3.1 General observations

Figure 2(a) shows a typical coating film observed for a concentrated suspension ( $\unicode[STIX]{x1D719}=0.40$ , $d=145~\unicode[STIX]{x03BC}\text{m}$ ) and a velocity $U$ much larger than the entrainment threshold. A typical film thickness profile obtained with the laser profilometry is shown in figure 2(b). The liquid/plate triple line is essentially pinned (at $x=0$ ) and a slowly draining suspension film is entrained from the onset of the motion. As for a pure liquid with $Ca\ll 1$ , the film is uniform at a sufficient distance from the triple line. To characterize the coating we consider the steady-state entrainment thickness of the film, $h$ , which we measure by averaging the thickness over the plateau close to the suspension bath (see figure 2 b). Figure 2(c) reports the evolution of $h$ with the withdrawal velocity for the same suspension. From low to high velocities, it reveals three distinct coating regimes, as also reported very recently by Gans et al. (Reference Gans, Dressaire, Colnet, Saingier, Bazant and Sauret2019): (i) a ‘no-particle-entrainment’ regime, where the film is much thinner than $d$ , (ii) a ‘monolayer’ regime, where the film thickness is of the order of the particle diameter $(h\simeq d)$ and is essentially independent of $U$ , and (iii) a ‘thick-film’ regime, where the film thickness is much larger than the particle size ( $h\gtrsim d$ ) and increases with increasing $U$ . These three regimes determine two boundary velocities $U_{I}$ and $U_{II}$ separating (i)/(ii) and (ii)/(iii), respectively.

Figure 2.

(a) Typical coating observed in the thick-film regime ( $d=145~\unicode[STIX]{x03BC}\text{m}$ , $\unicode[STIX]{x1D719}=0.40$ , $\unicode[STIX]{x1D702}_{0}=100~\text{mPa}~\text{s}$ , $U=6~\text{mm}~\text{s}^{-1}$ ). (b) Typical coating thickness profile measured from laser profilometry in the thick-film regime ( $d=145~\unicode[STIX]{x03BC}\text{m}$ , $\unicode[STIX]{x1D719}=0.40$ , $\unicode[STIX]{x1D702}_{0}=100~\text{mPa}~\text{s}$ , $U=0.9~\text{mm}~\text{s}^{-1}$ , the plate withdrawal length is 50 mm, and $x=0$ labels the initial position of the triple line on the plate). The red and blue lines represent the raw and smoothed signals, respectively. The dashed lines delimit the thickness plateau over which the average steady-state thickness $h$ is measured. (c) Steady-state thickness versus withdrawal velocity (same suspension as in (b)) illustrating the three different coating regimes. From low to high velocities: (i) no-particle-entrainment regime ( $h\lesssim d$ ), (ii) monolayer regime $(h\simeq d)$ , and (iii) thick-film regime ( $h\gtrsim d$ ).  $U_{I}$ and $U_{II}$ denote the transition velocities between the regimes. The dashed line indicates $h=d$ .

3.2 Thick-film regime (iii)

We consider first the thick film observed for large velocities. The inset of figure 3(a) presents the steady-state film thickness $h$ as a function of the withdrawal velocity $U$ for a particle volume fraction ranging from 0.20 to 0.50 and the same particles and liquid as in figure 2 ( $d=145~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D702}_{0}=100~\text{mPa}~\text{s}$ ). The reproducibility of the coating is illustrated by the three series of measurements presented for $\unicode[STIX]{x1D719}=0.40$ . On the right-hand side of the graph, where $h\gtrsim d$ , the thickness $h$ is found to increase with increasing $U$ , as for a pure viscous liquid, but also with increasing $\unicode[STIX]{x1D719}$ . These two trends are consistent with the effective bulk rheology of a suspension, valid for large-scale flows ( $\gg d$ ) and low Reynolds numbers, in which the relative viscosity $\unicode[STIX]{x1D702}/\unicode[STIX]{x1D702}_{0}$ of the suspension increases with the particle volume fraction (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). Therefore, it is natural to represent the thickness $h$ as a function of the capillary number $\unicode[STIX]{x1D702}U/\unicode[STIX]{x1D70E}$ , based on the effective shear viscosity $\unicode[STIX]{x1D702}$ of each suspension, which is measured independently in the shear cell (see § 2). In this representation, shown in figure 3(a), the experimental film thicknesses for the different volume fractions collapse and agree well with the Landau–Levich–Derjaguin (LLD) model

