratio. The parameter $b$ represents the volume occupied by the material's atoms and is chosen arbitrarily and kept constant. In this study, we use (3.6) with $a=0.05$, $b=4$ and $T=0.00472$ which permit a density ratio of O(1000); see Appendix A.3 for more details on $a$ and $b$. The macroscopic variables of density, pressure and velocity are calculated from the moments of the probability density function using the following equations, respectively:

(3.7)$$\begin{gather} \rho (\boldsymbol{x},t) = \sum_\alpha f_\alpha(\boldsymbol{x},t), \end{gather}$$

(3.8)$$\begin{gather}p(\boldsymbol{x},t) = c_s^{2} \rho(\boldsymbol{x},t) + \frac{c_s^{2}G}{2}\psi^{2}(\boldsymbol{x},t), \end{gather}$$

(3.9)$$\begin{gather}\rho(\boldsymbol{x},t)\boldsymbol{u} (\boldsymbol{x},t) = \sum_\alpha \boldsymbol{e}_\alpha f_\alpha(\boldsymbol{x},t) + 0.5\delta t \boldsymbol{F}, \end{gather}$$

while the kinematic viscosity is given by

(3.10)\begin{equation} \nu = c_s^{2}(s_\nu^{{-}1} - 0.5\delta t). \end{equation}

3.2. Wetting and CAH boundary condition

Various methods can be used to prescribe a contact angle at a wall boundary, such as introducing an interaction force between the solid and fluid nodes (see for e.g. Li et al. Reference Li, Luo, Kang and Chen2014) or prescribing a constant density at the wall to achieve a predetermined contact angle (see for e.g. Castrejón-Pita et al. Reference Castrejón-Pita, Kubiak, Castrejón-Pita, Wilson and Hutchings2013). These methods work well at relatively low density ratios but become unstable at higher values. The prescription of a density at the wall also requires calibration to give a specific contact angle whenever the density ratio or the equation of state is changed (Wilson & Kubiak Reference Wilson and Kubiak2016). An alternative is to use the geometric boundary condition developed by Ding & Spelt (Reference Ding and Spelt2007), which has been used to simulate inkjet printed droplets at high density ratio without loss of stability, and hence it is used in this study. This condition was originally developed for the volume of fluid method and adopted in the phase field multiphase LBMs by Connington & Lee (Reference Connington and Lee2013) and has been used for the pseudo-potential multiphase models by Zhang et al. (Reference Zhang, Cheng, Ku, Song and Zhang2018). A derivation detailed in Appendix A.2 results in the equation

(3.11)\begin{equation} \tan\left(\frac{\rm \pi}{2} - \theta\right) = \frac{-\boldsymbol{\nabla}\rho \boldsymbol{\cdot} \boldsymbol{n}}{|\boldsymbol{\nabla}\rho- (\boldsymbol{\nabla}\rho \boldsymbol{\cdot} \boldsymbol{n})\boldsymbol{n} | } , \end{equation}

where $\boldsymbol {n}$ is the unit normal to the solid surface and $\rho$ is the density. This can be discretised and used to calculate a density to assign to the solid wall locally to satisfy a given contact angle. The discrete form for our geometry is detailed in Appendix A.2.

A CAH was implemented by calculating the local contact angle using the inverse of (3.11). If the value of the local contact angle is lower than the receding contact angle ($\theta _R$) then $\theta$ is replaced with $\theta _R$ and similarly, if the local contact angle is higher than the advancing contact angle, it is replaced with $\theta _A$. Assigning the density on the wall controls the interaction pseudo-potential in (3.5) to satisfy the contact angle $\theta$. In order to capture CAH, the wall lattice sites are initialised with $\theta$ in (3.11) set to $\theta _A$. This allows the droplet to spread provided that the contact line forms a local contact angle of $\theta \geqslant \theta _A$. Once a lattice site has been wetted, the value of $\theta$ in (3.11) is replaced with $\theta _R$ for this lattice site. This will stop the droplet from dewetting or the contact line receding unless the local contact angle at the contact line is $\theta \leqslant \theta _R$. This is implemented by rearranging (3.11) for $\theta$ and calculating in every time step locally in every wall lattice site. Note that the values of $\theta$ assigned at the wall are used to control the interaction force, while the contact line can have any contact angle. The Shan & Chen (Reference Shan and Chen1993) model with these additions can capture a range of contact angles, CAH, coalescence and breakup, and contact-line dynamics at high density ratio up to $10^{3}$. The simulation is numerically stable for $45^{\circ } \leqslant \theta _A \leqslant 140^{\circ }$ in the smooth surface case and $55^{\circ } \leqslant \theta _A \leqslant 130^{\circ }$ in the case with the scratch.

