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

Ci a ion: Bo odich, F.M.; Gao, Z.; Go b, E.V.; Go b, S.N.; Jin, X. Wax P o usions on An i-Adhesi e Plan Su aces and Thei In e ac ions wi h Insec Adhesi e Pads: A Mechanical In e p e a ion. Biomime ics 2024,9, 442. h ps://doi.o g/10.3390/ biomime ics9070442 Academic Edi o : Bo Su Recei ed: 30 May 2024 Re ised: 13 July 2024 Accep ed: 16 July 2024 Published: 19 July 2024 Copy igh : © 2024 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). biomime ics Re iew Wax P o usions on An i-Adhesi e Plan Su aces and Thei In e ac ions wi h Insec Adhesi e Pads: A Mechanical In e p e a ion Feodo M. Bo odich 1,*, Zaida Gao 1, Elena V. Go b 2, S anisla N. Go b 2and Xiaoqing Jin 1,* 1College o Ae ospace Enginee ing, Chongqing Uni e si y, Chongqing 400030, China; [email p o ec ed] 2Depa men o Func ional Mo phology and Biomechanics, Zoological Ins i u e, Uni e si y o Kiel, Am Bo anischen Ga en 1-9, 24098 Kiel, Ge many; [email p o ec ed] (E.V.G.); [email p o ec ed] (S.N.G.) *Co espondence: [email p o ec ed] (F.M.B.); [email p o ec ed] (X.J.) Abs ac : Insec a achmen de ices enhance adhesion o complex-geome y subs a es by inc easing he eal con ac a ea. In na u e, insec s mainly in e ac wi h plan su aces ha a e o en co e ed by 3D wax s uc u es. He e, we desc ibe, discuss, and gi e a mechanical in e p e a ion o plan waxes and he possible ac u e mechanisms o hese wax s uc u es du ing hei in e ac ions wi h he adhesi e pads o insec s. I is a gued ha hese plan su ace mic os uc u es signi ican ly in luence insec adhesion h ough educing he con ac a ea and con amina ing he insec pads. Keywo ds: insec –plan in e ac ions; su ace; adhesion; bio-inspi ed echnology 1. In oduc ion Explo ing biomime ics in ela ion o insec a achmen de ices may help o p epa e a i icial adhesi es wi h nume ous a achmen –de achmen cycles. Indeed, in o de o adap o complex and changing en i onmen s, insec s, geckos, ee ogs, and o he animals ha e e ol ed complex mic o- and nanos uc u es on hei legs o con ol he adhesion unc ion on a ious na u al subs a es. On he o he hand, plan s ha e de eloped some mechanisms o p e en insec s om adhe ing o hei su aces. Many lea es and ui s a e co e ed by c ys alline wax s uc u es ha dec ease insec s’ abili ies o adhe e o hese su - aces. He e, we discuss he mechanical p ope ies o plan waxes and p esen a mechanical in e p e a ion o he mechanisms o ac u ing in hese 3D wax p ojec ions. These models explain he mic oscopic mechanisms o insec a achmen o plan su aces om a mul idis- ciplina y pe spec i e, p o iding a heo e ical basis o unde s anding he basic p inciples o biological a achmen and ans e ing hem o biomime ic applica ions. Hence, hey may be used o explain he beha io o biological and a i icial an i-adhesi e su aces wi h mic o- and nanos uc u es. The animals men ioned abo e can modula e a achmen s eng h ia shea -sensi i e adhesi e pads and manage de achmen by al e ing he angle o a achmen o he limb o he subs a e [ 1 – 4 ]. This apid abili y o o ganisms o es ablish and elease a achmen has signi ican ly inspi ed esea ch du ing las decades. Nume ous esea che s ha e elucida ed he mechanisms behind biological climbing. When o ganisms espond o complex en i on- men s wi h di e en modes o locomo ion, a combina ion o s ong adhesion (a achmen o ce pe pendicula o a subs a e) and s ong ic ion (a achmen o ce pa allel o a sub- s a e) is equi ed [ 5 , 6 ]. Biological adhesion de ices can be di ided in o wo ypes: we and d y. Fo example, spide s and geckos use d y adhesion, which is p ima ily achie ed by in e molecula o ces (Van de Waals) be ween de o mable se ae connec ed o he adhesi e pad and he subs a e [ 7 – 9 ]. We adhesion occu s when o ganisms sec e e luid on hei ee and use capilla y and iscous o ces o adhe e o he subs a e [ 10 – 12 ]. He e, we concen a e Biomime ics 2024,9, 442. h ps://doi.o g/10.3390/biomime ics9070442 h ps://www.mdpi.com/jou nal/biomime ics Biomime ics 2024,9, 442 2 o 15 on discussing he adhesion o insec s, spide s, and geckos o su aces con amina ed by pa icula e ma e ials o a ce ain shape, in pa icula by plan wax c ys als. In o de o unde s and he mechanical p inciple o s ong adhesion om some special- ized plan su aces, his e iew ocuses mainly on explo ing he in e ac ion be ween he insec a achmen o gans and he plan su ace. Plan su aces exhibi a di e se a ay o ex u es in he o m o mic o- and nanos uc u es. These can be ei he smoo h o s uc u ed, and he la e ones can be co e ed by di e en ypes o hai s ( ichomes) o mic oscopic c ys als o epicu icula waxes o e y di e se shapes [13–15]. Plan c ys alline waxes se e a ious unc ions, as discussed in e iews by Ba hlo [ 16 ] and Ba gel e al. [ 17 ]. Speci ically, hey p o ec plan s by inhibi ing insec a achmen o hei su aces. This is impo an because he majo i y o insec