4.3.1. Particle deposition after droplet evaporation experiencing single stick-slip process

Figure 10.

Final particle deposition profiles under different liquid and particle properties and surface wettability: (a) surface tension $\sigma \in (4.9 \times {10^{ - 4}},1.5 \times {10^{ - 2}})\ {\textrm{N}}\,\text{m}^{- 1}$ with initial contact angle ${\theta _{\textit{eq}}} = 35^\circ$ , liquid kinematic viscosity ${\mu _l} = 1.7 \times {10^{ - 2}} \ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ and particle diffusion coefficient ${D_{\!p}} = 1.7 \times {10^{ - 9}} \ {\textrm{m}}^{2}\,\text{s}^{- 1}$ ; (b) ${\mu _l} \in (3.4 \times {10^{ - 3}},1.71)\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ with ${\theta _{\textit{eq}}} = 35^\circ$ , $\sigma = 6.4 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} = 1.7 \times {10^{ - 9}} \ {\textrm{m}}^{2}\,\text{s}^{- 1}$ ; (c) ${D_{\!p}} \in (8.6 \times {10^{ - 10}},1.1 \times {10^{ - 7}})\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ with ${\theta _{\textit{eq}}} = 35^\circ$ , ${\mu _l} = 1.7 \times {10^{ - 2}} \ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ and $\sigma = 6.4 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ ; (d) ${\theta _{\textit{eq}}} \in (25^\circ ,85^\circ )$ with ${\mu _l} = 3.4 \times {10^{ - 2}} \ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , $\sigma = 7.5 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} = 1.7 \times {10^{ - 9}} \ {\textrm{m}}^{2}\,\text{s}^{- 1}$ .

The simulation results for the colloidal droplet evaporation experiencing single stick-slip process are compared in figure 10. In figure 10(a), the initial contact angle is ${\theta _{\textit{eq}}} = 35^\circ$ , liquid dynamic viscosity is ${\mu _l} = 1.71 \times {10^{ - 2}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ and particle diffusion coefficient is ${D_{\!p}} = 1.71 \times {10^{ - 9}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ , while the surface tension varies from $\sigma = 4.9 \times {10^{ - 4}}\ {\textrm{N}}\,\text{m}^{- 1}$ to $\sigma = 1.5 \times {10^{ - 2}}\ {\textrm{N}}\,\text{m}^{- 1}$ by increasing 30 times. We observe that the deposition pattern gradually transits from mountain-type to uniform, and finally to ring deposition, with the obtained deposition ratio varying from ${r_\phi } = 0.21$ to ${r_\phi } = 6.28$ . The reason lies in that the capillary transport is related to the pressure difference between droplet apex and periphery, which is dependent on the liquid–vapour surface tension $\sigma$ . When $\sigma$ increases, the capillary transport is strengthened, thus more particles accumulate and finally deposit at the periphery of the droplet. In figure 10(b), ${\theta _{\textit{eq}}} = 35^\circ$ , $\sigma = 6.4 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} = 1.71 \times {10^{ - 9}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ , while $\mu _l$ varies from $3.4 \times {10^{ - 3}}$ to $1.7 \,{\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ with an increase of 500 times. As explained in figure 5, high viscosity weakens the capillary transport from droplet apex to periphery, resulting in a transition from coffee ring ( ${r_\phi } = 92.1$ ) to mountain-type ( ${r_\phi } = 0.43$ ) deposition pattern. In figure 10(c), ${\theta _{\textit{eq}}} = 35^\circ$ , ${\mu _l} = 1.71 \times {10^{ - 2}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ and $\sigma = 6.4 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ , while $D_{\!p}$ increases by 128 times from $8.6 \times {10^{ - 10}}$ to $1.1 \times {10^{ - 7}} \ {\textrm{m}}^{2}\,\text{s}^{- 1}$ . Similar to figure 5, at higher $D_{\!p}$ , the diffusion of particles is stronger, making the distribution more uniform. Due to the slip of droplets in the final stage, the mountain-type deposition pattern could occur at high $D_{\!p}$ . The deposition ratios of resulting patterns in figure 10(c) are ${r_\phi } = 19.7,5.3,0.80,0.25$ , respectively. Finally in figure 10(d), ${\mu _l} = 3.4 \times {10^{ - 2}}$ , $\sigma = 7.5 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} = 1.71 \times {10^{ - 9}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ , while $\theta _{\textit{eq}}$ increases from $25^\circ$ to $85^\circ$ . Since the contact angle hysteresis range $\theta _{\textit{eq}} - \theta _{R}$ increases, the period of capillary transport is prolonged. Moreover, the contact radius (i.e. the transport path) is shorter, which is beneficial for deposition at the droplet periphery. As a result, the deposition ratios of coffee rings in figure 10(d) increase significantly from ${r_\phi } = 5.6$ to ${r_\phi } = 112$ . In the results of figure 10(d), the horizontal axis is normalised by the contact radius of the validation case with ${\theta _{\textit{eq}}} = 35^\circ$ , which is the reason why the deposited particles are not located at $(X - {X_m})/{R_{c,0}}({\theta _{\textit{eq}}} = 35^\circ ) = 1.0$ . In this parametric study, the overall ranges of the two dimensionless numbers for the evaporation experiencing single stick-slip process are $\textit{Pe} \in (0.38,70.0)$ and $\textit{Ca} \in (5.4 \times {10^{ - 4}},4.1 \times {10^{ - 3}})$ . While the small $\textit{Ca} \ll 1$ indicates diffusive evaporation, the wide range $\textit{Pe} \in ({10^{ - 1}},{10^2})$ is responsible for various deposition patterns.

