Re‐Entrant Microstructures for Robust Liquid Repellent Surfaces

Superhydrophobic surfaces have many interesting applications because of their self‐cleaning, waterproof, anti‐biofouling, anti‐corrosion, and low‐adhesion properties. Accordingly, numerous surfaces with hierarchical micro/nanostructures are designed and engineered to achieve superhydrophobicity. However, these surfaces have two major problems. First, they lose superhydrophobic properties over time, primarily because of environmental conditions such as vibration, external pressure, evaporation, and pollution. Second, most superhydrophobic surfaces fail to repel all types of liquids, especially those with low surface tensions. To address this bottleneck, microstructures with re‐entrant curvature have emerged, demonstrating excellent liquid‐repellent abilities and robustness. Additionally, microstructures with re‐entrant curvature have significant applications in designing surfaces with unidirectional wetting properties for passive liquid handling. Accordingly, this review systematically summarizes the design and fabrication strategies of these re‐entrant microstructures. The emphasis is given to wettability studies and other surface properties of re‐entrant microstructures and their applications, especially for liquid self‐transporting. This paper also highlights the potential applications and remaining technical challenges of fabricating these structures. Finally, the study is concluded by providing the future directions in this promising field.


Introduction
Both plants and animals in nature possess remarkable features of structured surfaces. [1] Recently, bioinspired materials and surfaces have been developed to mimic nature. [2] Hydrophobic surface is a surface that has the ability to repel water. In contrast, a hydrophilic surface attracts water and allows it to wet the surface. Hydrophobicity or water repellence is an intriguing property of surfaces in nature. The hydrophobicity results from the wetting. Another example from nature is the Araucaria leaf. [25] The leaf consists of re-entrant curvatures along its surface and exhibits directional liquid transport for low and high-surface tension liquids. A re-entrant structure has an oversized top and a smaller bottom, allowing it to repel a droplet. [26] At first, some straightforward "re-entrant" designs, such as inverse trapezoidal microstructures, showed improved omniphobicity but could not retain a stable hydrophobic state. [27] "Fractal structure" is another re-entrant geometry that exhibits self-similarity and non-integer dimensions in all scale lengths, Figure 2G. [28] Fractal structures, such as plants, clouds, or snowflakes, can be easily found in nature and exhibits super repellency to water and oils. [29] Different approaches employing chemically treated rigid or inorganic materials have been reported for obtaining oil-repellent surfaces. [30] Nevertheless, high cost, complexity, and surface contamination limit their use only to a small scale. [21a] Next, mushroom-like structures became a more feasible solution with efficient re-entrant profiles and a simpler fabrication surface. These structures allow for robust omniphobic surfaces even for liquids with low surface tension.
Many reviews have discussed the superhydrophobicity/ hydrophilicity as well as their robustness, designs, and practical applications. [22,26,31] Superoleophobic surfaces with re-entrant geometry have been reviewed regarding their design, fabrication progress, and applications. [32] An amphiphobic surface shows low affinity to both water and oils. Recently, Si and Guo investigated the factors affecting the lifespan and the durability of superamphiphobic surfaces and provided a general summary of the re-entrant structures. [33] However, most reviews in this field considered previous works with predetermined designs and did not provide a deep insight into how to achieve an optimal design for liquid repellency. In contrast to the previous reviews, this paper focuses on the need for re-entrant curvature for liquid repellency, especially low surface tension liquids. We then provide a comprehensive guideline for designing optimal re-entrant structures. Moreover, we discuss the advantages and disadvantages of each proposed fabrication method in detail to assist with the appropriate choice of manufacturing process. Figure 2 summarizes the different types of re-entrant structures: T-shaped, doubly re-entrant, triply re-entrant, duallevel re-entrant, matchstick-like, microspheres, fractal, inverted trapezoidal, microhoodoos, micro/nano hierarchical, mushroom-like structures and spherical cavities.
The first section provides the fundamentals of re-entrant structures. This section explains the wetting properties of a surface and their essential applications in real life. Next, Section 2 provides the fundamentals related to the wetting behavior of reentrant structures. Subsequently, experimental studies on the wetting behavior of these structures are discussed. Section 3 presents the design considerations of the re-entrant structures. Section 4 systematically reviews all relevant fabrication methods for re-entrant structures, including silicon micromachining, replica molding, template molding, hot embossing, additive processes, and bottom-up approaches. This section also highlights the advantages and limitations of both top-down and bottom-up approaches that are highlighted thoroughly in this section. Section 5 elaborates on technical challenges and applications of re-entrant structures. Finally, we conclude the paper with future perspectives on this field. The review provides comprehensive insights into better re-entrant structures for a targeted application.

