3.1. Effect of environmental conditions on colloidal patterns on compliant versus rigid materials

We investigated the particle deposition patterns formed from drying colloidal droplets on silicone elastomer coated glass substrates in both humid and dry environments and for film thicknesses of $h = 72\ \textrm {nm}$ (rigid substrate; see § 2.1 and appendix A) and $h = {35}\ \mathrm {\mu }\textrm {m}$ (compliant substrate; see § 2.1). This allowed us to examine the influence of substrate compliance on the deposits of drying colloidal droplets. In each experiment, we first deposited colloidal droplets with an initial volume, $V(t_0) = 1\ \mathrm {\mu } \textrm {l}$, and an initial concentration, $c(t_0) = 0.01\ \text {w/v}\%$, of polystyrene particles with a radius of $a = 0.92\ \mathrm {\mu } \textrm {m}$. The relative humidity was set to either $rh = 15.0 \pm 12.1\,\%$ (dry case) or $rh = 87.6 \pm 2.8\,\%$ (humid case) and the environmental temperature was set to $T = 23.0\pm 0.9\,^{\circ }\textrm {C}$. In the following, we will often refer to low $rh$ conditions, by which we mean $rh<35\,\%$ and high $rh$ conditions which means $rh>80\,\%$.

In figure 1(a) we show a side view image sequence of a drying colloidal droplet on a compliant substrate in a low-$rh$ environment. Under these conditions, the final deposit shape is a particle ring, figure 1(b). In figure 1(c), we show a representative final deposit formed on the compliant substrate at high $rh$ conditions. Interestingly, simply increasing the humidity causes the shape of the deposit to transition from a ring to a full disk, with a clear reduction in the deposit size. In figures 1(d) and 1(e) we present the final deposits again for low-$rh$ and high-$rh$ cases, respectively, but this time the substrate is rigid; however, we see no significant differences in shape between the deposits. For characterization of the final deposit shape, we fitted a circle to obtain the outer diameter of the final deposit, $d_o$. We also extracted the area enclosed within the deposit that did not contain any particles. We used this particle-free area to calculate the inner effective diameter, $d_{{i,eff}}$ (see appendix B). Figure 1(f) shows $d_o$ for the above four cases (low or high $rh$; compliant or a rigid substrate). The deposits that were formed on the compliant substrate at low $rh$ have an average value of $d_o = 396\ \mathrm {\mu } \textrm {m}\pm 47\ \mathrm {\mu } \textrm {m}$. This is clearly much larger than for the other three cases: $d_o = 190\ \mathrm {\mu } \textrm {m} \pm 36\ \mathrm {\mu } \textrm {m}$ on the compliant substrate in high $rh$ conditions; $d_o = 166\ \mathrm {\mu } \textrm {m} \pm 17\ \mathrm {\mu } \textrm {m}$ on the rigid substrate in low $rh$ conditions; and $d_o = 151\ \mathrm {\mu } \textrm {m} \pm 23\ \mathrm {\mu } \textrm {m}$ on the rigid substrate in high $rh$ conditions. In order to have a measure for the shape of the obtained deposits (either disk or ring), we defined the ratio between the inner effective diameter, $d_{{i,eff}}$, and $d_o$, as a shape parameter; $d_{{i,eff}}/d_o = 0$ represents a complete disk while values approaching unity signify a ring-like pattern.

Figure 1.

Wetting and drying of colloidal droplets on thick (compliant) and very thin (rigid) silicone coatings on glass substrates under varying humidity conditions. We deposited $1\ \mathrm {\mu } \textrm {l}$ droplets from a 0.01 w/v% particle dispersion in water (particle radius: $a = 0.92\ \mathrm {\mu }\textrm {m}$). (a) Side view image sequence showing a colloidal droplet drying on a soft silicone coating (relative humidity: $rh = 0.5\,\%$), also defining the droplet contact radius, $R$, and contact angle $\theta ^*$. Micrographs showing the final dry particle deposits from colloidal droplets drying on (b), compliant ($rh = 0\,\%$), (c), compliant ($rh = 90\,\%$), (d), rigid ($rh = 0\,\%$) and (e), rigid ($rh = 84\,\%$) silicone coatings. Also shown in (b) is the outer diameter of the final deposit, $d_o$, and the inner effective diameter of the final deposit, $d_{{i,eff}}$. Plots of (f) $d_o$ and (g) the ratio of $d_{{i,eff}}$ to $d_o$ for compliant and rigid substrates in high ($rh = 87.6 \pm 2.8\,\%$) and low $rh$ conditions ($rh = 15.0 \pm 12.1\,\%$). The experiments in (f,g) were conducted at least $N = 6$ times for each of the cases. Scale bars: (a), $0.5$ and (b–e), $50\ \mathrm {\mu }\textrm {m}$.