(3.1) $$\begin{eqnarray}h\simeq 0.95\ell \left(\frac{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})U}{\unicode[STIX]{x1D70E}}\right)^{2/3}=h_{0}\left(\frac{\unicode[STIX]{x1D702}}{\unicode[STIX]{x1D702}_{0}}\right)^{2/3},\end{eqnarray}$$

over the thick-film range, $h\gtrsim d$ , and provided $(\unicode[STIX]{x1D702}/\unicode[STIX]{x1D702}_{0})Ca$ remains smaller than 1 (as also reported by Gans et al. Reference Gans, Dressaire, Colnet, Saingier, Bazant and Sauret2019). This suggests that, for the present case of fully wetted particles and for the range of particle volume fraction considered, the global surface stresses that limit the entrainment of the suspension are not very different from those expected from the surface tension of the pure suspending liquid. Finally, for $(\unicode[STIX]{x1D702}/\unicode[STIX]{x1D702}_{0})Ca$ larger than 1, the effective behaviour persists: the thickness agrees with the purely viscous regime $h\sim \sqrt{\unicode[STIX]{x1D702}U/\unicode[STIX]{x1D70C}g}\sim \ell (\unicode[STIX]{x1D702}Ca/\unicode[STIX]{x1D702}_{0})^{1/2}$  (Derjaguin Reference Derjaguin1943).

Figure 3.

(a) Dimensionless thickness $h/d$ versus effective capillary number $\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})U/\unicode[STIX]{x1D70E}$ for different particle volume fractions $\unicode[STIX]{x1D719}$ . The error bars indicate the resolution of the laser profilometry (see § 2). The solid line is (3.1). The dashed line is the Derjaguin asymptotic regime $h=0.67\,\ell (\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})U/\unicode[STIX]{x1D70E})^{1/2}$ expected for $\unicode[STIX]{x1D702}(\unicode[STIX]{x1D719})U/\unicode[STIX]{x1D70E}\gg 1$ (Kizito et al. Reference Kizito, Kamotani and Ostrach1999). The inset shows the raw data and indicates the particle size $d=145~\unicode[STIX]{x03BC}\text{m}$ . The three series of measurements presented for $\unicode[STIX]{x1D719}=0.40$ illustrate the coating reproducibility. (b) Transition capillary number between the thick-film ( $h\gtrsim d$ ) and the monolayer regime ( $h\simeq d$ ) versus particle volume fraction for various particle sizes. The solid lines represent (3.3).

The above measurements concern the total quantity of suspension entrained and do not distinguish between the solid and liquid contents. To complete the characterization of the coating in the thick-film regime, additional dip-coating experiments have been carried out to measure the particle volume fraction in the film. Five different withdrawal velocities have been considered for a fixed particle volume fraction of the bath, $\unicode[STIX]{x1D719}=0.40$ . The measurement procedure is as follows. Immediately after the plate has been fully withdrawn from the bath following the standard protocol, the whole of the suspension that has been entrained with the plate is weighed, rinsed, cleaned and dried thoroughly before being weighed a second time in order to measure the quantity of particles it contains. The ratio of the latter to the former weights yields the average particle volume fraction in the coating, denoted $\unicode[STIX]{x1D719}_{film}$ . For the five effective capillary numbers ( $5.1\times 10^{-3}\leqslant \unicode[STIX]{x1D702}U/\unicode[STIX]{x1D70E}\leqslant 3.1\times 10^{-2}$ ) we obtain $\unicode[STIX]{x1D719}_{film}=0.398\pm 0.003$ , without significant variation with $U$ . This shows that particle migration effects are negligible in the present case. The solid volume fraction of the coating is essentially the same as that of the bath:

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D719}_{film}\simeq \unicode[STIX]{x1D719},\end{eqnarray}$$

which is consistent with the use of the effective viscosity of the bath in equation (3.1).

