Film Dynamics and Lubricant Depletion by Droplets Moving on Lubricated Surfaces

Lubricated surfaces have shown promise in numerous applications where impinging foreign droplets must be removed easily; however, before they can be widely adopted, the problem of lubricant depletion, which eventually leads to decreased performance, must be solved. Despite recent progress, a quantitative mechanistic explanation for lubricant depletion is still lacking. Here, we first explain the shape of a droplet on a lubricated surface by balancing the Laplace pressures across interfaces. We then show that the lubricant film thicknesses beneath, behind, and wrapping around a moving droplet change dynamically with the droplet’s speed—analogous to the classical Landau-Levich-Derjaguin problem. The interconnected lubricant dynamics results in the growth of the wetting ridge around the droplet, which is the dominant source of lubricant depletion. We then develop an analytic expression for the maximum amount of lubricant that can be depleted by a single droplet. Counterintuitively, faster-moving droplets subjected to higher driving forces deplete less lubricant than their slower-moving counterparts. The insights developed in this work will inform future work and the design of longer-lasting lubricated surfaces.

Popular Summary

Surfaces infused with suitable lubricant can repel liquids and have shown promise in a wide range of applications, from biomedical to anti-icing. However, before such surfaces can be widely adopted, the problem of lubricant depletion, which leads to decreased performance, must be solved. Various strategies have been proposed to retain the lubricant, but without a solid understanding of the depletion process, this is very much like the proverbial “cart before the horse.” Here, we provide a quantitative description of how lubricant depletion works and identify new, nonintuitive strategies to avoid it.

We recognize the growth of the wetting ridge—a ring of lubricant around a droplet—as the main source of lubricant depletion. As a droplet sweeps across a surface, it entrains lubricant flow around it, and as the lubricant is being depleted, the size of the wetting ridge surrounding the moving droplet increases. We model the process of lubricant entrainment and explicitly show that the volume of the wetting ridge is equal to the volume of the lubricant lost during the entrainment process. More importantly, we provide an analytic solution for the maximum amount of lubricant that can be depleted by a single droplet.

We believe that the insights developed in this work will aid in the design of longer-lasting lubricated surfaces.

Figure 1

(a) Schematic showing the geometry of a droplet when a wrapping layer is present ($S_{\mathrm{LD}}>0$). (b) Confocal image confirming the profile of the wetting ridge (scale $\mathrm{bar}=25\mu \mathrm{m}$) around a droplet on fluorescently dyed silicone oil. (c) Measured apparent contact angle vs predicted apparent contact angle based on Eq. (4). The red line indicates a slope of 1.

Figure 2

(a) Time-lapse images of the wetting ridge for static and moving droplets (scale $\mathrm{bar}=0.5\mathrm{mm}$). (b) Growth of $r_{\mathrm{ext}}$ over time for static and moving droplets. As can be seen in both (a) and (b), the wetting ridge grows more quickly for the moving droplets. In all cases, $25\text{−}\mu \mathrm{l}$ droplets are placed on a surface infused with 20-cP silicone oil with an initial film thickness of $4\mu \mathrm{m}$.

Figure 3

(a) Schematic of the experimental setup used to study the lubricant dynamics. The substrate is moved while the droplet is held in place by a capillary tube, allowing for measurement of the wetting ridge and lubricant thicknesses in various positions. (b) RICM image demonstrating the lubricant profile around a moving droplet (scale $\mathrm{bar}=1\mathrm{mm}$). See the Supplemental Material Video S1 [27] for lubricant dynamics visualized using RICM. (c) Typical experimental measurement of $h_f$ and $r_{\mathrm{ext}}$ for a droplet moving on a lubricant-infused surface. The data for the first two open circles refer to initial thickness $h_i\approx 4\mu \mathrm{m}$.

Figure 4

(a) Schematic showing the LLD film geometry of thickness $h_f$ at the trail behind the moving droplet. (b) $h_f/r_{\mathrm{ext}}$ vs distance for droplets moving on 20-cP silicone-oil-infused surfaces under various experimental conditions. (c) Scaling of $h_f/r_{\mathrm{ext}}$ with $C{a}_{\mathrm{LV}}$ for droplets moving on surfaces infused with silicone and perfluorinated oils of various viscosities. (d) Schematic showing the LLD film geometry of thickness $h_d$ beneath the moving droplet. (e) $h_d/r_{\mathrm{int}}(1+r_{\mathrm{int}}/R{)}^{2/3}$ vs distance for droplets moving on 20-cP silicone-oil-infused surfaces under various experimental conditions. (f) Scaling of $h_d/r_{\mathrm{int}}(1+r_{\mathrm{int}}/R{)}^{2/3}$ with $C{a}_{\mathrm{LD}}$ for droplets moving on surfaces infused with silicone and perfluorinated oils of various viscosities. In all cases, the droplet volume is $25\mu \mathrm{l}$ with initial thickness $h_i=4\mu \mathrm{m}$. Data points in (c) and (f) are averages of the scaling arguments measured during the entire experiment, and error bars indicate the standard deviation. Lines of best fit are indicated in red.

