3.1. Radius growth

Figure 1 shows the typical time evolution of the radius of drops consisting of pure fluid and granular suspensions. In figure 1(a), the raw data are plotted versus time $t$ for suspensions made of 40 $\mathrm {\mu }$m particles. The spreading at late times follows a gravity-driven dynamics, i.e. $R\sim t^{1/8}$. Figure 1(b) displays the normalised radius $R^\star = R(\eta _f/V_0^3)^{1/8}$ versus $t+t_0$ for pure fluid and suspensions made of 40 and 550 $\mathrm {\mu }$m particles, with $\eta _f$ the dynamic viscosity of the PEG copolymer measured by a capillary viscometer for each experiment. This radius normalisation removes volume and viscosity effects, as evidenced by the tight collapse of the curves for a given fluid, but it should be stressed that this quantity possesses a dimension. The time offset corrects the initial bending seen in figure 1(a), as already explained at the end of § 2. This figure also displays the prediction for radius growth with the bulk viscosity of the suspensions. This trend fails at capturing any of our datasets, suggesting that the suspensions have an apparent viscosity during spreading that differs from its bulk value. We will address this observation in § 3.2.

Figure 2 compiles the values for the exponent $n$ and the prefactor $A$ obtained from fitting power law (2.1) to the raw data. The exponents for pure fluid and granular suspensions are plotted in figures 2(a,c), respectively, and show no dependence on drop volume $V_0$. The values of the exponents are consistent with the conclusion drawn from figure 1, indicating that $n\simeq 1/8$. This is evidenced by the mean and standard deviation of $n$ being $(0.125,0.005)$ and $(0.125, 0.009)$ in the cases of the pure fluid and granular suspension, respectively. Furthermore, the confirmation of a gravity-driven spreading is also supported by the variations of $A$ with respect to the drop volume $V_0$. This is demonstrated by the relationship $A\sim V_0^{3/8}$, which is shown in figures 2(b,d) for the pure fluid and the granular suspensions, respectively.

Figure 2.

Spreading exponent $n$ and factor $A$ versus drop volume $V_0$ for (a,b) pure fluid, and (c,d) granular suspensions at $\phi =40\,\%$. Symbol shape corresponds to the drop volume: $100\ \mathrm {\mu }$l ($\circ$), $300\ \mathrm {\mu }$l ($\triangle$), and $1000\ \mathrm {\mu }$l ($\square$). Symbol colour correspond to the particle diameter, from 20 $\mathrm {\mu }$m (blue) to 550 $\mathrm {\mu }$m (gold). Solid and dashed lines correspond to the theoretical predictions of the $V_0$ dependency in the gravity-driven ($n=1/8$, $A\sim V_0^{3/8}$) and capillary-driven ($n=1/10$, $A\sim V_0^{3/10}$) spreading, respectively.

In summary, we observe that the radius of spreading of large drops of suspensions of any volume and concentration follows the power-law growth established for continuous Newtonian fluids in the gravity-driven regime. However, there are two main differences evidenced in figure 1(b). First, the curves of the log-log plot for suspension drops exhibit a small vertical shift compared to those for the pure fluid, which indicates that the factor $A$ of the power law differs when adding particles to the fluid, but with a value that is not the one expected from the bulk suspension viscosity. Second, for the largest particles ($d=550\ \mathrm {\mu }{\rm m}$), the dynamic slows down for $t\gtrsim 200$ s. Deviations from the power law will be discussed in §§ 3.2 and 3.4.

