Wax Protrusions on Anti-Adhesive Plant Surfaces and Their Interactions with Insect Adhesive Pads: A Mechanical Interpretation

Citation: Borodich, F.M.; Gao, Z.; Gorb, E.V.; Gorb, S.N.; Jin, X. Wax Protrusions on Anti-Adhesive Plant Surfaces and Their Interactions with Insect Adhesive Pads: A Mechanical Interpretation. Biomimetics 2024,9, 442. https://doi.org/10.3390/ biomimetics9070442 Academic Editor: Bo Su Received: 30 May 2024 Revised: 13 July 2024 Accepted: 16 July 2024 Published: 19 July 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). biomimetics Review Wax Protrusions on Anti-Adhesive Plant Surfaces and Their Interactions with Insect Adhesive Pads: A Mechanical Interpretation Feodor M. Borodich 1,*, Zaida Gao 1, Elena V. Gorb 2, Stanislav N. Gorb 2and Xiaoqing Jin 1,* 1College of Aerospace Engineering, Chongqing University, Chongqing 400030, China; [email protected] 2Department of Functional Morphology and Biomechanics, Zoological Institute, University of Kiel, Am Botanischen Garten 1-9, 24098 Kiel, Germany; [email protected] (E.V.G.); [email protected] (S.N.G.) *Correspondence: [email protected] (F.M.B.); [email protected] (X.J.) Abstract: Insect attachment devices enhance adhesion to complex-geometry substrates by increasing the real contact area. In nature, insects mainly interact with plant surfaces that are often covered by 3D wax structures. Here, we describe, discuss, and give a mechanical interpretation of plant waxes and the possible fracture mechanisms of these wax structures during their interactions with the adhesive pads of insects. It is argued that these plant surface microstructures significantly influence insect adhesion through reducing the contact area and contaminating the insect pads. Keywords: insect–plant interactions; surface; adhesion; bio-inspired technology 1. Introduction Exploring biomimetics in relation to insect attachment devices may help to prepare artificial adhesives with numerous attachment–detachment cycles. Indeed, in order to adapt to complex and changing environments, insects, geckos, tree frogs, and other animals have evolved complex microand nanostructures on their legs to control the adhesion function on various natural substrates. On the other hand, plants have developed some mechanisms to prevent insects from adhering to their surfaces. Many leaves and fruits are covered by crystalline wax structures that decrease insects’ abilities to adhere to these surfaces. Here, we discuss the mechanical properties of plant waxes and present a mechanical interpretation of the mechanisms of fracturing in these 3D wax projections. These models explain the microscopic mechanisms of insect attachment to plant surfaces from a multidisciplinary perspective, providing a theoretical basis for understanding the basic principles of biological attachment and transferring them to biomimetic applications. Hence, they may be used to explain the behavior of biological and artificial anti-adhesive surfaces with microand nanostructures. The animals mentioned above can modulate attachment strength via shear-sensitive adhesive pads and manage detachment by altering the angle of attachment of the limb to the substrate [ 1 – 4 ]. This rapid ability of organisms to establish and release attachment has significantly inspired research during last decades. Numerous researchers have elucidated the mechanisms behind biological climbing. When organisms respond to complex environments with different modes of locomotion, a combination of strong adhesion (attachment force perpendicular to a substrate) and strong friction (attachment force parallel to a substrate) is required [ 5 , 6 ]. Biological adhesion devices can be divided into two types: wet and dry. For example, spiders and geckos use dry adhesion, which is primarily achieved by intermolecular forces (Van der Waals) between deformable setae connected to the adhesive pad and the substrate [ 7 – 9 ]. Wet adhesion occurs when organisms secrete fluid on their feet and use capillary and viscous forces to adhere to the substrate [ 10 – 12 ]. Here, we concentrate Biomimetics 2024,9, 442. https://doi.org/10.3390/biomimetics9070442 https://www.mdpi.com/journal/biomimetics Biomimetics 2024,9, 442 2 of 15 on discussing the adhesion of insects, spiders, and geckos to surfaces contaminated by particulate materials of a certain shape, in particular by plant wax crystals. In order to understand the mechanical principle of strong adhesion from some specialized plant surfaces, this review focuses mainly on exploring the interaction between the insect attachment organs and the plant surface. Plant surfaces exhibit a diverse array of textures in the form of microand nanostructures. These can be either smooth or structured, and the latter ones can be covered by different types of hairs (trichomes) or microscopic crystals of epicuticular waxes of very diverse shapes [13–15]. Plant crystalline waxes serve various functions, as discussed in reviews by Barthlott [ 16 ] and Bargel et al. [ 17 ]. Specifically, they protect plants by inhibiting insect attachment to their surfaces. This is important because the