Basic Theories
Surface wetting behavior is the interaction between a liquid phase and a solid. Understanding the wetting behavior of liquids on solids is of great importance for real-life applications such as icephobicity (the surface ability to repel ice or prevent ice formation), [34] droplet-based microfluidics, [35] anti-corrosion, [36] DNA extraction, [37] oil/water separation, [38] and inkjet printing. [39] Surface wettability is characterized by the contact angle (CA), which is the intersection between the  solid-liquid interface and the liquid-gas interface. The two models describing the wetting state of a droplet of a rough surface are the Wenzel model and the Cassie-Baxter model. [40] The Wenzel state assumes that the liquid droplet wets the entire surface, Figure 3B. [40a] Recently, the hemiwicking state has been introduced to describe the phenomenon of liquid spreading on a rough hydrophilic surface driven by capillarity and imbibition. [41] The presence of a liquid film around a hemiwicking droplet is the primary difference compared to Wenzel and hemiwicking states. [42] Unlike the Wenzel state, the textured solid in the hemiwicking state can absorb the liquid. A liquid film penetrates the solid texture with a drop resting on a solid/ liquid composite structure, Figure 3C. [43] On the other hand, the Cassie-Baxter model assumes that the liquid droplet only partially penetrates the surface because of air entrapment created by surface roughness, Figure 3D. [40b] Two key parameters are used to characterize a surface wettability: contact angle (CA) and contact angle hysteresis (CAH). CA is measured at the three-phase contact line, where the solid surface meets the liquid-vapor interface. CAH is the difference between the advancing and the receding CA on a solid surface, Figure 3A. A surface is defined as hydrophobic  if its CA is higher than 90° and is defined as hydrophilic if water CA (WCA) is lower than 90°. The surface is considered superhydrophobic when the WCA is higher than 150° and a low CAH (CAH < 10°). [45] If the CA is higher than 150° and CAH is smaller than 10° for a low-surface-tension liquid (e.g., oil or hexadecane), the surface is defined as superoleophobic. The surface is characterized as superamphiphobic when the CAs of both water and low-surface-tension liquid are larger than 150°, and the CAH is lower than 10°. In general, the chemical composition and physical properties determine the wetting state of a surface. [46] The droplet mobility on a surface is also of great interest. [44b] The mobility is determined by the sliding angle, which is the minimum tilted angle of the substrate when the droplet starts to slide or move. Dai et al. demonstrated high droplet mobility with both Wenzel and Cassie-Baxter states. The team engineered hierarchical nanoand microscale textures and infused liquid lubricant into the nanotextures to form a highly slippery surface, Figure 3H,I,L.
Slippery lubricant-infused porous surfaces (SLIPS) enhance droplet movement and will be discussed in the following sections, Figure 3G,K.
For a smooth and chemically homogeneous surface, Young's equation can be used to describe the wetting behavior, which expresses the CA as a function of the relative surface tensions among the solid, liquid, and vapor: cos Y sv sl lv θ γ γ γ = − , [47] where sv γ , sl γ , and lv γ are the surface tensions of the solid-vapor, solidliquid, and liquid-vapor interfaces, respectively, while Y θ is the static CA on a smooth homogeneous surface.
However, such an ideally smooth and chemically homogeneous surface does not exist in real life. Therefore, Wenzel used the term "roughness factor" (r) to describe the effect of roughness on the wettability of microstructured surfaces. [40a] The roughness factor is the ratio between the planar and the actual surface areas. In Wenzel model, the apparent CA of a rough surface ( W θ ) can be described as: [48] Adv. Mater. Technol. 2023, 8, 2201836 The lubricant is filled inside the cavity of the nanotextures to create a slippery surface, I) slippery Wenzel state, J) Re-entrant structure with the Cassie-Baxter state, K) the combination of re-entrant structure with SLIPS, and L) the liquid droplet moves when the slippery substrate is tilted at a very low angle. [44] ( where f s is the ratio of the top surface area and the projected area. [49] The r and f s depend on the surface roughness and structures.
Depending on the desired application, researchers find ways to drive the transition from Wenzel to Cassie-Baxter state or the other way around. [19c,50] Although many efforts have been placed into understanding droplet stability, the underlying mechanism behind the activation of Wenzel to Cassie-Baxter state is still controversial owing to debatable experimental data. [51] On the other hand, the reverse transition is clearly understood. This transition can take place under external energy inputs such as condensation, [50] laser heating, [52] and electrical voltage. [53] The original equation of Wenzel and Cassie-Baxter models calculates the static CA with a solid phase area fraction over the whole surface. However, its assumption is not necessarily satisfied for all surfaces. [22] McHale implied that Wenzel's roughness ratio and Cassie's fractional contact area are valid only if the surface is isotropic with uniform morphology. [54] Therefore, other systems have been established to precisely describe the relationship between the microstructure and the static CA. One is the re-entrant geometry description system, which was introduced by Tuteja et al. [23b] The re-entrant structure can be observed in several geometries, such as T-shape, doubly re-entrant, triply re-entrant, dual-level re-entrant, mushroom-like, matchstick-like, microspheres, inverted trapezoidal, microhoodoos, or micro/nano hierarchical structures, Figure 2. [22] The capillary force at the liquid-air interface between reentrant structures creates an energy barrier that stops the droplet from forming the irreversible transition despite the minimum surface energy of a system and an energetically favorable Cassie-Baxter to Wenzel transition. [22] Considering design parameters that estimate the stability of the composite interface, the CA of micro-hoodoo arrays can be described as: where D R D R * = + is the spacing ratio with R as the radius of the micro-hoodoo, D is the half distance between micro-hoodoos, θ is the intrinsic CA of the material.
The apparent CA C-B θ needs to be larger than the local geometric angle texture to suspend liquid droplets on top of a re-entrant structure. However, if the breakthrough pressure needed to cause the transition to the fully wetted state is relatively low, a surface with a re-entrant texture might not be able to maintain the desired composite interface. Therefore, the simultaneous improvement of two other design parameters is carried out. The first one is called robustness angle, T*, and can be formulated as: [22,30a] sin where P θ is the robustness pressure, P ref is the reference pressure determined by balancing surface forces and body forces on the fluid interface, l cap is the capillary length of liquid, and min ψ is the local geometric angle of texture.
The other design parameter is robustness height, H*: where P H is the required pressure to force the liquid-gas interface to reach the maximum pore depth of a microhoodoo structure. These parameters, including the D*, H*, and T*, allow us to engineer a solid surface with high apparent CA and a robust composite interface. [23b] In addition, another factor that should be of interest is the robustness of any composite interface, A* = 1/H*+1/T*. [30a] The robustness factor A* increases as the size of the robustness parameters (H* and T*) increases.
According to thermodynamic equilibrium, a droplet must eventually pass through a microstructured surface if the surface is made up of a naturally wetting material. Therefore, it is essential to create a microstructured surface that traps air for longer periods in a metastable condition without transition. [22] A more extreme case of a re-entrant design is doubly reentrant microstructure, which further improves the resistance of the Cassie-Baxter state and liquid repellency even for liquids with low surface tension such as perfluorohexane. [55] The liquid would come into contact with the surface and wet the top surface before flowing down the sidewall of the vertical overhangs. The liquid would stop moving at the bottom of the vertical overhangs, where the surface tension starts to point upward. According to thermodynamic analyses dealing with the minimization of Gibbs energy of a system, this phenomenon helps the overhang surfaces suspend any liquid, even for those with θ Y close to 0°. [24b,56] Not only non-wetting state is favorable, but wetting state is also desirable for applications such as microfluidics, antifogging, liquid separation, and heat transfer enhancement. [13] The so-called Wenzel and hemiwicking states, in which liquid fills the surface structures and causes a droplet to exhibit a low CA, were formerly the only ways to achieve highly wetting behavior for intrinsically wetting liquid/material combinations. Even non-wetting liquids can achieve a metastable hemiwicking condition due to roughness comprised of re-entrant structures. The surface energy model shows that when the structure is filled with liquid, the re-entrant feature generates a local energy barrier that inhibits liquid depletion from surface structures, regardless of the intrinsic wettability.
Unlike high-surface-tension liquids such as water, low-surface-tension liquids usually spread on most surfaces. However, some specially engineered structures like re-entrant structures allow for the formation of a low-surface-tension droplet with high CA. The two transport methods for low-surface-tension droplets are spread and slip. [57] During spreading, the wet surface beneath the droplet stays wet while the contact line moves along the surface. On the other hand, slip is characterized by a retraction of the contact line at the droplet's trailing end. According to the Wenzel model, rough surfaces prefer spreading over slipping. [40a] Roughness will not increase the surface's ability to repel low-surface-tension liquids because their equilibrium CAs are smaller than 90° on smooth surfaces. [57] However, re-entrant topography has been demonstrated to have the ability to pin the liquid/air interface, resulting in a metastable Cassie-Baxter state for low surface tension liquids. This composite interface has a higher capillary resistance due to its larger intrinsic CA. Droplet transportation velocity and maximum travel distance are two parameters that were used to evaluate the effectiveness of spread and slip modes. [57] The re-entrant structures possess low solid-liquid contact area and substantial wettability gradient over a short length produced by organized microarrays, resulting in fast droplet transport yet limited travel distance.

Chemically Modified Re-Entrant Microstructures
Even though having a re-entrant structure, the surface still needs to be chemically modified to achieve high resistance against wetting. For instance, Kang et al. deposited octafluorocyclobutane C 4 F 8 gas on the surface in an ICP chamber to make the surface superomniphobic. [21a] The water CA measurements were carried out with the fabricated mushroom-like structures with varying space/diameter ratios and different materials and showed high CA (ranging from ≈140° to ≈160°) and low CAH. In the case of an ethanol drop, Cassie Baxter state can be achieved for some specific ratios. This study indicates the essential role of dimensions in designing re-entrant structures. Furthermore, the result also confirms the importance of chemical treatment. Without the chemical coating, the surface would experience a dramatic change from a non-wetting to a wetting state. Some other groups conducted the same method of measuring CA and CAH since it is a simple yet effective way to assess the wetting behavior of the re-entrant str uctures. [20,24a,58]

Flexible Superomniphobic Surfaces
Kim et al. considered flexibility of superomniphobic surfaces. [59] Interestingly, the surface remained superomniphobic even under high strain (≈50%), bending, twisting, as well as repeated stretching. Even after ultraviolet (UV) and oxygen plasma treatments, no degradation of wettability was observed.