species in e ac wi h plan s, and he e o e hey ypically need o adhe e e ec i ely o plan su aces [18–20]. I is known ha plan waxes a e made up o a numbe o chemical subs ances [ 21 – 23 ]. Unde s anding he physical and chemical p ope ies o plan waxes helps o imp o e ou unde s anding o he mechanisms behind he an i-insec -a achmen abili y o plan su aces. On one hand, unde s anding he apid and e e sible a achmen mechanisms in biological sys ems c ea es new oppo uni ies o he de elopmen o biomime ic ap- plica ions, such as climbing obo s, g ippe s, g een adhesi es, e c. On he o he hand, unde s anding he e ec i e an i-adhesi e mechanisms o plan su aces migh help in he de elopmen o no el g een an i-adhesi e coa ings o swi chable con ollable a achmen de ices employing swi chable changes in he su ace mic os uc u e, as desc ibed in [ 24 , 25 ]. 2. Plan Waxes, Thei P ope ies, and F ac u e Beha io 2.1. S uc u e o Plan Waxes Plan s ha e de eloped cu icles o p o ec hei in e nal issues. These cu icles exhibi complex ul as uc u es and chemical composi ions in esponse o a ious en i onmen al s esses and in e ac ions wi h mic oo ganisms, insec s, and o he abio ic and bio ic ac o s. The cu icle may be imp egna ed wi h in acu icula waxes, o waxes may be anspo ed ac oss he cu icle and deposi ed on i s su ace as epicu icula waxes. Epicu icula waxes accumula e in o ms anging om amo phous ilms o mic oc ys alline s uc u es [ 13 ]. Elec on mic oscopy and X- ay di ac ion analyses [ 13 , 21 , 26 , 27 ] ha e e ealed he di e se s uc u es o epicu icula waxes, such as massi e c us s, ilamen s, odle s, pla es, e c. (Figu e 1). The di e si y o hese shapes a ises om molecula sel -assembly on he cu icle su ace [26,28–31]. Epicu icula waxes include se e al majo classes o alicyclic and long-chain alipha ic compounds, ypically wi h homologous chain leng hs in he ange o C 16 o C 35 [ 21 , 22 ]. Waxes exhibi di e ences in hei composi ion, abundance, ela i e dis ibu ion o classes, and homolog chain leng hs, which a y among plan species, plan pa s, de elopmen al s ages, and en i onmen al ac o s. Mic oscopically small c ys als equen ly p o ude om he wax ilm, which o e lays he plan cu icle, gi ing ise o he p uinose o powde y appea ance obse ed on he su aces o nume ous plan species. The leng h o hese wax c ys als anges om a ew hund ed nanome e s o se e al mic ome e s. These p uinose waxy su aces a e p esen on he s ems, lea es, lowe s, seeds, and ui s o nume ous plan species (Figu e 2). The s uc u e o he epicu icula wax on Nepen hes ala a comp ises wo dis inc laye s and wa an s a de ailed examina ion. The uppe laye is composed o sepa a e, easily dis inguishable, i egula pla ele s ha co e he su ace [ 13 ] (Figu e 3A). The c ys als in he uppe laye a e b i le and can easily ex olia e o b eak in o iny pieces. Bo h whole c ys als and small agmen s can adhe e o insec ee (Figu e 3B), con amina ing he a achmen o gans and impeding p ope con ac be ween adhesi e pads and plan su aces, which signi ican ly educes he a achmen o ce. The lowe laye is composed o in e connec ed memb anous pla ele s esembling a oam (see Figu e 3C). The c ys als in he lowe laye a e no easily de ached and can emain in ac e en a e he emo al o he uppe laye . The c ys al ne wo k can wi hs and la e al o ces om climbing insec s. Howe e , i s mic o- Biomime ics 2024,9, 442 3 o 15 oughness signi ican ly dec eases he con ac a ea wi h he insec s’ adhesi e o gans, as demons a ed in Figu e 3D. This educes he insec ’s abili y o a ach. sssssBiomime ics 2024, 9, x FOR PEER REVIEW 3 o 16 Figu e 1. Scanning elec on mic oscopy (SEM) mic og aphs o waxy plan su aces in a young s em o Ace negundo (a) and in adaxial (uppe ) lea sides o Aloe e a (b), Aquilegia ulga is (c), B assica ole acea (d), Chelidonium majus (e), Chenopodium album ( ), I is ge manica (g), Lac uca se iola (h), and T i olium mon anum (i). PL, wax pla ele s; RD, wax odle s; TU, wax ubules. A ows in (d) deno e ilamen -like b anches on op o he ubules. Scale ba s: 2 μm (a,b,d,g,h) and 1 μm (c,e, ,i). Rep oduced wi h pe mission om [32]. Epicu icula waxes include se e al majo classes o alicyclic and long-chain alipha ic compounds, ypically wi h homologous chain leng hs in he ange o C16 o C35 [21,22]. Waxes exhibi di e ences in hei composi ion, abundance, ela i e dis ibu ion o classes, and homolog chain leng hs, which a y among plan species, plan pa s, de elopmen al s ages, and en i onmen al ac o s. Mic oscopically small c ys als equen ly p o ude om he wax ilm, which o e lays he plan cu icle, gi ing ise o he p uinose o powde y appea ance obse ed on he su aces o nume ous plan species. The leng h o hese wax c ys als anges om a ew hund ed nanome e s o se e al mic ome e s. These p uinose waxy su aces a e p esen on he s ems, lea es, lowe s, seeds, and ui s o nume ous plan species (Figu e 2). Figu