Figure 11.

Comparison of particle deposition profiles after colloidal droplet evaporation experienced symmetrical multiple stick-slip processes. (a) Illustration of the evaporation process with different evaporation conditions, where $\theta _{\textit{eq}}$ and $\theta _R$ represent the equilibrium and receding contact angles, respectively. (b–d) Particle deposition profiles at different liquid and particle properties. Cases 1, 1^′ and 1^′′: liquid dynamic viscosity ${\mu _l} = 1.7 \times {10^{ - 2}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , surface tension $\sigma = 6.4 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ and particle diffusion coefficient ${D_{\!p}} = 1.7 \times {10^{ - 9}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ . Cases 2, 2^′ and 2^′′: ${\mu _l} = 3.4 \times {10^{ - 1}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , $\sigma = 1.2 \times {10^{ - 2}}\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} = 1.7 \times {10^{ - 9}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ . Cases 3, 3^′ and 3^′′: ${\mu _l} = 1.7 \times {10^{ - 1}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , $\sigma = 3.1 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} = 1.7 \times {10^{ - 9}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ . Cases 4, 4^′ and 4^′′: ${\mu _l} = 3.4 \times {10^{ - 1}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , $\sigma = 1.2 \times {10^{ - 2}}\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} = 1.7 \times {10^{ - 8}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ . The corresponding average deposition ratios are ${r_{\phi ,a}} = 23.3,3.53,0.94,0.12$ , ${r_{\phi ,a}} = 13.8,2.84,0.66,0.11$ and ${r_{\phi ,a}} = 5.77,1.31,0.47,0.11$ , respectively.

4.3.2. Particle deposition after droplet evaporation experiencing multiple stick-slip processes

The above-explained mechanisms are also valid for colloidal droplet evaporation experiencing multiple stick-slip processes, with/without considering space-dependent contact angle hysteresis. As shown in figure 11(a), we first conduct three series of simulations with different contact angle hysteresis $({\theta _R},{\theta _{\textit{eq}}}) \in (30^\circ ,60^\circ )$ (cases 1–4), $(25^\circ ,50^\circ )$ (cases 1^′–4’) and $(25^\circ ,35^\circ )$ (cases 1^′′–4^′′), respectively, while the hysteresis does not vary in space. In each series, the simulation parameters of liquid and particle vary. For cases 1, 1^′ and 1^′′, the liquid dynamic viscosity is ${\mu _l} = 1.71 \times {10^{ - 2}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , surface tension is $\sigma = 6.4 \times {10^{ - 3}}\ {\textrm{N}}\,\text{m}^{- 1}$ and particle diffusion coefficient is ${D_{\!p}} = 1.71 \times {10^{ - 9}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ , where multi-ring deposition patterns are shown as red curves in figure 11(b–d). For cases 2, 2^′ and 2^′′, the viscosity and surface tension are increased by 20 and 2 times to ${\mu _l} = 3.4 \times {10^{ - 1}}\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ and $\sigma = 1.2 \times {10^{ - 2}}\ {\textrm{N}}\,\text{m}^{- 1}$ , respectively, and the resulting multi-ring deposition is weakened, as seen in the green dashed curves in figure 11(b–d). The reason is that the effect of high viscous drag increment dominates the small capillary transport enhancement. For cases 3, 3^′ and 3^′′, the viscosity $\mu _l$ and surface tension $\sigma$ decrease by 50 % and 75 %, respectively, compared to cases 2, 2^′ and 2^′′. We can see that the deposition patterns transit from coffee ring to mountain-type, as shown in the blue curves. The reason lies in the decrease of surface tension being higher than that of the viscosity, thus the viscous drag dominates the capillary transport. Still compared to cases 2, 2^′ and 2^′′, if we only increase the particle diffusion coefficient $D_{\!p}$ by 10 times, then we can see that strong mountain-type deposition patterns are formed as in the grey curves, since the particle diffusion is prevailing. If we compare the simulations with different contact angle hysteresis ranges, then we can see that the average deposition ratio $r_{\phi ,a}$ decreases generally. For instance, from case 1 with $({\theta _R},{\theta _{\textit{eq}}}) \in (30^\circ ,60^\circ )$ to case 1^′ with $({\theta _R},{\theta _{\textit{eq}}}) \in (20^\circ ,35^\circ )$ , $r_{\phi ,a}$ decreases from 23.3 to 5.8. On the other hand, with the decrease of hysteresis range, the stick-slip cycles increase from four in case 1 to seven in case 1^′, since the slip period is shorter due to the smaller hysteresis range. The overall ranges of the two dimensionless numbers for the evaporation experiencing multiple symmetrical stick-slip processes are $\textit{Pe} \in (0.84,43.6)$ and $\textit{Ca} \in (7.4 \times {10^{ - 4}},5.6 \times {10^{ - 3}})$ .

Figure 12.

Comparison of particle deposition profiles after colloidal droplet evaporation experienced unsymmetrical multiple stick-slip processes. (a) Illustration of the evaporation process with different evaporation conditions, where $\theta _{\textit{eq}}$ , $\theta _{R,L}$ and $\theta _{R,R}$ represent the equilibrium contact angle and receding contact angles at the left and right half-domains, respectively. (b–d) Particle deposition profiles at different liquid and particle properties. The parameters in cases 5–8, 5^′–8^′ and 5^′′–8^′′ are the same as those in cases 1–4, 1^′–4^′ and 1^′′–4^′′ in figure 11. The corresponding average deposition ratios are ${r_{\phi ,a}} = 44.1,5.06,0.92,0.20$ , ${r_{\phi ,a}} = 29.4,2.61,0.90,0.25$ and ${r_{\phi ,a}} = 7.31,1.49,0.65,0.22$ , respectively.