Figure 1(g) shows a plot of $d_{{i,eff}}/d_o$ for the four cases. Again, we see a distinct difference between the deposits formed on the compliant substrate at low $rh$ and the other three cases, with the compliant low-$rh$ case appearing ‘ring like’ and the other three cases being ‘disk like’. Thus, by using a compliant substrate and adjusting the humidity, one can control the shape of the dried colloidal patterns – going from $d_{{i,eff}}/d_o = 0$ to 0.64 – which is not achievable with rigid substrates.

Next, we examined the evolution of $\theta ^*$ and $R$ by performing suspension droplet drying experiments on a goniometer. Figures 2(a) and 2(b) show $\theta ^*$ versus $t$ and figures 2(c) and 2(d) show $R$ versus $t$ for suspension droplets drying on a rigid and a compliant silicone substrate and for a high and low relative humidity environment, respectively. From this we can see that $\theta ^*$ remains at a value of approximately $100^{\circ }$ on the rigid substrate over a large portion of the evaporation process, similarly for both conditions. However, for the compliant substrate, $\theta ^*$ stabilizes at a value of approximately $85^{\circ }$ at high relative humidity, whereas at low relative humidity, there is a further pronounced decrease of $\theta ^*$ during drying. This transition towards lower apparent contact angles at higher evaporative flux has been investigated in a previous work (Gerber et al. Reference Gerber, Lendenmann, Eghlidi, Schutzius and Poulikakos2019), and was found to occur for contact line velocities approaching and exceeding the rate at which substrate relaxation occurs. This relaxation velocity can be estimated by taking the ratio of the substrate deformation length scale, $\gamma _{{LV}}/E$, to the characteristic relaxation time scale of the viscoelastic material, $\tau$, where $\gamma _{{LV}}$ is the liquid surface tension. From previous work we know that $E = {12.6}\ \textrm {kPa}$ (Bergert et al. Reference Bergert, Lendenmann, Zündel, Ehret, Panozzo, Richner, Kim, Kress, Norris and Sorkine-Hornung2016) and $\tau = 5.8\ \textrm {s}$ (Gerber et al. Reference Gerber, Lendenmann, Eghlidi, Schutzius and Poulikakos2019), where $\tau$ was measured by means of step strain tests and by fitting a Maxwell solid to the stress versus time curves. For the material used $\gamma _{{LV}}/(E\tau )= 0.98\ \mathrm {\mu } \textrm {m}$. When $\textrm {d}R/\textrm {d}t \ll \gamma _{{LV}}/(E\tau )$, the wetting ridge can re-adapt to the imposed strain rate. However, when $\textrm {d}R/\textrm {d}t >\gamma _{{LV}}/(E\tau )$, the substrate is not able to re-adapt on this short time scale, leading to a surge in stress in the substrate and the wetting ridge starts to bend inwards asymmetrically, in the direction of the contact line motion, and thus allows for lower contact angles (Gerber et al. Reference Gerber, Lendenmann, Eghlidi, Schutzius and Poulikakos2019). We propose here that the transition between the shapes of the colloidal patterns is caused through the same mechanism, namely a combination of evaporative flux, viscoelastic braking and inwards tilt of the wetting ridge.

Figure 2.