e 1. Scanning elec on mic oscopy (SEM) mic og aphs o waxy plan su aces in a young s em o Ace negundo (a) and in adaxial (uppe ) lea sides o Aloe e a (b), Aquilegia ulga is (c), B assica ole acea (d), Chelidonium majus (e), Chenopodium album ( ), I is ge manica (g), Lac uca se iola (h), and T i olium mon anum (i). PL, wax pla ele s; RD, wax odle s; TU, wax ubules. A ows in (d) deno e ilamen -like b anches on op o he ubules. Scale ba s: 2 µ m (a,b,d,g,h) and 1 µ m (c,e, ,i). Rep oduced wi h pe mission om [32]. sssssBiomime ics 2024, 9, x FOR PEER REVIEW 3 o 16 Figu e 1. Scanning elec on mic oscopy (SEM) mic og aphs o waxy plan su aces in a young s em o Ace negundo (a) and in adaxial (uppe ) lea sides o Aloe e a (b), Aquilegia ulga is (c), B assica ole acea (d), Chelidonium majus (e), Chenopodium album ( ), I is ge manica (g), Lac uca se iola (h), and T i olium mon anum (i). PL, wax pla ele s; RD, wax odle s; TU, wax ubules. A ows in (d) deno e ilamen -like b anches on op o he ubules. Scale ba s: 2 μm (a,b,d,g,h) and 1 μm (c,e, ,i). Rep oduced wi h pe mission om [32]. Epicu icula waxes include se e al majo classes o alicyclic and long-chain alipha ic compounds, ypically wi h homologous chain leng hs in he ange o C16 o C35 [21,22]. Waxes exhibi di e ences in hei composi ion, abundance, ela i e dis ibu ion o classes, and homolog chain leng hs, which a y among plan species, plan pa s, de elopmen al s ages, and en i onmen al ac o s. Mic oscopically small c ys als equen ly p o ude om he wax ilm, which o e lays he plan cu icle, gi ing ise o he p uinose o powde y appea ance obse ed on he su aces o nume ous plan species. The leng h o hese wax c ys als anges om a ew hund ed nanome e s o se e al mic ome e s. These p uinose waxy su aces a e p esen on he s ems, lea es, lowe s, seeds, and ui s o nume ous plan species (Figu e 2). Figu e 2. Su aces o plan s co e ed by epicu icula wax p ojec ions: (A)Lac uca se iola; (B)Chelido- nium majus; (C)Chenopodium album; (D)B assica ole acea; (E)Ace negundo; (F)P unus domes ica. Biomime ics 2024,9, 442 4 o 15 sssssBiomime ics 2024, 9, x FOR PEER REVIEW 4 o 16 Figu e 2. Su aces o plan s co e ed by epicu icula wax p ojec ions: (A) Lac uca se iola; (B) Chelidonium majus; (C) Chenopodium album; (D) B assica ole acea; (E) Ace negundo; (F) P unus domes ica. The s uc u e o he epicu icula wax on Nepen hes ala a comp ises wo dis inc laye s and wa an s a de ailed examina ion. The uppe laye is composed o sepa a e, easily dis inguishable, i egula pla ele s ha co e he su ace [13] (Figu e 3A). The c ys als in he uppe laye a e b i le and can easily ex olia e o b eak in o iny pieces. Bo h whole c ys als and small agmen s can adhe e o insec ee (Figu e 3B), con amina ing he a achmen o gans and impeding p ope con ac be ween adhesi e pads and plan su aces, which signi ican ly educes he a achmen o ce. The lowe laye is composed o in e connec ed memb anous pla ele s esembling a oam (see Figu e 3C). The c ys als in he lowe laye a e no easily de ached and can emain in ac e en a e he emo al o he uppe laye . The c ys al ne wo k can wi hs and la e al o ces om climbing insec s. Howe e , i s mic o- oughness signi ican ly dec eases he con ac a ea wi h he insec s’ adhesi e o gans, as demons a ed in Figu e 3D. This educes he insec ’s abili y o a ach. Figu e 3. Laye ing o wax c ys als o he pi che plan Nepen hes ala a educes insec a achmen capaci y: (A) SEM image o epicu icula wax o he uppe laye ; (B) schema ic image o con amina ion o an insec adhesi e mic os uc u e ( ed) by wax c ys als o he uppe wax laye (yellow); (C) SEM image o epicu icula wax o he lowe laye ; (D) schema ic image o in e ac ion be ween an insec adhesi e mic os uc u e and mic o- oughness o a plan . Rep oduced wi h pe mission om [33]. 2.2. Mechanical P ope ies o Plan Waxes The ollowing sec ion p o ides a discussion o he mechanical p ope ies o plan waxes. Un o una ely, he a ailable expe imen al in o ma ion abou wax s ess–s ain cu es is a he limi ed, e en in he case o axially loaded samples. To he bes o ou knowledge, he e is no a ailable in o ma ion abou mul iaxial s ess s a es. I is unde s ood ha o mode a e loads, speci ically when he applied s esses a e equal o a speci ic s ess, he one-dimensional ension–comp ession p opo ional limi p σ o c ys alline ma e ials obeys Hooke’s law, which es ablishes a linea ela ionship be ween s ess σ and s ain ε , o be ween ensile elonga ion o educ ion δ and he o ce P applied o a es ed specimen: == o PL δ σ Eε AE (1) whe e A ep esen s he c oss-sec ional a ea o he specimen, E deno es a ma e ial p ope y (Young’s modulus o elas ic modulus), and L is he leng h o he sample. In a Figu e 3. Laye ing o wax c ys als o he pi che plan Nepen hes ala a educes insec a achmen capaci y: (A) SEM image o epicu icula wax o he uppe laye ; (B) schema ic image o con amina ion o an insec adhesi e mic os uc u e ( ed) by wax c ys als o he uppe wax laye (yellow); (C) SEM image o epicu icula wax o he lowe laye ; (D) schema ic image o in e ac ion be ween an insec adhesi e mic os uc u e and mic o- oughness o a plan . Rep oduced wi h pe mission om [33]. 