3.2. Effective viscosity

We start by discussing the impact of adding particles on the factor $A$. In figure 3(a), the normalised factor

(3.1)\begin{equation} A^\star{=}A \left(\frac{\eta_f}{V_0^3}\right)^{1/8} \end{equation}

is plotted as a function of particle diameter $d$ for all the experiments, using the experimental drop volume $V_0$ and the pure fluid dynamic viscosity $\eta _f$ for the normalisation. The normalisation eliminates the $V_0$ dependence in $A \propto V_0^{3/8}$ found in figures 2(b,d), as well as the effects of pure fluid viscosity variations due to temperature or humidity. Note that this new quantity $A^\star$ is not dimensionless. In figure 3(a), we see that the value of $A^\star$ is smaller for suspensions than for the pure continuous phase, and its value is approximately independent of $d$. However, the viscosity of the continuous phase $\eta _f$ used to compute $A^\star$ underestimates the effective viscosity of the suspensions. Seeking to compare the effective viscosity derived from the power law with its bulk counterpart $\hat {\eta }_s\,\eta _f$, with $\hat {\eta }_s$ the relative viscosity of the bulk suspension, we write the suspension spreading viscosity of the drop as $\hat {\eta }_d\,\eta _f$. The non-dimensional quantity $\hat {\eta }_d$ is thus the relative effective viscosity during drop spreading, and it quantifies the additional dissipation in the flow due to the addition of particles. To shift the suspension data onto the solid line corresponding to the pure fluid data in figure 3(a), the relative viscosity $\hat {\eta }_d$ should be computed as

(3.2)\begin{equation} \hat{\eta}_d = \frac{\rho g}{\eta_f V_0^3}\left(\frac{k}{A}\right)^8=\left(\frac{A^\star_{f}}{A^\star}\right)^8, \end{equation}

where $k=0.61\pm 0.02$ has been inferred from the pure fluid data (open symbols) using $A^\star _{f} = k(\rho g)^{1/8}=1.94$. This relative viscosity extracted from the power law is reported in figure 3(b) for the different suspensions and drop volumes. It is found to be independent of $d$ and $V_0$, with value $\hat {\eta }_d= 2.2 \pm 0.5$, where the uncertainty is equal to the standard deviation.

Figure 3.

(a) Normalised factor $A^\star =A(\eta _f/V_0^3)^{1/8}$ versus particle diameter $d$ (from 20 $\mathrm {\mu }$m to 550 $\mathrm {\mu }$m) for drop volumes 100, 300 and $1000\ \mathrm {\mu }$l. The white symbols correspond to experiments with pure fluid. (b) Ratio of the relative effective viscosity obtained from the power law, $\hat {\eta }_d$, to the relative bulk viscosity $\hat {\eta }_s$ versus particle diameter $d$. The red horizontal line corresponds to this ratio for a polydisperse suspension made of particles of 20, 40, 80 and $140\ \mathrm {\mu }$m, each size representing 10 % of the suspension volume. The star symbols correspond to the relative effective wetting viscosity of the same suspensions obtained from the Cox–Voinov law, $\hat {\eta }_w$ (Zhao et al. Reference Zhao, Oléron, Pelosse, Limat, Guazzelli and Roché2020).

The spreading viscosity $\hat {\eta }_d$ is much smaller than the bulk viscosity of the suspension as the relative bulk viscosity is $\hat {\eta }_s \simeq 10$ at the same $\phi =40\,\%$. This result is in line with the discrepancy observed in figure 1(b) between the experimental radius growth and that anticipated using the suspension bulk viscosity shown by the red dashed line. In other words, granular suspensions spread faster than would be expected from their bulk properties.

This relative drop spreading viscosity $\hat {\eta }_d$ that describes the global dynamics of the drop differs from the relative wetting viscosity of the contact line $\hat {\eta }_w$ extracted from the Cox–Voinov law found in the viscous–capillary region corresponding to the very close vicinity of the contact line; see the star symbols in figure 3(b). In contrast with the constant behaviour of $\hat {\eta }_d$, $\hat {\eta }_w$ decreases with increasing $d$, and reaches value 1 (corresponding to the pure fluid) for a cut-off size ($\simeq 100\ \mathrm {\mu }$m) above which particles are too large to affect dissipation in this region close to the contact line (Zhao et al. Reference Zhao, Oléron, Pelosse, Limat, Guazzelli and Roché2020; Pelosse et al. Reference Pelosse, Guazzelli and Roché2023). This difference between $\hat {\eta }_w$ and $\hat {\eta }_d$ may not come as a surprise as it derives from two different energy balances at the local and global scales, i.e. from a balance of capillary and viscous forces at the local scale of the contact line, and of gravity and viscous forces at the global scale of the drop.