Contact angle and contact line dynamics of suspension droplets drying on rigid (black) and compliant (blue) silicone substrates. We used droplets consisting of a 0.01 % w/v suspension ($a = 0.92\ \mathrm {\mu } \textrm {m}$) in water on a layer of Silicone CY 52–276, $\textrm {A}\,{:}\,\textrm {B} = 9\,{:}\,10$. Measured apparent contact angle versus time, $t$, at (a), high humidity ($rh = 87.6 \pm 1.4\,\%$) and, (b), low humidity environment ($rh = 13.0 \pm 2.9\,\%$). Contact radius, $R$, versus time, $t$, at (c) high humidity ($rh = 87.6 \pm 1.4\,\%$) and (d) low humidity ($rh = 13.0 \pm 2.9\,\%$). The plots in (a–d) represent at least 3 individual experiments and the lines represent a fit of a Fourier series with four terms to the data. Receding contact line velocities, $-\textrm {d}R/\textrm {d}t$, in (e), high humidity ($rh = 87.6 \pm 1.4\,\%$) and (f) low humidity ($rh = 13.0 \pm 2.9\,\%$), which are the derivatives of the fitted Fourier series in (c,d), respectively. The dashed line represents the threshold of the substrate relaxation velocity, $\gamma _{{LV}}/(E\tau ) = 0.98\ \mathrm {\mu } \textrm {m}$.

Based on the above, we examined the contact line velocity in all cases by time differentiation of the contact radius line of best fit in figures 2(c) and 2(d). The receding contact line velocities, $-\textrm {d}R/\textrm {d}t$, of suspension droplets drying on a rigid and a compliant silicone substrate for high and low relative humidity environment, respectively, are presented in figures 2(e) and 2(f). The comparison of $-\textrm {d}R/\textrm {d}t$ to $\gamma _{{LV}}/(E\tau )$ shows that only in the case of low relative humidity the contact line velocity exceeds the relaxation velocity (at $t \approx 200\ \textrm {s}$), signifying that we can expect significant viscoelastic braking, inwards tilt of the wetting ridge and significant decrease in contact angles (see figure 2b) in comparison to the slow evaporation (Gerber et al. Reference Gerber, Lendenmann, Eghlidi, Schutzius and Poulikakos2019). For an exceedingly fast contact line, the wetting ridge does not instantaneously adapt, due to its viscoelastic nature. In this case, the wetting ridge acts as a non-responsive (quasi-static) defect, and in the extreme case the contact line can actually pin on the wetting ridge (Kajiya et al. Reference Kajiya, Daerr, Narita, Royon, Lequeux and Limat2013, Reference Kajiya, Brunet, Royon, Daerr, Receveur and Limat2014). We observe pinning in our experiments in the last 2 s of evaporation at low $rh$ conditions on the compliant substrate (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2020.855), which explains the larger deposit diameters, $d_o$, compared to the high $rh$ conditions, and also compared to the rigid, hydrophobic substrate. Note, that the resolution of the camera on the goniometer is insufficient to capture the events a few seconds before complete evaporation (such as the deceleration of $-\textrm {d}R/\textrm {d}t$ to zero for the droplets on the compliant substrate in figure 2f).

3.2. Internal flows during droplet evaporation on a compliant material

In order to understand the formation of these distinct colloidal deposits on a compliant material resulting from a change in relative humidity, we measured the transport of particles within the droplet during evaporation. We recorded videos of the fluorescent particles using an inverted confocal fluorescence microscope, and we always made sure that we focused on the contact line region (at the height of the wetting ridge, defined as $z = 0\ \mathrm {\mu }\textrm {m}$). Using a commercial software (IMARIS, Bitplane), we were then able to track the motion of a considerable share of the visible particles over time (see supplementary movies 2 and 3; note that, for clarity, the playback speed of both videos is slower compared to real time, and also that we move the stage several times to capture the moving contact line motion, which we note each time as ‘stage readjustment’), and transform their velocity to a cylindrical coordinate system. Furthermore, we calculated the distance of the particles to the contact line, $d_{{CL}}$, for every time step (see § 2.3).

Figure 3(a) shows the trajectory of the particles in a drying colloidal droplet on a compliant substrate at low $rh$ for three time intervals: $t = {-301}\ \textrm {s}$ to $t = -271\ \textrm {s}$, $t = {-95}\ \textrm {s}$ to $t = {-65}\ \textrm {s}$ and $t = {-50}\ \textrm {s}$ to $t = {-20}\ \textrm {s}$ (we define $t = {0}\ \textrm {s}$ as the moment when evaporation is complete). Also shown are the definitions of the radial velocity, $v_r$, and $d_{{CL}}$. We see that particles that are located near the contact line are dragged inwards during all three time intervals. However, the trajectories of the particles which are not trapped by the contact line change over time. Initially, their motion is random, indicating a dominance of Brownian motion. However, as drying progresses, these particles start to be pulled towards the contact line. Figure 3(b) shows a consolidated plot of $t$ vs. $d_{{CL}}$ vs. the Péclet number, $Pe = v_rR/D$, from three individual droplet evaporation experiments at low $rh$ on the compliant substrate; $D$ is the diffusion coefficient of the particles in the bulk, calculated by the Stokes–Einstein relation, $D={k_BT}/{6{\rm \pi} \eta a}$. The Péclet number describes the importance of advective transport to diffusive transport, and was defined as positive for transport towards the contact line and as negative for transport towards the droplet centre.