2.2. Mechanical P ope ies o Plan Waxes The ollowing sec ion p o ides a discussion o he mechanical p ope ies o plan waxes. Un o una ely, he a ailable expe imen al in o ma ion abou wax s ess–s ain cu es is a he limi ed, e en in he case o axially loaded samples. To he bes o ou knowledge, he e is no a ailable in o ma ion abou mul iaxial s ess s a es. I is unde s ood ha o mode a e loads, speci ically when he applied s esses a e equal o a speci ic s ess, he one-dimensional ension–comp ession p opo ional limi σp o c ys alline ma e ials obeys Hooke’s law, which es ablishes a linea ela ionship be ween s ess σ and s ain ε , o be ween ensile elonga ion o educ ion δand he o ce Papplied o a es ed specimen: δ=PL AE o σ=Eε(1) whe e A ep esen s he c oss-sec ional a ea o he specimen, E deno es a ma e ial p op- e y (Young’s modulus o elas ic modulus), and L is he leng h o he sample. In a one- dimensional p oblem, he s ain εis calcula ed as δ/L, and σis calcula ed as P/A. When s esses all be ween he p opo ional limi σp and he yield s ess σpl (σp ≤ σ<σpl) , a small egion o nonlinea elas ic beha io may be obse ed. This means ha a e unloading, he sample e ains i s o iginal shape, al hough he ma e ial’s beha io de ia es om he linea ela ion. I he ensile (o comp essi e) s ess exceeds he yield s ess σpl , he sample unde goes plas ic de o ma ion upon unloading, esul ing in a shape di e en om i s p e-loading s a e. Typically, σp ∼ =σpl , sugges ing hey a e equi alen . B i le ma e ials ail wi h minimal elonga ion o educ ion (jus a ew pe cen ) pos yield s ess (poin B in Figu e 4). Figu e 4depic s he s anda d s ess–s ain cu e o a b i le c ys alline ma e ial. I has been obse ed ha mechanical p ope ies such as he Young’s modulus E , p opo ional limi σp , and comp essi e s eng h σs o waxes a e gene ally empe a u e- dependen . The de ails o hese a e shown in Table 1. Biomime ics 2024,9, 442 5 o 15 sssssBiomime ics 2024, 9, x FOR PEER REVIEW 5 o 16 one-dimensional p oblem, he s ain ε is calcula ed as /δL , and σ is calcula ed as /PA . When s esses all be ween he p opo ional limi p σ and he yield s ess < ( ) pl p pl σ σ σ σ , a small egion o nonlinea elas ic beha io may be obse ed. This means ha a e unloading, he sample e ains i s o iginal shape, al hough he ma e ial’s beha io de ia es om he linea ela ion. I he ensile (o comp essi e) s ess exceeds he yield s ess pl σ , he sample unde goes plas ic de o ma ion upon unloading, esul ing in a shape di e en om i s p e-loading s a e. Typically, p pl σ σ , sugges ing hey a e equi alen . B i le ma e ials ail wi h minimal elonga ion o educ ion (jus a ew pe cen ) pos yield s ess (poin B in Figu e 4). Figu e 4 depic s he s anda d s ess–s ain cu e o a b i le c ys alline ma e ial. Figu e 4. Schema ic ypical s ess–s ain diag am. Poin A ep esen s he p opo ional limi p σ , poin B indica es he yield poin pl σ , and poin C signi ies he ac u e s ess s σ ( he ma e ial’s s eng h). I has been obse ed ha mechanical p ope ies such as he Young’s modulus E , p opo ional limi p σ , and comp essi e s eng h s σ o waxes a e gene ally empe a u e-dependen . The de ails o hese a e shown in Table 1. Table 1. Mechanical p ope ies o ca nauba wax a wo di e en empe a u es [34]. Tempe a u e E (×106 Pa) Pa 6 ( 10 ) p σ  Pa 6 ( 10 ) s σ  23 C 1806.5 772.2 10.94 18.775 37 C 5.72 9.03 Fo example, ca nauba wax, which is ex ac ed om he lea es o he ca nauba wax palm (Cope nicia p uni e a), exhibi s his empe a u e sensi i i y. I has a Poisson’s a io 0.49ν= and a densi y 3 951 kg / mρ= . C aig e al. [34] no ed ha he s ain be o e ac u e unde comp ession o all es ed waxes anged om 2.7% o 4.3%. The samples exhibi ed b i le ac u e beha io . This obse a ion o he b i le ac u e cha ac e is ics o c ys alline waxes aligns wi h he indings om expe imen s on plan waxes epo ed in [33]. Shellhamme e al. [35] ound ha na u al candelilla and ca nauba waxes beha ed simila ly o ha d and elas ic ma e ials a 2% comp essi e s ain, wi h candelilla wax exhibi ing g ea e iscosi y han ca nauba wax. Howe e , bo h waxes demons a ed Figu e 4. Schema ic ypical s ess–s ain diag am. Poin A ep esen s he p opo ional limi σp , poin B indica es he yield poin σpl, and poin C signi ies he ac u e s ess σs ( he ma e ial’s s eng h). Table 1. Mechanical p ope ies o ca nauba wax a wo di e en empe a u es [34]. Tempe a u e E (×106Pa) σp (×106Pa)σs (×106Pa) 23 ◦C 1806.5 10.94 18.775 37 ◦C 772.2 5.72 9.03 Fo example, ca nauba wax, which is ex ac ed om he lea es o he ca nauba wax palm (Cope nicia p uni e a), exhibi s his empe a u e sensi i i y. I has a