majority of insect species interact with plants, and therefore they typically need to adhere effectively to plant surfaces [18–20]. It is known that plant waxes are made up of a number of chemical substances [ 21 – 23 ]. Understanding the physical and chemical properties of plant waxes helps to improve our understanding of the mechanisms behind the anti-insect-attachment ability of plant surfaces. On one hand, understanding the rapid and reversible attachment mechanisms in biological systems creates new opportunities for the development of biomimetic applications, such as climbing robots, grippers, green adhesives, etc. On the other hand, understanding the effective anti-adhesive mechanisms of plant surfaces might help in the development of novel green anti-adhesive coatings or switchable controllable attachment devices employing switchable changes in the surface microstructure, as described in [ 24 , 25 ]. 2. Plant Waxes, Their Properties, and Fracture Behavior 2.1. Structure of Plant Waxes Plants have developed cuticles to protect their internal tissues. These cuticles exhibit complex ultrastructures and chemical compositions in response to various environmental stresses and interactions with microorganisms, insects, and other abiotic and biotic factors. The cuticle may be impregnated with intracuticular waxes, or waxes may be transported across the cuticle and deposited on its surface as epicuticular waxes. Epicuticular waxes accumulate in forms ranging from amorphous films to microcrystalline structures [ 13 ]. Electron microscopy and X-ray diffraction analyses [ 13 , 21 , 26 , 27 ] have revealed the diverse structures of epicuticular waxes, such as massive crusts, filaments, rodlets, plates, etc. (Figure 1). The diversity of these shapes arises from molecular self-assembly on the cuticle surface [26,28–31]. Epicuticular waxes include several major classes of alicyclic and long-chain aliphatic compounds, typically with homologous chain lengths in the range of C 16 to C 35 [ 21 , 22 ]. Waxes exhibit differences in their composition, abundance, relative distribution of classes, and homolog chain lengths, which vary among plant species, plant parts, developmental stages, and environmental factors. Microscopically small crystals frequently protrude from the wax film, which overlays the plant cuticle, giving rise to the pruinose or powdery appearance observed on the surfaces of numerous plant species. The length of these wax crystals ranges from a few hundred nanometers to several micrometers. These pruinose waxy surfaces are present on the stems, leaves, flowers, seeds, and fruits of numerous plant species (Figure 2). The structure of the epicuticular wax on Nepenthes alata comprises two distinct layers and warrants a detailed examination. The upper layer is composed of separate, easily distinguishable, irregular platelets that cover the surface [ 13 ] (Figure 3A). The crystals in the upper layer are brittle and can easily exfoliate or break into tiny pieces. Both whole crystals and small fragments can adhere to insect feet (Figure 3B), contaminating the attachment organs and impeding proper contact between adhesive pads and plant surfaces, which significantly reduces the attachment force. The lower layer is composed of interconnected membranous platelets resembling a foam (see Figure 3C). The crystals in the lower layer are not easily detached and can remain intact even after the removal of the upper layer. The crystal network can withstand lateral forces from climbing insects. However, its micro- Biomimetics 2024,9, 442 3 of 15 roughness significantly decreases the contact area with the insects’ adhesive organs, as demonstrated in Figure 3D. This reduces the insect’s ability to attach. sssssBiomimetics 2024, 9, x FOR PEER REVIEW 3 of 16 Figure 1. Scanning electron microscopy (SEM) micrographs of waxy plant surfaces in a young stem of Acer negundo (a) and in adaxial (upper) leaf sides of Aloe vera (b), Aquilegia vulgaris (c), Brassica oleracea (d), Chelidonium majus (e), Chenopodium album (f), Iris germanica (g), Lactuca serriola (h), and Trifolium montanum (i). PL, wax platelets; RD, wax rodlets; TU, wax tubules. Arrows in (d) denote filament-like branches on top of the tubules. Scale bars: 2 μm (a,b,d,g,h) and 1 μm (c,e,f,i). Reproduced with permission from [32]. Epicuticular waxes include several major classes of alicyclic and long-chain aliphatic compounds, typically with homologous chain lengths in the range of C16 to C35 [21,22]. Waxes exhibit differences in their composition, abundance, relative distribution of classes, and homolog chain lengths, which vary among plant species, plant parts, developmental stages, and environmental factors. Microscopically small crystals frequently protrude from the wax film, which overlays the plant cuticle, giving rise to the pruinose or powdery appearance observed on the surfaces of numerous plant species. The length of these wax crystals ranges from a few hundred nanometers to several micrometers. These pruinose waxy surfaces are present on the stems, leaves, flowers, seeds, and fruits of numerous plant species (Figure 