Figure 3.

Flow within evaporating colloidal droplets on a thick (compliant) silicone coating on a glass substrate under varying humidity conditions. (a) Tracked particles during evaporation of a colloidal droplet near the solid–liquid interface ($\sim z = {0}\ \mathrm {\mu }\textrm {m}$) in a dry environment ($rh = 33.2\,\%$) on a compliant material at the intervals of $t = -301\ \textrm {s}$ to $t = -271\ \textrm {s}$, $t = -95\ \textrm {s}$ to $t = -65\ \textrm {s}$ and $t = -50\ \textrm {s}$ to $t = -20\ \textrm {s}$. (b) Diagram of time, $t$, versus distance from contact line, $d_{{CL}}$, versus Péclet number, $Pe$, for colloidal droplets evaporating on a compliant material in low $rh$ conditions ($rh = 33.2 \pm 0.9\,\%$) (see § 2.3). (c) Tracked particles during evaporation of a colloidal droplet near the solid–liquid interface ($\sim z = 0\ \mathrm {\mu } \textrm {m}$) in a humid environment ($rh = 83.0\,\%$) on a compliant material at the same time intervals as in (a). (d) Diagram of $t$ versus $d_{{CL}}$ versus $Pe$ for colloidal droplets evaporating on a compliant material in high $rh$ conditions ($rh = 85.2 \pm 2.2\,\%$) (see § 2.3). In (b,d) the values of $Pe$ have been interpolated from $N = 3$ individual experiments each. Positive values of $Pe$ signify motion towards the contact line. Negative values of $Pe$ signify motion towards the droplet centre. Scale bars: (a), (c) $30\ \mathrm {\mu } \textrm {m}$.

Figure 3(b) confirms that particles which were not trapped by the contact line are dragged towards the contact line. Despite the fact that the contact line is not pinned until very late in the drying process (see supplementary movie 1), we still observe the outward replenishing flow that is typical for the coffee ring effect. This signifies that the speed with which the contact line is receding cannot fully compensate for the loss of liquid, so this liquid is replenished from the bulk.

The reason for this is that the velocity of the contact line is decreased through viscous dissipation in the compliant solid (Carré et al. Reference Carré, Gastel and Shanahan1996; Karpitschka et al. Reference Karpitschka, Das, van Gorcum, Perrin, Andreotti and Snoeijer2015; Zhao et al. Reference Zhao, Dervaux, Narita, Lequeux, Limat and Roché2018; Gerber et al. Reference Gerber, Lendenmann, Eghlidi, Schutzius and Poulikakos2019). Zhao et al. (Reference Zhao, Dervaux, Narita, Lequeux, Limat and Roché2018) found that on thin layers the dissipated power scales with $P_{diss}\propto l^{(1-m)}$, with $l=(\gamma _{{s}} h^3/G)^{(1/4)}$, with the solid surface energy, $\gamma _{{s}}$, the shear modulus, $G$, and $m$ an exponent that is in the range 0.5–0.7 (Shanahan & Carré Reference Shanahan and Carré1995; Zhao et al. Reference Zhao, Dervaux, Narita, Lequeux, Limat and Roché2018). One thus expects significantly lower dissipation in the case of the thinner layers ($P_{{diss,72\ \textrm {nm}}}/P_{{diss,35 \ \mathrm {\mu }\textrm {m}}}\approx (72/35\,000)^{3(1-m)/4}\approx 0.16$), explaining why ring formation is not observed on the samples with a thin silicone layer. The work here thus nicely illustrates the relation between the deposit geometry and the amplitude of dissipation in the substrate.

