Deformed liquid marble formation: Experiments and computational modeling

The formation of deformed liquid marbles via impact of drops onto powder beds is analyzed using experimental and computational modeling approaches. Experimentally, particular attention is paid to determining a relationship between the maximum contact area of the spreading drops, which determines how much powder the drop's surface is able to harvest, and the drop's surface area when the powder (potentially) encapsulates and then immobilises (“freezes”) the surface of the drop to form a liquid marble. Comparisons between impacts on powder beds to those on rigid and impermeable superhydrophobic substrates show good agreement for a range of parameters and motivate the development of the first mathematical model for the process of liquid marble formation via drop impact. The model utilizes experimentally determined functions to capture encapsulation and freezing thresholds and accounts for the powder's influence on the drop via a surface viscous mechanism. Simulations in the volume-of-fluid framework qualitatively recover many features of the experiments and highlight physical effects that should be incorporated into future analyses.

Figure 1

Photograph and diagram of the experimental setup for drop impacts onto a superhydrophobic powder bed.

Figure 2

Spreading factor versus impact Weber number for drop impact onto a superhydrophobic powder bed. A power-law approximation is provided for $\text{We}>20$.

Figure 3

Liquid marbles formed with different impact Weber numbers. (a) We = 46 spherical marble, (b) We = 52 slightly deformed liquid marble, (c) We = 73 elongated deformed marble, (d) We = 78 elongated deformed marble maintaining the same shape as at the moment of encapsulation.

Figure 4

Maximum visible drop radius and maximum contact line radius, divided by the initial spherical drop radius $R_0$, both taken at the moment of maximum spread for drop impact simulations, as a function of the impact Weber number. The inset shows how these two quantities are defined

Figure 5

Drop impact experiments on a superhydrophobic powder bed showing the ejection of a satellite drop with a liquid-gas interface that is: (a) clean, (b) partially coated, (c) fully coated. Cases (b) and (c) reduce the total mass of powder adsorbed to the liquid-gas interface of the primary drop.

Figure 6

Dependence of ${\alpha }_{\text{encap}}$ and ${\alpha }_{\text{freeze}}$ on the spreading ratio $\gamma$. A linear fit is given for the ${\alpha }_{\text{encap}}$ data points, and a constant fit for the deformed ${\alpha }_{\text{freeze}}$ data points. Values of ${\alpha }_{\text{encap}}$ are given for simulations of drop impact with We = 35, 40, 45, 51, 56, 61, 66, 71, 77, 87, shown in terms of $\gamma$, which increases monotonically with increasing We. For We = 87, ${\alpha }_{\text{encap}}$ is set equal to ${\alpha }_{\text{freeze}}$.

Figure 7

Spreading factor versus impact Weber number for drop impacts onto a superhydrophobic powder bed and rigid impermeable superhydrophobic substrate.

Figure 8

Comparison of drop impacts on (a) a rigid impermeable superhydrophobic substrate and (b) a superhydrophobic powder bed. (a) $D_0=1.94\mathrm{mm}$ and $\text{We}=14$, with spreading factor $\gamma =1.62$, (b) $D_0=1.99\mathrm{mm}$ and $\text{We}=15$, with spreading factor $\gamma =1.67$. Scale bar has a width of 2 mm.

Figure 9

Comparison of drop impacts on (a) a rigid impermeable superhydrophobic substrate, and (b) a superhydrophobic powder bed. (a) $D_0=1.94\mathrm{mm}$ and $\text{We}=24$, with spreading factor $\gamma =1.87$, (b) $D_0=1.99\mathrm{mm}$ and $\text{We}=25$, with spreading factor $\gamma =1.87$. Scale bar has a width of 2 mm.

Figure 10

Comparison of drop impacts on (a, b) a rigid impermeable superhydrophobic substrate, and (c, d) a superhydrophobic powder bed. (a, b) $D_0=1.94\mathrm{mm}$ and $\text{We}=35$, with spreading factor $\gamma =2.06$, (c) $D_0=1.99\mathrm{mm}$ and $\text{We}=36$, with spreading factor $\gamma =2.07$, (d) $D_0=1.99\mathrm{mm}$ and $\text{We}=36$, with spreading factor $\gamma =2.14$. Scale bar has a width of 2 mm.

Figure 11

Diagram showing (a) the drop at maximum spread, with maximum contact line radius ${\overline{r}}_{\text{CL}}$ and a uniform coating of powder on its liquid-solid interface ${\mathcal{S}}_{\text{LS}}$, followed by retraction of the drop with the contact line reducing to $r_{\text{CL}}\left(t\right)<{\overline{r}}_{\text{CL}}$ at some time $t$. We have (b) ${\alpha }_{\text{encap}}<1$, for more densely packed powder on the liquid-gas interface ${\mathcal{S}}_{\text{LG}}$, (c) ${\alpha }_{\text{encap}}=1$, for the same packing on ${\mathcal{S}}_{\text{LG}}$, or (d) ${\alpha }_{\text{encap}}>1$, for more sparsely packed powder on ${\mathcal{S}}_{\text{LG}}$.