Apart from the above three series of simulations experiencing multiple symmetrical stick-slip processes, we further conduct another three series of simulations with the consideration of space-dependent contact angle hysteresis range. As shown in figure 12(a), in cases 5–8, 5^′–8^′ and 5^′′–8^′′, the receding contact angles in the left ( $\theta _{R,L}$ ) and right ( $\theta _{R,R}$ ) half-domains are different, i.e. $\theta _{R,L}= 20^\circ ,17^\circ ,12^\circ$ are smaller than $\theta _{R,R}= 30^\circ ,25^\circ ,20^\circ$ . Similar to the simulation shown in figures 7(b) and 7(f), the contact point remains pinned at the left half-domain before the right-hand contact point recedes to the left half-domain after experiencing several stick-slip processes. As explained above in the discussion of figure 11(b–d), the variations of liquid and particle parameters lead to different deposition patterns ranging from multiple ring to uniform and mountain-type, as shown in figure 12(b–d). Moreover, the stick-slip cycles increase with the decrease of receding contact angle, for both the left- and right-hand contact points. For instance, they increase from two and seven to three and eleven, for the contact angle ranges varying from $\theta _{\textit{eq}} =60^\circ ,\ \theta _{R,L} /\theta _{R,R} =20^\circ /30^\circ$ in case 6 to $\theta _{\textit{eq}} =35^\circ ,\ \theta _{R,L} /\theta _{R,R} =12^\circ /20^\circ$ in case 6^′′. The differences of particle deposition in figure 12(a) lie in the unsymmetrical deposition, i.e. more deposition occurs in the left half-domain, as shown in figure 12(b–d). Similarly, the overall ranges of the two dimensionless numbers for the evaporation experiencing multiple unsymmetrical stick-slip processes are $\textit{Pe} \in (0.72,101.8)$ and $\textit{Ca} \in (4.4 \times {10^{ - 4}},2.2 \times {10^{ - 2}})$ .

In the parametric study above, we have qualitatively analysed the individual and coupled influences of liquid, particle and substrate properties on the deposition patterns, considering colloidal droplet evaporation experiencing single and multiple symmetrical/unsymmetrical stick-slip processes. However, quantitative analyses on how the deposition patterns are related to these parameters are still unclear and need to be understood. In next subsection, we desire to theoretically derive the influence of each parameter on the deposition patterns, and to finally quantify various deposition patterns using a single dimensionless variable.

4.4.1. The proposed scaling parameter and validation by simulations

As we explained in the results of figure 5, the two mechanisms that account for the particle transport during isothermal evaporation are droplet internal capillary flow and particle diffusion under different liquid/particle conditions, which determine the deposition ratio ${r_\phi } = {\phi _p}/{\phi _c}$ . To quantify the competition of the two processes considering all patterns studied above, we try to relate it to the dimensionless Péclet number $\textit{Pe} = {u_c}{R_{c,0}}/{D_{\!p}}$ , where $u_c$ is the characteristic fluid velocity due to internal capillary transport, which is shown by simulations related to liquid properties such as viscosity and surface tension, $R_{c,0}$ is the initial contact radius, and $D_{\!p}$ is the particle diffusion coefficient. The estimation or scaling of $u_c$ is key to the calculation of $\textit{Pe}$ , which is determined based on lubrication theory (Oron, Davis & Bankoff Reference Oron, Davis and Bankoff1997; Kaplan & Mahadevan Reference Kaplan and Mahadevan2015). We define the droplet aspect ratio $\varepsilon = h/{r_c}$ , where $h$ and $r_c$ are the droplet height and contact radius during the evaporation process, respectively. According to classical lubrication theory, under a thin film assumption $\varepsilon \ll 1$ , the balance between radial pressure gradient and vertical viscous dissipation can be written as $\partial p/\partial {r_c} = {\rho _l}{v_l}\,{\partial ^2}u/\partial {h^2}$ , and scaled as $\Delta p/{r_c} \sim {\mu _l}{u_c}/{h^2}$ . For the pressure difference $\Delta p$ , we estimate it using the capillary pressure $\Delta p \approx \sigma\, {\partial ^2}h/\partial r_c^2$ , which is scaled as $\Delta p \sim \sigma h/r_c^2$ . By combining the two scaling relations $\Delta p/{r_c} \sim {\mu _l}{u_c}/{h^2}$ and $\Delta p \sim \sigma h/r_c^2$ , we simply obtain the scaled characteristic velocity as ${u_{\textit{c,scale}}} \sim {{\sigma {\varepsilon ^3}}}/{{{\mu _l}}}$ . We emphasise that this derivation is under the lubrication theory $\varepsilon \ll 1$ . In our simulations, we also consider cases including higher aspect ratio up to $\varepsilon \approx 0.9$ . Therefore, we have to modify the balance between radial pressure gradient and viscous dissipation, i.e. we use $\partial p/\partial {r_c} = {\rho _l}{\nu _l}({\partial ^2}u/\partial {h^2} + {\partial ^2}u/\partial r_c^2)$ instead of $\partial p/\partial {r_c} = {\rho _l}{v_l}\,{\partial ^2}u/\partial {h^2}$ , in order to consider the viscous dissipation in both height and radial directions. With this modification, the more accurate scaled characteristic velocity is obtained as ${u_{\textit{c,scale}}} \sim ( {\sigma }/{{{\mu _l}}}) ( {{{\varepsilon ^3}}}/{{(1 + {\varepsilon ^2})}})$ . The reduced capillary number is also modified from $C{a^*} = {{Ca}}/{{{\varepsilon ^3}}} = {{{\mu _l}{u_c}}}/{{({\varepsilon ^3}\sigma) }}$ to $C{a^*} = Ca\,( {{1 + {\varepsilon ^2}}})/{{{\varepsilon ^3}}} = {{(1 + {\varepsilon ^2}){\mu _l}{u_c}}}/({{{\varepsilon ^3}\sigma }})$ . Using the scaled velocity, the reduced capillary number is calculated as $\textit{Ca}_{\textit{scale}}^* = {{(1 + {\varepsilon ^2}){\mu _l}{u_{\textit{c,scale}}}}}/({{{\varepsilon ^3}\sigma }}) \sim ( {{(1 + {\varepsilon ^2}){\mu _l}}}/({{{\varepsilon ^3}\sigma }}))( {{\sigma {\varepsilon ^3}}}/({{{\mu _l}(1 + {\varepsilon ^2})}})) = 1$ . We note that $\textit{Ca}_{\textit{scale}}^* \sim 1$ is more like a qualitative description that the viscous diffusion is balanced by the internal transport by capillary convection. Quantitatively, it is only valid when using the scaled velocity $u_{\textit{c,scale}}$ , and it does not mean that the real reduced capillary number in our simulations is also at the magnitude of $O(1)$ . For instance, by using the real averaged velocity $u_{c,LBM}$ in the simulations in § 4.1, the calculated reduced capillary number $C{a^*} = {{(1 + {\varepsilon ^2}){\mu _l}{u_{c,LBM}}}}/({{{\varepsilon ^3}\sigma }})$ for ring to mountain-type depositions is in the range $C{a^*} \in (1.9 \times {10^{ - 2}},6.1 \times {10^{ - 2}})$ , which is far less than $O(1)$ . The huge difference in the values of $C{a^*}$ originates from the scaling of radial liquid pressure difference $\Delta p$ by capillary pressure, since the latter strongly overestimates the former. To avoid any confusion, we note that for all the comparisons between our simulations and experiments in the literature given in previous sections, we use the real simulated velocity $u_{c,LBM}$ to calculate (reduced) capillary number $\textit{Ca}$ ( $C{a^*}$ ) and Péclet number $\textit{Pe}$ .