Poisson’s a io ν= 0.49 and a densi y ρ= 951 kg/m3 . C aig e al. [ 34 ] no ed ha he s ain be o e ac u e unde comp ession o all es ed waxes anged om 2.7% o 4.3%. The samples exhibi ed b i le ac u e beha io . This obse a ion o he b i le ac u e cha ac e is ics o c ys alline waxes aligns wi h he indings om expe imen s on plan waxes epo ed in [ 33 ]. Shellhamme e al. [ 35 ] ound ha na u al candelilla and ca nauba waxes beha ed simila ly o ha d and elas ic ma e ials a 2% comp essi e s ain, wi h candelilla wax exhibi ing g ea e iscosi y han ca nauba wax. Howe e , bo h waxes demons a ed beha io mo e akin o ha o elas ic ma e ials compa ed o ha o beeswax, which exhibi ed signi ican ly mo e iscosi y and less elas ic beha io . The widesp ead claim ha ca nauba wax is he ha des known wax is suppo ed by e idence ha i has he highes mechanical p ope ies a he ele an empe a u e o all o he waxes es ed by C aig e al. [ 34 ]. Howe e , ou p e ious expe imen s [ 33 ] indica ed ha o he plan waxes may exhibi highe ha dness and elas ic modulus alues compa ed o ca nauba wax. Speci ically, dep h-sensing nanoinden a ion on he wax o he ca ni o ous plan Nepen hes ala a e ealed an elas ic modulus o app oxima ely E= 2.5 GPa and a ha dness o H=0.1 GPa a T=22 ◦C. The elas ici y o eal ma e ials signi ican ly a ec s he elas ic–plas ic inden a ion p o- cess. Ini ially, when he yield poin is exceeded, he plas ic zone is small and comple ely su ounded by ma e ial ha emains elas ic. This esul s in plas ic s ains ha a e compa a- ble o he su ounding elas ic s ains. In such cases, he ma e ial displaced by he inden e is abso bed by he elas ic expansion o he su ounding solid. Wi h inc easing inden a ion dep h, he plas ic zone e en ually ex ends o he ee su ace, allowing displaced ma e ial o low plas ically o he sides o he inden e . Johnson [ 36 ] sugges ed he ollowing ela ion be ween he mean p essu e pm ac ing no mal o he o iginal su ace as pm=cσpl , whe e he cons an c, which depends on he geome y o he inden e and he in e acial ic ion, ypically has a alue o abou 3. Ini ially, when yielding i s occu s, he cons an c is app oxima ely one. In he mechanics o elas ic–plas ic con ac , i is gene ally accep ed ha ha dness is he a e age s ess unde he inden e a which he en i e ma e ial yields. Using Johnson’s calcula ion me hod, one can app oxima e H≈cσpl , 2.8 ≤c≤ 3. Hence, we Biomime ics 2024,9, 442 6 o 15 can es ima e o he wax o he ca ni o ous plan Nepen hes ala a, ha ing H= 0.1 GPa a T= 22 ◦C, ha σpl is app oxima ely 35 MPa. We can assume ha he waxes ha e a bilinea diag am (Figu e 5), whose e e ence poin s A, B, and C a y wi h empe a u e. Figu e 5depic s an idealized s ess–s ain diag am in wo cases: (a) a bilinea diag am wi h s ain ha dening, and (b) linea elas ic– ideal plas ic diag am. When s esses ha e absolu e alues less han he yield s ess σpl , he ma e ial obeys Hooke’s law (1) wi h E= an α . I he absolu e alue o s ess exceeds he yield s ess, s ain ha dening is app oxima ed by a line wi h ano he slope: E1= an β . I β= 0, hen σ=σpl . This indica es ha he wax yields wi hou any u he inc ease in he ex e nal load. Wi h Hooke’s law, calcula ions can p oceed as ollows: εpl =σpl E=35 ·106 25 ·109=0.014 (2) sssssBiomime ics 2024, 9, x FOR PEER REVIEW 6 o 16 beha io mo e akin o ha o elas ic ma e ials compa ed o ha o beeswax, which exhibi ed signi ican ly mo e iscosi y and less elas ic beha io . The widesp ead claim ha ca nauba wax is he ha des known wax is suppo ed by e idence ha i has he highes mechanical p ope ies a he ele an empe a u e o all o he waxes es ed by C aig e al. [34]. Howe e , ou p e ious expe imen s [33] indica ed ha o he plan waxes may exhibi highe ha dness and elas ic modulus alues compa ed o ca nauba wax. Speci ically, dep h-sensing nanoinden a ion on he wax o he ca ni o ous plan Nepen hes ala a e ealed an elas ic modulus o app oxima ely 2.5 GPaE= and a ha dness o 0.1 GPaH= a =22 CT . The elas ici y o eal ma e ials signi ican ly a ec s he elas ic–plas ic inden a ion p ocess. Ini ially, when he yield poin is exceeded, he plas ic zone is small and comple ely su ounded by ma e ial ha emains elas ic. This esul s in plas ic s ains ha a e compa able o he su ounding elas ic s ains. In such cases, he ma e ial displaced by he inden e is abso bed by he elas ic expansion o he su ounding solid. Wi h inc easing inden a ion dep h, he plas ic zone e en ually ex ends o he ee su ace, allowing displaced ma e ial o low plas ically o he sides o he inden e . Johnson [36] sugges ed he ollowing ela ion be ween he mean p essu e m p ac ing no mal o he o iginal su ace as m pl pcσ= , whe e he cons an c, which depends on he geome y o he inden e and he in e acial ic ion, ypically has a alue o abou 3. Ini ially, when yielding i s occu s, he cons an c is app oxima ely one. In he mechanics o elas ic–plas ic con ac , i