2). Figure 1. Scanning electron microscopy (SEM) micrographs of waxy plant surfaces in a young stem of Acer negundo (a) and in adaxial (upper) leaf sides of Aloe vera (b), Aquilegia vulgaris (c), Brassica oleracea (d), Chelidonium majus (e), Chenopodium album (f), Iris germanica (g), Lactuca serriola (h), and Trifolium montanum (i). PL, wax platelets; RD, wax rodlets; TU, wax tubules. Arrows in (d) denote filament-like branches on top of the tubules. Scale bars: 2 µ m (a,b,d,g,h) and 1 µ m (c,e,f,i). Reproduced with permission from [32]. sssssBiomimetics 2024, 9, x FOR PEER REVIEW 3 of 16 Figure 1. Scanning electron microscopy (SEM) micrographs of waxy plant surfaces in a young stem of Acer negundo (a) and in adaxial (upper) leaf sides of Aloe vera (b), Aquilegia vulgaris (c), Brassica oleracea (d), Chelidonium majus (e), Chenopodium album (f), Iris germanica (g), Lactuca serriola (h), and Trifolium montanum (i). PL, wax platelets; RD, wax rodlets; TU, wax tubules. Arrows in (d) denote filament-like branches on top of the tubules. Scale bars: 2 μm (a,b,d,g,h) and 1 μm (c,e,f,i). Reproduced with permission from [32]. Epicuticular waxes include several major classes of alicyclic and long-chain aliphatic compounds, typically with homologous chain lengths in the range of C16 to C35 [21,22]. Waxes exhibit differences in their composition, abundance, relative distribution of classes, and homolog chain lengths, which vary among plant species, plant parts, developmental stages, and environmental factors. Microscopically small crystals frequently protrude from the wax film, which overlays the plant cuticle, giving rise to the pruinose or powdery appearance observed on the surfaces of numerous plant species. The length of these wax crystals ranges from a few hundred nanometers to several micrometers. These pruinose waxy surfaces are present on the stems, leaves, flowers, seeds, and fruits of numerous plant species (Figure 2). Figure 2. Surfaces of plants covered by epicuticular wax projections: (A)Lactuca serriola; (B)Chelidonium majus; (C)Chenopodium album; (D)Brassica oleracea; (E)Acer negundo; (F)Prunus domestica. Biomimetics 2024,9, 442 4 of 15 sssssBiomimetics 2024, 9, x FOR PEER REVIEW 4 of 16 Figure 2. Surfaces of plants covered by epicuticular wax projections: (A) Lactuca serriola; (B) Chelidonium majus; (C) Chenopodium album; (D) Brassica oleracea; (E) Acer negundo; (F) Prunus domestica. The structure of the epicuticular wax on Nepenthes alata comprises two distinct layers and warrants a detailed examination. The upper layer is composed of separate, easily distinguishable, irregular platelets that cover the surface [13] (Figure 3A). The crystals in the upper layer are brittle and can easily exfoliate or break into tiny pieces. Both whole crystals and small fragments can adhere to insect feet (Figure 3B), contaminating the attachment organs and impeding proper contact between adhesive pads and plant surfaces, which significantly reduces the attachment force. The lower layer is composed of interconnected membranous platelets resembling a foam (see Figure 3C). The crystals in the lower layer are not easily detached and can remain intact even after the removal of the upper layer. The crystal network can withstand lateral forces from climbing insects. However, its micro-roughness significantly decreases the contact area with the insects’ adhesive organs, as demonstrated in Figure 3D. This reduces the insect’s ability to attach. Figure 3. Layering of wax crystals of the pitcher plant Nepenthes alata reduces insect attachment capacity: (A) SEM image of epicuticular wax of the upper layer; (B) schematic image of contamination of an insect adhesive microstructure (red) by wax crystals of the upper wax layer (yellow); (C) SEM image of epicuticular wax of the lower layer; (D) schematic image of interaction between an insect adhesive microstructure and micro-roughness of a plant. Reproduced with permission from [33]. 2.2. Mechanical Properties of Plant Waxes The following section provides a discussion of the mechanical properties of plant waxes. Unfortunately, the available experimental information about wax stress–strain curves is rather limited, even in the case of axially loaded samples. To the best of our knowledge, there is no available information about multiaxial stress states. It is understood that for moderate loads, specifically when the applied stresses are equal to a specific stress, the one-dimensional tension–compression proportional limit pr σ of crystalline materials obeys Hooke’s law, which establishes a linear relationship between stress σ and strain ε , or between tensile elongation or reduction δ and the force P applied to a tested specimen: == or PL δ σ Eε AE (1) where A represents the cross-sectional area of the specimen, E denotes a material property (Young’s modulus or elastic modulus), and L is the length of the sample. In a Figure 3. Layering of wax crystals of the pitcher plant Nepenthes alata reduces insect attachment capacity: (A) SEM image of epicuticular wax of the upper layer; (B) schematic image of contamination of an insect adhesive microstructure (red) by wax crystals of the upper wax layer (yellow); (C) SEM image of epicuticular wax of the lower layer; (D) schematic image of interaction between an insect adhesive microstructure and micro-roughness of a plant. Reproduced with permission from [33]. 