This decrease in viscosity and therefore in dissipation may be due to particle slip along the very smooth substrate. Another difference may come from the suspension microstructure, which certainly is dissimilar in the present non-viscometric flow and in a pure shearing flow. Besides, confinement by both the substrate and the mobile interface may create layering that could span over 10 particle diameters at $\phi =40\,\%$ (Gallier et al. Reference Gallier, Lemaire, Lobry and Peters2016) and end up in a strong dip in dissipation (Ramaswamy et al. Reference Ramaswamy, Lin, Leahy, Ness, Fiore, Swan and Cohen2017). However, experiments with polydisperse suspensions do not support this explanation. We use mixtures of 20, 40, 80 and 140 $\mathrm {\mu }$m particles with a total solid volume fraction $\phi =40\,\%$, equal volume fractions of each particle diameter, and drop volumes 100, 300 and $1000\ \mathrm {\mu }$l. It is anticipated that such polydispersity in the suspension will impede crystallisation. If the decrease in dissipation observed in monomodal suspensions, $\hat {\eta }_d^{mono}\simeq 0.22\hat {\eta }_s$, were due to crystallisation, then the relative viscosity $\hat {\eta }_d^{poly}$ of these polydisperse suspensions should be much closer to the relative bulk viscosity $\hat {\eta }_s$. However, the polymodal effective relative viscosity does not show any significant increase in dissipation as $\hat {\eta }_d^{poly}\simeq 0.27 \hat {\eta }_s$; see the red line in figure 3(b).

To discriminate between possible causes of this smaller value of $\hat {\eta }_d$, we examine in detail the particulate flow during spreading in § 3.3.

3.3. Visualisation of the particulate flow in the one-dimensional configuration

To gain further insight into particle motion during spreading, particle imaging is performed in a more convenient configuration, i.e. for spreading along only one direction, using a density and index-matched suspension described in § 2. As sketched in figure 4(a), imaging without optical distortion is achieved using two orthogonal microscope slides. On the horizontal slide, a 2 cm long thread of liquid is deposited with a spatula so that one end touches the vertical wall. Care is taken to ensure a uniform thickness. A green laser sheet (Coherent lasers, $\lambda = 532$ nm, 50 mW) illuminates the fluid through the bottom slide, and a side-view camera images the flow. The distance between the vertical slide and the laser sheet is 5 mm. The laser sheet is not deflected across the flat walls or in the index-matched suspension, and induces fluorescence of Rhodamine 6G in the illuminated slice (approximately 30 $\mathrm {\mu }$m thick) which is excited by the green light and emits red wavelengths. This light is filtered by a high-pass red filter (Thorlabs, $\lambda >550$ nm) and collected by the monochrome camera (Imaging Source, DMK33UX174, 2.3 MP) mounted with a zero distortion lens (Opto Engineering, MC100X), orthogonal to the laser sheet. The spatial resolution is $3\ \mathrm {\mu }{\rm m}\ {\rm pixel}^{-1}$, and the acquisition frame rate is 1 s$^{-1}$. With this apparatus, the non-fluorescing particles appear dark in the illuminated slice, as seen in figure 4(b) and the accompanying movie. In figure 4(c), the horizontal speed of the particulate phase is coming from the particle image velocimetry (PIV) analysis performed with the openPIV module of Python on two consecutive frames.

Figure 4.

(a) Sketch of the apparatus used to visualise the particle flow in a one-dimensional spreading with a laser sheet. A fluorescent dye suspended in the fluid is excited by the green light and fluoresces red light collected by a side-view camera. (b) Imaging for a suspension made of 60 $\mathrm {\mu }$m particles at $\phi =40\,\%$ (scale bar 1 mm). See accompanying supplementary movie 1, available at https://doi.org/10.1017/jfm.2024.568. (c) Horizontal speed $v_x$ from PIV analysis of the particle flow in (b). The speed is normalised by the contact-line velocity $U$. The red arrow indicates the occurrence of slip. (d) Sum of raw pictures between $t$ and $t+\Delta t$, with $\Delta t = 2d/U$ in the corresponding red (left) and blue (right) squares in (b).