In the experiment, the characteristic velocity is normally calculated by liquid mass loss due to diffusive evaporation, i.e. ${u_{c,{\textit{diff}}}} = {{{J_c}R}}/({{{\rho _l}H}}) = {{{J_c}}}/({{{\rho _l}\varepsilon }})$ , with ${J_c} = {{D({c_{\textit{sat}}} - {c_\infty })}}/{R}$ representing the diffusive flux. This method of calculating characteristic velocity is accurate for diffusive evaporation. For instance, for the experiment in Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011), the calculated velocity is ${u_{c,{\textit{diff}}}} \approx 1.9 \times {10^{ - 6}}\ {\textrm{m}}\,\text{s}^{-1}$ , leading to the reduced capillary number $\textit{Ca}_{{\textit{diff}}}^* \approx 1 \times {10^{ - 5}} \ll 1$ , as expected. However, the calculation of $u_{c,{\textit{diff}}}$ by diffusive mass loss is based on the assumption that the capillary convection is much faster than the mass loss by vapour diffusion. That is, the liquid is almost ‘immediately’ transported to the droplet surface without considering the internal fluid dynamics. But in reality, the transport takes a certain period of time, and the dynamic process is affected by liquid properties such as viscosity and surface tension, as shown in our simulations. These influences could not be considered when using the expression ${u_{c,{\textit{diff}}}} = {{D({c_{\textit{sat}}} - {c_\infty })}}/({{{\rho _l}\varepsilon R}})$ . Instead, the scaled velocity ${u_{\textit{c,scale}}} \sim ( {\sigma }/{{{\mu _l}}})({{{\varepsilon ^3}}}/({{1 + {\varepsilon ^2}}}))$ takes into consideration all the variables, including liquid properties (surface tension $\sigma$ and viscosity $\mu _l$ ) and surface wettability (droplet aspect ratio $\varepsilon = H/R$ ). In the current work, since we mainly consider the influences of liquid, surface and particle properties on the particle deposition patterns, we choose to use $u_{\textit{c,scale}}$ instead of $u_{c,{\textit{diff}}}$ . Another reason is related to current model limitations. As we introduced in § 2, our modelling system is based on a single-component multiphase model with liquid and its vapour, but without air component. The evaporation is induced by the difference between vapour pressure at the liquid–vapour interface (saturation vapour pressure) and vapour pressure at the outlet boundary (a little lower than saturation). In this system, there is no definition of vapour diffusion coefficient $D$ or vapour concentration ${c_{\textit{sat}}},{c_\infty }$ . Therefore, it is currently not possible for us to calculate $u_{c,{\textit{diff}}}$ . We will work on introducing an air component to our simulation system for more precise modelling of vapour diffusion in the future. In the current work, by using the scaled characteristic velocity ${u_{\textit{c,scale}}} \sim ( {\sigma }/{{{\mu _l}}})({{{\varepsilon ^3}}}/({{1 + {\varepsilon ^2}}}))$ with ${\mu _l}={\rho _l}{\nu _l}$ , and using as a simplification of the initial aspect ratio ${\varepsilon _0} = {H_0}/{R_{c,0}}$ , we obtain the dimensionless Péclet number scaled as

