Micro–Nano Hierarchical Structure Enhanced Strong Wet Friction Surface Inspired by Tree Frogs

Abstract Superior wet attachment and friction performance without the need of special external or preloaded normal force, similar to the tree frog's toe pad, is highly essential for biomedical engineering, wearable flexible electronics, etc. Although various pillar surfaces are proposed to enhance wet adhesion or friction, their mechanisms remain on micropillar arrays to extrude interfacial liquid via an external force. Here, two‐level micropillar arrays with nanocavities on top are discovered on the toe pads of a tree frog, and they exhibit strong boundary friction ≈20 times higher than dry and wet friction without the need of a special external or preloaded normal force. Microscale in situ observations show that the specific micro–nano hierarchical pillars in turn trigger three‐level liquid adjusting phenomena, including two‐level liquid self‐splitting and liquid self‐sucking effects. Under these effects, uniform nanometer‐thick liquid bridges form spontaneously on all pillars to generate strong boundary friction, which can be ≈2 times higher than for single‐level pillar surfaces and ≈3.5 times higher than for smooth surfaces. Finally, theoretical models of boundary friction in terms of self‐splitting and self‐sucking are built to reveal the importance of liquid behavior induced by micro–nano hierarchical structure.


Theoretical analysis of liquid self-splitting effect on bioinspired pillar surface
Since each channel belongs to two pillars, the volume of liquid in channels around each pillar which will be drawn onto the pillar top can be represented as ) where a, w and h are the side length of pillar, the channel width, and the pillar height, respectively.
The maximum volume of liquid can be reserved on top of each pillar is

√ )
The liquid self-splitting effect is influenced by

√ )
When V P < V C , liquid in channels cannot be all drawn out and gathered into a large liquid bridge, which leads to self-splitting failure. While V P : V C is close to or higher than 1, self-splitting appears.

In-situ observation of nano-liquid bridges on smooth pillar and concave pillar
The bioinspired pillar surface is placed upside-down and adhered to a glass slide with 1.5 μL deionized water ( Figure S8a). The microscope (BX51, Olympus) is set to reflect status to achieve thinfilm interference between pillar and substrate with a wavelength λ = 600 nm light source. A highspeed camera (X100, Photron) is used to record the movement of interference fringes during water evaporation. To enhance the clearance of TFI, a layer of chromium film with thickness of ~10 nm is coated onto glass slide by magnetron sputtering. The light intensity of two points (~10 μm × 10 μm) on smooth pillar (at rim and center) and concave pillar (at rim and in cavity) have been achieved from one high-speed video simultaneously to allow the light intensity comparable.
Define relative light intensity as ̅ , which is where I max and I min are the max and min light intensity at a point. [1] During liquid evaporating, pillar is gradually adhered to substrate with interference fringes continuously appearing from the center of pillar. When solid-solid contact forms with d decreasing to minimum gap distance d M , a uniform light intensity appears on pillar, which forms the first level of interference. To make the gap distance on smooth pillar and concave pillar comparable, the change of TFI on both pillars are recorded at the same time in one shot. ̅ of first level interference at the rim of concave pillar, ̅ , is higher than on smooth pillar, which means the minimum gap distance on concave pillar rim is thinner than on smooth pillar. Thus, minimum gap distance at concave pillar rim is regarded as the basic gap distance and represented as . The change of gap distance on smooth pillar and concave pillar during evaporation can all be presented as where n = 0, 1, 2… and s is the refractive index of materials in gap; 1.33 for water and 1.00 for air. [1] The measured gap distance at rim and center of smooth pillar and concave pillar are shown in Figure   S9.

Theoretical analysis of boundary friction increase induced by self-sucking enhancing effect
Based on capillary theory, [2] the capillary force between a pillar and substrate can be presented where P C is capillary pressure, γ the surface tension of liquid, A P the pillar area, θ S and θ P ' are the intrinsic contact angle of substrate and apparent contact angle of pillar, respectively; d is the thickness of liquid bridge, i.e., the gap distance between interfaces; K is the ratio of rim area to pillar area on concave pillar ( Figure S12). Specially, K = 1 is the situation on a smooth pillar.
) is the real contact area ratio. If excessive liquid exists between the interfaces, W C drops to almost zero when liquid spreads to the edge of pillar without any solid-solid contacts between interfaces, which results in wet friction. With liquid decreasing by FLC steps or evaporating, the capillary force gradually increases and drags surfaces into solid-solid contact and a resistant force is generated by its elastic contact deformation, which leads to boundary liquid film state and boundary friction ( Figure   S12). According to GW theory, [3] when the heights of rough substrate asperities follow an exponential distribution: ) , the real contact area ratio ) is where η, σ and β describe the surface properties of substrate, which are the density of asperities, the standard deviation of the height distribution, and the radius of asperity summits, respectively. Then, the elastic contact force W E can be presented as ) ) ) ) ) ) Since the pillar is much softer than substrate, the effective elastic modulus ), where is the Poisson's ratio and E the elastic modulus of pillar. During evaporating, the highest capillary adhesion ) appears when d declines to the minimum gap distance d M , and accompanied by decreases to pillar's intrinsic contact angle ( Figure S12). The force balance can be presented as On concave pillar, d M is minimum gap distance at rim area since it forms solid-solid contact and liquid at rim edge determines the capillary pressure. Based on Equation S6, S8 and S9, the relationship between K and d M is where can be achieved with K =1.
) decreases with K decreasing, which indicates larger cavity leads to thinner liquid film. In general, friction generated by solid-solid contact with normal force W C can be described as