3.3 Transition from thick-film to monolayer regimes (II)

The experimental data from figures 2 and 3 show that the LLD effective trend of the thick-film regime stops for small withdrawal velocities. The transition is rather sharp and occurs when the average film thickness plateaus at a value which is of the order of the particle diameter. Making use of the LLD effective law (3.1) and of the thickness $h\simeq d$ observed at the transition, one can readily express the transition capillary number $Ca_{II}\equiv \unicode[STIX]{x1D702}_{0}U_{II}/\unicode[STIX]{x1D70E}$ according to

(3.3) $$\begin{eqnarray}Ca_{II}\simeq \frac{\unicode[STIX]{x1D702}_{0}}{\unicode[STIX]{x1D702}}\left(\frac{d}{0.95\ell }\right)^{3/2},\end{eqnarray}$$

and as also proposed by Gans et al. (Reference Gans, Dressaire, Colnet, Saingier, Bazant and Sauret2019).

The value of the transition velocity $U_{II}$ can be extracted from the profilometry measurements. Figure 3(b) shows the experimental transition capillary number as a function of the particle volume fraction for a viscosity $\unicode[STIX]{x1D702}_{0}=100~\text{mPa}~\text{s}$ and two different particle sizes, $d=145$ and $550~\unicode[STIX]{x03BC}\text{m}$ . Over the whole range of $\unicode[STIX]{x1D719}$ , equation (3.3), which is represented by the solid lines, is in good agreement with the data.

Figure 4.

(a) Dimensionless particle surface concentration versus capillary number for $\unicode[STIX]{x1D719}$ ranging from 0.10 to 0.50 and $d=145~\unicode[STIX]{x03BC}\text{m}$ . The value $\tilde{c}_{s}\equiv \unicode[STIX]{x03C0}c_{s}d^{2}/6\unicode[STIX]{x1D719}\equiv h\unicode[STIX]{x1D719}_{film}/d\unicode[STIX]{x1D719}$ equals 1 for a monolayer with a flat liquid interface and the same particle volume fraction as in the bulk (as sketched in the inset). The two independent data series for $\unicode[STIX]{x1D719}=0.20$ and 0.30 show the reproducibility of the transition. (b) Schematics of the sampling technique using a plate with a removable circular central part to measure the liquid content of the monolayer by weighing. (c,d) Typical coating structure observed across transition I for $\unicode[STIX]{x1D719}=0.10$ (c) and $\unicode[STIX]{x1D719}=0.40$ (d). The pictures correspond to the large symbols in (a), with withdrawal velocity increasing from left to right.