The algorithm uses interrogation window size 16 pixels, search area size 18 pixels, and overlap 9 pixels. The interrogation box typically contains 7–8 particles. The velocity is normalised by the speed of the advancing contact line (black arrow in figure 4c). In figure 4(d), the sum of the pictures between $t$ and $t+\Delta t$ is shown for $\Delta t =2d/U$ in two specific regions. With this time interval, particles moving at a speed close to that of the contact line should not appear as fixed dark dots but should be blurred.

Visualisation of the particulate flow during the one-dimensional spreading of the suspension unveils several significant elements. Importantly, figure 4(b) and its accompanying movie do not exhibit any clear development of layering or ordering during spreading, meaning that particle self-organisation cannot explain the strong decrease in viscosity as that reported by e.g. Ramaswamy et al. (Reference Ramaswamy, Lin, Leahy, Ness, Fiore, Swan and Cohen2017). However, PIV measurements presented in figure 4(c), and the sum of consecutive pictures in figure 4(d), reveal two crucial features of the particle flow. First, a strong particle slip develops along the wall near the advancing contact line, as shown by the velocity signal in figure 4(c) or by the smearing of dark dots near the edge in figure 4(d) (right-hand image). Second, in the centre, particles barely move, as illustrated by the weak velocity signal from the PIV in figure 4(c) or by the sum of the pictures in figure 4(d) (left-hand image). In this latter figure, sharp dots at the centre (i.e. on the left-hand image) indicate that the particles have not moved within the time interval $\Delta t = 2d/U$, in contrast with the observations near the contact line (i.e. on the right-hand image) where the particulate phase undergoes a strong flow.

These observations of the structure of the particulate flow can explain the low value of the effective spreading viscosity displayed in figure 3(b). In rheometers, slip velocity of the particles leads to a smaller stress on the particles and results in an effective smaller viscosity (Yoshimura & Prud'homme Reference Yoshimura and Prud'homme1988; Jana, Kapoor & Acrivos Reference Jana, Kapoor and Acrivos1995; Ahuja & Singh Reference Ahuja and Singh2009). In spreading experiments, particles close to the tip exhibit a strong slip at roughly the contact-line speed. This slip is not surprising under such strong confinement and given the very smooth surface of the solid substrate. Yet slip by itself cannot account for such a large decrease in effective viscosity, and should also vary with particle size. These last two points suggest that at least one other mechanism contributes to the reduction in dissipation in drop spreading flows. In particular, this decrease could also be attributed to the near-frozen particulate phase at the centre, which is such that the network of frictional particles in contact (as the particles are nearly jammed) acts more like a porous medium than like a flowing suspension in this region. Interestingly, the effective viscosity of a porous medium (Brinkman Reference Brinkman1949; Kim & Russel Reference Kim and Russel1985; Breugem Reference Breugem2007) and similarly of a granular sediment bed (Vowinckel et al. Reference Vowinckel, Biegert, Meiburg, Aussillous and Guazzelli2021) is found to be independent of particle size and to possess a value close to the Einstein viscosity ($\hat {\eta }_s = 1+2.5 \phi =2$ at $\phi =0.4$), i.e. a value much smaller than that of a sheared dense suspension due to the screening of the long-range hydrodynamic interactions. This bears a striking resemblance to the behaviour of $\hat {\eta }_d$ in figure 3(b).

In summary, slip at the tip of the drop and porous-like behaviour at the centre of the drop could both account for the observed small value of the effective relative viscosity of drop spreading, $\hat {\eta }_d$. However, it should be noted that this analogy with a granular fixed bed to account for the decrease in dissipation seems less relevant in the case of the smallest particles, for which particle rearrangements can be observed in most of the drop until the end of the experiment.