Figure 12

Plots from the moment of encapsulation to when the drop reaches the apex of its rebound from the substrate. Simulations are shown for two different We numbers (a, b) and (c, d) with the decay of surface area for different $\overline{\beta }=0.05,0.5,5$ shown in panels (a) and (c) and a comparison of the aspect ratio of the drops to experiments, for the chosen value of $\overline{\beta }=0.5$, shown in panels (b) and (d). Pink dotted lines in panels (a) and (c) show the surface area of the sphere, the minimum surface energy state, which differs between the two plots because of prior satellite droplet ejections. Encapsulation in experiments is matched to occur at the start of the plot. Surface area and time are given in dimensionless units, using a length-scale of $R_0$ and capillary time-scale of $\tau =\sqrt{\rho R_0^3/\sigma }$.

Figure 13

Comparison of drop profiles for simulations (yellow line) overlaid onto images taken from our experiments at We $\approx 40,51,61,71$, at multiple instances in (dimensionless) time. Snapshots are provided shortly after contact is made with the powder bed, leading up to and passing through maximum spread, also during retraction, and immediately prior to the formation of a vertical jet which leads to the drop rebounding from the powder bed.

Figure 14

Drop profiles at encapsulation for $35\le$ We $\le 87$. The red outlines indicate the primary drop at the moment of encapsulation—any other drops in the simulation are ignored in the subsequent dynamics. The impact Weber number for each simulation or experiment is provided above each drop profile, with the spreading factor provided below.

Figure 15

Drop profiles for deformed liquid marbles obtained for $51\le$ We $\le 78$. The impact Weber number for each simulation or experiment is provided above each drop profile, with the spreading factor provided below.

Figure 16

The dilatational surface viscous coefficient ${\lambda }^s$ (log-scale) and effective surface tension ${\sigma }_{\text{eff}}$ over the course of the We = 45 simulation.

Figure 17

Drop profile comparisons from the moment of encapsulation between the simulation with We = 45 and $\gamma =2.31$, and a powder bed experiment with We = 57 and $\gamma =2.45$. The dimensionless time taken between images in the experiment is the same as that in the simulation, and $t=3.38$ is the time of encapsulation for the primary drop in the simulation.

Figure 18

Drop profile comparison from the moment of encapsulation to the moment of deformed liquid marble formation between the simulation with We = 51 and $\gamma =2.40$, and a powder bed experiment with We = 52 and $\gamma =2.60$. The dimensionless time $t=0.00$ corresponds to the moment the drop first comes into contact with the substrate.

Figure 19

Comparisons of interesting phenomena observed in the We = 61 ($\gamma =2.53$) simulation, with examples of the same phenomena occurring in our drop impact experiments: (a) We = 62 ($\gamma =2.53$) experiment showing ejection of a clean satellite followed by a continuous reduction in surface area to encapsulation in the right-most image. (b) We = 62 ($\gamma =2.53$) experiment showing ejection of a satellite post-encapsulation. (c) We = 68 ($\gamma =2.56$) experiment showing the “squashing” of the drop surface leading to deformed liquid marble formation in the right-most image.

Figure 20

Plots for the We = 71 simulation. (Top panel) Surface area for the clean simulation with overlaid surface viscous simulation post-encapsulation, with an inset focusing on the period between the moment of encapsulation and (deformed) liquid marble formation in the surface viscous simulation. The encapsulation and freezing areas reduce following reductions in primary drop surface area due to satellite drop ejections in the surface viscous simulation. (Bottom panel) Dilatational surface viscous parameter ${\lambda }^s$ and effective surface tension ${\sigma }_{\text{eff}}$ from the moment of encapsulation to the moment of deformed liquid marble formation.

Figure 21

Drop profile comparison from the moment of encapsulation to the moment of deformed liquid marble formation between the simulation with We = 71 and $\gamma =2.66$, and a powder bed experiment with We = 78 and $\gamma =2.69$. Encapsulation is on the far left, with the deformed liquid marble on the far right, and intermediary images between them.

Figure 22

Examples of deformed liquid marbles created at the moment of encapsulation in powder bed impact experiments, and the We = 87 ($\gamma =2.83$) simulation. The deformed liquid marble in the simulation (outlined in red) forms as an immediate result of a satellite drop ejection.

Figure 23

Liquid marbles with overlaid smooth boundaries and centroids. The impact Weber number for each experiment is: (a) We = 49, (b) We = 59, (c) We = 69, (d) We = 79.

Figure 24

Example of the process of approximating the surface area of a liquid marble: (a) Identify the boundary (red) of the two-dimensional image of the drop and calculate the centroid of the filled shape; (b) draw a line (blue) through the centroid that can reasonably act as an axis of symmetry for the drop shape—this splits the boundary into a left and right component (green and pink); (c, d) construct a surface-of-revolution using the left and right components of the drop boundary. The “true” surface area is approximated as the average of the approximations in panels (c) and (d).

Figure 25

Drop profiles showing the drop at encapsulation for We = 71 ($\gamma =2.66$) on the left; the remaining images show the liquid marble created using $\overline{\beta }=0.5$, $\overline{\beta }=5$, and $\overline{\beta }=10$.