2.2. Mechanical Properties of Plant Waxes The following section provides a discussion of the mechanical properties of plant waxes. Unfortunately, the available experimental information about wax stress–strain curves is rather limited, even in the case of axially loaded samples. To the best of our knowledge, there is no available information about multiaxial stress states. It is understood that for moderate loads, specifically when the applied stresses are equal to a specific stress, the one-dimensional tension–compression proportional limit σpr of crystalline materials obeys Hooke’s law, which establishes a linear relationship between stress σ and strain ε , or between tensile elongation or reduction δand the force Papplied to a tested specimen: δ=PL AE or σ=Eε(1) where A represents the cross-sectional area of the specimen, E denotes a material property (Young’s modulus or elastic modulus), and L is the length of the sample. In a onedimensional problem, the strain εis calculated as δ/L, and σis calculated as P/A. When stresses fall between the proportional limit σpr and the yield stress σpl (σpr ≤ σ<σpl) , a small region of nonlinear elastic behavior may be observed. This means that after unloading, the sample retains its original shape, although the material’s behavior deviates from the linear relation. If the tensile (or compressive) stress exceeds the yield stress σpl , the sample undergoes plastic deformation upon unloading, resulting in a shape different from its pre-loading state. Typically, σpr ∼ =σpl , suggesting they are equivalent. Brittle materials fail with minimal elongation or reduction (just a few percent) post yield stress (point B in Figure 4). Figure 4depicts the standard stress–strain curve for a brittle crystalline material. It has been observed that mechanical properties such as the Young’s modulus E , proportional limit σpr , and compressive strength σst of waxes are generally temperaturedependent. The details of these are shown in Table 1. Biomimetics 2024,9, 442 5 of 15 sssssBiomimetics 2024, 9, x FOR PEER REVIEW 5 of 16 one-dimensional problem, the strain ε is calculated as /δL , and σ is calculated as /PA . When stresses fall between the proportional limit pr σ and the yield stress < ( ) pl pr pl σ σ σ σ , a small region of nonlinear elastic behavior may be observed. This means that after unloading, the sample retains its original shape, although the material’s behavior deviates from the linear relation. If the tensile (or compressive) stress exceeds the yield stress pl σ , the sample undergoes plastic deformation upon unloading, resulting in a shape different from its pre-loading state. Typically, pr pl σ σ , suggesting they are equivalent. Brittle materials fail with minimal elongation or reduction (just a few percent) post yield stress (point B in Figure 4). Figure 4 depicts the standard stress–strain curve for a brittle crystalline material. Figure 4. Schematic typical stress–strain diagram. Point A represents the proportional limit pr σ , point B indicates the yield point pl σ , and point C signifies the fracture stress st σ (the material’s strength). It has been observed that mechanical properties such as the Young’s modulus E , proportional limit pr σ , and compressive strength st σ of waxes are generally temperature-dependent. The details of these are shown in Table 1. Table 1. Mechanical properties of carnauba wax at two different temperatures [34]. Temperature E (×106 Pa) Pa 6 ( 10 ) pr σ  Pa 6 ( 10 ) st σ  23 C 1806.5 772.2 10.94 18.775 37 C 5.72 9.03 For example, carnauba wax, which is extracted from the leaves of the carnauba wax palm (Copernicia prunifera), exhibits this temperature sensitivity. It has a Poisson’s ratio 0.49ν= and a density 3 951 kg / mρ= . Craig et al. [34] noted that the strain before fracture under compression for all tested waxes ranged from 2.7% to 4.3%. The samples exhibited brittle fracture behavior. This observation of the brittle fracture characteristics of crystalline waxes aligns with the findings from experiments on plant waxes reported in [33]. Shellhammer et al. [35] found that natural candelilla and carnauba waxes behaved similarly to hard and elastic materials at 2% compressive strain, with candelilla wax exhibiting greater viscosity than carnauba wax. However, both waxes demonstrated Figure 4. Schematic typical stress–strain diagram. Point A represents the proportional limit σpr , point B indicates the yield point σpl, and point C signifies the fracture stress σst (the material’s strength). Table 1. Mechanical properties of carnauba wax at two different temperatures [34]. Temperature E (×106Pa) σpr (×106Pa)σst (×106Pa) 23 ◦C 1806.5 10.94 18.775 37 ◦C 772.2 5.72 9.03 For example, carnauba wax, which is extracted from the leaves of the carnauba wax palm (Copernicia prunifera), exhibits this temperature sensitivity. It has a Poisson’s