3.4. Long-time behaviour: validity of the power law

We now turn to the slowdown of the spreading dynamics observed above 200 s for suspensions made of the largest 550 $\mathrm {\mu }$m particles. Top-view pictures of drop spreading evidence different patterns for suspensions having different particle sizes 40, 250 and $550\ \mathrm {\mu }$m at $\phi =40\,\%$ in figures 5(a), 5(b), and 5(c), respectively. For the largest particles, top views at a higher volume fraction, namely $\phi =47\,\%$, are also presented in figure 5(d). Clearly, the particle phase initially spreads with the fluid but then does not expand further beyond a critical radius that decreases with increasing particle size, as depicted in figures 5(a,b,c), and with increasing particle volume fraction, as shown in figures 5(c,d). There is a striking difference between the behaviour for the smallest particles where the particle arrest is barely seen (figure 5a) and that for the largest particles (figure 5d), where particle freeze happens at a very early stage. These observations suggest that the departure from the power law can be associated with the freezing of the particle matrix while the contact line still moves.

Figure 5.

Top view pictures of the spreading of different suspensions: (a) $d=20\ \mathrm {\mu }$m, $\phi = 40\,\%$; (b) $d=250\ \mathrm {\mu }$m, $\phi = 40\,\%$; (c) $d=550\ \mathrm {\mu }$m, $\phi = 40\,\%$; (d) $d=550\ \mathrm {\mu }$m, $\phi = 47\,\%$ (scale bar 5 mm). See accompanying supplementary movie 2.

In figure 6(a), the fluid radius $R$ of a $100\ \mathrm {\mu }$l drop of a suspension consisting of 550 $\mathrm {\mu }$m particles at $\phi =40\,\%$ is plotted as a function of time (black solid line). The particle spreading radius $R_p$, defined in the top inset, is also plotted, with a yellow solid line. Side-view snapshots numbered from (i) to (iv) taken during this spreading illustrate two different regimes. First, for pictures (i) and (ii), the slope of the drop profile is decreasing with increasing distance to the contact-line $R_p$ increases, meaning that particles move with the fluid, in agreement with the early-time dynamics in figure 5. In this regime, the drop radius growth obeys the power law with exponent $n=1/8$, and the suspension behaves like a continuous fluid. Second, while the contact line still progresses from (ii) to (iv), $R_p$ saturates, i.e. the particles do not move and remain at the centre of the drop, also in agreement with observations from figure 5. This translates into a significant slowdown of the drop radius growth $R$ and departure from the power law (black solid line). In this second regime, the continuous approach fails and the fluid drains out of the porous matrix consisting of the large particles. Then the slope of the drop profile increases with increasing distance to the edge, as seen in the thumbnails (iii) and (iv). In these pictures, the particle protrusions become more significant, and one can identify in picture (iv), away from the contact line, a region of constant height strongly distorting the free surface, which corresponds to a monolayer of particles.

Figure 6.

(a) Drop of suspension consisting of 550 $\mathrm {\mu }$m particles with $V_0=100\ \mathrm {\mu }$l. Main graph: fluid radius $R$ (black) and particle radius $R_p$ (yellow) as a function of time. The dashed line corresponds to the power law. Thumbnails (i) to (iv): side-view pictures corresponding to the black dots in the main graph. Top inset: top-view picture with fluid ($R$) and particle ($R_p$) radii. (b) Particle radius $R_p$ as a function of time for different suspensions (see legend), with drop volume $V_0\simeq 100\ \mathrm {\mu }$l. Error bars are computed from standard deviations over different experiments.

Time evolutions of normalised particle radii for drops consisting of different particle sizes and volume fractions are plotted in figure 6(b). The same normalisation to remove volume and viscosity effects is used, and the drop volumes are kept similar at approximately $100\ \mathrm {\mu }$l. This graph confirms that particle freezing happens earlier with large particles and large volume fractions. Conversely, for particle diameters ${\lesssim }100\ \mathrm {\mu }$m, $R_p$ follows a power-law growth with exponent $1/8$, i.e. the same dynamics as that predicted for a continuous fluid for gravity-driven spreading.