To enable the extensive parametric study presented in § 4, the code was accelerated using CUDA to run on GPUs. A simulation (of size $256\times 256 \times 576$ lattice cells) that would take up to three days on a single CPU then ran in under two hours. On a 24-core CPU high-performance computing node (Intel Xeon Gold 6138 @ 2 GHz) speeds of 5 million lattice updates per second (MLUPS) were achieved, compared with 200 MLUPS on an NVIDIA P100 GPU and 500 MLUPS on an NVIDIA V100 GPU.

3.3. Validation of simulations against experimental data

For the impact of a single droplet falling onto a smooth solid surface we compare our simulations with experimental data from Lim et al. (Reference Lim, Han, Chung, Chung, Ko and Grigoropoulos2009), who examined droplets with in-flight diameter $D_0=48.1\ {\mathrm {\mu }}\textrm {m}$ hitting a smooth surface at speed $u=1.9\ \textrm {m}\ \textrm {s}^{-1}$ corresponding to a Reynolds number $Re = uD_0/\nu =107$, Weber number $We = \rho u^{2}D_0/\gamma = 2.4$ and Ohnesorge number $Oh = \sqrt {We}/Re=0.015$, where $\gamma$ is the surface tension. The advancing and receding contact angles were reported to be $\theta _A=60^{\circ }$ and $\theta _R=40^{\circ }$, respectively. The normalised spreading diameter of the droplet $D_s/D_0$ and height $H_s/D_0$ were tracked over time. Lim et al.'s (Reference Lim, Han, Chung, Chung, Ko and Grigoropoulos2009) data are compared with equivalent simulations in figure 4. To give an indication of the sensitivity of the simulations to the resolution of the lattice, several different lattices were tested, with the resolution expressed in terms of the number of cells per initial droplet radius. Note that in the lattice Boltzmann method, testing the sensitivity to lattice resolution is not as straightforward as for direct discretisations of the Navier–Stokes equations, since the lattice discretises both coordinate space and the molecular velocity space. Hence modifications of the lattice node spacing require adjustments of other parameters to ensure that the same physical system is being simulated.

Figure 4.

Comparison between the experimental results by Lim et al. (Reference Lim, Han, Chung, Chung, Ko and Grigoropoulos2009) and corresponding simulations of a $48.1\ {\mathrm {\mu }}\textrm {m}$ droplet impacting a flat surface at $1.9\ \textrm {m}\ \textrm {s}^{-1}$ ($Re=107$; $We=2.4$) using different lattice resolutions. The data show the spreading diameter (in red) and height (in blue) of the droplet, all scaled by the initial droplet diameter.

In general terms, good agreement is achieved between the simulations and both sets of experimental data as the lattice resolution increases; oscillations in height agree well particularly for the first few periods, while the spreading rate and final diameter are close, with some small variation in experimental data due to experimental noise. There is still some sensitivity in the time scales of the simulations using different lattices, which becomes more evident at later times, but the same equilibrium state is reached in each case. In the non-axisymmetric simulations presented in § 4, the finest resolution was used and, for a random sample of the conditions considered, simulations were repeated with the other lattice resolutions. The same equilibrium shapes were obtained with each lattice.