Slippery Liquid-Infused Porous Surfaces
Slippery surface is another new method to reduce the adhesion of a droplet. [2] SLIPS provide outstanding repellency and drop mobility to a wide range of liquids, including low-surface-tension liquids and complex fluids such as blood or cell medium. [23c] The lubricant layer is superior to the trapped air pockets in other cases of superhydrophobic surfaces as it is more robust toward pressure and can be designed for various applications such as self-cleaning, anti-fogging, anti-biofouling, and anti-icing. [44c] Dong et al. introduced a novel slippery superoleophobic surface by fabricating the doubly re-entrant and then infusing a hydrophobic lubricant over the top of the nanostructures. [44c] The doubly re-entrant structure keeps the surface superoleophobic and prevents the lubricant from impregnating into the cavities. Concurrently, the slippery layer decreases the adhesive force between the liquid and the pillars drastically. However, the CAH did not reduce considerably as expected from the slippery property of the surfaces since flat SLIPS exhibited low CAs for low surface tension liquids. [23c]

Underwater Oleophobicity
Underwater oleophobicity is a fascinating and distinct field of study employed for self-cleaning and anti-fouling. [60] Superhydrophilic and superhydrophobic or oleophilic abilities are also for oil-water separation. Hensel's group immersed a polymer membrane into a liquid-flooded chamber to investigate the robustness and the breakdown pressure of re-entrant structures. [24a] By gradually increasing hydrostatic pressure, the breakdown pressure was determined and plotted as a graph versus cavity diameter. The highest breakdown pressure was observed with a cavity diameter smaller than 1 µm. The nucleation and growth of liquid inside the cavities were investigated. The bottom of the cavity or the overhang edges are energetically advantageous locations for nucleation. Most condensates, however, stabilized and stopped growing after about 2 h. These condensates' state remained steady for at least 7 days.

Design Considerations of Re-Entrant Microstructures
Several criteria need to be considered to engineer robust superamphiphobic surfaces effectively. For superhydrophobic surfaces, it is sufficient to have only non-hierarchical, microscale topography on a hydrophobic polymer that is not necessarily coated. [61] However, when it comes to superamphiphobic surfaces, other designs and implementations are necessary. [45] Studying plant leaves leads to the discovery that micropatterns frequently have hierarchical surface topography with nanoroughness superimposed on microroughness. [60] Therefore, re-entrant structures with hierarchical organization and welldesigned spacing and height are crucial for the omniphobicity of the surface, as they help improve the stability of a composite interface. [62] Even though micro/nanofabrication methods for re-entrant structures have improved substantially, their designs still need more optimization. [63]

Ratio of Pillar Width and Pillar Spacing
Kang et al. varied the spacing ratio between pillars from 1 to 7 to investigate the effect of the width to the spacing ratio of mushroom-like micropillar structures. [58b] The team found that a space/pillar width ratio ranging from 1 to 7 was appropriate for achieving superhydrophobicity. In contrast, a ratio from 4 to 7 was needed for ethanol droplets to achieve the Cassie-Baxter state on mushroom-like arrays.

Design Model
Wu et al. proposed four design criteria to achieve and sustain the Cassie-Baxter state for both high and low surface tension liquids: pressure balance, pinning effects, surface curvature, and suspending condition. [58g] Figure 4 illustrates typical design models with corresponding parameters to further elaborate the optimal criteria. First, Laplace pressure and imposed pressure on liquid should be balanced to maintain the Cassie-Baxter state.
[58g] The topography is often re-entrant and consists of concave and convex segments. In Figure 4A, P max is the maximum pressure that the surface can resist. The solid angle δ plays an essential role in pressure balance. For the design in Figure 4A,B with δ > π/2, the liquid-air interface of water has the shape of a convex meniscus. The spreading oil penetrates the gap between the two pillars in the shape of the concave meniscus. The sharp edges and round curvatures in Figure 4C,D can sufficiently balance the imposed pressure for both water and low-surfacetension liquids such as oil. However, the T-shaped structure in Figure 4E,F showed the best performance in pinning both high-and low-surface-tension liquids on top and bottom corners, respectively.
The pinning condition is the second criterion for achieving the Cassie-Baxter state for overhang structures with sharp edges. [58g] Gibbs inequalities are used to calculate the maximum and minimum pinning CAs on edges with the following expression: where Θ r and Θ a are respectively the minimum and maximum pinning CAs on edges before the three-phase contact line moves. In Figure 4, pinning CAs θ pin should be in the range of minimum and maximum pinning CAs for liquids to be pinned. The pinning condition is a requirement for all micro/nanostructures with sharp edges to achieve lyophobicity, especially for surfaces made of hydrophilic materials. This result agrees with the work of Li et al. and Wang et al., who pointed out the importance of sidewall angles for microtextured Figure 4. Schematic illustrations of different criteria for superoleophobic surfaces. A) Sharp corner with angle δ ≥ π/2 pins the water and convex meniscus appears in a microcavity; B) corners with δ ≥ π/2 cannot pin oil due to Pmax < 0; C) oil is pinned on sharp edges with solid angle δ < π/2 while pinning condition θpin < Θa is satisfied; D) oil can be pinned on round curvatures with angle δ < π/2, θpin < Θa, and Pmax > 0; E) water is pinned on T-shape structure with perpendicular corner and has convex meniscus; F) oil is pinned on lower corners of T-shape structure. Reproduced (Adapted) with permission. [58g] Copyright 2011, Elsevier.
design. [64] From their thermodynamic explanation, the low sidewall angle (e.g., inverse-trapezoidal) can amplify the stability of the Cassie-Baxter state. Therefore, the teams came up with the same T-shaped structure as the best design for preparing superomniphobic surfaces like that of Wu et al., since this structure can reduce the sidewall angle to zero and keep the pillar height high enough. The high energy barrier and simultaneous avoidance of the "sag" transition provided by this structure improved the stability of the Cassie state. The third criterion is surface curvature, especially for round or curved surfaces, as shown in Figure 4D. Bumps rather than grooves should be the curvature topography to pin liquids. [58g] However, such geometry is challenging to fabricate with micro/nanotechnologies. These structures also have a small depth, which typically results in less pressure robustness. Nosonovsky stated that the formation of a concave (re-entrant) surface by nanoscale roughness onto microscale asperities leads to local energy minima and consequently stabilizes the Cassie-Baxter composite interface. [65] A convex surface, on the other hand, would lead to energy maxima and, thus, unstable equilibrium states. Since the local surface curvature mostly depends on the value of the sidewall angle, it also influences the wettability of the surface. [64a] Therefore, we can conclude that T-shaped structures exhibit the highest performance in repelling low and high-surface tension liquids concerning pressure balance, pinning effects, and surface curvature conditions.

Pillar Height
The last criterion proposed by Wu et al. is suspending condition, which relates to the liquid penetration and the cavity height.
[58g] The liquid will penetrate the cavity and completely wet the surface when the edge of the liquid-air interface touches the bottom of the cavity. [32d] The nadir of the curvature, however, cannot even penetrate the bottom of the gap if the pillar height is high enough. The cavity height should be optimized by considering the trade-off between the stability of the Cassie-Baxter state and the mechanical robustness of pillars since large height makes supporting pillars thin and easy to collapse. [58g] As a result, the T-shaped overhang structure with a cap and a stiff pillar ( Figure 4E,F) is regarded as the best choice for a reliable, high-performance superamphiphobic surface for engineering applications. Furthermore, it was reported that microstructured geometries show lower pressure stability than nano-scale ones. [58g]

Array Arrangement
It has been shown that the simple micropillars' array arrangement can affect pinning of the contact line and subsequently affect both static and dynamic wettability. [66] Accordingly, Wu et al. examined different array arrangements, including linear, square, and hexagonal, of re-entrant microstructures and varied the vertical and diagonal spacing and tip size. [58g] The experimental results indicate that the wettability of the surface mainly depends on the solid fraction and not array arrangement.