From figure 3(b), at late times ($-20\ \textrm {s} < t < 0\ \textrm {s}$), we can also see that many particles are suddenly moving inwards. This is due to the fact that, actually, many particles have already aggregated at the droplet edge (but above or below the focal plane), and that this broad total peripheral aggregate then captures particles which are now relatively far from the contact line; however, during this time interval we still observe individual particles in the bulk liquid moving outwards, indicated by positive values of $Pe$ (see supplementary movie 2). In figure 3(c) we show tracked particles within an evaporating droplet for the high $rh$ case on the compliant substrate, again for three time instances (each representing a time duration of 30 s).

Figure 3(d) presents the diagram of the flow within a droplet evaporating from a compliant material in high $rh$ conditions (showing $t$ vs. $d_{{CL}}$ vs $Pe$) consisting of three individual experiments. The flow pattern in this case is markedly different from the low $rh$ conditions, such that we practically only observe the particles at the contact line being dragged along with the contact line. Also, the particles, which are not trapped by the contact line, have negative values of $Pe$, signifying a slow transport towards the droplet centre, with a significantly lower magnitude of $Pe$ than at the low $rh$ conditions. We do not observe any strong outward replenishing flow. As shown, the outward flow only establishes itself for the case of high evaporative flux, while it is absent at low evaporative flux, which indicates that the flow is actually triggered through the evaporation at the droplet edge, replenishing the liquid that is lost near the contact line, such as in the case of the classical coffee ring effect.

3.3. Colloidal transport and assembly at the liquid–vapour interface for low $rh$ conditions on a compliant material

In our particle tracking experiments at low $rh$ conditions, in which we focused near the solid–liquid interface, we observed that many particles were only brought into focus during the last stages of drying (see supplementary movie 2). Thus, we were interested in how these particles were transported within the droplet, which interface they assembled at and how this impacted the final deposit. To determine the assembly of particles at the interfaces, we recorded three-dimensional fluorescence images during colloidal droplet evaporation on a compliant material, detected the positions of the particles and calculated two-dimensional projections on the $r$–$z$ plane (see appendix C for a description of the detection method).

In figure 4(a), we overlaid these two-dimensional projections with the macroscopically measured contact angles and contact line position. From figure 4(a) we see that many particles assemble at the liquid–vapour interface and that the concentration of particles at this interface, especially at the droplet edge, increases over time, which is a result of the replenishing outward flow (see figure 3b). However, there was almost no particle assembly at the solid–liquid interface. A particle scaffold self-assembled at the droplet free surface, from the contact line region inward, as shown in figure 4(b). Furthermore, we observed that aggregates of particles (mostly self-assembled in hexagonal ordering) were forming at the liquid–vapour interface during evaporation (see figures 4c and 4d). Figure 4(e) presents an image focusing on the free surface of the colloidal droplet in a high humidity environment, showing that there are only very few particles and no particle aggregation at the liquid–vapour interface under these conditions (figures 4c and 4e are both taken at times that are when 90 % of the drying time has elapsed so they can be compared). The lower concentration of particles at the vapour interface at high humidity conditions indicates that there is a threshold differential velocity above which the particles are captured at the interface, in accordance with Bigioni et al. (Reference Bigioni, Lin, Nguyen, Corwin, Witten and Jaeger2006). We excluded floatation forces as the sole driver for colloidal aggregation at the interface, because the time scale for aggregation due to floatation forces alone is ${\sim } 3\times 10^{7}$ higher than the time scale of evaporation (see appendix D). In order to understand whether Brownian motion is the main driver for the aggregation of particles – and which are then held through capillarity at the liquid–vapour interface – we calculated the binary collision frequency at the interface. For this we made certain simplifying assumptions, e.g. that all the particles contained in a certain volume evaporated from the droplet would reach the liquid–vapour interface (due to outwards flow detected in the experiments at low $rh$), and that the particles are held at the interface through capillarity without re-migrating into the bulk liquid. Also, we assumed a homogeneous distribution of particles at the liquid–vapour interface.

Figure 4.