3.4. Comparison with analytical models

In addition to the models for the maximum spreading diameter (see table 1), theoretical models have been developed to predict the equilibrium diameter $\beta _{eq}$ for a given static contact angle $\theta$ (in the absence of hysteresis). Using conservation of volume, Van Dam & Le Clerc (Reference Van Dam and Le Clerc2004) identified an expression for $D_{eq}$ in terms of the in-flight diameter of the droplet $D_0$ and $\theta$

(3.12)\begin{equation} \beta_{eq}=\frac{D_{eq}}{D_0} = \left( \frac{8}{\tan\dfrac{\theta}{2}\left(3+\tan^{2}\dfrac{\theta}{2}\right)} \right)^{1/3}. \end{equation}

As further validation for the simulation methodology, and to provide a baseline simulation for more complex topographies, a typical inkjet droplet impacting a smooth surface was simulated. The droplet was $48.8\ {\mathrm {\mu }}\textrm {m}$ in diameter falling at $3.74\ \textrm {m}\ \textrm {s}^{-1}$, with density $1000\ \textrm {kg}\ \textrm {m}^{-3}$, surface tension $26\ \textrm {N}\ \textrm {m}^{-1}$ and dynamic viscosity $9\times 10^{-4}\ \textrm {Pa}\ \textrm {s}$ (i.e. $Re=204$ and $We=26$). Several simulations were run without contact angle hysteresis for various $\theta$ and using different lattice resolutions. The resulting equilibrium diameters are compared with values predicted by (3.12) in figure 5(a), which shows very good convergence of the numerical simulations to the analytical result.

Figure 5.

Comparison of simulation predictions with analytical models for droplet spreading diameters on a flat surface. In (a), the equilibrium diameter is determined (without CAH) using different lattice resolutions and compared with (3.12) for different contact angles. The inset plot shows the $L_\infty$-norm of the error. In (b), the maximum spreading diameter from simulations using the finest lattice is shown for different Weber numbers, with $\theta _A=75^{\circ }$ and $\theta _R=1^{\circ }$. This is compared with the predictions using the models in table 1. Eq., equation.

Figure 5(b) shows the maximum spreading diameter of a droplet (scaled by its initial diameter) obtained from numerical simulations with CAH included ($\theta _A=75^{\circ }$; $\theta _R=1^{\circ }$) for different values of the Weber number. For comparison, the models from table 1 are plotted, showing that the simulations are consistent with the collective predicted behaviour. The more recent models (Roisman Reference Roisman2009; Laan et al. Reference Laan, De Bruin, Bartolo, Josserand and Bonn2014; Lee et al. Reference Lee, Laan, De Bruin, Skantzaris, Shahidzadeh, Derome, Carmeliet and Bonn2015; Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016) produce similar values to each other, and – despite being developed for larger droplets – agree quite well with the numerical simulations of micro-droplets, particularly for the larger values of Weber number. For the lowest Weber numbers considered, the model by Chandra & Avedisian (Reference Chandra and Avedisian1991) fits our data most closely, but agreement with this model is poor for larger $We$. Overall, based on our micro-droplet simulations, the model of Roisman (Reference Roisman2009) is arguably the best both in terms of its agreement but also because it is an explicit model that is easier to use.

Further comparison with the models in table 1 is given in figure 6 for the specific case $We=26$. The figure highlights the effect of CAH on the spreading. With hysteresis accounted for (again with $\theta _A=75^{\circ }$ and $\theta _R=1^{\circ }$), the contact line expands as the droplet spreads and becomes pinned essentially at the maximum spread diameter since the receding contact angle is so low. Such pinning is important in printing applications and is seen in practice with the colloidal inks used in the printed electronics industry (Duineveld Reference Duineveld2003). The grey shaded band again corresponds to the predictions of maximum spreading diameter from the models in table 1. In contrast, when CAH is not included in the simulation, the droplet recoils and contracts after reaching its maximum extent, then overshoots and oscillates in diameter as it settles to an equilibrium diameter consistent with (3.12).

Figure 6.

The spreading diameter as a function of time for a single droplet on a smooth surface. These are results from simulations with and without CAH compared with predictions of the maximum spread diameter from the models in table 1 (grey shaded area) and the equilibrium diameter given by (3.12) (grey dashed line).

4.1. Printed droplet morphologies

Figure 7 presents a map of the scratch width–depth parameter space showing the outcomes of droplet impact simulations that were run until an equilibrium morphology was formed. Different symbols indicate the morphology obtained at each width–depth combination, and corresponding examples of the equilibrium shapes for the same $Re$, $We$, $\theta _A$ and $\theta _R$ are given in figure 3. In addition, figure 7 shows an example of an edge-pinned final droplet, where the droplet has spread over the tops of the side ridges, reaching the outer edge with insufficient momentum to spill onto the original substrate surface. The contact line remains pinned on the outer edge as the very low receding contact angle prevents recession and the Gibbs pinning criterion is satisfied.