is gene ally accep ed ha ha dness is he a e age s ess unde he inden e a which he en i e ma e ial yields. Using Johnson’s calcula ion me hod, one can app oxima e , 2.8 3 pl Hcσc   . Hence, we can es ima e o he wax o he ca ni o ous plan Nepen hes ala a, ha ing H = 0.1 GPa a T = 22 °C, ha pl σ is app oxima ely 35 MPa. We can assume ha he waxes ha e a bilinea diag am (Figu e 5), whose e e ence poin s A, B, and C a y wi h empe a u e. Figu e 5 depic s an idealized s ess–s ain diag am in wo cases: (a) a bilinea diag am wi h s ain ha dening, and (b) linea elas ic– ideal plas ic diag am. When s esses ha e absolu e alues less han he yield s ess pl σ , he ma e ial obeys Hooke’s law (1) wi h anEα= . I he absolu e alue o s ess exceeds he yield s ess, s ain ha dening is app oxima ed by a line wi h ano he slope: 1 anEβ= . I 0β= , hen pl σ σ= . This indica es ha he wax yields wi hou any u he inc ease in he ex e nal load. Wi h Hooke’s law, calcula ions can p oceed as ollows:  = = =  6 9 35 10 0.014 25 10 pl pl σ εE (2) (a) (b) Figu e 5. (a) Bilinea s ess–s ain diag am and (b) linea elas ic–ideal plas ic diag am. I is no ed ha an α=E and an β=E1 , whe e E and E1 a e Young’s modulus and he plas ic ha dening modulus, espec i ely. Based on he expe imen s conduc ed by C aig e al. [ 34 ], i can be assumed ha he s ain a ac u e is 2.7%, i.e., εs = 0.027. To plo he bilinea diag am, we need ei he he slope o he linea ha dening pa ( he angen modulus E1 ) o he s ess a ac u e σs . F om his bilinea diag am, we de i e he ela ion εs −εpl =σs −σpl/E1. We may assume ha he a io σs /σpl o he Nepen hes ala a wax is he same as ha o ca nauba wax, i.e., 1.575. The e o e, we ob ain εs −εpl = 0.013 = 0.575 σpl/E1 . Thus, E1 can be calcula ed as E1=20.125 ·106/0.013 =1.548 ·109Pa. 2.3. De o ma ion and F ac u e o Plan Wax S uc u es 2.3.1. Eule Buckling o Wax P ojec ions In he mechanics o ma e ials, long and slende s uc u es, as plan wax p ojec ions, subjec ed o axial comp ession a e e e ed o as columns. Fi s , le us analyze he s abili y o ubula columns, which esemble ubula wax c ys als, unde comp essi e loading. When he applied comp essi e load P inc eases, a slende elas ic column will buckle a he c i ical load Pc , de e minable using he Eule o mula: Pc =π2EI (KL)2=π2EI Le 2(3) Biomime ics 2024,9, 442 7 o 15 whe e K ep esen s he e ec i e leng h ac o , which a ies depending on he bounda y condi ions a he end o he column, and whe e e ec i e leng h is de ined as Le =KL . When a column has pinned ends, K is 1. When he base is buil -in ( ixed) and he op end is ee (Figu e 6A), K is 2. When he base is ixed and he op is pinned (Figu e 6C), K is 0.7. When bo h ends a e buil -in (Figu e 6B), K is 0.5. I is plausible o assume ha one end o he wax p ojec ion is a ached o he plan su ace, indica ing a buil -in s a e, while he o he end is in ol ed in adhesi e in e ac ions wi h he insec ’s a achmen o gan (adhesi e pad). Insec s usually ely on capilla y adhesion esul ing om he sec e ion o luid (pad sec e ion) in o he con ac zone be ween he pad and he subs a e [ 10 ]. These condi ions sugges ha he column may be modeled as a od wi h an elas ically es ained (clumped) end, which may be ep esen ed as an elas ic o sional sp ing a his end (Figu e 6D). The o sion sp ing model ep esen s he adhesi e in e ac ion be ween an insec ’s se a and a wax column. Consequen ly, Le = 0.6 L can be conside ed a easonable app oxima ion o he e ec i e leng h o he column. sssssBiomime ics 2024, 9, x FOR PEER REVIEW 7 o 16 Figu e 5. (a) Bilinea s ess–s ain diag am and (b) linea elas ic–ideal plas ic diag am. I is no ed ha anαE= and 1 an βE= , whe e E and 1 E a e Young’s modulus and he plas ic ha dening modulus, espec i ely. Based on he expe imen s conduc ed by C aig e al. [34], i can be assumed ha he s ain a ac u e is 2.7%, i.e., 0.027 s ε= . To plo he bilinea diag am, we need ei he he slope o he linea ha dening pa ( he angen modulus 1 E ) o he s ess a ac u e s σ . F om his bilinea diag am, we de i e he ela ion ( ) 1 / s pl s pl ε ε σ σ E− = − . We may assume ha he a io / s pl σ σ o he Nepen hes ala a wax is he same as ha o ca nauba wax, i.e., 1.575. The e o e, we ob ain 1 0.013 0.575 / s pl pl ε ε σ E− = = . Thus, 1 E can be calcula ed as 69 120.125 10 / 0.013 1.548 10 PaE=  =  . 