Colloidal transport and assembly at the liquid–vapour interface during droplet evaporation in an undersaturated environment on a compliant substrate. (a) Locations of particles at the edge of the droplet ($-10\ \mathrm {\mu }\textrm {m} < d_{{CL}} < {80}\ \mathrm {\mu }\textrm {m}$ and $-5\ \mathrm {\mu }\textrm {m} < z < 20\ \mathrm {\mu } \textrm {m}$) at various time steps during colloidal droplet evaporation on a compliant substrate at low $rh$ conditions ($rh = 13.8\,\%$), also showing the macroscopically measured contact angle; $z = 0\ \mathrm {\mu }\textrm {m}$ is at the height of the contact line (tip of the wetting ridge). Each black dot represents the location of a detected particle. See appendix C for details on the detecting method. (b) Image showing the particles assembling at the droplet edge in a low humidity environment ($t = -60\ \textrm {s}$, $rh = 19.0\,\%$, at $z = 0\ \mathrm {\mu }\textrm {m}$). (c) Image showing the particles assembling at the liquid–vapour interface in a low humidity environment ($t = -60\ \textrm {s}$, $rh = 24.5\,\%$, at $z = 100\ \mathrm {\mu }\textrm {m}$). (d) Image showing the hexagonally ordered aggregates on the liquid–vapour interface in a low humidity environment ($t = -5.8\ \textrm {s}$, $rh = 13.8\,\%$). (e) Image showing the particles at the liquid–vapour interface in a high humidity environment ($t = -425\ \textrm {s}$, $rh = 89\,\%$, at $z = 100\ \mathrm {\mu }\textrm {m}$). For (a–e) we used a $1\ \mathrm {\mu }\textrm {l}$ drop of 0.01 w/v% of a dispersion of particles with $a = 0.92\ \mathrm {\mu }\textrm {m}$ on a soft silicone substrate ($h = 35\ \mathrm {\mu }\textrm {m}$). (f) Normalized number of particles at the liquid–vapour interface, $N_A/N_p$, vs. time, $t$, and normalized total binary collision events, $N_{Z,b}/N_p$, vs. time, $t$, during colloidal droplet evaporation on a compliant substrate at low $rh$ conditions. We have to note that the droplet was placed at $t_0$, which was before $t = -500\ \textrm {s}$, but we disregarded earlier times in this plot. Scale bars: (b, c) $200$, (d) $50$ and (e) $200\ \mathrm {\mu } \textrm {m}$.

The first step in the calculation requires one to determine the number of particles at the liquid–vapour interface, $N_A(t)$. Assuming that all the particles dispersed in a certain volume of liquid which is then evaporating will end up at the liquid–vapour interface, one can write

(3.1)\begin{equation} N_A(t)=\frac{c(t_0)}{\tfrac{4}{3}{\rm \pi} a^3\rho_p}V\left(t_0\right)-V(t) , \end{equation}

with the initial concentration of particles in the bulk, $c(t_0)$ in $\textrm {kg}\,\textrm {m}^{-3}$, the radius of the particle, $a$, the density of the particle, $\rho _p$, the initial droplet volume, $V(t_0)$, and the droplet volume at time $t$, $V(t)$, which is

(3.2)\begin{equation} V\left(t\right)=V\left(t_0\right)-\frac{1}{\rho}\int_{t_0}^{t}\frac{\textrm{d}m}{\textrm{d}\tilde{t}}\,\textrm{d}\tilde{t}. \end{equation}

We find the mass loss due to evaporation, $-{\textrm {d}m}/{\textrm {d}t}$, through Fick's law and in accordance to Popov (Reference Popov2005) and Brutin (Reference Brutin2015) as

(3.3)\begin{equation} -\frac{\textrm{d}m}{\textrm{d}t}={\rm \pi} R\left(t\right)D_v\left(c_s-c_\infty\right)f\left(\theta\left(t\right)\right), \end{equation}