Pillar Width and Pillar Spacing
Apart from the four criteria mentioned above, the effect of pillar width and pillar spacing was also elaborated to optimize the design of microstructures. Li et al. showed that the smaller the pillars are, the more robust the composite interface would be. [64a] Nevertheless, the pillar width and pillar spacing exhibit no effect on the transition from hydrophilicity to superhydrophobicity.

Doubly Re-Entrant Structure
In addition, doubly re-entrant geometries, which have both horizontal and vertical overhangs to the surface, can support a favorable liquid-vapor interface shape with an upward-pointing surface tension even with a fully wetting liquid. [67] In order to ensure that the droplet is resting stably on air pockets for the surface to be repellent, such structures must be constructed to limit the liquid-solid contact fraction. Therefore, the structures must be as thin as feasible. [55] However, singly re-entrant geometries are more frequently used for liquid repellency because these surface characteristics are fragile and challenging to manufacture on an industrial scale.

Slippery Liquid-Infused Porous Surfaces
Alternatively, minimizing the liquid-solid contact fraction can be done by increasing the gaps between pillars. However, this leads to a low breakthrough pressure inducing the transition from Cassie-Baxter to Wenzel state. [30a] Therefore, a novel method with SLIPS has been introduced to lower the adhesion force of a droplet. These surfaces are created by infusing a lubricating liquid into nano/microstructured surfaces that possess a smooth and reconfigurable liquid-liquid interface, which is useful for anti-biofouling, anti-icing, or drag reduction properties. [68] Dong et al. fabricated doubly re-entrant structures with nanoroughness on the mushroom cap. The lubricating liquid was impregnated into the cavity of nanostructured caps, Figure 5. [44c] The combination of doubly re-entrant structures with nanoroughness and SLIPS guarantees the entrapment of the lubricant layer on top of the micropillars. Scanning droplet adhesion microscopy revealed a reduction in adhesion between the liquid and the pillars with the lubricant layer.
Combining two extremes of superoleophobicity and SLIPS can achieve superoleophobic and superhydrophobic surfaces with superior liquid-repellent, anti-icing, or anti-fouling capabilities. However, this area still lacks research and needs more theoretical and experimental data to comprehensively develop a proper re-entrant structure with SLIPS.

Topological Liquid Diode
Topological liquid diodes allow the directional and spontaneous flow of liquid on a surface. [68] It is worth mentioning that the re-entrant structure is also a key factor in designing liquid self-transport. [69] Li et al. utilized the special convex shape of re-entrant structure inside the cavity of U-shaped island arrays to unidirectionally transport liquid with both relatively high velocity and long distance. [70] The topological surface can also transport a variety of liquids, including low-surface-tension liquids, such as hexane, and viscous liquids, like ethylene glycol. The presence of a re-entrant structure enhances the pinning effect of the advancing liquid since this structure stops the liquid from penetrating the cavities. Therefore, the liquid can only transport in one direction. However, to the best of our knowledge, the specific design parameters of this re-entrant structure in a liquid diode have not been optimized for droplet transport in terms of velocity, length, and robustness. In the future, more efforts shall be devoted to elucidating the effects of design parameters on the efficiency of liquid diodes.

Design Parameters for Specific Applications
Panter et al. used computational methods to explain the mechanisms of the wetting models, Figure 6. [63] Three essential wetting properties were investigated: CAH, critical pressure, and sustainable pressure. The CAH depends on the area fraction F r of the cap [F r = (W/B) 2 ] and the total cap height D (the subscript r indicates the dimension is relative to system size B, for example, W r = W/B with B = 60 lattice spacings). Meanwhile, the team reported that critical pressure ΔP c was only dependent on the area fraction F r and the pillar height H r . Increasing the pillar height leads to higher critical pressure, thus, more robust omniphobicity. However, if the pillar is too tall, structure will be much weaker. Looking into the minimum energy transition  . Schematic illustration of the 3D re-entrant topography with T shape (left), with 2D cross-section of doubly re-entrant structure together with labeled structural parameters (right). Reproduced (Adapted) under the terms of the Creative Commons Attribution license. [63] Copyright 2019, Science Advances, American Association for the Advancement of Science. mechanisms, three transition states were demonstrated: base contact, pillar contact, and cap contact.
In a re-entrant geometry, the optimal structure design involves maximizing the cap width W r and minimizing the pillar width A r at a critical pillar height, as it increases the liquid-vapor interfacial area to maximize the energy barrier between wetted and unwetted states. For the case of doubly re-entrant geometry, the liquid-vapor interfacial area could be maximized by increasing W r together with decreasing lip depth L r and A r, also at an optimal pillar height. The team optimized the re-entrant geometry by simultaneously considering the three wetting parameters for two practical applications: membranes for water purification via membrane distillation and droplet-based digital microfluidics. They created an application-specific scoring algorithm that rates a design against the specified wetting properties. The scoring function was then improved using two techniques. First, the team selected the best structure by evaluating the scoring function throughout the whole parameter range examined. In the second way, they showed that genetic algorithms could be effectively employed to carry out the simultaneous optimization for designs when wetting property assessments might be too complicated. Finally, the team attained the optimal values for six parameters, that is, A r (pillar width), H r (pillar height), L r (lip depth), T r (cap thickness), W r (cap width), and the system scale Although the results were very specific for the two applications with specific conditions, this computational approach is flexible and can be applied to other structured surfaces.
The optimal conditions for designing different parameters of re-entrant structures are listed in Table 1.

Fabrication Methods of Re-Entrant Microstructures
Many micro/nanofabrication methods have been suggested in the literature to make re-entrant micro/nanostructures. Figure 7 summarizes these techniques, which are discussed in detail in the following sections.

Silicon Micromachining Approach
In silicon (Si) micromachining approach, silicon dioxide (SiO 2 ) is usually grown on top of the silicon wafer. Photolithography, followed by dry etching, is then used to produce the re-entrant microstructures. In this approach, the over-hang cap is typically made from SiO 2 , and the supporting pillars are made of bulk Si.  Table 1. Summary of optimal parameters for the design of re-entrant structures.

Design parameters
Optimal conditions Comments Design model [55,58g] T-shape re-entrant structure with nanoscale Singly or doubly T-shape overhang structure with a cap and stiff pillar. Microstructured geometries show lower pressure stability than nano-scale ones. Singly re-entrant structure is simpler to fabricate than doubly reentrant structure.
Pillar height [58g] The liquid cannot penetrate the cavity if the pillar height is tall enough The cavity height should be optimized by considering the trade-off between the stability of the Cassie-Baxter state and the mechanical robustness of pillars since large height makes supporting pillars thin and easy to collapse.
Ratio of pillar width (d)  The smaller the pillars are, the more robust the composite interface would be. Increasing the gaps between pillars leads to a low breakthrough pressure inducing the transition from Cassie-Baxter to Wenzel state The pillar width and pillar spacing exhibit no effect on the transition from hydrophilicity to superhydrophobicity Array arrangement [58g] Do not affect wettability Can be linear, square, or hexagonal