ratio ν= 0.49 and a density ρ= 951 kg/m3 . Craig et al. [ 34 ] noted that the strain before fracture under compression for all tested waxes ranged from 2.7% to 4.3%. The samples exhibited brittle fracture behavior. This observation of the brittle fracture characteristics of crystalline waxes aligns with the findings from experiments on plant waxes reported in [ 33 ]. Shellhammer et al. [ 35 ] found that natural candelilla and carnauba waxes behaved similarly to hard and elastic materials at 2% compressive strain, with candelilla wax exhibiting greater viscosity than carnauba wax. However, both waxes demonstrated behavior more akin to that of elastic materials compared to that of beeswax, which exhibited significantly more viscosity and less elastic behavior. The widespread claim that carnauba wax is the hardest known wax is supported by evidence that it has the highest mechanical properties at the relevant temperature of all of the waxes tested by Craig et al. [ 34 ]. However, our previous experiments [ 33 ] indicated that other plant waxes may exhibit higher hardness and elastic modulus values compared to carnauba wax. Specifically, depth-sensing nanoindentation on the wax of the carnivorous plant Nepenthes alata revealed an elastic modulus of approximately E= 2.5 GPa and a hardness of H=0.1 GPa at T=22 ◦C. The elasticity of real materials significantly affects the elastic–plastic indentation process. Initially, when the yield point is exceeded, the plastic zone is small and completely surrounded by material that remains elastic. This results in plastic strains that are comparable to the surrounding elastic strains. In such cases, the material displaced by the indenter is absorbed by the elastic expansion of the surrounding solid. With increasing indentation depth, the plastic zone eventually extends to the free surface, allowing displaced material to flow plastically to the sides of the indenter. Johnson [ 36 ] suggested the following relation between the mean pressure pm acting normal to the original surface as pm=cσpl , where the constant c, which depends on the geometry of the indenter and the interfacial friction, typically has a value of about 3. Initially, when yielding first occurs, the constant c is approximately one. In the mechanics of elastic–plastic contact, it is generally accepted that hardness is the average stress under the indenter at which the entire material yields. Using Johnson’s calculation method, one can approximate H≈cσpl , 2.8 ≤c≤ 3. Hence, we Biomimetics 2024,9, 442 6 of 15 can estimate for the wax of the carnivorous plant Nepenthes alata, having H= 0.1 GPa at T= 22 ◦C, that σpl is approximately 35 MPa. We can assume that the waxes have a bilinear diagram (Figure 5), whose reference points A, B, and C vary with temperature. Figure 5depicts an idealized stress–strain diagram in two cases: (a) a bilinear diagram with strain hardening, and (b) linear elastic– ideal plastic diagram. When stresses have absolute values less than the yield stress σpl , the material obeys Hooke’s law (1) with E=tan α . If the absolute value of stress exceeds the yield stress, strain hardening is approximated by a line with another slope: E1=tan β . If β= 0, then σ=σpl . This indicates that the wax yields without any further increase in the external load. With Hooke’s law, calculations can proceed as follows: εpl =σpl E=35 ·106 25 ·109=0.014 (2) sssssBiomimetics 2024, 9, x FOR PEER REVIEW 6 of 16 behavior more akin to that of elastic materials compared to that of beeswax, which exhibited significantly more viscosity and less elastic behavior. The widespread claim that carnauba wax is the hardest known wax is supported by evidence that it has the highest mechanical properties at the relevant temperature of all of the waxes tested by Craig et al. [34]. However, our previous experiments [33] indicated that other plant waxes may exhibit higher hardness and elastic modulus values compared to carnauba wax. Specifically, depth-sensing nanoindentation on the wax of the carnivorous plant Nepenthes alata revealed an elastic modulus of approximately 2.5 GPaE= and a hardness of 0.1 GPaH= at =22 CT . The elasticity of real materials significantly affects the elastic–plastic indentation process. Initially, when the yield point is exceeded, the plastic zone is small and completely surrounded by material that remains elastic. This results in plastic strains that are comparable to the surrounding elastic strains. In such cases, the material displaced by the indenter is absorbed by the elastic expansion of the surrounding solid. With increasing indentation depth, the plastic zone eventually extends to the free surface, allowing displaced material to flow plastically to the sides of the indenter. Johnson [36] suggested the following relation between the mean pressure m p acting normal to the original surface as m pl pcσ= , where the constant c, which depends on the geometry of the indenter and the interfacial friction, typically has a value of about 3. Initially, when yielding first occurs, the constant c is approximately one. In the mechanics of elastic–plastic contact, it is