with the diffusivity of water vapour in air, $D_v = 0.282\times 10^{-4}\ \textrm {m}^{2}\,\textrm {s}^{-1}$; the saturated water vapour concentration, $c_s = 2.14\times 10^{-2}\ \textrm {kg}\,\textrm {m}^{-3}$; the water vapour concentration far away from the droplet, $c_{\infty }=({rh}/{100})c_s$; and a function which takes the droplet geometry into account, $f(\theta )$ (dimensionless). We determined $R(t)$ and $\theta (t)$ empirically, by conducting droplet evaporation experiments on a goniometer (Dataphysics OCA 35, $rh = 13.4 \pm 3.0\,\%$, $T = 24.3\,^{\circ }\textrm {C}\pm 1.3\,^{\circ }\textrm {C}$, 3 times repetition), and then fitting a Fourier series with four terms (with a coefficient of determination of 0.981 for $\theta$ and 0.989 for $R$). We approximate the dependence of the droplet geometry according to Hu & Larson (Reference Hu and Larson2002) with $f(\theta )\approx 1.3+0.27\theta ^2$. According to Brutin (Reference Brutin2015), the difference between the analytical solution and the approximation for $f(\theta )$ lies within roughly 10 % accuracy for contact angles of ${0}^{\circ } < \theta < 110^{\circ }$. Figure 4(f) shows $N_A$ divided by the total number of particles in the droplet, $N_p={c(t_0)V(t_0)}/{({4}/{3}){\rm \pi} a^3\rho _p}$, vs. $t$, which correctly describes an increase of particles at the liquid–vapour interface over time.

The concentration of particles per unit droplet surface area, $n_A$, which will be needed for the calculation of the binary collision frequency, is then simply defined as $n_A(t)={N_A(t)}/{A(t)}$, with the liquid–vapour surface area, $A(t)$, which is calculated with the assumption that the droplet takes the form of a hemi-spherical cap.

In order to calculate the binary collision frequency of particles bound to an interface, we transformed the arguments made in Philipse (Reference Philipse2018), from three to two dimensions, which is necessary to calculate collisions along a surface. In this manner, we defined the mean free area, $\langle A_f \rangle$, as opposed to the mean free volume, as

(3.4)\begin{equation} \langle A_f \rangle= \frac{A}{N_A}=\frac{1}{n_A}. \end{equation}

The collision cross-section for a particle being constrained to a surface is $4a$, which signifies that the mean free area is equal to

(3.5)\begin{equation} \langle A_f \rangle = 4a\lambda, \end{equation}

with $\lambda$ being the mean free path of a particle. Considering that the particles move with respect to each other, it is the relative rather than the absolute velocity that matters, and a factor of $\sqrt {2}$ is introduced (see Moore Reference Moore1972). From (3.4) and (3.5) and considering the velocity factor, we thus get

(3.6)\begin{equation} \lambda=\frac{1}{4\sqrt2 a n_A}. \end{equation}

The mean free path on the other hand is the product of the mean velocity, $\langle v_p \rangle$, and the time between two collisions, $t_B$, which in turn is the inverse of the collision frequency, $Z$,

(3.7)\begin{equation} \lambda=\langle v_p \rangle t_B=\langle v_p \rangle /Z. \end{equation}

Thus,

(3.8)\begin{equation} Z=4\sqrt{2} a n_A \langle v_p \rangle, \end{equation}

which is the collision frequency of one specific particle at an interface. From the Einstein–Smucholski equation for Brownian motion (Einstein Reference Einstein1905; von Smoluchowski Reference von Smoluchowski1906), we know that,

(3.9)\begin{equation} \langle v_p \rangle=\sqrt{2D_A/t_B} , \end{equation}

where $D_A$ is the diffusivity of the particles on the liquid–vapour interface. According to Danov, Dimova & Pouligny (Reference Danov, Dimova and Pouligny2000), this surface diffusivity is equal to $D_A=\bar {f}\, D$, with $D$, being the diffusivity of the particles in the bulk obtained through Stokes–Einstein equation ($D={k_B T}/{6{\rm \pi} \eta a}$, with Boltzmann constant, $k_B$, and liquid dynamic viscosity, $\eta$) and $\bar {f}\cong 1$ (Danov et al. Reference Danov, Dimova and Pouligny2000), (for a particle that is immersed to ${\sim }1/2$ in the liquid phase, with the assumption of a non-viscous interface and an interface curvature to particle radius ratio of ${\approx }400$). Thus, the difference between bulk and surface diffusivities is negligible and the factor $\bar {f}$ was not considered in the following equations.