3.4 Monolayer regime (ii)

Below $U_{II}$ , the profilometry measurements indicate that the average film thickness remains of the order of $d$ over an extended range of velocities. This is consistent with direct observations revealing that the particles form a monolayer. In this regime, the particle surface concentration $c_{s}$ becomes a relevant quantity to characterize the coating, which also has the advantage of being experimentally accessible with a great accuracy from single-particle tracking (see, for example, figure 1 d). Figure 4(a) presents the dependence of $c_{s}$ upon $U$ and $\unicode[STIX]{x1D719}$ in the steady-state coating regime, as measured by averaging over a central zone of the plate of $20\times 30~\text{mm}^{2}$ located 5 mm above the bath surface. We use the non-dimensional form $\tilde{c}_{s}\equiv \unicode[STIX]{x03C0}c_{s}d^{2}/6\unicode[STIX]{x1D719}\equiv h\unicode[STIX]{x1D719}_{film}/d\unicode[STIX]{x1D719}$ , which is by definition equal to 1 for a film having a thickness $h$ and the same particle volume fraction $\unicode[STIX]{x1D719}$ as the bath. For the smallest volume fraction, $\unicode[STIX]{x1D719}=0.10$ , the surface concentration decreases regularly with decreasing $U$ below the transition velocity $U_{II}$ . However, for higher volume fractions, the dimensionless surface concentration $\tilde{c}_{s}$ collapses on $1$ (i.e. $h\unicode[STIX]{x1D719}_{film}/d\unicode[STIX]{x1D719}\simeq 1$ ) over a significant range of velocities, which further extends towards lower velocities for higher values of $\unicode[STIX]{x1D719}$ . This suggests that over this plateau $h\simeq d$ and $\unicode[STIX]{x1D719}_{film}\simeq \unicode[STIX]{x1D719}$ . However, since other combinations of $h$ and $\unicode[STIX]{x1D719}_{film}$ also yield $\tilde{c}_{s}=1$ we have conducted additional measurements to determine unambiguously the solid and liquid contents of the coating. To this end, we have used a PMMA plate with a removable circular portion (with diameter 12 mm and removed when it is 35 mm above the bath surface), which allows one to collect and weigh a sample of the steady-state coating and to determine its volume fraction $\unicode[STIX]{x1D719}_{film}$ from the comparison of its weight and the number of particles obtained from particle tracking. For a fixed bath volume fraction $\unicode[STIX]{x1D719}=0.40$ and different withdrawal velocities ( $3.0~\text{mm}~\text{s}^{-1}\leqslant U\leqslant 10.5~\text{mm}~\text{s}^{-1}$ ) we obtained $\unicode[STIX]{x1D719}_{film}=0.389\pm 0.003$ without a significant dependence on $U$ , which means that the monolayer regime observed at high particle volume fractions is characterized well by

(3.4a,b ) $$\begin{eqnarray}h\simeq d\quad \text{and}\quad \unicode[STIX]{x1D719}_{film}\simeq \unicode[STIX]{x1D719}.\end{eqnarray}$$

3.5 Transition from monolayer to no-particle-entrainment regimes (I)

Finally, we consider the transition from the monolayer regime to the no-particle-entrainment regime, labelled I, which represents the threshold for the steady entrainment of particles in the film. This transition shows in the measurements of figure 4(a) (for $d=145~\unicode[STIX]{x03BC}\text{m}$ and various $\unicode[STIX]{x1D719}$ ) as the dimensionless surface concentration drops from $\tilde{c}_{s}\simeq 1$ to $\tilde{c}_{s}\simeq 0$ with decreasing entrainment velocities. For high particle volume fractions the transition is sharp, whereas for low $\unicode[STIX]{x1D719}$ it is broad – extending over almost two decades for $\unicode[STIX]{x1D719}=0.10$ – and shifted towards higher entrainment velocities. The typical structure of the coating across the transition is illustrated in figure 4(a,b) for a low ( $\unicode[STIX]{x1D719}=0.10$ ) and a high ( $\unicode[STIX]{x1D719}=0.40$ ) volume fraction. Close to the monolayer regime the coating is uniform for both diluted and concentrated suspensions. However, close to the lower end of the transition, isolated clusters of particles form, which can be more easily observed for diluted suspensions because of the broadness of the transition. These evolutions are summarized in figure 5(a), which presents the capillary number $Ca_{I}\equiv \unicode[STIX]{x1D702}_{0}U_{I}/\unicode[STIX]{x1D70E}$ , around which the transition occurs, as a function of $\unicode[STIX]{x1D719}$ . As shown in figure 5(b) the raw transition velocity $U_{I}$ varies as the inverse of the suspending liquid viscosity $\unicode[STIX]{x1D702}_{0}$ , which suggests that for a given particle size and volume fraction it is relevant to express the transition in terms of $Ca_{I}$ . Experimentally, we define the latter as the geometrical mean of the ends of the range over which $\tilde{c}_{s}$ varies significantly ( $10^{-6}\lesssim \tilde{c}_{s}\lesssim 1$ ), whereas the vertical bars show the range itself (which is much broader than the typical variability in the measurements, see the repeated measurements in figure 4 a). It is found to decrease with increasing $\unicode[STIX]{x1D719}$ below 0.30. These measurements at low $\unicode[STIX]{x1D719}$ are consistent with the entrainment threshold expected in the diluted limit $\unicode[STIX]{x1D719}\simeq 0$ . Indeed, in this limit particles should not interact with each other and the criteria $h\sim \ell {Ca_{I}}^{2/3}\sim d$ , i.e. $Ca_{I}\sim (d/\ell )^{3/2}$ , derived by Colosqui et al. (Reference Colosqui, Morris and Stone2013) for an isolated particle should apply. The latter limit, indicated in figure 5(a), is found to be in line with our measurements for $\unicode[STIX]{x1D719}\leqslant 0.30$ .