Figure 7.

Numerically determined regime map for morphologies formed by printing a single droplet centred on the idealised scratch shown in figure 2. Simulation parameters: $Re=204$, $We=26$, $\theta _A=75^{\circ }$, $\theta _R=1^{\circ }$. The overlaid lines and curves give the theoretically estimated regime boundaries developed in § 2. Equations (2.3) and (2.4) employ predictions of maximum spreading diameters from table 1; the three curves for each equation represent the mean and standard deviation due to the models.

The theoretically estimated regime boundaries developed in § 2 are superimposed in figure 7 and show generally very good agreement with the outcomes of the numerical simulations. Recall that the three curves corresponding to (2.3) and (2.4) give the mean and standard deviations of the predictions based on the five explicit maximum spreading models in table 1. There is slight deviation between (2.1) and the boundary of the capillary regime identified from simulations because (2.1) assumes an infinite source of liquid and does not capture the force due to the Laplace pressure resulting from curvature of the droplet, which becomes significant when the source droplet is of commensurate width to the groove. However, the criterion (2.3) for pinning of the contact line on the edges of the side ridges captures very well the numerical predictions. Note that similar pinning has been observed even on rounded edges (Kant et al. Reference Kant, Hazel, Dowling, Thompson and Juel2017).

The splitting boundary of (2.4) also fits well with the simulation results but has a slight deviation at larger depths. This is likely due to the assumption that the entire depth of the scratch is filled when deriving (2.4), i.e. the simple geometrical argument does not account for the non-trivial shape of the liquid free surface within the scratch. Similarly, the conditions for full imbibition of the droplet into the scratch are quite well identified by the criteria in (2.5a,b), with some discrepancy in the critical depth due to the crude approximation of the free-surface shape within the scratch. The delineation of the quasi-spherical cap region of figure 7 is somewhat subjective. Here, we define this to be where the final droplet shape deviates from a spherical cap by less than 10 % in the lateral dimensions.

4.2. Droplet spreading dynamics

Although seven different equilibrium morphologies have been identified in figures 3 and 7, essentially only two types of flow dominate the spreading dynamics, namely inertia-driven spreading and capillary flow. Front views of the droplet spreading process at different times are shown in figure 8 for a selection of scratch widths and depths that lead to each of the seven equilibrium morphologies, while figure 9 illustrates how the horizontal and vertical dimensions of the droplet change in time for different morphologies. There, $D_{Along}$ refers to the length of the liquid in the direction of the scratch, $D_{Across}$ is the diameter perpendicular to the scratch, and $H$ is the droplet height measured at the centre of the scratch from the bottom of the scratch to the free surface. These quantities are normalised by the equilibrium spreading diameter of an equivalent droplet on a flat surface, $D_{flat}$ and height $H_{flat}$ of a corresponding droplet after impact on a smooth flat surface, i.e. with no scratch, from § 3.4. Note that in most cases considered here $D_{flat}\approx D_{max} = \beta_{max}D_0$ (the expected maximum spreading diameter of the droplet on a flat substrate with the same advancing and receding contact angles) since the very low receding contact angle prevents contraction of the contact line, but for other values of $\theta _R$ these values are generally different.

Figure 8.

Front-view snapshots of the different regimes evolving in time measured in milliseconds. The images are to scale, the zoom scale is constant and the droplet diameter (hence also volume) is kept constant. The last row contains 3-D views to demonstrate the capillary flow along the scratch.

Figure 9.

Evolution with time of the morphology height and dimensions along and across a scratch for the (a) quasi-spherical cap, (b) edge-pinned and (c) capillary flow cases shown in figure 8.

In the quasi-spherical cap regime, the scratch is filled and covered very quickly, the droplet spreads to reach a maximum diameter and then oscillates, as seen in the $H$ curve in figure 9(a), before relaxing more slowly to its equilibrium shape. The resulting horizontal dimensions $D_{Along}$ and $D_{Across}$ are similar to the corresponding spherical cap formed on a smooth flat surface. However, the slight extension of the droplet along the scratch results in a lower equilibrium height.