SLIPS, liquid diode
Have not yet been explored in detail No optimized conditions proposed Dimensions of singly/doubly reentrant structure [63] Pillar height H r ≥ 0.17 Dimensions of singly/doubly re-entrant structure simultaneously optimized for the application: membranes for water purification via membrane distillation. The purpose is to maximize the critical pressure, minimize the energy barrier, and maximize the CAH.
Dimensions of doubly re-entrant structure [63] Cap height D r = 0.22 Dimensions of singly/doubly re-entrant structure simultaneously optimized for the application: droplet-based digital microfluidics. The purpose is to maximize the critical pressure, minimize the energy barrier, and maximize the CAH. The pillar width A r and ratio of L r to t r are freely chosen. B = 100 µm.
hybrid SiO 2 -Si microhoodoos that could repel both polar and nonpolar liquids.
[23b] The re-entrant curvature was mainly fabricated in a two-step process that included reactive ion etching (RIE) and isotropic etching using Xenon difluoride (XeF 2 ). [21b] The structure robustness and liquid repellency could be optimized by varying the design parameters for water and various organic liquids. Liu et al. used similar approach to fabricate doubly re-entrant microstructures with SiO 2 as a sacrificial substrate. [55] The team established the process flow to fabricate a doubly re-entrant structure on Si in three main steps (vertical posts, re-entrant posts, and doubly re-entrant posts) with RIE and deep reactive ion etching (DRIE). After several etching and deposition steps, the doubly re-entrant structures were used for various wettability studies. Figure 8 shows the fabrication process and SEM images of these doubly re-entrant microposts. The surface of these re-entrant microstructures was further coated with a hydrophobic layer of perfluorocyclobutane (C₄F₈). Intriguingly, these doubly re-entrant microposts showed excellent stability against high temperatures up to 1100 °C and robust repellency up to 150 °C. The same group later replaced XeF 2 etching with DRIE to produce a smaller cell size of 15 µm. [34a] This new design not only presented better repellency but also higher dynamic pressure resistance under normal and freezing temperatures. Although the re-entrant structure showed a remarkable ability to repel liquids, the costly, time-consuming, and complex process is the major limitation. Additionally, only mechanically stable silicon wafers are suitable for this process, which makes it undesirable for practical application.
Wu and Suzuki employed Si pillars on Si substrate to assess the design of superoleophobic surfaces using different T-shaped microstructures.
[58g] Several techniques were used to achieve various topological geometries on silicon. The first structure was created by making a polymer cap, while the second structure used standard photolithography to make a SiO 2 cap with subsequent C 4 F 8 surface treatment. The third structure utilized DRIE to create better undercut patterns below the cap. Self-assembled monolayer (SAM) of 1H, 1H, 2H, 2H-perfluorooctyltrichlorosilane was coated on the third structure. The structures demonstrated the best nonwetting ability and thermal stability, with a static CA of 167° and a hysteresis CA of 8° with hexadecane droplets. The group stated that a T-shaped structure with well-controlled geometries was ideal for achieving small solid fractions and increasing superoleophobicity. At the same time, chemical surface treatment reduces the surface energy of Si and SiO 2 .  As discussed above, doubly re-entrant structures are the best candidates for repelling any type of liquid, as the intrinsic contact angle of the liquid can be as small as zero. However, the height of these micropillars still affects the stability of the Cassie-Baxter state, and it is more desirable to use shallow, doubly re-entrant micropillars. The main challenge is control-ling the etching time to reduce these heights. Rontu et al. proposed an efficient silicon-on-insulator (SOI) micromachining approach. [71] To this aim, the team utilized repetitive steps of photolithography and etching on a SOI wafer to fabricate doubly re-entrant micropillars with 5-10 µm height. Without chemical coating or modification, the team showed that shallow  doubly re-entrant micropillar arrays could significantly improve the superomniphobic stability.
Other groups have also attempted to fabricate these structures using semiconductor-based techniques yet combining different materials and structures. For example, Wang and coworkers used thermo-responsive shape memory polymer to fabricate metamorphic superomniphobic surfaces (MorphS) to study the wetting transition. [72] The team used plasmaenhanced chemical vapor deposition (PECVD) followed by photolithography and RIE to generate MorphS. Hu et al. fabricated re-entrant shapes combined with hierarchical structures to explore superhydrophobicity and wetting hysteresis. [73] Kim et al. improvised the procedure slightly by using conventional lithography to form circular disks and subsequent reactiondiffusion-based lithography to form vertical pillars, which were flipped to create re-entrant structures. [74] Telecka et al. utilized block-copolymer self-assembly combined with a selective RIE method to fabricate overhang superamphiphobic nanostructured surfaces. [75]

Soft Lithography from Etched Si Microholes
Soft lithography from etched Si microholes is a popular and effective method for fabricating robust re-entrant micro/ nanostructures. Polymers, such as poly(dimethylsiloxane) (PDMS) or polyurethane acrylate, are cast into a silicon (Si) master mold, which had been fabricated by combined photolithography and etching. The foundation of this method is the precise insertion of an etch-stop layer of SiO 2 , combined with deep and isotropic RIE, to produce a silicon master with undercut microholes on a poly-Si substrate. [76] A demolding procedure can then produce the re-entrant micro/nanostructures by either carefully peeling the cured polymer from the master [77] or by sacrificing the master mold using selective chemical etching. [77a,78] To date, soft lithography is the most widely used method for designing this kind of surface due to its remarkable advantages such as reliability, high resolution even to nanoscales, simple procedure for mass production, repeatable polymer replicas with the same master mold, durable and optically transparent surfaces. [21,24a,58b,77a,79] However, some limitations need to be addressed: i) the polymer-based materials are susceptible to mechanical forces, UV, or other harsh conditions, ii) the peeling-off step can lead to distortion and damage to the replicated structures and iii) fabrication of the master mold requires cleanroom facilities and involves high-cost Si-on-insulator wafers and (DRIE) process.
To solve the demolding error and achieve a more robust omniphobicity without chemical coating, Kim Figure 9A illustrates the fabrication process of this technique. The master mold was first fabricated by combined photolithography, RIE and DRIE on a SOI wafer. To facilitate the demolding process, the surface energy of the master mold was reduced by coating it with octafluorocyclobutane (C 4 F 8 ). Finally, the doublering mushroom-like arrays were formed after PDMS casting and demolding. The team demonstrated that master mold coating and over-etching both sides of the concentric microholes were crucial for successful demolding of PDMS.
Inspired by the hierarchical micro/nanoscale foot hairs of geckos with excellent self-cleaning ability, Kim et al. reported a method to form flat mushroom tips of more complex 3D geometries using a master mold patterned by DRIE and the notching effect. [77a] Starting with photolithography, the substrate was patterned with isotropic and DRIE to obtain a spatulate tip at the end of a hole. Finally, a polymer liquid filled the template and was subsequently released by removing the Si template with XeF 2 . This approach exhibits some benefits: i) any liquid or gas phase polymer can be used; ii) many arrays of fibers can be fabricated using one mold; iii) high efficiency up to 100%; iv) higher resolution down to nanometer can be achieved. The mushroom-shaped tip ends of the microfiber array are hydrophobic enough to have a persistent wet self-cleaning ability, even if the fiber material is intrinsically hydrophilic. However, demolding by sacrificing the master is undesirable in the industrial economy. In addition, this work only focused on the vertical fibers; therefore, angled fibers for anisotropic properties need to be explored in the future.
Accordingly, Jeong et al. presented a way to design angled slanted nanopillars with spatulate tips by forming tilted holes with undercuts through slanting etching. In this work, parameters such as leaning angles, sizes, tip shapes, and hierarchical structures are well-controlled to show excellent adhesion properties. [80]

Replica Molding without Etching
As an alternative to the above approach, using a stack of two photosensitive polymers [79b,81] or double-step UV exposures is a suitable approach. [82] For example, Wang's group proposed a simple technique based on a unique process of double-side photolithography and molding for creating bioinspired dry adhesives with mushroom microstructures. [82b] The main difference from other conventional molding techniques was utilizing both masked and maskless exposures. These exposures were carried out on the top and bottom of a positive photoresist to obtain an array of microholes on a glass substrate with undercut features at the bottom following the development step. Slanted micropillars could also be generated by inclined angled exposure. Without using any complicated etching step, this method is less complex and more cost-effective.
Alternatively, utilizing a bilayer stack made up of a top SU-8 layer and a bottom sacrificial lift-off resist layer, Yi et al. used conventional photolithography to create a master with mushroom-shaped micropillars. [79b] By adjusting the spincoating thickness and lift-off resist layer development time, it was possible to regulate the thickness and diameter of the mushroom-shaped tips. With the help of this technique, mushroom-shaped micropillars with precisely controlled tip geometry could be produced quickly, consistently, and in large quantities. Wang [20] Figure 8B schematically shows the fabrication steps of this technique. In contrast to reliance on the timing of wet or dry etching, the advantage of this fabrication technique is the precise control of the geometrical parameters of the mushroom-like pillars during the photolithography processes.