Conversely, for higher volume fractions, $Ca_{I}$ plateaus and becomes almost independent of $\unicode[STIX]{x1D719}$ . For a given particle size, all concentrated suspensions ( $\unicode[STIX]{x1D719}\geqslant 0.30$ ) are found to have a very similar particle-entrainment velocity threshold. This contrasts sharply with the evolution of $U_{II}\propto 1/\unicode[STIX]{x1D702}$ , which depends strongly upon $\unicode[STIX]{x1D719}$ , in particular close to the critical volume fraction $\unicode[STIX]{x1D719}_{max}$ , where it is expected to vanish (Guazzelli & Pouliquen Reference Guazzelli and Pouliquen2018). The value, denoted $Ca_{I,\unicode[STIX]{x1D719}\geqslant 0.30}$ , of this high- $\unicode[STIX]{x1D719}$ plateau is determined as the mean of $Ca_{I}$ over the range $0.30\leqslant \unicode[STIX]{x1D719}\leqslant 0.50$ (over which the relative standard deviation is smaller than 7 % for each value of $d$ ). It increases with increasing particle size, which means that, similarly to the diluted case, small particles are more easily entrained than large particles. As shown in the inset of figure 5(a), this dependence of $Ca_{I}$ on $d$ is found to be pronounced. For $d$ ranging from $80~\unicode[STIX]{x03BC}\text{m}$ to $550~\unicode[STIX]{x03BC}\text{m}$ , the value of $Ca_{I,\unicode[STIX]{x1D719}\geqslant 0.30}$ increases by two decades, which suggests a dependence close to $Ca_{I,\unicode[STIX]{x1D719}\geqslant 0.30}\propto d^{2}$ , as highlighted in the inset of figure 5(a). Although no model could be derived to explain these experimental results, the latter suggest that the dependence of $Ca_{I}$ on $d$ at high $\unicode[STIX]{x1D719}$ is stronger than for the diluted case ( $Ca_{I}\propto d^{3/2}$ ). Also, in the present case where a liquid film is always observed below $U_{I}$ (see § 3.6), the same transition velocity ( $U_{I}$ ) is obtained whether the plate is made of PMMA or glass.

Figure 5.