In the inertial regime, the droplet touches the side ridges, then penetrates to the bottom surface of the scratch. The liquid spreads on the side ridges and spills over onto the original substrate surface, as seen in the second row of figure 8. Compared with the quasi-spherical cap case, there is greater spreading along the scratch and slightly shorter spreading in the direction perpendicular to the scratch. The wider scratch results in a greater volume of liquid occupying the scratch, and consequently a slightly reduced final droplet height. In the edge-pinned regime, i.e. for wider scratches, the droplet contact line reaches the outer edges of the side ridges, but, as noted in § 4.1, there is insufficient momentum to carry the free surface past the edges, and the contact line remains pinned as seen in figure 8. Figure 9(b), which corresponds to $w=0.5$, shows the increased inertia-driven spreading along the scratch and the subsequent halting of the contact line producing a constant $D_{Along}$ as a result of the very low receding contact angle. The extent of the droplet in the direction across the scratch is reduced by the enhanced flow along the direction of the scratch and the pinning on the ridge edges.

Under conditions leading to the semi-imbibed morphology, the droplet impacts the bottom of the scratch and spreads along and across the scratch and then impacts onto the side ridges, see figure 8. The droplet spreads on the top of the side ridges but the contact line does not reach the outer edges; it becomes pinned somewhere on the top of the ridges because of the very low receding contact angle. In the split semi-imbibed regime, the droplet impacts the side ridges first, and then penetrates into the scratch, reaching the bottom and spreading along it. As the droplet spreads into the scratch, it splits along the inner edges of the side ridges, leaving separated droplets sitting on the top of the side ridges. These droplets are almost flat because of the very low receding contact angle. In the fully imbibed regime, the droplet never impacts on the top surface of the side ridges but falls into the scratch and spreads until an equilibrium is reached.

Finally, as shown in the 3-D view included in the bottom row of figure 8, for scratches producing capillary flow, the droplet initially spills over the side ridges onto the original substrate as in the quasi-spherical cap and inertial regimes. However, the liquid then spreads along the scratch by capillary action until there is no more liquid to supply further flow. This can be clearly seen in the evolution of both $D_{Along}$ and the droplet height in figure 9(c). Note the different scale on the vertical axis in this plot compared with the others in figure 9.

The rate of capillary propagation has been studied extensively starting with work by Washburn (Reference Washburn1921), Bell & Cameron (Reference Bell and Cameron1906) and Lucas (Reference Lucas1918) on cylindrical capillaries. The main finding is that the propagation length $x$ increases as the square root of time, that is $x\propto \sqrt {t}$. More recently this analysis has been extended to other geometries including capillary flow in open rectangular micro-channels, studied theoretically and experimentally by Yang et al. (Reference Yang, Krasowska, Priest, Popescu and Ralston2011). They followed a similar approach to that used to derive the Washburn model except that they applied the procedure to an open rectangular micro-channel. The reservoir droplet in their experiments was large enough relative to the micro-channel to ignore its Laplace pressure in the model. Although the capillary flow seen here initially follows the propagation rate predicted by their model, the limited volume of fluid in the droplet supplying the flow soon results in a more rapid decrease in the speed of propagation and the corresponding flattening of the $D_{Along}$ curve in figure 9(c).

4.3. Groove without ridges

Removing the side ridges from the topography in figure 2 reduces the number of possible equilibrium morphologies from seven to five, and alters the combinations of groove width and depth at which the morphologies arise. The corresponding regime map, constructed from simulations for groove widths ranging from 0.1–1 and depths of 0.05–0.75 with increments of 0.1 and 0.05, respectively, is shown in figure 10. The morphologies caused by the presence of the ridges, namely edge pinned and inertial (where the droplet spills over the ridges), are not seen with this topography. These both merge into the semi-imbibed morphology, where the equilibrium shape of the droplet occupies both the groove and the nearby region of the substrate surface, and is the dominant morphology for sufficiently shallow grooves. For narrow and shallow grooves, the morphology can still be classed as a quasi-spherical cap, and for narrow and deep grooves, the capillary morphology is still seen. The fully imbibed, and split semi-imbibed morphologies are again seen for sufficiently wide and deep grooves.