generally accepted that hardness is the average stress under the indenter at which the entire material yields. Using Johnson’s calculation method, one can approximate , 2.8 3 pl Hcσc   . Hence, we can estimate for the wax of the carnivorous plant Nepenthes alata, having H = 0.1 GPa at T = 22 °C, that pl σ is approximately 35 MPa. We can assume that the waxes have a bilinear diagram (Figure 5), whose reference points A, B, and C vary with temperature. Figure 5 depicts an idealized stress–strain diagram in two cases: (a) a bilinear diagram with strain hardening, and (b) linear elastic– ideal plastic diagram. When stresses have absolute values less than the yield stress pl σ , the material obeys Hooke’s law (1) with tanEα= . If the absolute value of stress exceeds the yield stress, strain hardening is approximated by a line with another slope: 1tanEβ= . If 0β= , then pl σ σ= . This indicates that the wax yields without any further increase in the external load. With Hooke’s law, calculations can proceed as follows:  = = =  6 9 35 10 0.014 25 10 pl pl σ εE (2) (a) (b) Figure 5. (a) Bilinear stress–strain diagram and (b) linear elastic–ideal plastic diagram. It is noted that tan α=E and tan β=E1 , where E and E1 are Young’s modulus and the plastic hardening modulus, respectively. Based on the experiments conducted by Craig et al. [ 34 ], it can be assumed that the strain at fracture is 2.7%, i.e., εst = 0.027. To plot the bilinear diagram, we need either the slope of the linear hardening part (the tangent modulus E1 ) or the stress at fracture σst . From this bilinear diagram, we derive the relation εst −εpl =σst −σpl/E1. We may assume that the ratio σst/σpl of the Nepenthes alata wax is the same as that of carnauba wax, i.e., 1.575. Therefore, we obtain εst −εpl = 0.013 = 0.575 σpl/E1 . Thus, E1 can be calculated as E1=20.125 ·106/0.013 =1.548 ·109Pa. 2.3. Deformation and Fracture of Plant Wax Structures 2.3.1. Euler Buckling of Wax Projections In the mechanics of materials, long and slender structures, as plant wax projections, subjected to axial compression are referred to as columns. First, let us analyze the stability of tubular columns, which resemble tubular wax crystals, under compressive loading. When the applied compressive load P increases, a slender elastic column will buckle at the critical load Pcr, determinable using the Euler formula: Pcr =π2EI (KL)2=π2EI Le f 2(3) Biomimetics 2024,9, 442 7 of 15 where K represents the effective length factor, which varies depending on the boundary conditions at the end of the column, and where effective length is defined as Le f =KL . When a column has pinned ends, K is 1. When the base is built-in (fixed) and the top end is free (Figure 6A), K is 2. When the base is fixed and the top is pinned (Figure 6C), K is 0.7. When both ends are built-in (Figure 6B), K is 0.5. It is plausible to assume that one end of the wax projection is attached to the plant surface, indicating a built-in state, while the other end is involved in adhesive interactions with the insect’s attachment organ (adhesive pad). Insects usually rely on capillary adhesion resulting from the secretion of fluid (pad secretion) into the contact zone between the pad and the substrate [ 10 ]. These conditions suggest that the column may be modeled as a rod with an elastically restrained (clumped) end, which may be represented as an elastic torsional spring at this end (Figure 6D). The torsion spring model represents the adhesive interaction between an insect’s seta and a wax column. Consequently, Le f = 0.6 L can be considered a reasonable approximation for the effective length of the column. sssssBiomimetics 2024, 9, x FOR PEER REVIEW 7 of 16 Figure 5. (a) Bilinear stress–strain diagram and (b) linear elastic–ideal plastic diagram. It is noted that tanαE= and 1 tan βE= , where E and 1 E are Young’s modulus and the plastic hardening modulus, respectively. Based on the experiments conducted by Craig et al. [34], it can be assumed that the strain at fracture is 2.7%, i.e., 0.027 st ε= . To plot the bilinear diagram, we need either the slope of the linear hardening part (the tangent modulus 1 E ) or the stress at fracture st σ . From this bilinear diagram, we derive the relation ( ) 1 / st pl st pl ε ε σ σ E− = − . We may assume that the ratio / st pl σ σ of the Nepenthes alata wax is the same as that of carnauba wax, i.e., 1.575. Therefore, we obtain 1 0.013 0.575 / st pl pl ε ε σ E− = = . Thus, 1 E can be calculated as 69 120.125 10 / 0.013 1.548 10 PaE=  =  . 