Template Molding with Additional Tip-Forming Step
In this approach, template molding techniques such as injection molding and hot embossing are used to generate micro/ nanopillars with mushroom-shaped tips. Template molding is commonly used to first generate the supporting pillars (or pillars' stems in the case of mushroom-like pillars), and   ) is spin-coated on the glass substrate to define the cap thickness of the mushroom-like arrays. c) After baking the first photoresist, it is exposed without any photo mask. d) A second layer of the same photoresist is spin-coated on top of the first layer. e) The cap diameter of the mushroom-like arrays is defined by photolithography. f) After developing the exposed positive photoresist layers, the microhole arrays are formed. g,h) After PDMS casting and demolding, the mushroom-like micropillar arrays are formed. Reproduced (Adapted) with permission. [20] Copyright 2017, American Institute of Physics (AIP) Publishing.
subsequently, the pillars' tips or caps are formed with an additional step. [59,83] Simplicity and cost-effectiveness are the two main advantages of this approach.
For instance, mechanical pressing on uncured precursors after capillary UV molding is a straightforward method for generating nanopillars with controllable mushroom-shaped tips. [84] Using this approach, a well-controlled modification of the head tip shape is possible by varying the UV exposure time and magnitude of the pressure. Similarly, mushroomshaped tips were created by submerging the fully formed and cured polymer pillars in a liquid-state prepolymer, pressing them onto a substrate while applying a constant load, and then allowing the prepolymer to cure on top of the polymer pillars at room temperature. [83c] Nonetheless, since these methods rely on mechanical pressing to modify prefabricated pillars, they possess the risk of structural deformation of the partially-cured polymer, particularly when applying a significant force in the nanoscale regime. Furthermore, problems with controlling the lateral shape and spatial uniformity of the tips also arise due to the lack of lateral confinement for the material reflow. [82b] Utilizing a flexible nylon-mesh template, an imprinting method is sufficient to create the re-entrant structures without damaging the imprinted structures after the demolding procedure. [85] The mechanical robustness of the re-entrant structures is maintained due to the flexibility of the nylon-mesh templates. After imprinting time, silica nanoparticles were bound onto the re-entrant micropillars producing a robust nano-on-micron geometry. The main advantage of this method is that the nylonmesh template is inexpensive and available commercially.
Hu et al. demonstrated an electrically induced polymer deformation method to prepare mushroom-like structures with different morphologies. [76] Figure 10 shows low-aspect-ratio features prepared by hot embossing on a PMMA (polymethyl methacrylate) surface. An electrical voltage was applied to pull the pre-formed micropillars upward to touch the upper plate. The fibers were flattened and spread to form a mushroomshaped tip with a high aspect ratio and large diameter by electrowetting effect. This technique provides a facile way for a more straightforward demolding step, which keeps the surfaces intact and suitable for high aspect ratio fabrication; however, it is only limited to thermoplastic polymers.
While arrays of micropillars were prepared initially by photolithography, spherical photoresist nanoparticles were crosslinked spontaneously to micropillars by emulsification and UV exposure. [58d] The nucleation of photoresist nanoparticles occurred all over the surface but mainly concentrated on top of the pillars to form mushroom tips rather than filling the gaps between the pillars. Then, the surfaces were treated with fluorination to obtain superhydrophobicity. An opposite procedure was demonstrated by Lee et al., where silica nanoparticles were poured into micron-sized holes first to form the tip of the pillars. [58a] Epoxy resin was backfilled afterward to generate the pillars before being treated with oxygen plasma to reduce the micropillar diameter. This technique showed precise control of the tip size and shape, leading to better hydrophobicity and omniphobicity.
Since a doubly re-entrant structure is challenging to fabricate, very few attempts have been made to engineer this structure. Choi et al. demonstrated a method using field-gradient photofluidization with the help of an azobenzene-containing polymer. [86] The first step produces a micropillar array by solvent-assisted soft-molding. The second step utilized localized Adv. Mater. Technol. 2023, 8, 2201836   Figure 10. Process flow for producing mushroom-shaped micropillars with three steps from left to right. a) The polymer is prestructured into low-aspect ratio micropillar arrays on a conductive substrate using hot embossing as the initial step. b) The second stage involves applying a voltage to electrohydrodynamically force the micropillars upward into contact with the upper electrode. c) The third step is the electrowetting-induced deformation of the micropillar tip. d) Mushroom-like arrays are formed by removing the upper electrode. e,f) Illustrate how an electric field can lead to the formation of high-aspect-ratio micropillars. Reproduced (Adapted) with permission. [76] Copyright 2014, American Chemical Society. photofluidization to bend the polymer into mushroom-like heads through irradiation. The deformation of the polymer produced a doubly re-entrant structure and stopped when no illumination was present. To lower the solid surface energy, successive coatings of a perfluorocarbon substance were applied to the surfaces. This method possesses numerous benefits, such as straightforward procedure, cost-efficient, reducibility, and scalability.

Laser-Based 3D Printing Techniques of Photopolymers
Laser-based 3D printing techniques are among the high-resolution 3D photolithography techniques for fabricating complex structures using ultrashort laser pulses. [87] A recent direct laser writing (DLW) technology called multi-photon polymerization (MPP) 3D printing can generate 3D objects with sub-100 nm resolution at scan speeds of 10 000 ms −1 . [88] The technique relies on nonlinear photon absorption by photopolymers to create tiny features in photosensitive material. DLW via MMP is a highly precise method for generating hierarchical micro/ nanostructures with re-entrant surface curvatures. [89] Recently, Liu et al. and Dong et al. successfully engineered those doubly re-entrant structures with embedded nanostructures using the 3D DLW based on the two-photon polymerization method. [44c,90] Figure 11A shows the SEM images of these micropillars. This two-photon polymerization technology can fabricate singly, doubly, and even triply re-entrant structures on rigid silicon wafers and flexible substrates. Due to this unique structure, the triply re-entrant arrays exhibit super repellence toward liquids with high surface tension such as water (γ = 72.8 mNm −1 ) as well as liquids with low surface tension such as organic liquids (γ = 12.0-27.1 mNm −1 ). However, it is worth noting that twophoton polymerization does not show high throughput and faces difficulties in tailoring large-area samples. [58e] Yin et al. synthesized omniphobic mushroom-like structures from hydrophilic photo-curable resin by 3D printing technique. [91] The team precisely tuned the geometric features of the surface inspired by the structure of omniphobic springtail skin. Haghanifar et al. presented a new 3D-printed re-entrant structure with high robustness and super-repellency via direct laser writing. [92] The structure exhibited triply re-entrant shapes with a dual-level geometry. Sun et al. fabricated multilayered, doubly re-entrant posts with increasing diameter from top to bottom. [93] The structures are defined as stepwise wetting structures that display a favored wetting transition in a stepwise manner. The multiple energy barriers between layers greatly enhanced the resistance against wetting of the surface. Since the design of multilayered re-entrant structure has been recently established, not many applications have been proposed. However, the robustness and superomniphobicity of this structure may lead to promising applications in medical, electronic devices, and many more. [92,94] DLW can also be used to fabricate fractal structures. Davis et al. reported 3D nanostructured fractal surfaces were created using two-photon photolithography for functionalized surface applications. [28] The fabrication process produced three arbitrarily shaped structures in a quick manner with high resolution: a cubic fractal surface, a broccoli fractal surface, and a sphere flake fractal surface.