Figure 10.

Regime map of equilibrium morphologies formed by printing a single droplet centred on a groove of width $w$ and depth $d$ with no side ridges. Simulation parameters: $Re=204$, $We=26$, $\theta _A=75^{\circ }$ and $\theta _R=1^{\circ }$. The overlaid lines and curves are theoretical estimates of the regime boundaries based on conditions (2.1), (2.4) and (2.5a,b). Equation (2.4) employs predictions of maximum spreading diameters from table 1, the three curves representing the mean and standard deviation of the predictions.

The theoretical estimates (2.1), (2.4) and (2.5a,b) for the boundaries of the regions in the regime map are readily adapted by using the appropriate depth $\bar {d}$, and these are included in figure 10. Again, good agreement is seen between these estimates and the results of the simulations.

5.1. Effect of Reynolds number and Weber number

Increasing $Re$ and/or $We$ promotes greater spreading of the droplet on impact and the droplet is therefore able to spill over wider side ridges. Hence $w_{pin}$ – the critical scratch width for edge pinning in (2.3) – increases and the inertial region of figure 7 will expand to the right. This effect is illustrated in the specific case shown in figure 11, for a scratch geometry given by $d=0.15$ and $w=0.5$. A droplet impact at $Re=62$ results in a semi-imbibed morphology, whereas an impact on the same scratch at $Re=102$ produces the edge-pinned morphology, and at $Re=204$ the inertial morphology arises, consistent with the increase in the value of $w_{pin}$.

Figure 11.

Effect of $Re$ on the morphology resulting from a droplet impact on a scratch of dimensions $d=0.15$ and $w=0.5$ at $We=26$, with $\theta _A=75^{\circ }$ and $\theta _R=1^{\circ }$.

The greater inertia of the droplet will also result in greater penetration into and along the scratch, resulting in a smaller liquid height within the scratch and consequently a greater tendency for the droplet to split along the inner edges of the scratch and leave separate small droplets on the outer surface. From the form of (2.4) it is clear that the boundary of the split semi-imbibed region in both figures 7 and 10 will expand upwards and to the left – i.e. towards narrower, shallower scratches. The critical condition (2.1) for the onset of capillary flow is independent of $Re$ and $We$ since capillary flow is not inertia-driven and continues long after the initial spreading of the droplet. It is of course greatly influenced by $\theta _A$, and this is discussed below.

5.2. Effect of the advancing contact angle, $\theta _A$

The effect of $\theta _A$ on the initial spreading of a droplet on a flat surface is captured in some of the models of maximum spreading diameter in table 1. It is well known that contact angles below $90^{\circ }$ promote spreading and those above hinder it. Hence increasing $\theta _A$ results in reduced spreading both along and perpendicular to the scratch, and a consequently increased droplet height (unless of course the droplet is fully imbibed). However, $\theta _A$ also influences the shape of the liquid free surface within the scratch, as can be seen in figure 12, which shows just the liquid volume with the confining solid made invisible.

Figure 12.

Effect of $\theta _A$ on final droplet shape for $Re=204$, $We=26$ and $\theta _R=1^{\circ }$. The images show just the liquid volume(s) within and around each scratch.

The contact line on the bottom of the scratch, which is generally concave at $\theta _A=75^{\circ }$, becomes convex when $\theta _A=115^{\circ }$ as the sidewalls of the scratch then act to hinder rather than assist spreading. The non-trivial shape of the liquid free surface in the semi-imbibed and split semi-imbibed cases illustrate why there is a small discrepancy between the predicted regime boundary given by (2.4) and that observed from the simulations.

In the quasi-spherical cap regime, the liquid volume that runs along the scratch decreases with increasing $\theta _A$ until it no longer extends beyond the diameter perpendicular to the scratch, and for $\theta _A=115^{\circ }$ (figure 12) the liquid inside the scratch does not extend as far as that above the scratch. Moreover, capillary flow will no longer occur for $\theta _A\geqslant 90^{\circ }$, and as a consequence the region in figures 7 and 10 where capillary flow exists moves to smaller values of $w/d$ as $\theta _A$ increases, until the capillary flow region vanishes.