To gain insight into the transition between these two regimes, the shape and thickness of the central part of the drop have been investigated. For gravity-driven spreading, the theoretical prediction is given by Lopez et al. (Reference Lopez, Miller and Ruckenstein1976)

(3.3)\begin{equation} h(r,t)=\frac{4V_0}{3{\rm \pi}\, R^2(t)}\left[1-\left(\frac{r}{R(t)}\right)^2\right]^{1/3},\end{equation}

where $r$ is the radial distance from the drop centre. Figure 7(a) shows at different times the profiles extracted from side views (solid lines) of a spreading droplet of suspensions made of 140 $\mathrm {\mu }$m particles at $\phi =40\,\%$ and with a volume $V_0=215\ \mathrm {\mu }$l. The profiles at the centre of the droplet agree well with the theoretical prediction (3.3) (dashed lines), especially at long times. This agreement suggests that side views are not required, and drop profiles in the central region can be inferred using the drop radius $R(t)$ from top-view pictures. The reader should note that the prediction (3.3) conflicts with experimental data at the droplet edge. This discrepancy was already highlighted by Hocking (Reference Hocking1983) and arises from the fact that the central region of the droplet is not matched to the region near the contact line where capillary forces must be accounted for.

Figure 7.

Spreading of a drop of suspension made of 140 $\mathrm {\mu }$m particles with $V_0=215\ \mathrm {\mu }$l and $\phi =40\,\%$. (a) Experimental profiles from side views (solid lines) and theoretical shapes. Dashed lines: predictions from (3.3) given by Lopez et al. (Reference Lopez, Miller and Ruckenstein1976). (b) Dots: experimental radius $R$ as a function of time. Red line: power-law growth $R = A(t+t_0)^{1/8}$, with the fitted parameters $A=4.79\times 10^{-3}\ {\rm m\ s}^{-1/8}$, $t_0=10.8$ s. (c) Central thickness $h_0$ as a function of time. Dots: measurements from side views. Dashed line: prediction of (3.3) using $R(t)$ fitted in (b). The colour gradation indicates the time progression, and the large dots in (b) and (c) correspond to the moments of the profiles plotted in (a).

In figure 7(b), the radius data (computed from top views) are fitted with the power law (red solid line) and used in figure 7(c) to compute the drop central thickness $h_0$ according to (3.3) (blue dashed line). In figure 7(c), measurement of $h_0$ from side views shows excellent agreement with the theoretical prediction from $R(t)$ data.

The simplest criterion that can be proposed is that the breakdown of the power-law radius growth for granular suspensions happens when the drop thickness reaches approximately a particle diameter, as suggested by the side-view thumbnails in figure 6(a), where drop thickness is roughly two diameters thick at the transition for the 550 $\mathrm {\mu }$m particles. This can be estimated by evaluating the time $t_c$ (or equivalently the drop radius $R_c$) for which the central thickness is $h_0(R_c)\simeq d$ with $R_c=At_c^{1/8}$. Figure 8 shows that a nice collapse of drop radius versus time can be obtained for different particle sizes and drop volumes by normalising the radius by the critical radius $R_c$, and the time by the critical time $t_c$. Spreading deviates from the power law (solid line) for $t\simeq t_c$ for the 250 and 550 $\mathrm {\mu }$m particles when $R\simeq R_c$. This slowing down of the dynamics is accompanied by a change of convexity of the liquid/gas interface, as seen in figure 6(a). This change may stem from various origins, such as the constraint imposed by the edge of the particle layer on the interface height, or the ability of the capillary stress driving the motion of the contact line to drain liquid out of the particle layer. For the smallest particle diameter, the experiments are too short to observe the deviation as the critical thickness $h_0=d$ is far from being reached.

Figure 8.

Radius as a function of time, normalised by $R_c$ and $t_c$, respectively, with $R_c=At_c^{1/8}$ and $h_0(R_c)=d$ (see legend in figure 3 for the symbols). Inset shows full time and radius range; main graph shows blow-up exhibiting the slowing dynamics.