The time interval over which this average Brownian velocity is computed is the time between two collisions, which is $1/Z$, thus

(3.10)\begin{equation} \langle v_p \rangle_{1/Z}= \sqrt{2D_A Z}. \end{equation}

By inserting (3.10) into (3.7) and solving for $Z$, we obtain the binary collision frequency (of one specific particle with the others),

(3.11)\begin{equation} Z\approx 64 a^2n_a^2D_A=\frac{32k_B T}{3{\rm \pi}\eta}n_a^2 a. \end{equation}

Since there are $N_A$ particles on an area of $A$, the total number of binary collisions at the interface per time per unit area, $Z_b$, is (in accordance with Philipse Reference Philipse2018),

(3.12)\begin{equation} Z_b=\frac{1}{2}Z\frac{N_A}{A}\approx\frac{16k_BT}{3{\rm \pi}\eta}n_a^3a. \end{equation}

The total number of binary collision events, $N_{Z,b}$, is the integration of the binary collision frequency over time multiplied by the droplet free surface area,

(3.13)\begin{equation} N_{Z,b}(t)\approx A(t)\int_{t_0}^{t}{Z_b(\tilde{t}})\,\textrm{d}\tilde{t}. \end{equation}

We have to mention that we did not take into account higher-order collisions, we only considered the collision between single particles, hence the number of collision events in total is expected to be higher than estimated here. Also, the integrals were computed numerically.

We show in figure 4(f) the total number of binary collision events, $N_{Z,b}$, normalized by $N_p$. We see from figure 4(f) that $N_{Z,b}/N_p$ is increasing exponentially over time and that at the end of the evaporation roughly 30 % of the particles contained in the droplet are likely to have experienced at least one collision event with another particle at the interface. This makes it more likely that particles come into so close proximity, in order for attractive forces, such as van der Waals interactions and floatation forces, to then cause aggregation. We did not observe that particle assemblies break off once aggregates are formed; therefore, the total number of collision events is a good measure for the probability of aggregation. Over the course of evaporation, as the droplets become thinner, these aggregates were affected by the coffee ring effect, being transported towards the droplet edge (see figure 4a at $t = -23\ \textrm {s}$), and ultimately contributing to the ring shape.

3.5. Bifurcation of printed lines on a compliant material

Having elucidated the mechanism of formation of the particle deposits on compliant substrates and having guided their shape through controlling the evaporation rate at different environmental humidities, we take a further step toward applications and exploit this knowledge in the area of functional contact printing. We designed an experiment in which we contact printed a colloidal ink ($a = 0.92\ \mathrm {\mu }\textrm {m}$, $c(t_0) = 0.1\ \text {w/v}\%$ in water) onto a compliant material (see § 2.1, $h=35\ \mathrm {\mu }\textrm {m}$), controlling the printing location, $x(t)$, by a motorized stage while adjusting the sample temperature, $T_s$, and matching the flow rate, $Q$, to the evaporative flux (see § 2.4). Figure 6(a) shows a schematic of the set-up used in this experiment. The evaporation rate is controlled through the substrate temperature in this application. In figure 6(b) we show a printed deposit resulting from straight constant stage motion ($\textrm {d}x/\textrm {d}t = 0.04\ \textrm {mm}\ \textrm {s}^{-1}$) while decreasing both $T_s$ and $Q$, and keeping the air temperature, and $rh$ roughly constant (parameters are shown in figures 6c and 6d). Calculations showed that, in this regime, Marangoni flows are not expected to be significant (see appendix E). From figure 6(b) we see that, at high $T_s$, the particles are dragged outwards, creating two separate lines, whereas when the substrate temperature is decreased towards room temperature the deposited particle lines converge until the particles are deposited in the centre creating a single line. Hence, we have shown that the shape changes in colloidal deposits on compliant materials as a result of environmental conditions can be exploited to obtain two very distinct printing modes, transitioning from two to a single deposited particle line, with simple variation of the substrate temperature.

Figure 6.

Compliant materials enable shape changes of footprints in a printing configuration, from a bifurcated to a single line. (a) Experimental set-up used for contact printing experiments. (b) Footprint resulting from straight stage motion under decreasing sample temperature, $T_s$, and flow rate, $Q$, profiles, dispersion of 0.1 w/v% of particles ($a = 0.92\ \mathrm {\mu }\textrm {m}$). (c) Plot of $rh$ vs. $t$ and $T$ vs. $t$ during the printing experiment shown in (b). (d) Plot of $Q$ vs. $t$ during the printing experiment in (b). Scale bar: (b) 2 mm.