(4.3) \begin{equation} Pe = \frac {{{u_{\textit{c,scale}}}{R_{c,0}}}}{{{D_{\!p}}}} \sim \frac {{\sigma {R_{c,0}}}}{{{\rho _l}{\nu _l}{D_{\!p}}}}\frac {{\varepsilon _0^3}}{{1 + \varepsilon _0^2}}. \end{equation}

Equation (4.3) shows that high liquid–vapour surface tension $\sigma$ , high initial aspect ratio $\varepsilon _0$ (corresponding to high contact angle), low liquid viscosity $v_l$ , and low particle diffusion coefficient $D_{\!p}$ lead to high $\textit{Pe}$ . On the other hand, through the results in the parametric study in § 4.3, we also find that high $\sigma$ , low $v_l$ and small $D_{\!p}$ are beneficial for coffee ring deposition patterns with high deposition ratio $r_\phi$ . That is, the dependence of $\textit{Pe}$ and $r_\phi$ on these parameters is compatible, and $\textit{Pe}$ may be used to quantify different deposition patterns. Compared with the dimensionless number $\alpha = {\varepsilon ^4}/\textit{Ca}$ (Kaplan & Mahadevan Reference Kaplan and Mahadevan2015), which quantifies the competition between capillary convection $u_c$ and evaporation rate $E_0$ , our proposed effective $\textit{Pe}$ quantifies the competition between capillary convection $u_c$ and particle diffusion at a specific evaporation rate $E_0$ . In other words, the cases that we studied focus on how different droplet properties such as $\mu ,\sigma ,\varepsilon$ affect $u_c$ , and how the particle diffusion coefficient $D_{\!p}$ affects the diffusion at a constant $E_0$ . However, the scaling (4.3) only works for the deposition patterns with single stick-slip process. To extend it to deposition patterns after consecutive stick-slip processes as shown in figures 11 and 12, we introduce the average value of each $\textit{Pe}_i$ as the average dimensionless number $\textit{Pe}_a$ :

(4.4) \begin{equation} P{e_a} = \frac {1}{n}\sum _{i = 0}^{n - 1} {P{e_i}} ,\quad P{e_i} = P{e_{i,sk}} - P{e_{i,sp}}, \end{equation}

where the subscripts $sk$ and $sp$ represent the $i$ th stick and slip phases as shown in figure 7(a,b). The corresponding $\textit{Pe}_{i,sk}$ and $\textit{Pe}_{i,sp}$ can be calculated with (4.3). The difference in calculating $\textit{Pe}_{i,sk}$ and $\textit{Pe}_{i,sp}$ lies in $\varepsilon _0$ being dependent on the contact angles of equilibrium $\theta _{\textit{eq}}$ and receding $\theta _{R}$ , respectively. The calculations of $P{e_i}$ and $P{e_a}$ are straightforward for droplet evaporation experiencing single or symmetrical multiple stick-slip processes. For droplet evaporation with unsymmetrical multiple stick-slip processes – as shown in figure 7(b), for instance – the estimation of $\textit{Pe}_i$ at contact point $A_0$ is rather complex, since it is always pinned before the right-hand contact point reaches the left half-domain, during which both the contact radius and the contact angle vary, and there is extra capillary flow from the right-hand contact area as shown in figure 7(c). Since the extra capillary flow is dynamic and difficult to predict, we simplify this situation by neglecting it and using the summation of all the $\textit{Pe}_i$ of the right-hand contact point to estimate that for the left-hand one, before the left-hand one recedes, e.g. to contact point $A_0$ in figure 7(b), ${\textit{Pe}_{{A_0}}} = \sum \nolimits _{i = 0}^5 {{{Pe}_{B_i}}}$ . In the following, we use $r_{\phi ,a}$ in (4.2) and $\textit{Pe}_a$ in (4.4) to quantify the deposition, since they are universal for droplet evaporation experiencing both single ( $i=0$ ) and multiple ( $i=0,1,\ldots ,n-1\gt 0$ ) stick-slip processes. In figure 13, we plot $r_{\phi ,a}$ against $\textit{Pe}_a$ using the 48 simulations conducted in §§ 4.1–4.3, with the liquid, particle and surface parameters spanning ${\mu _l} \in (3.4 \times {10^{ - 3}},1.71)\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , $\sigma \in (4.9 \times {10^{ - 4}},1.2 \times {10^{ - 2}})\ {\textrm{N}}\,\text{m}^{- 1}$ and ${D_{\!p}} \in (8.6 \times {10^{ - 10}},1.1 \times {10^{ - 7}})\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ , ${\varepsilon _{0}} \in (0.24,0.93)$ ( ${\theta _{\textit{eq}}} \in (25^\circ ,85^\circ )$ ) and ${\theta _{R,L}},{\theta _{R,R}} \in (12^\circ ,30^\circ )$ . A very neat linear relation (the slope is 1 for the log–log axes)

(4.5) \begin{equation} {r_{\phi ,a}} \approx k*P{e_a},\quad k = 6.7 \times {10^{ - 3}} \end{equation}

Figure 13.