2.3. De o ma ion and F ac u e o Plan Wax S uc u es 2.3.1. Eule Buckling o Wax P ojec ions In he mechanics o ma e ials, long and slende s uc u es, as plan wax p ojec ions, subjec ed o axial comp ession a e e e ed o as columns. Fi s , le us analyze he s abili y o ubula columns, which esemble ubula wax c ys als, unde comp essi e loading. When he applied comp essi e load P inc eases, a slende elas ic column will buckle a he c i ical load c P , de e minable using he Eule o mula: ( ) ( ) == 22 22 c e πEI πEI P KL L (3) whe e K ep esen s he e ec i e leng h ac o , which a ies depending on he bounda y condi ions a he end o he column, and whe e e ec i e leng h is de ined as e L KL= . When a column has pinned ends, K is 1. When he base is buil -in ( ixed) and he op end is ee (Figu e 6A), K is 2. When he base is ixed and he op is pinned (Figu e 6C), K is 0.7. When bo h ends a e buil -in (Figu e 6B), K is 0.5. I is plausible o assume ha one end o he wax p ojec ion is a ached o he plan su ace, indica ing a buil -in s a e, while he o he end is in ol ed in adhesi e in e ac ions wi h he insec ’s a achmen o gan (adhesi e pad). Insec s usually ely on capilla y adhesion esul ing om he sec e ion o luid (pad sec e ion) in o he con ac zone be ween he pad and he subs a e [10]. These condi ions sugges ha he column may be modeled as a od wi h an elas ically es ained (clumped) end, which may be ep esen ed as an elas ic o sional sp ing a his end (Figu e 6D). The o sion sp ing model ep esen s he adhesi e in e ac ion be ween an insec ’s se a and a wax column. Consequen ly, 0.6 e LL= can be conside ed a easonable app oxima ion o he e ec i e leng h o he column. Figu e 6. Some bounda y condi ions o comp essed columns: (A) a column wi h buil -in and ee ends; (B) a column wi h bo h ends buil -in; (C) a column wi h buil -in and pinned ends; (D) a column wi h buil -in and elas ically clumped ends. K is he e ec i e leng h ac o o di e en bounda y condi ions a he end o he column. 2.3.2. Elas oplas ic Buckling o Wax Columns The Eule o mula in Fo mula (4) assumes a linea ly elas ic s ess–s ain ela ionship and becomes in alid when comp essi e s esses in he column exceed he yielding s ess. Consequen ly, inelas ic buckling mus be conside ed. Engesse and Jasinski de eloped his heo y, as ci ed in [ 37 ]. Acco ding o his heo y, he c i ical load Pc a which a slende elas ic–plas ic column buckles can be calcula ed as ollows: Pc =π2E I (KL)2(4) whe e E ep esen s he column’s educed modulus. Fo a ec angula c oss-sec ion, we can calcula e his modulus using Fo mula (5). This se es as an app oxima ion o columns wi h ci cula c oss-sec ions. E =4EE1 √E+√E12(5) Gi en he moduli o elas ici y, E= 2.5 GPa and E1= 1.548 GPa , o he plan wax, he educed modulus o he column can be calcula ed as ollows: E =4·2.5 ·1.548 √2.5 +√1.5482=1.94 GPa (6) Biomime ics 2024,9, 442 8 o 15 These o mulas a e applicable o slende columns. To cha ac e ize he beha io o a column, i is essen ial o in oduce he slende ness a io λ=Le / , whe e =√I/A , he adius o gy a ion o he column’s c oss-sec ion in he bending plane. He e, A deno es he c oss-sec ion’s a ea, and I ep esen s he momen o ine ia o he c oss-sec ional a ea. Fo a ci cula c oss-sec ion wi h diame e D , A=πD2/ 4 and I=πD4/ 64, he adius o gy a ion is calcula ed below: = I A= 4D4 64D2=D 4(7) I is impo an o unde s and he di e ence be ween he aspec a io and he slen- de ness o a column. The aspec a io is a pu ely geome ic cha ac e is ic, de ined as he a io be ween he la ges and smalles dimensions o he column, whe eas he slende ness depends on bo h he geome y, ep esen ed by he leng h L and he adius o gy a ion o he column’s c oss-sec ion, and he loading condi ions, indica ed by he e ec i e leng h ac o K . Fo ins ance, al hough all cases A–D in Figu e 6sha e he same aspec a io L/D , hei slende ness alues di e signi ican ly. The Eule o mula in Fo mula (3) is applicable solely o pu ely linea ly elas ic ma e ials; he e o e, he ansi ion om elas ic o he elas ic–plas ic case, i.e., om Fo mula (4) o Fo mula (3), should be a he c i ical o ce such ha i causes he c i ical s ess o be equal o he yield s ess, as ollows: σc =Pc A=π2EI AL2 e =σpl (8) Consequen ly, he c i ical slende ness a io can be calcula ed as ollows: λc=Le c =sπ2E σpl (9) Hence, he ela ionship be ween he a e age comp essi e s ess σ and slende ness a io λ o plan waxes may be ep esen ed by he g aph shown in Figu e 7. sssssBiomime ics 2024, 9, x FOR PEER REVIEW 9 o 16  ==    2 e c pl c LπE λ σ (9) Hence, he ela ionship be ween he a e age comp essi e s ess σ and slende ness a io λ o plan waxes may be ep esen ed by he g aph shown in Figu e 7. Figu e 7. Rela ionship be ween plan wax comp essi e s ess and slende ness a io. In egion BC, he c i ical s ess a he yield limi pl σ is gi en by 2 2 c πE σλ = . In egion AB, he c i ical s ess a he s eng h limi s σ is gi en by 2 2 c πE σλ = . When cpl λ λ , he ma e ial will no buckle, as i has al eady exceeded he s eng h limi , and will ins ead be c ushed. Subs i u ing he alues o plan wax, we ob ain he ollowing c i ical slende ness a io:  ==  9 6 9.87 2.5 10 26.5 35 10 c λ (10) Hence, he slende ness a io o columns wi h 0.6K= and gy a ion adius /4 D= is as ollows: == 0.6 2.4LL λ D (11) Go b e al. [38] epo ha ce ain plan s, such as Aquilegia ulga is (Eu opean columbine), Be be is ulga is (common ba be y), Chelidonium majus (whi e goose oo ), and P unus