)
where is the frictional coefficient. [4,5] The self-sucking enhancing coefficient of boundary friction on concave pillar ζ Sk can be presented as well with the measured friction on different concave pillar surface that friction rises ~33% with K decreasing from 1 to 0.5 (Figure 4c).
Based on Equation S10 and S12, the increase of surface roughness (i.e. lower η, σ and β) can also lead to larger minimum gap distance d M and weaker boundary friction, i.e. smaller , and ( Figure S14). For a certain substrate, the elastic modulus E of bioinspired surface exhibits an optimum value that can generate the highest boundary friction ( Figure S15). With surface roughness increasing, the optimum E decreases which indicates softer bioinspired surface is more suitable for rougher substrate. These results have also been proved by liquid film characterization ( Figure S8b

Theoretical analysis of boundary friction duration increase induced by self-sucking enhancing effect
Smaller minimum gap distance on concave pillar could also decrease the liquid evaporation, thus increasing the capillary bridge and boundary friction duration. On smooth pillar with liquid bridge evaporating from edge, the evaporating speed is considered to be proportional to its liquid film thickness d and liquid film circumference C C . Define the liquid vaporization coefficient , and the liquid evaporating speed can be presented as [ )] . The C C has a relation to the liquid bridge area A C , √ . The evaporation time for liquid bridge area decreasing For liquid bridge area A C shrinking from pillar area A P to 0, the evaporation time can be presented as On concave pillar, the decrease of d M and increase of real contact area ratio ) lead to stronger sealing effect, which effectively reduces the liquid evaporation on concave pillar. Based on Equation S6 and S8, force equilibrium relationship during liquid evaporation is ) ) ) and the relationship between and d can be obtained. The cavity deforming pressure P Con results from capillary pressure P C , and its relationship with cavity volume V can be simplified as

) )
where P 0 and V 0 are the initial pressure and volume of cavity, and k the rate of cavity pressure to cavity volume, which can be achieved by

) )
Based on Equation S6, S8, S14 and S15, the relationship between d and cavity volume V during evaporating is The evaporation time on concave pillar is Based on Equation S13 and S18, the boundary friction duration enhancing coefficient of self-sucking on concave pillar is For the case the numerical solutions based on Equation S16, S17 and S19 are shown in Table S2, and increases with K decreasing. With K = 0.65, which indicates the boundary states on concave pillar is ~4 times longer than on smooth pillar. It agrees well with the evaporation time on smooth pillar and concave pillar ( Figure S13 and Movie S8).
And this boundary friction duration can even be enhanced about 30 times higher as K = 0.4.

Theoretical analysis of boundary friction increase induced by self-splitting enhancing effect
Considering a smooth surface contact with a spherical rough bump on a rough substrate, liquid gathers around the bump and forms a thick liquid bridge with large minimum gap distance d M Va at bump valley ( Figure S17a). Based on Equation S6 and S11, the friction generated by smooth surface on rough substrate F SS can be described as where A C is the area of liquid bridge. [3][4][5] When bioinspired surface contacts the bump, pillars at bump peak have much smaller minimum gap distance d M Pe ( Figure S17b). The thickness of nth liquid bridge from bump center is where N is the number of pillars per rough bump along radius direction, [6] which can be presented as where λ R is the wavelength of substrate roughness, here can be regarded as the diameter of spherical bump section, and λ B is the diameter of pillar. Specifically, N = 1 when λ R is larger than 2λ B .

d M Va can be represented as ) )
where r is the radius of the spherical bump. The area proportion of the nth liquid bridge ) is

) )
The capillary pressure generated by the nth liquid bridge is ) ) ) ) The total friction generated by bioinspired concave pillar surface on rough substrate F BS can be described as Define ζ Sp (N) as the friction enhancing coefficient of self-splitting effect, which is            On concave pillar during continuously evaporating, the pillar is gradually pulled to substrate, and solid-solid contact first forms at pillar rim (Left). Since the rim contact area K * A P is smaller than capillary force acting area A P , solid-solid contact stress concentrates at rim area and leads to gap distance d lower with higher capillary pressure than on smooth pillar. With liquid keeps being sucked from cavity to rim area by capillarity to fill the evaporating vacancy, the cavity bottom is gradually stretched to substrate and forms the self-sucking effect (Middle). This self-sucking process further increases the capillary pressure and pressure concentration at rim area to form an even lower gap distance d M CP , which eventually leads to stronger boundary friction on concave pillar (Right). The theoretical analysis is presented in Section S3. During liquid evaporation, the apparent contact angle on pillar θ P ' varies with gap distance d, and eventually decreases to the intrinsic contact angle of pillar θ P .          Movie S1. Tree frog toe pad peeling off from substrate during a crawling step (400 fps).
Movie S2. The flow of liquid on smooth surface and bioinspired surface during separation from substrate. A continuous liquid film appears on smooth surface while self-split liquid films appear on bioinspired surface.
Movie S3. Liquid self-splitting in successive FLC steps.
Movie S5. Different TFI on smooth pillar and concave pillar induce by liquid strong capillarity.
Movie S6. The quick-separation on micropillar during liquid evaporation. Since liquid volume remains almost unchanged during quick-separation, liquid film thickness before quick-separation can be obtained by comparing the area changing before and after quick-separation.
Movie S7. Different lateral deformation on pillars in wet, boundary and dry states. Boundary state pillar with nanometer-thick liquid film forms the largest lateral deformation, i.e the strongest friction.
Movie S8. Comparison of boundary friction duration on smooth pillar and concave pillar. The duration of boundary state on concave pillar is ~4 times higher than on smooth pillar.