The approximately linear relation ${r_{\phi ,a}} \approx k*{\textit{Pe}_a},k = 6.7 \times {10^{ - 3}}$ between average deposition ratio $r_{\phi ,a}$ (ratio of particles at droplet peripheries over particles at centre) and estimated average dimensionless number $\textit{Pe}_a$ for a broad range of parameters, i.e. droplet initial contact angle ${\theta _{\textit{eq}}} \in (25^\circ ,85^\circ )$ , droplet receding contact angle ${\theta _R} \in (5^\circ ,30^\circ )$ , liquid dynamic viscosity ${\mu _l} \in (3.4 \times {10^{ - 3}},1.71)\ {\textrm{kg}}\,\textrm{m}^{-1} \, \textrm{s}^{-1}$ , surface tension $\sigma \in (4.9 \times {10^{ - 4}},1.2 \times {10^{ - 2}})\ {\textrm{N}}\,\text{m}^{- 1}$ and particle diffusion coefficient ${D_{\!p}} \in (8.6 \times {10^{ - 10}},1.1 \times {10^{ - 7}})\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ .

is observed, within a large range of $\textit{Pe}_a$ values covering over three orders of magnitude. The prefactor $k \ll 1$ is explained in the discussions in the next subsubsection. We note that the prefactor $k$ is related to the overall evaporation rate determined by ambient vapour pressure. In the current work, we only consider isothermal evaporation with the same ambient vapour pressure, so that $k$ is a constant. To validate this argument, we conducted two simulations with different ambient pressures to realise different $k$ , while keeping all the other parameters constant. The results in Appendix A show that the ambient vapour pressure affects the evaporation flux distribution, which further affects the capillary convection, leading to different particle deposition profiles. More analysis about the influence of evaporation rate on the deposition patterns is subject to future study. To further validate the linear relation, we conducted 24 more simulations within the above-mentioned parameter ranges, considering the evaporation process experiencing symmetrical/unsymmetrical single/multiple stick-slip processes. As shown in figure 13, for all the studied cases, from simple mountain-like, uniform, single ring to unsymmetrical/symmetrical multiple rings, the overall coefficient of determination is up to ${R^2} = 0.969$ , indicating very good reliability to estimate $r_{\phi ,a}$ by (4.5) through $\textit{Pe}_a$ in (4.4). We note that relatively high deviations may occur in the results of complex evaporation processes with multiple stick-slip motions. But the general accuracy is acceptable for engineering applications.

In terms of incorporating some experimental data in the scaling result of figure 13, we are incapable of accomplishing it under current circumstances, for the following three reasons. First, our current simulations are conducted in two dimensions, while the experiments are based on three dimensions. The difference in dimension leads to different deposition ratios and bandwidths, making it inappropriate to plot them in the same figure. Second, the evaporation conditions are different. In our model, we only consider liquid and vapour without air, and we realise slow evaporation by using small vapour pressure difference. This condition may not match exactly with the isothermal evaporation, which affects the slope of the scaling relation. Third, as far as we know, there are very limited experimental results providing the deposition ratio results (Yunker et al. Reference Yunker, Still, Lohr and Yodh2011), due to the difficulty in measurement. In the future, we will further develop the model to three dimensions, and make the evaporation conditions more realistic by incorporating the air component. Moreover, we will conduct some experiments to obtain deposition ratios to validate our proposed scaling relation.

4.4.2. Discussions of the proposed scaling parameter

We have numerically shown that (4.5) is valid for various deposition patterns. In this subsubsection, we give some discussions on the scaled $\textit{Pe}$ based on the transport equation of particles. We consider the case of single coffee ring deposition pattern as an example. For uniform or mountain-type depositions, the influence of the contact line receding in the final evaporation period has to be taken into account, which is very complex and subject to future study. According to the convection–diffusion equation (2.23), the change of local concentration with time, $\partial \phi /\partial t$ , consists of two parts, i.e. the convection part $\boldsymbol{\nabla }\boldsymbol{\cdot }(\phi {\boldsymbol {u}_c}) \approx {{\partial (u\phi ) }}/{{\partial x}}$ and the diffusion part $\boldsymbol{\nabla }\boldsymbol{\cdot }[{D_{\!p}}\boldsymbol{\nabla }\phi ] \approx {D_{\!p}}( {{{\partial ^2}\phi }}/{{\partial {x^2}}})$ , when we mainly consider the particle transport in the $x$ direction. At a certain evaporation time $t_e$ , the particle concentration at the droplet periphery ${\phi _p}({t_e})$ consists of three different parts:

1. (i) initial particle concentration $\phi _0$

2. (ii) concentration increment with time due to capillary convection from the droplet central area $\int _0^{{t_e}} ({{{\partial (u(t)\,\phi (t))}}/{{\partial x}}})\, {\textrm{d}}t$

3. (iii) concentration reduction with time due to diffusion from the droplet periphery towards the central area $-\int _0^{{t_e}} {{D_{\!p}}( {{{\partial ^2}\phi (t)}}/{{\partial {x^2}}}})\, {\textrm{d}}t$ ; by simple summation, we obtain

   (4.6) \begin{equation} {\phi _p}({t_e}) = {\phi _0} + \int _0^{{t_e}} {\frac {{\partial (u(t)\,\phi (t))}}{{\partial x}}}\, {\textrm{d}}t - \int _0^{{t_e}} {{D_{\!p}}\frac {{{\partial ^2}\phi (t)}}{{\partial {x^2}}}}\, {\textrm{d}}t. \end{equation}

Figure 14.

Explanation of particle accumulation and the length scale by capillary convection and diffusion.