2.3. Deformation and Fracture of Plant Wax Structures 2.3.1. Euler Buckling of Wax Projections In the mechanics of materials, long and slender structures, as plant wax projections, subjected to axial compression are referred to as columns. First, let us analyze the stability of tubular columns, which resemble tubular wax crystals, under compressive loading. When the applied compressive load P increases, a slender elastic column will buckle at the critical load cr P , determinable using the Euler formula: ( ) ( ) == 22 22 cr ef πEI πEI P KL L (3) where K represents the effective length factor, which varies depending on the boundary conditions at the end of the column, and where effective length is defined as ef L KL= . When a column has pinned ends, K is 1. When the base is built-in (fixed) and the top end is free (Figure 6A), K is 2. When the base is fixed and the top is pinned (Figure 6C), K is 0.7. When both ends are built-in (Figure 6B), K is 0.5. It is plausible to assume that one end of the wax projection is attached to the plant surface, indicating a built-in state, while the other end is involved in adhesive interactions with the insect’s attachment organ (adhesive pad). Insects usually rely on capillary adhesion resulting from the secretion of fluid (pad secretion) into the contact zone between the pad and the substrate [10]. These conditions suggest that the column may be modeled as a rod with an elastically restrained (clumped) end, which may be represented as an elastic torsional spring at this end (Figure 6D). The torsion spring model represents the adhesive interaction between an insect’s seta and a wax column. Consequently, 0.6 ef LL= can be considered a reasonable approximation for the effective length of the column. Figure 6. Some boundary conditions for compressed columns: (A) a column with built-in and free ends; (B) a column with both ends built-in; (C) a column with built-in and pinned ends; (D) a column with built-in and elastically clumped ends. K is the effective length factor for different boundary conditions at the end of the column. 2.3.2. Elastoplastic Buckling of Wax Columns The Euler formula in Formula (4) assumes a linearly elastic stress–strain relationship and becomes invalid when compressive stresses in the column exceed the yielding stress. Consequently, inelastic buckling must be considered. Engesser and Jasinski developed this theory, as cited in [ 37 ]. According to this theory, the critical load Pcr at which a slender elastic–plastic column buckles can be calculated as follows: Pcr =π2ErI (KL)2(4) where Er represents the column’s reduced modulus. For a rectangular cross-section, we can calculate this modulus using Formula (5). This serves as an approximation for columns with circular cross-sections. Er=4EE1 √E+√E12(5) Given the moduli of elasticity, E= 2.5 GPa and E1= 1.548 GPa , for the plant wax, the reduced modulus of the column can be calculated as follows: Er=4·2.5 ·1.548 √2.5 +√1.5482=1.94 GPa (6) Biomimetics 2024,9, 442 8 of 15 These formulas are applicable to slender columns. To characterize the behavior of a column, it is essential to introduce the slenderness ratio λ=Le f /r , where r=√I/A , the radius of gyration of the column’s cross-section in the bending plane. Here, A denotes the cross-section’s area, and I represents the moment of inertia for the cross-sectional area. For a circular cross-section with diameter D , A=πD2/ 4 and I=πD4/ 64, the radius of gyration is calculated below: r=rI A=r4D4 64D2=D 4(7) It is important to understand the difference between the aspect ratio and the slenderness of a column. The aspect ratio is a purely geometric characteristic, defined as the ratio between the largest and smallest dimensions of the column, whereas the slenderness depends on both the geometry, represented by the length L and the radius of gyration r of the column’s cross-section, and the loading conditions, indicated by the effective length factor K . For instance, although all cases A–D in Figure 6share the same aspect ratio L/D , their slenderness values differ significantly. The Euler formula in Formula (3) is applicable solely to purely linearly elastic materials; therefore, the transition from elastic to the elastic–plastic case, i.e., from Formula (4) to Formula (3), should be at the critical force such that it causes the critical stress to be equal to the yield stress, as follows: σcr =Pcr A=π2EI AL2 e f =σpl (8) Consequently, the critical slenderness ratio can be calculated as follows: λc=Le f rc =sπ2E σpl (9) Hence, the relationship between the average compressive stress σ and slenderness ratio λfor plant waxes may be represented by the graph shown in Figure 7. sssssBiomimetics 2024, 9, x FOR PEER REVIEW 9 of 16  ==    2 ef c pl c LπE λrσ (9) Hence, the relationship between the average compressive stress σ and slenderness ratio λ for plant waxes may be represented by the graph shown in Figure 7. Figure 7. Relationship between plant wax compressive stress and slenderness ratio. In region BC, the critical stress at the yield limit pl σ is given by 2 2 cr πE σλ = . In region AB, the critical stress at the strength limit st σ is given by 2 2 r cr πE σλ = . When cpl λ λ , the material will not buckle, as it has already exceeded the strength limit, and will instead be crushed. Substituting the values for plant wax, we obtain the following critical slenderness ratio:  ==  9 6 9.87 2.5 10 26.5 35 10 c λ (10) Hence, the slenderness ratio for columns with 0.6K= and gyration radius /4rD= is as follows: == 0.6 2.4LL λrD (11) Gorb et al. [38] report that certain plants, such as Aquilegia vulgaris (European columbine), Berberis vulgaris (common barberry), Chelidonium majus (white goosefoot), and Prunus domestica (European plum), exhibit tubular epicuticular wax crystals. Aristolochia fimbriata (white-veined Dutchman’s pipe) crystals, as well as those of many other Aristolochia species, also exhibit this tubular shape, but at a very high slenderness ratio, as shown in Figure 8. Table 2 presents the average lengths and diameters of these crystal columns, along with the calculated slenderness ratios ( / ef λLr= ) for all five plants mentioned above. Figure 7. Relationship between plant wax compressive stress and slenderness ratio. In region BC, the critical stress at the yield limit σpl is given by σcr =π2E λ2 . In region AB, the critical stress at the strength limit σst is given by σcr =π2Er λ2 . When λ≤λcpl , the material will not buckle, as it has already exceeded the strength limit, and will instead be crushed. Biomimetics 2024,9, 442 9 of 15 Substituting the values for plant wax, we obtain the following critical slenderness ratio: λc=r9.87 ·2.5 ·109 35 ·106=26.5 (10) Hence, the slenderness ratio for columns with K= 0.6 and gyration radius r=D/ 4 is as follows: λ=0.6L r=2.4L D(11) Gorb et al. [ 38 ] report that certain plants, such as Aquilegia vulgaris (European columbine), Berberis vulgaris (common barberry), Chelidonium majus (white goosefoot), and Prunus domestica (European plum), exhibit tubular epicuticular wax crystals. Aristolochia fimbriata (white-veined Dutchman’s pipe) crystals, as well as those of many other Aristolochia species, also exhibit this tubular shape, but at a very high slenderness ratio, as shown in Figure 8. Table 2presents the average lengths and diameters of these crystal columns, along with the calculated slenderness ratios (λ=Le f /r) for all five plants mentioned above. sssssBiomimetics 2024, 9, x FOR PEER REVIEW 10 of 16 Figure 8. (A) White-veined Dutchman’s pipe (Aristolochia fimbriata) flower; (B) scanning electron microscopy micrographs of trichomes on the inner surface of the flower trap; (C) wax crystals covering the trichome surface that may buckle. Table 2. Geometrical parameters of the wax columns in selected plant species: L—length; D— diameter; λ—slenderness ratio. Plant Species Aquilegia vulgaris Berberis vulgaris Chelidonium majus Prunus domestica Aristolochia fimbriata ( ) nmL 580 730 830 580 6310 ( ) nmD 170 160 180 260 92 / ef λLr= 8.18 10.94 11.06 5.35 164.6 Clearly, among the tubular-shaped plant wax crystals studied [39,40], only Aristolochia fimbriata can elastically buckle, as the slenderness ratios for the other plants fall below the critical value of 26.5. Therefore, the crystals from these four plants do not meet the length criteria for the Euler formula approximation. Formula (3) applies solely under stresses below the material’s ultimate compressive stress st σ . Consequently, the critical slenderness ratio cpl λ for discontinuing the use of Formula (3) due to elastic–plastic buckling is identified as follows:  ==    2 ef r cpl st cpl LπE λrσ (12) The plant critical slenderness ratio cpl λ can be obtained by substituting the specified plant wax values into Formula (12), as follows:  ==  9 6 9.87 1.94 10 18.66 55 10 cpl λ (13) Plotting a diagram of the average stress against the slenderness ratio (Figure 7) reveals that, aside from Aristolochia fimbriata, which buckles in the elastic regime, crystals from all other studied plants do not buckle in either the elastic or elastic–plastic regimes given the calculated slenderness ratios. Indeed, these short columns are not susceptible to failure through a pure buckling mechanism. Failure occurs only when compressive stress reaches the wax’s strength limit. Therefore, it is necessary to account for the bending of wax crystals under simultaneous axial and orthogonal loading. There are two restrictions in the application of the above Euler formula: One is that the default compression column is perfectly straight before the load is applied, but in reality, the presence of tubular columnar plant wax crystals does not guarantee perfect Figure 8. (A) White-veined Dutchman’s pipe (Aristolochia fimbriata) flower; (B) scanning electron microscopy micrographs of trichomes on the inner surface of the flower trap; (C) wax crystals covering the trichome surface that may buckle. Table 2. Geometrical parameters of the wax columns in selected plant species: L—length; D—diameter; λ—slenderness ratio. Plant Species Aquilegia vulgaris Berberis vulgaris Chelidonium majus Prunus domestica Aristolochia fimbriata L(nm)580 730 830 580 6310 D(nm)170 160 180 260 92 λ=Le f /r8.18 10.94 11.06 5.35 164.6 Clearly, among the tubular-shaped plant wax crystals studied [ 39 , 40 ], only Aristolochia fimbriata can elastically buckle, as the slenderness ratios for the other plants fall below the critical value of 26.5. Therefore, the crystals from these four plants do not meet the length criteria for the Euler formula approximation. Formula (3) applies solely under stresses below the material’s ultimate compressive stress σst . Consequently, the critical slenderness ratio λcpl for discontinuing the use of Formula (3) due to elastic–plastic buckling is identified as follows: λcpl =Le f rcpl =sπ2Er σst (12)