domes ica (Eu opean plum), exhibi ubula epicu icula wax c ys als. A is olochia imb ia a (whi e- eined Du chman’s pipe) c ys als, as well as hose o many o he A is olochia species, also exhibi his ubula shape, bu a a e y high slende ness a io, as shown in Figu e 8. Table 2 p esen s he a e age leng hs and diame e s o hese c ys al columns, along wi h he calcula ed slende ness a ios ( / e λL = ) o all i e plan s men ioned abo e. Figu e 7. Rela ionship be ween plan wax comp essi e s ess and slende ness a io. In egion BC, he c i ical s ess a he yield limi σpl is gi en by σc =π2E λ2 . In egion AB, he c i ical s ess a he s eng h limi σs is gi en by σc =π2E λ2 . When λ≤λcpl , he ma e ial will no buckle, as i has al eady exceeded he s eng h limi , and will ins ead be c ushed. Biomime ics 2024,9, 442 9 o 15 Subs i u ing he alues o plan wax, we ob ain he ollowing c i ical slende ness a io: λc= 9.87 ·2.5 ·109 35 ·106=26.5 (10) Hence, he slende ness a io o columns wi h K= 0.6 and gy a ion adius =D/ 4 is as ollows: λ=0.6L =2.4L D(11) Go b e al. [ 38 ] epo ha ce ain plan s, such as Aquilegia ulga is (Eu opean columbine), Be be is ulga is (common ba be y), Chelidonium majus (whi e goose oo ), and P unus domes ica (Eu opean plum), exhibi ubula epicu icula wax c ys als. A is olochia imb ia a (whi e- eined Du chman’s pipe) c ys als, as well as hose o many o he A is olochia species, also exhibi his ubula shape, bu a a e y high slende ness a io, as shown in Figu e 8. Table 2p esen s he a e age leng hs and diame e s o hese c ys al columns, along wi h he calcula ed slende ness a ios (λ=Le / ) o all i e plan s men ioned abo e. sssssBiomime ics 2024, 9, x FOR PEER REVIEW 10 o 16 Figu e 8. (A) Whi e- eined Du chman’s pipe (A is olochia imb ia a) lowe ; (B) scanning elec on mic oscopy mic og aphs o ichomes on he inne su ace o he lowe ap; (C) wax c ys als co e ing he ichome su ace ha may buckle. Table 2. Geome ical pa ame e s o he wax columns in selec ed plan species: L—leng h; D— diame e ; λ—slende ness a io. Plan Species Aquilegia ulga is Be be is ulga is Chelidonium majus P unus domes ica A is olochia imb ia a ( ) nmL 580 730 830 580 6310 ( ) nmD 170 160 180 260 92 / e λL = 8.18 10.94 11.06 5.35 164.6 Clea ly, among he ubula -shaped plan wax c ys als s udied [39,40], only A is olochia imb ia a can elas ically buckle, as he slende ness a ios o he o he plan s all below he c i ical alue o 26.5. The e o e, he c ys als om hese ou plan s do no mee he leng h c i e ia o he Eule o mula app oxima ion. Fo mula (3) applies solely unde s esses below he ma e ial’s ul ima e comp essi e s ess s σ . Consequen ly, he c i ical slende ness a io cpl λ o discon inuing he use o Fo mula (3) due o elas ic–plas ic buckling is iden i ied as ollows:  ==    2 e cpl s cpl LπE λ σ (12) The plan c i ical slende ness a io cpl λ can be ob ained by subs i u ing he speci ied plan wax alues in o Fo mula (12), as ollows:  ==  9 6 9.87 1.94 10 18.66 55 10 cpl λ (13) Plo ing a diag am o he a e age s ess agains he slende ness a io (Figu e 7) e eals ha , aside om A is olochia imb ia a, which buckles in he elas ic egime, c ys als om all o he s udied plan s do no buckle in ei he he elas ic o elas ic–plas ic egimes gi en he calcula ed slende ness a ios. Indeed, hese sho columns a e no suscep ible o ailu e h ough a pu e buckling mechanism. Failu e occu s only when comp essi e s ess eaches he wax’s s eng h limi . The e o e, i is necessa y o accoun o he bending o wax c ys als unde simul aneous axial and o hogonal loading. The e a e wo es ic ions in he applica ion o he abo e Eule o mula: One is ha he de aul comp ession column is pe ec ly s aigh be o e he load is applied, bu in eali y, he p esence o ubula columna plan wax c ys als does no gua an ee pe ec Figu e 8. (A) Whi e- eined Du chman’s pipe (A is olochia imb ia a) lowe ; (B) scanning elec on mic oscopy mic og aphs o ichomes on he inne su ace o he lowe ap; (C) wax c ys als co e ing he ichome su ace ha may buckle. Table 2. Geome ical pa ame e s o he wax columns in selec ed plan species: L—leng h; D—diame e ; λ—slende ness a io. Plan Species Aquilegia ulga is Be be is ulga is Chelidonium majus P unus domes ica A is olochia imb ia a L(nm)580 730 830 580 6310 D(nm)170 160 180 260 92 λ=Le / 8.18 10.94 11.06 5.35 164.6 Clea ly, among he ubula -shaped plan wax c ys als s udied [ 39 , 40 ], only A is olochia imb ia a can elas ically buckle, as he slende ness a ios o he o he plan s all below he c i ical alue o 26.5. The e o e, he c ys als om hese ou plan s do no mee he leng h c i e ia o he Eule o mula app oxima ion. Fo mula (3) applies solely unde s esses below he ma e ial’s ul ima e comp essi e s ess σs . Consequen ly, he c i ical slende ness a io λcpl o discon inuing he use o Fo mula (3) due o elas ic–plas ic buckling is iden i ied as ollows: λcpl =Le cpl =sπ2E σs (12)