In the following, we try to simplify (4.6) based on some simple assumptions. First, the initial particle concentration is very small ( ${\phi _0} = 0.5\,\%$ ), thus it is directly ignored. Then we assume that the average capillary convection velocity is ${u_{\textit{c,real}}} \approx {k_1}{u_{\textit{c,scale}}}$ , where $u_{\textit{c,scale}}$ is the scaled characteristic velocity derived above, which considers the influence of liquid properties and substrate wettability, i.e. liquid dynamic viscosity $\mu _l$ and surface tension $\sigma$ as well as contact angle $\theta$ . Here, ${k_1} = {k_1}({E_{\!p}})$ is the coefficient related to the overall evaporation rate $E_{\!p}$ determined by the ambient vapour pressure, as discussed in Appendix A and also summarised in Wilson & D’Ambrosio (Reference Wilson and D’Ambrosio2023). As we explained at the beginning of § 4.4.1, the scaled characteristic convection velocity $u_{\textit{c,scale}}$ is overestimated. We obtain $k_1$ in ${u_{\textit{c,real}}} = {k_1}{u_{\textit{c,scale}}}$ by comparing the simulated convective velocity ${u_{\textit{c,real}}}(\text{LBM})$ with the scaled velocity ${u_{\textit{c,scale}}}(\text{LBM})$ . For the conducted simulations, we find that $k_1$ is situated in the range ${k_1} \in (0.015,0.12)$ . As shown in figure 14, the convection is from the droplet internal area ( $u_{\textit{centre}}$ ), and the particles cannot penetrate out of the droplet from the periphery ( ${u_{\textit{peri}}} = 0$ ), i.e. the change of local concentration by convection is written as $ {{\partial (u(t)\,\phi (t))}}/{{\partial x}} \sim ({u_{\textit{c,real}}}{\phi _c} - {u_{\textit{peri}}}{k_3}{\phi _p})/{(k_2 R_{c,0})} \sim {u_{\textit{c,real}}}{\phi _c}/{(k_2 R_{c,0})}$ . In this way, the convection part is simplified as ${k_1}{u_{\textit{c,scale}}}{t_e}{\phi _c}/(k_2 {R_{c,0}})$ . Third, for the diffusion part, the Laplacian can be simplified as $({k_3}{\phi _p} - {\phi _c})/{({k_2}{R_{c,0}})^2} \sim {k_3}{\phi _p}/{({k_2}{R_{c,0}})^2}$ , where $k_3$ is a coefficient less than 1 since the particle concentration increases from $\phi _0$ to $\phi _p$ continuously before deposition. In our simulations, at approximately 40 % of the time of total evaporation, the maximum concentration $\phi _p$ is reached. If we simply assume linear increase, then we can obtain ${k_3} \approx 40\,\% \times (0 + 1)/2 + (1 - 40\,\% ) \times 1 = 0.8$ .

We have that $k_2$ is a coefficient also less than 1 since the effective convection/diffusion distance is smaller than the droplet radius. Moore et al. (Reference Moore, Vella and Oliver2021) pioneer proposing a similar parameter $w_{1/2}$ called the full width at half-maximum to quantify the width of a nascent coffee ring. They proposed the scaling relations for both kinetic and diffusive evaporation modes, i.e. ${w_{1/2}} \sim (1 - t)\,Pe^{ - 1}$ and ${w_{1/2}} \sim (1 - t)\,Pe^{ - 2}$ , respectively, where $t$ is the normalised evaporation time. In our simulations with diffusive evaporation, we obtained the scaling ${w_{1/2}} \sim Pe^{ - 1.2}$ . The discrepancy might originate from two possible reasons. First, the theoretical scaling relation is proposed assuming singular diffusive flux at the contact line, which overestimates the convection. In our simulations, the diffusive flux is a fixed value instead of a singularity (see figure 15), leading to the scaling power between $-1$ (kinetic model) and $-2$ (diffusive mode with singularity). Second, the scaling theory is based on dilute concentration regime with very small initial particle concentration $\phi _0$ . In Moore et al. (Reference Moore, Vella and Oliver2021), the authors declared that at $\textit{Pe} = 10$ , from ${\phi _0} = 0.0001\,\%$ to ${\phi _0} = 1\,\%$ , the validity of their theoretical prediction decreases significantly from 82 % to 4 % of the drying time. In our simulations, the initial concentration is ${\phi _0} = 0.5\,\%$ , and $\textit{Pe}$ ranges from $O(10^{-1})$ to $O(10^{2})$ or even higher, which may exceed the applicable range of their theory. Considering ring deposition in this discussion, the Péclet number is of the order of $O(10)$ or higher, and the corresponding parameter $k_2$ obtained is in the range approximately ${k_2} = 2{w_{1/2}} \in (0.02,0.14)$ . Figure 14 shows an example where the particle concentration quickly decreases from 0.96 to approximately 0.1 at the droplet periphery by a distance approximately $0.07R_{c,0}$ ( $ {\textrm{d}}x =k_2R_{c,0}\approx 7\,\%\,R_{c,0}$ (see also figure 4 b,d). Thus the diffusion part is simplified as ${D_{\!p}}{t_e}{\phi _p}/{({k_2}{R_{c,0}})^2}$ . Summarising the different parts, we obtain

(4.7) \begin{equation} {\phi _p} \approx \frac {{{k_1}{u_{\textit{c,scale}}}{t_e}}}{{{k_2 R_{c,0}}}}{\phi _c} - \frac {{{k_3}{D_{\!p}}{t_e}}}{{k_2^2R_{_{c,0}}^2}}{\phi _p}. \end{equation}

By rearrangement, we obtain the simplified deposition ratio

(4.8) \begin{equation}{r_\phi } = \frac{{{\phi _p}}}{{{\phi _c}}} \sim {k_1}k_2\frac{{{u_{\textit{c,scale}}}{R_{c,0}}}}{{\left(\dfrac{{k_2^2R_{_{c,0}}^2}}{{{t_e}}} + {k_3}{D_{\!p}}\right)}}.\end{equation}