For a scratch that produces the inertial morphology when $\theta _A=75^{\circ }$, the reduced spreading that occurs when $\theta _A$ is increased can mean that the droplet no longer has sufficient momentum to spill over the side ridges, and the droplet becomes edge-pinned or semi-imbibed. A similar effect is seen for wider scratches, and we conclude that the boundaries within the regime map in figure 7 shift towards lower widths as $\theta _A$ increases, with the capillary flow region eventually disappearing.

5.3. Effect of the receding contact angle, $\theta _R$

If, as the droplet shape changes, the contact angle falls below the receding contact angle, the contact line will recede. Changes in $\theta _R$ are therefore be expected to have a significant influence on the droplet morphologies discussed above. Figure 13 highlights this via top-view plots of the final droplet shape for five different values of $\theta _R$ from the $1^{\circ }$ value used to create figures 7 and 10 up to $\theta _R=75^{\circ }$, which corresponds to the case where there is no CAH. For scratch dimensions leading to the quasi-spherical cap morphology, the droplet remains quasi-spherical, but its footprint becomes smaller as $\theta _R$ increases because the contact line recedes until an equilibrium is reached where the contact angle is everywhere equal to or greater than $\theta _R$. The conditions for capillary flow to occur are independent of $\theta _R$, but as $\theta _R$ increases, the contraction of the droplet above the scratch means that more liquid is available to feed the capillary flow and the final extent of the capillary flow is therefore increased. However, for intermediate $\theta _R$, as the droplet recedes from the top of the side ridges and original surface, small amounts of liquid are left behind in the corners between the outer wall of the side ridges and the original surface of the substrate. When there is no contact angle hysteresis ($\theta _R=\theta _A=75^{\circ }$), the entire droplet is pulled into the scratch by capillary action.

Figure 13.

Effect of $\theta _R$ on final droplet shape. The images show top views for $\theta _A=75^{\circ }$, $Re=204$ and $We=26$.

The inertial morphology in figure 7 arises when the droplet spills over the side ridges of the scratch onto the original surface, but no capillary action occurs. When $\theta _R$ is larger, the contact line recedes after reaching its maximum spread, and climbs back onto the side ridges to reach a different equilibrium. Hence the inertial morphology becomes edge pinned and semi-imbibed for $\theta _R$ values of $50^{\circ }$ and $75^{\circ }$, respectively. The liquid inside the scratch also recedes with higher $\theta _R$. The semi-imbibed morphology becomes fully imbibed due to the recession of the footprint outside the groove into it. The edge-pinned morphology turns semi-imbibed and then fully imbibed as $\theta _R$ approaches $\theta _A$. The fully imbibed morphology only changes in its extent along the scratch. Increasing the receding contact angle generally results in an equilibrium morphology that is less spread in the direction perpendicular to the scratch, and also shorter in the direction along the scratch. The exceptions are the capillary and edge-pinned regimes, where the lack of contact-line pinning on the upper substrate releases more liquid to penetrate along the scratch. All the regime map boundaries shift to lower widths, except the capillary regime because it is governed by $\theta _A$.

6.1. Line printing across a scratch

With the prospect of a break in line continuity, a key quantity to consider is the extent of single-droplet spreading in the direction perpendicular to the scratch. Figure 14 shows the behaviour of $D_{Across}$, i.e. the length of the resulting single-droplet morphology in the direction perpendicular to the scratch (see figure 7), as a function of scratch width and depth. The plots show $D_{Across}$ normalised by $D_{flat}$, the equilibrium spreading diameter of an equivalent droplet on a flat surface. Recall that the scratch width and depth are scaled by the impacting droplet's in-flight diameter. While narrow, shallow scratches cause only a small change in the extent of spreading, it is clear that as the droplet and scratch become similar in size, a significant shortfall in the spread length occurs. Depending on the degree of overlap between consecutive droplets, which determines the printed line width (Stringer & Derby Reference Stringer and Derby2010), such a reduction in spreading could prevent coalescence of a droplet with the rest of the line and hence a break in continuity. Note that the convergence of the lines in figure 14(a) and the sudden drop in the $w=0.4$ line in figure 14(b) are a result of pinning of the contact line on the outer edge of the side ridges. The upturn between $w=0.9$ and $w=1.0$ in figure 14(a) is because for the fully imbibed morphology, $D_{Across}=w$.