Electroplating
Electroplating is a process where metal ions in an electrolyte solution are transferred to coat a layer onto a surface. [95] In electroplating, electroforming is the process of electrodepositing material onto a (temporary) master pattern to achieve a specific geometric shape. For instance, Grigoryev et al. used electroforming to grow 3D nickel (Ni) on the master pattern, forming Ni micronails with a spherical cap on top, Figure 11B. [58c] The cap size is precisely engineered by controlling the deposition time and current. Despite the random allocation of Ni micronails on the substrate, the repellency of the surfaces was barely affected. Interestingly, by exploiting the magnetic properties of Ni, manipulation of wetting behavior was explored by applying an external magnetic field to bend Ni micronails, thereby leading to the transition between Cassie-Baxter and Wenzel states. The same approach was conducted by Kehagias et al., using electrodeposition process to produce Ni mushroom-like structures. [96] Controlling the electroplating time could adjust the growth of mushroom-like Ni features in terms of height and uniformity.

Bottom-Up Approach
Another strategy to fabricate re-entrant curvature involves growing randomly-distributed re-entrant nanostructures on flat or shallow microstructured surfaces. [97] Many nanostructures have been employed to grow such re-entrant geometries, including nanowires, nanofibers, nanorods, nanoflakes, nanoflowers, nanograss, nano dendritic structures, nano reticula structures, and nanoporous structures. [97] Herein we summarize these fabrication methods.

Solvent-Induced Phase Transformation
Brown and Bhushan reported a method for preparing superoleophobic polymer-nanoparticle composite nanomaterials with re-entrant geometries. [67] In their work, the nanoparticles were impregnated into the polymer surface to produce a robust superoleophobic polycarbonate. When exposed to acetone, polycarbonate undergoes a solvent-induced phase transition that forms a superhydrophobic surface. The nanoparticles agglomerates incorporated on the surface and near-surface area acted as nucleation sites for polymer crystallization. The surface was then treated with UV exposure and fluorosilane to achieve oil repellency. Additionally, factors like treatment time, temperature, drying conditions, nanoparticle size, shape, and chemistry affected the development of composite surfaces and the surface attributes.

Electrospinning
Microfibers and nanofibers are among the re-entrant structures suitable for industrial use since they are simple to mass manufacture. [98] Electrospinning may create microfibers and nanofibers whose diameter can be easily modulated with process parameters such as the solvent, viscosity, surface tension, and electrical conductivity to ensure adequate robustness against wetting from low surface tension liquids. Additional secondary structures, such as wrinkles, beads, and nanopores, can increase the amphiphobicity of electro-spun fibers. Electrostatic spinning has emerged as one of the most efficient methods for creating superamphiphobic nanofiber-structured surfaces due to its benefits of simple equipment, controllable processes, cost-efficient, and adaptability to varied materials. [32b] Choi et al. used an unconventional electrospinning technique to fabricate a web of PTFEMA (poly(2,2,2-trifluoroethyl methacrylate)) nanofibers without further surface functionalization. [98] PTFEMA was electrospun after being created by standard emulsion polymerization. The polymer solution's concentration was varied, but the team kept the solvent, the applied voltage, the gap between the tip and the collector, and the flow  The setup of three-electrode electrodeposition including 1) a track-etch template made of polycarbonate membranes for the growth of Ni micronails, 2) a metal working electrode (WE) with one side covered with gold to form a conductive electrode, another thick layer of copper was deposited on top of the gold layer to support the Ni micronails, 3) a Ni electroplating bath with controlled pH at 3.7, 4) a Ni counter electrode (CE) with a current to fill the pore with Ni, 5) and a reference electrode (RE). 6) The enlarged schematic illustrates the 1D electrochemical growth of Ni wires within a template pore. 7) The gradual formation of Ni vertical pillar inside the pores as Ni fills the pores. c) After Ni reached the upper surface of the template, the Ni growth was no longer restricted by the cylindrical pore. 8) The enlarged graphic demonstrates Ni's unhindered 3D growth on the template surface, which leads to the development of a hemispherical cap. The cap size was controlled by deposition current and time. d) Immersing the template into dimethylformamide (DMF) dissolved the template and exposed the Ni micronails on the electrode surface. e) The SEM images of the Ni micronails which show the random distribution of them due to the random position of pores in the polymer template. f) Enlarged SEM image of a single micronail with a large hemispherical cap representing the re-entrant structure. g) SEM image of an epoxy resin droplet resting on a bed of Ni micronails. Reproduced (Adapted) with permission. [58c] Copyright 2012, American Chemical Society.
velocity of the solutions unchanged. Even though the process was straightforward, it entailed some limitations as the surface could not achieve high robustness against wetting due to the inhomogeneity of the fibers. Similarly, Tuteja's group used the electrospinning method to prepare hierarchical re-entrant curvature by employing a mix of polymers PMMA or PDMS (polydimethylsiloxane) and 50 wt% fluorodecyl POSS (polyhedral oligomeric silsesquioxanes). [99]

Other Methods
Han's group proposed a novel method for creating ordered microstructures with 3D re-entrant nanostructures. [97] Hierarchical surfaces with 3D re-entrant CuO nanograss on Cu microcones were created using a combination of "top-down" and "bottom-up" hybrid approaches that utilized ultrafast laser ablation and chemical bath processing. First, the microcones array was precisely produced by repetitive laser pulses in a wellcontrolled manner. Subsequently, the nanograss formed mainly from the redeposition of nanoparticle clusters on the microcones during laser ablation. When immersed in a chemical bath, those nanoparticles grew all over the surface to become re-entrant nanostructures. This method appears to be a simple yet efficient method to fabricate hierarchical surfaces with large surface roughness.
Colloidal lithography can also attain re-entrant structures using templates with densely packed nanoparticles and postremoval/modification processes. [100] For instance, Chang's group formed an omniphobic triangular post array with re-entrant geometry on a substrate with silica nanoparticles. The highly packed multilayers of silica nanoparticles were embedded into a negative photoresist and underwent thermal annealing and selective RIE followed by UV exposure.
Yang et al. utilized femtosecond laser and shape memory polymer to produce a laser-induced self-growing mushroomlike structures on a flat surface. [101] The group manipulated localized femtosecond laser heating and ablation on the poly(ethyleneterephthalate) tape/heat-shrinkable polystyrene bilayer surface to induce the spontaneous and fast growth of re-entrant micropillars within 0.36 s.
Other methods such as non-solvent induced phase separation and sol-gel process, [102] self-assembly, [100,103] spray coating, [104] nanoparticle-encapsulation, [105] or anodization [106] can also be used for obtaining the re-entrant structures and prove to be easy-to-implement techniques for rapid fabrication, especially for large scale production.
To conclude this section, Table 2 summarizes and compares the advantages and disadvantages of the overviewed methods for the fabrication of re-entrant micro/nanostructures.