We first discuss (4.8) from the view of current simulations of quasi-isothermal evaporation. To be more straightforward, we simply use lattice units here. In the performed simulations, the droplet radius $R_{c,0}$ is approximately $50{-}80$ lattices, the total evaporation time $t_e$ is approximately $84\,000{-}110\,000$ iterations, and the range of the diffusion coefficient is ${D_{\!p}} \in (1.5 \times {10^{ - 3}},0.1)$ . With the coefficient ${k_2} \in (0.02,0.14)$ and $k_3\approx0.8$ , we obtain $( {{k_2^2R_{c,0}^2/{t_e}}})/({{{k_3}{D_{\!p}}}}) \in (0.03,0.24)$ in our simulations. Thus the term $k_2^2R_{_{c,0}}^2/{t_e}$ in the denominator can be ignored for most of the cases, i.e. the term ${u_{\textit{c,scale}}}{R_{c,0}}/({{k_2^2R_{c,0}^2}}/{{{t_e}}} + {k_3}{D_{\!p}})$ is simplified to be ${u_{\textit{c,scale}}}{R_{c,0}}/({k_3}{D_{\!p}})$ . Then (4.8) is rewritten as

(4.9) \begin{align} {r_\phi } = \frac {{{\phi _p}}}{{{\phi _c}}} \approx k*Pe,\quad \textrm{where}\quad k = {k_1}{k_2}/{k_3},\ Pe = \frac {{{u_{\textit{c,scale}}}{R_{c,0}}}}{{{D_{\!p}}}},\ {u_{\textit{c,scale}}} \sim \frac {\sigma }{{{\mu _l}}}\frac {{\varepsilon _0^3}}{{1 + \varepsilon _0^2}}. \end{align}

Finally, we obtain the simplified relation between deposition ratio ${r_\phi } = {\phi _p}/{\phi _c}$ and the dimensionless number $\textit{Pe}$ , i.e. ${r_\phi } = k * Pe$ . Given that ${k_1} \in (0.015,0.12)$ , ${k_2} \in (0.02,0.14)$ and ${k_3} \approx 0.8$ , the parameter $k = {k_1}{k_2}/{k_3}$ is situated in the range $k \in (3.8 \times {10^{ - 4}},2.1 \times {10^{ - 2}})$ . As shown in figure 13, $k = 6.7 \times {10^{ - 3}}$ in (4.5) is the best-fitted value for all the simulations that we conducted in the current work.

One may ask whether $k_2^2R_{_{c,0}}^2/{t_e} \ll k_3{D_{\!p}}$ is still valid in physical experiments for the isothermal evaporation of a colloidal droplet at room temperature. Here, we discuss this, considering the experimental condition in Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011) to explain it, where the initial droplet radius is approximately $1.4\,{\textrm{mm}}$ , and the quasi-isothermal evaporation of the colloidal droplet takes $2070\,{\textrm{s}}$ . The effective diffusion length from droplet periphery is found to be approximately $1.5\,\%\, {R_{c,0}}$ , i.e. ${k_2} \approx 0.015$ . In this way, we obtain $k_2^2R_{_{c,0}}^2/{t_e} = 2.1 \times {10^{ - 13}} \ {\textrm{m}}^{2}\,\text{s}^{-1}$ . In Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011), the particle diameter is $d_p=1300\,{\textrm{nm}}$ . Using the Stokes–Einstein equation, the diffusion coefficient is calculated as ${D_{\!p}} = 3.7 \times {10^{ - 13}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ . With $k_3$ close to 1 in Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011), it is clearly seen $k_2^2R_{c,0}^2/{t_e}$ is still smaller than $k_3{D_{\!p}}$ , but its influence may not be entirely neglected. In this case, it will overestimate the calculated $\textit{Pe}$ by approximately 56 %. However, we must mention that the particle diameter ${d_p} = 1300\,{\textrm{nm}}$ in Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011) is very large, leading to a very low diffusion coefficient ${D_{\!p}} = 3.7 \times {10^{ - 13}}\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ . As we estimated in § 3.2, the Péclet number reaches a very high value $\textit{Pe} \approx 5.5 \times {10^3} \gg 1$ , where the particle diffusion is very weak and almost negligible. Under this condition, an overestimation of $\textit{Pe}$ by 56 % may affect much on the deposition pattern. We also note that for general particle diameter ${d_p} \in (10,200)\,{\textrm{nm}}$ normally used in evaporation of colloid suspension, the range of the diffusion coefficient is ${D_{\!p}} \in (2.4 \times {10^{ - 12}},4.8 \times {10^{ - 11}})\ {\textrm{m}}^{2}\,\text{s}^{- 1}$ , which satisfies $k_2^2R_{c,0}^2/{t_e} \ll k_3{D_{\!p}}$ . Therefore, the scaling relation ${r_\phi } = k * Pe$ generally works under experimental conditions with normal particle diameter not exceeding $200\,{\textrm{nm}}$ .

From § 4.4, we can see that the scaling relation works for diverse particle deposition patterns under different droplet and particle properties. This relation could be potentially applied to assist researchers or engineers to better analyse and study evaporation-induced particle deposition. For instance, if some preliminary experiments are already conducted to obtain the slope $k$ in the scaling relation, then it could help the researchers to quickly analyse how the deposition would develop when the liquid or particle properties ( ${\mu _l},\sigma ,{D_{\!p}},\varepsilon$ ) change.