Figure 14.

Value of $D_{Across}$, the length of the final single-droplet morphology in the direction perpendicular to a scratch following deposition of a droplet on the scratch shown in figure 2 at $Re=204$, $We=26$, with $\theta _A=75^{\circ }$ and $\theta _R=1^{\circ }$. The plots show variation with scratch width and depth, and the coloured symbols indicate the type of morphology following the same labelling as in figure 7.

Beside the prevention of coalescence due to the reduction in spreading, another mechanism by which line continuity could be broken is through the splitting of the droplet along the inner edge of the scratch. This is illustrated in figure 15, which shows simulations of printing a series of five droplets across two scratches with the same depth but different widths. Both scratches appear in the split semi-imbibed region of figure 7, and the third droplet, landing on the scratch centre, splits along the inner edges of the scratch, as expected – see figure 15(b,f). For the wider of the two scratches, this splitting is irrecoverable and the printing continues with a separate line on the other side of the scratch. However, for the slightly narrower scratch, when the fourth droplet is printed, it pushes back on the pre-existing liquid on the substrate, making the liquid coalesce and become continuous again, see figure 15(g). This illustrates that the dynamics of the consecutively printed and coalescing droplets can be subtly different from the single-droplet dynamics. Note that while the narrower scratch does not cause a break in continuity, the thinner parts of the printed line could cause problems such as higher resistance or local heat generation.

Figure 15.

Simulations of printing consecutive droplets across scratches of depth $\bar {d}=0.4$ ($d=0.6$) and widths $w=0.6$ (a–d) and $w=0.45$ (e–h).

The extent of liquid spreading along a scratch due to a single droplet impact, $D_{Along}$ (see figure 7), is also an important consideration in line printing, and this is captured in figure 16. For a line crossing the scratch, spreading along the scratch would create a variation in the thickness of the line that could potentially lead to instability and formation of bulges along the line. As can be seen in figure 16, all dimensions of scratch lead to enhanced spreading along the scratch direction, with the most significant extent being of course that corresponding to the deep, narrow scratches where capillary flow occurs. The long filaments of the capillary morphology could be problematic if, for instance, two parallel lines are being printed in close proximity for a printed circuit: they could cause the two lines to connect unintentionally, resulting in a short circuit. However, because of the slower time scale of the capillary flow, as seen in figure 9(c), this issue could perhaps be avoided using a fast enough curing mechanism or by creating a larger advancing contact angle. On the other hand, this morphology can be exploited to connect two lines by designing such a feature and using a slowly evaporating ink.

Figure 16.

Value of $D_{Along}$, the length of the final single-droplet morphology in the direction along a scratch following deposition of a droplet on the scratch shown in figure 2 at $Re=204$, $We=26$, with $\theta _A=75^{\circ }$ and $\theta _R=1^{\circ }$. The plots show variation with scratch width and depth, and the coloured symbols indicate the type of morphology following the same labelling as in figure 7.

6.2. Printing along a scratch

The edge-pinned morphology is an example of how structured substrates can be exploited to print lines with sharp edges. To demonstrate this effect, five droplets are printed into a groove with depths of $\bar {d}=0.3$ and $\bar {d}=0.45$ and width $w=0.4$, both corresponding to the edge-pinned morphology. The results are visualised in figure 17. In the first simulation, ($\bar {d}=0.3$ and $w=0.4$), although the first droplet forms a sharp edge as expected, subsequent droplets spill over as seen in figure 17(c,d). This occurs because the precursor droplet inside the groove is in the spreading path of the subsequent droplet; this causes it to spill over. Increasing the depth to $0.45$, however, allows more volume for droplet spreading inside the groove, so overspill does not occur and a sharp line is formed along the outer edges of the side ridges, as seen in figure 17(h). These two simulations demonstrate how challenging printing a sharp line can be.

Figure 17.

Simulations of printing five consecutive droplets along scratches with width $w=0.4$ and depths $\bar {d}=0.3$ (a–d) and $\bar {d}=0.45$ (e–h).

The simulations of printing along and across a scratch show that topographical features of commensurate size to droplets have a significant effect not only on a single droplet but also a series of droplets.