Applications and Technical Challenges of Fabricating Re-Entrant Microstructures
With interesting wettability, re-entrant microstructured surfaces have a wide range of essential applications in fields such as liquid manipulation, [113] self-cleaning, [114] anti-corrosion, [114] oil-water separation, [114] food industry, [115] anti-bacterial, [116] drag reduction, [117] and anti-fouling. [118] Adv. Mater. Technol. 2023, 8,2201836  Currently, the food industry raises the need for hydrophobic/ oleophobic surfaces in food bottles, containers, and bags, which can be self-cleaned thoroughly. Re-entrant surfaces with their superamphiphobicity can easily satisfy that need. Yamaguchi et al. proposed a facile microfabrication method for re-entrant texturing for such applications in liquid foods. [115] The as-prepared sample with spherical curvature exhibited excellent repellence against some liquids, such as Newtonian liquid foods. However, non-Newtonian liquid foods with high viscosities still need to be explored further since the pinning effects become more significant as the viscosity increases.
A simple method for creating inexpensive superhydrophobic coatings based on fly ash, an industrial waste product, was provided by Sow et al. [114] The fly ash particles possess an inherent spherical shape that puts them in the re-entrant geometry category. With the re-entrant structure of the fly ash particles, the coatings exhibited high robustness of hydrophobicity against several environmental impacts such as temperature, pH, and UV exposure. Then, the applications of this coating in corrosion prevention, self-cleaning, and oil-water separation have been demonstrated. The ability of flying ash coating results from the surface roughness characteristics that trap the air pockets in their gaps. Because of this, there is less contact between the contaminated particle and the surface, resulting in less adhesion force between the contaminant and the superhydrophobic surface. When a water droplet comes into contact with a polluted surface, the decreased adhesion force and high CA (>90°) ensure that the particles are lifted off the surface and adhere to the water droplet under the influence of surface tension forces. [21b] Additionally, the water droplets' low adherence to the superhydrophobic surface made it easy for them to roll off, catching and carrying any contaminants in their route. In addition, these coatings also showed high efficiency in corrosion inhibition as well as oil-water separation.
Reduction of frictional drag can significantly improve energy saving and environmental protection in some fields, such as marine ships or pipeline systems. [117] Superhydrophobic/ superoleophobic surfaces have shown great promise in drag reduction due to their ability to create air pockets underneath the textures. However, the robustness of the air pocket remains challenging for real-life applications. Recently, re-entrant structures in lubricant-infused surface (LIS) has been introduced, resulting in non-zero slip velocity at the liquid-liquid interface.
[117a] Lee et al. fabricated a biomimetic LIS with reentrant shaped cavities, reducing 18% frictional drag compared to the corresponding no-slip surface. Moreover, the sustainability of the drag reduction ability remained for a long time under external high shear stress because of re-entrant shaped surface topography. The proposed LIS would have enough potential to be used in actual applications soon, even though it is still insufficient for practical applications in extreme conditions. More study is needed to comprehend the hydrodynamics of surfaces with lubricant infusions fully.
Water harvesting is a practical application of the novel hydrophilic re-entrant SLIPS, which can solve the problem of water scarcity in remote areas around the world. [119] Guo et al. demonstrated how to obtain efficient water harvesting by integrating a flow-separation mode on slippery re-entrant channels.
During condensation, tiny droplets were rapidly removed from the top surface, which was coated with a slippery lubrication layer. Underneath the overhang structure of re-entrant microchannels, liquids were locked and transported to the end of each channel. The high energy barrier of the overhang structure was the main factor in refraining the liquids from flooding the upper surfaces. Additionally, this work could guide how to enhance condensation heat transfer by using cutting-edge reentrant SLIPS.
Boiling heat transfer is an intriguing application of reentrant structures. It relates to the mechanism of removing heat from a hot surface, which plays a crucial role in controlling the operating temperature of electronic devices. [120] Das et al. utilized open micro-channels with re-entrant cavities to achieve a high boiling heat transfer coefficient for water. [121] The inclined micro-channels with circular end pockets exhibited the highest efficiency for boiling heat transfer. The implementation of re-entrant cavities can increase artificial nucleation site density, which enhances boiling heat transfer. Another similar example is the work of Deng et al. [122] They developed porous structures with re-entrant cavities, which demonstrate considerably improved result for boiling heat transfer for ethanol and water. The reasons can be the increase in nucleation sites, enlargement of overall surface area, and improved liquid replenishment.
Other applications are the antibacterial and anti-fouling surfaces, where a superhydrophobic re-entrant surface is a potential candidate. [116] The adhesion of micro-organisms and pathogens has caused detrimental impacts on the environment, human health, and the economy. Atthi et al. fabricated a wellcontrolled hydrophobic surface with minimal surface adhesion to address this problem. [116,118] The team showed that a robust PDMS surface with an inverted trapezoidal form is efficient in antibacterial and anti-fouling due to its hydrophobicity, oleophobicity, air pocket generation, and low contact area. However, a careful design of pattern spacing, pillar height, surface charge density, and oxide formation is needed for better optimization of bacterial anti-adhesion properties.
Despite significant benefits and potential applications, several technical challenges still exist in the fabrication of reentrant micro/nanostructures. The fabrication of re-entrant microstructures, which involves lithography and etching steps, is complicated, time-consuming, and requires experienced technicians and expensive facilities. This strategy usually only serves for research purposes and is currently not applicable to large-scale production. On the other hand, some bottom-up methods relate to the random growth of materials on a substrate are more straightforward and cost-effective. However, since the growth and roughness of those nanostructured surfaces are not as well-controlled as those fabricated with lithography and etching, their superamphiphobicity might be less effective.
In summary, current fabrication techniques for re-entrant structures typically need complicated process conditions, expensive, specialized equipment, or expert personnel. Furthermore, re-entrant surfaces are still prone to harsh conditions that limit their applications in real life. [32b] To keep up, straightforward and affordable fabrication techniques should be developed based on recent theoretical findings.

Conclusions and Future Perspectives
Re-entrant structures have received increasing attention due to their unique properties. In this review, cutting-edge advances for the designing and fabrication of the re-entrant structures were systematically discussed. The fundamental theories were also briefly presented to give general audiences a basic understanding of the topic. The present review discussed different methods of evaluating the wetting behavior of liquid droplets on this re-entrant structure. We also compared the advantages and disadvantages of various methods for fabricating re-entrant micro/nanostructures. Finally, we highlighted the applications and technical challenges of re-entrant structures in different areas.
The discussed topics might facilitate the decision-making for future applications of these interesting structures. However, a broad future perspective for potential research on the wetting dynamics of this structure is still lacking. The in situ analyses of superhydrophobicity breakdown are still limited, which leads to the inevitable need for future study on state-of-the-art testing methods such as optical method, [16b,123] confocal laser scanning microscopy, [124] or Fourier-transform infrared spectroscopy. [14] Moreover, high-resolution imaging and improved image processing techniques are required to better understand the wetting dynamics of re-entrant surfaces. One possible solution is using high spatial resolution synchrotron X-ray radiography as reported by Yu et al. . [125] However, it required a highly customized light source, that is, upgraded Pohang Light Source (PLS)-II with 6D beam line. One simpler solution is using non-volatile polymer droplets to better visualize the three-phase contact lines and static contact angles of droplets on re-entrant micro/ nanostructures. For instance, Wang et al. placed the droplets of a UV-curable resin with low surface tension on top of the fabricated mushroom-like structures and used SEM to observe the wetting states, especially the meniscus at the liquid-solid interface. [20] The main concern was that the resin droplets contracted after UV exposure and during solidification. This might adversely affect the accuracy of this technique to capture the true wettability of uncured polymers. Nevertheless, the combination of photosensitive polymers and high-resolution microscopies, such as field-emission SEM with over 300 000 times magnification (300 times higher than the magnification of light microscopy) and few nanometers spatial resolution, is a very promising approach to better visualize the wetting state on such surfaces.
The combination of hierarchical micro and nanoscale roughness, re-entrant surface curvature, and the emerging concept of SLIPS open up new avenues in this field. Therefore, more fundamental investigations are required to understand the interaction of the solid-liquid interface on such complex surfaces with both organic and inorganic liquids.
Finally, the emerging field of topological liquid diodes with the combination of re-entrant micro/nanostructures could be used to precisely control the wettability, droplet mobility, and spreading of different liquids. Accordingly, it is an excellent approach for controlling droplet movement and passively pumping liquids without external sources. Therefore, the synergistic combination of this field with flexible and stretchable microfluidics for liquid handling could accelerate the commercialization of on-skin wearable biosensors.