(a) Transition capillary number between the monolayer and no-entrainment regimes versus $\unicode[STIX]{x1D719}$ for different particle sizes. The shape of the symbols indicates from which measurements ( $c_{s}$ , $l$ or $\unicode[STIX]{x1D6FC}$ ) the transition capillary number is determined. The vertical bars represent the range over which $\tilde{c}_{s}$ varies significantly ( $10^{-6}\lesssim \tilde{c}_{s}\lesssim 1$ ), whereas the symbols indicate the geometrical mean of the range limits. The coloured ticks at $\unicode[STIX]{x1D719}=0$ indicate the entrainment threshold $Ca_{I}\approx (0.35d/\ell )^{3/2}$ derived by Colosqui et al. (Reference Colosqui, Morris and Stone2013) for isolated particles. The dashed lines indicate the plateau value of $Ca_{I,\unicode[STIX]{x1D719}\geqslant 0.30}$ observed for $\unicode[STIX]{x1D719}\geqslant 0.30$ . Inset: $Ca_{I,\unicode[STIX]{x1D719}\geqslant 0.30}$ versus $d/\ell$ . The solid line shows $Ca_{I,\unicode[STIX]{x1D719}\geqslant 0.30}\sim (d/\ell )^{2}$ . (b) Evolution of $U_{I}$ with the suspending liquid viscosity for $d=145~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D719}=0.40$ . The solid line represents the capillary number $Ca_{I}\simeq 1.0\times 10^{-4}$ reported in (a). (c) Co-evolution of the dimensionless surface concentration $\tilde{c}_{s}$ and the particle-free length $l$ (defined in the inset) with $Ca$ for $d=550~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D719}=0.40$ and 0.50 ( $\unicode[STIX]{x1D702}_{0}=100~\text{mPa}~\text{s}$ ). (d) Co-evolution of $\tilde{c}_{s}$ and the narrowing angle $\unicode[STIX]{x1D6FC}$ (defined in the inset) with $Ca$ for $d=145~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D719}=0.40$ ( $\unicode[STIX]{x1D702}_{0}=12~\text{mPa}~\text{s}$ ). The insets present typical pictures of the transients (each of the two triangular coatings in (d) is on a different side of the plate).

So far, the content of the coating has been characterized and discussed on the basis of averaged measurements performed at a large distance ( ${\sim}30~\text{mm}$ ) from the initial position of the triple line on the plate ( $x=0$ ). However, further qualitative information about the no-particle-entrainment transition can be obtained by observing the structure of the film that is formed at the onset of the plate withdrawal. As illustrated in the insets of figure 5(c,d), transition I is actually found to proceed differently depending on the particle size. For the largest particles ( $d=550~\unicode[STIX]{x03BC}\text{m}$ ), a transitory film containing no particles is observed close to $x=0$ , whose length $l$ increases as $U$ decreases around $U_{I}$ . By contrast, for smaller sizes ( $d\leqslant 145~\unicode[STIX]{x03BC}\text{m}$ ), the first line of particles appears to be pinned at $x\simeq 0$ . As a consequence a transitory entrainment of particles is observed even below $U_{I}$ , but this entrainment is not steady for $U<U_{I}$ : the width of the particle monolayer shrinks with $x$ with a characteristic angle $\unicode[STIX]{x1D6FC}>0$ depending on $U$ (see the inset of figure 5 d). The reason for this transitory entrainment is not known. It might be related to differences between the static and dynamic shapes of the meniscus, but we cannot exclude influences of slight evaporation at the triple line for the few seconds the plate is immobile before it is withdrawn. Nevertheless, although the transitory coating actually differs between different particle sizes, it is consistent in all cases with the particle content observed at large distances (large $x$ ). As shown by comparison with the measurements of the dimensionless surface concentration $\tilde{c}_{s}$ in figure 5(c,d), both the particle-free length $l$ and the narrowing angle $\unicode[STIX]{x1D6FC}$ increase abruptly around the value of $U_{I}$ determined previously (figure 5 a), which explains why $\tilde{c}_{s}$ vanishes at large distances for low withdrawal velocities.

3.6 No-particle-entrainment regime (i)

The no-particle-entrainment regime has not been characterized quantitatively, but an important remark has to be made. In all the cases studied a continuous liquid film was formed below the no-particle-entrainment transition ( $U$ slightly lower than $U_{I}$ ). This observation suggests that the no-particle-entrainment transition discussed above is not directly related to the threshold velocity for the wetting of the plate by the suspending liquid (de Gennes Reference de Gennes1985; Bonn et al. Reference Bonn, Eggers, Indekeu, Meunier and Rolley2009; Snoeijer & Andreotti Reference Snoeijer and Andreotti2013).