Figure 5

(a) The wrapping layer imaged using color interferometry at different positions on the surface of a moving droplet. Note, the complex lubricant dynamics limit the extent to which LLD can be applied to $h_w$. The thickness scale on the left is calculated computationally and should be used only as a guide. $V_{\mathrm{drop}}=25\mu \mathrm{l}$, $\mu =50\mathrm{cP}$, $U=150\mu \mathrm{m}/\mathrm{s}$, $h_i=4\mu \mathrm{m}$, and $L=40\mathrm{mm}$. Scale $\mathrm{bar}=1\mathrm{mm}$. Note, the “side (front)” image is reflected to match the orientation of the other images. (b) Experimental schematic. The yellow dot indicates the position of the optical probe used to measure $h_w$. (c) $h_w/r_{\mathrm{ext}}$ as a function of the position for different capillary number experiments. (d) Scaling of $h_w/r_{\mathrm{ext}}$ with $C{a}_{\mathrm{LV}}$. The line of the best fit is indicated in red.

Figure 6

(a) RICM of a droplet moving on SLIPS, including the geometric components used to describe the shape of the lubricant behind the droplet (scale $\mathrm{bar}=1\mathrm{mm}$). (b) Normalized thickness profiles behind the droplet calculated using RICM intensity profiles [for example, along the red line in (a)] in combination with white-light interference measurements. In all cases, $h_i=4\mu \mathrm{m}$, $V=25\mu \mathrm{l}$, and $U=300\mu \mathrm{m}/\mathrm{s}$. (c) $V_{\mathrm{lost}}$ vs distance for droplets moving with various capillary numbers over silicone oil SLIPS with $h_i=4\mu \mathrm{m}$. (d) $V_{\text{ridge}}$ vs $V_{\mathrm{lost}}$ for experiments on silicone oil with various capillary numbers. The sole fitting parameter $\alpha =0.52$. (e),(f) Nondimensional plots of the growth of $r_{\mathrm{ext}}$ and $h_{f,0}$ with distance for droplets moving at $700\mu \mathrm{m}/\mathrm{s}$ on 50-cP silicone oil with various initial thicknesses (insets show dimensional quantities). Dashed lines indicate best-fit lines predicted by Eq. (10).

Figure 7

The apparent contact angle ${\theta }_{\mathrm{app}}$ given by a force balance of solely liquid-liquid and liquid-vapor interfacial tensions.

Figure 8

(a) Schematic showing the geometry of a droplet when a wrapping layer is present ($S_{\mathrm{LD}}>0$). (b) Geometric construction used to calculate ${\theta }_{\mathrm{app}}$ when $S_{\mathrm{LD}}>0$. (c) Schematic of a droplet without a wrapping layer ($S_{\mathrm{LD}}<0$). (d) Geometric construction used to calculate ${\theta }_{\mathrm{app}}$ when $S_{\mathrm{LD}}<0$.

Figure 9

Comparison between the force measurements performed in our previous paper using a cantilever force sensor (purple circles) [18] and the tilting experiments performed by Keiser et al. (red circles) [31].

Figure 10

(a) Comparison of the full and simplified scaling parameter for a $3\text{−}\mu \mathrm{l}$ droplet moving at $300\mu \mathrm{m}/\mathrm{s}$ on a 20-cP silicone-oil-infused surface with $h_i=4\mu \mathrm{m}$, and (b) comparison of the averaged scaling parameter for droplets of different volumes under the same conditions as (a).

Figure 11

Scaling of $h_f/r_{\mathrm{ext}}$ with $C{a}_{\mathrm{LV}}$ for various initial thicknesses. Line of best fit in red.

Figure 12

Evolution of the wrapping layer thickness and profile in time as measured using dual-wavelength RICM. (a) Silicone oil forms thick films on a droplet of water due to the presence of ambient airflow leading to fluctuating thickness measurements. (b) Because of the positive Hamaker constant, thin films of silicone oil on water dewet in the absence of airflow. (c) Silicone oil thin films reach a stable equilibrium thickness on a glycerine droplet due to the negative Hamaker constant, even in the absence of ambient airflow.

Figure 13

Plot of $h_f/r_{\mathrm{ext}}$ vs capillary number. The red line indicates the line of best fit for $h_f/r_{\mathrm{ext}}\sim C{a}^{2/3}$ using pinned droplets. The dashed line shows the approximate slope at high capillary numbers ($h_f/r_{\mathrm{ext}}\sim C{a}^{1/4}$).