Fractional viscoelastic models of porcine skin and its gelatin-based surrogates

Mechanics o Ma e ials 177 (2023) 104559 A ailable online 13 Janua y 2023 0167-6636/© 2023 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). F ac ional iscoelas ic models o po cine skin and i s gela in-based su oga es R. Mouˇ cka a , b , M. Sedlaˇ cík a , c , * , Z. P´ a íko ´ a d a Cen e o Polyme Sys ems, Uni e si y Ins i u e, Tomas Ba a Uni e si y in Zlín, Tˇ ída T. Ba i 5678, 760 01, Zlín, Czech Republic b Polyme Cen e, Facul y o Technology, Tomas Ba a Uni e si y in Zlín, Va eˇ cko a 275, 760 01, Zlín, Czech Republic c Depa men o P oduc ion Enginee ing, Facul y o Technology, Tomas Ba a Uni e si y in Zlín, Va eˇ cko a 275, 760 01, Zlín, Czech Republic d Depa men o Ma hema ics, Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín, Nad S ´ anˇ emi 4511, 76005, Zlín, Czech Republic ARTICLE INFO Keywo ds: Viscoelas ici y C eep F ac ional models Gela in Po cine skin ABSTRACT Viscoelas ici y o po cine skin and i s ma e ial subs i u e, modelled by a iously concen a ed bo ine gela in, was de e mined in s a ic (c eep es ) and dynamic (oscilla o y es ) mode by he means o o a ional heome y o ob ain c eep compliance and complex shea modulus. Mechanical p ope ies cha ac e iza ion was also supple- men ed wi h la ge de o ma ion comp ession es in o de o de e mine and co ela e shea and comp ession moduli o gela in wi h i s concen a ion dependence. Ob ained da a was i ed wi h ac ional iscoelas ic models (Poyn ing-Thomson, Maxwell) in o de o quan i y in de ail gela in’s ansi ion om iscous-like beha io o- wa ds solid-like s a e wi h inc easing gela in concen a ion and hence c osslinking densi y. Po en ial o gela in as bioma e ial o skin su oga e was iden i ied as well as a concen a ion egion in which gela in exhibi s closes iscoelas ic beha io o na i e po cine skin used. 1. In oduc ion Du ing i s li espan skin has o be able o adap i sel o and esis he load/impac o many ex e nal mechanical s imuli a ying signi ican ly in he mode o de o ma ion (Whi e e al., 2013) ( ensile (Yang e al., 2015), comp ession, ic ion (Bhushan e al., 2010; Ley a-Mendi il e al., 2017)), i s in ensi y and ime pe iod o hei impac . Addi ionally, skin has o main ain i s p ope ies in ce ain empe a u e and humidi y window (Wu e al., 2006). All hese equi emen s can be me o he la ge ex en owing o unique iscoelas ic p ope ies s emming om skin inne s uc u e/mo phology (Depalle e al., 2015). Skin comp ises o h ee laye s epide mis, de mis and hypode mis ( op o bo om) which a e oge he abou 1–3 mm hick. De ailed ma e ial knowledge o human skin in e ms o i s mechan- ical and s ess- esponse p ope ies has always been undamen al o many esea ch ields anging om cosme ics o e de ma ology and senso s applied di ec ly on o skin (Zhou e al., 2019) o biomechanics and issue enginee ing (Auge e al., 1998; Lee, 2000; Mansb idge, 2002; Vig e al., 2017). Ano he signi ican mo i a ion ueling he esea ch o iscoelas ic p ope ies o he skin is o be ul ima ely able o design ma e ial su oga e o he skin (Chanda, 2018). Al hough qui e di e se ma e ials can be used (Dab owska e al., 2016; Mo ales-Hu ado e al., 2015), hyd ogels in pa icula a e a g oup o ma e ials wi h high po- en ial in his ega d (Yi e al., 2021). Among hem gela in, a na u al polyme ex ac ed om collagen, is a p omising candida e owing o i s biocompa ibili y and mechanical p ope ies ailo able h ough concen- a ion and chemical c osslinking. The e o e, gela in has been used in a numbe o biomedical applica ions (Ala con-Sego ia e al., 2021; Auge e al., 1995) including a componen in addi i e echnology o 3D p in ing o human skin (Jin e al., 2021). I s mechanical p ope ies can be u he imp o ed h ough c osslinking, which is mos equen ly pe o med using glu a aldehyde (GTA) due o i s easy a ailabili y and inexpensi eness (Bigi e al., 2001). In o de o in de ail de e mine skin’s iscoelas ic p ope ies a ie y o echniques (Pissa enko and Meye s, 2020) such as inden ome y (Jachowicz e al., 2007), ension es (Yazdi and Baqe sad, 2022), cu ome y a e employed, howe e heological measu emen emains a key me hod in his ega d (Hol e al., 2008; Cheng e al., 2008; Pail- le -Ma ei e al., 2014; Ve die e al., 2009). Viscoelas ici y o he ma- e ial is ypically de e mined h ough c eep (Higgs and Ross-Mu phy, 1990; No mand and Ra ey, 1997) and elaxa ion es s and h ough dynamical measu emen o i s complex mechanical modulus, G*. All o hese es s a e designed o cap u e ime esponse o he ma e ial o an impulse (aka o cing) which is s a ic o c eep (de o ma ion esponse o * Co esponding au ho . Cen e o Polyme Sys ems, Uni e si y Ins i u e, Tomas Ba a Uni e si y in Zlín, Tˇ ída T. Ba i 5678, 760 01, Zlín, Czech Republic. E-mail add ess: [email p o ec ed] (M. Sedlaˇ cík). Con en s lis s a ailable a ScienceDi ec Mechanics o Ma e ials jou nal homepage: www.else ie .com/loca e/mechma h ps://doi.o g/10.1016/j.mechma .2023.104559 Recei ed 25 Augus 2022; Recei ed in e ised o m 13 No embe 2022; Accep ed 12 Janua y 2023 Mechanics o Ma e ials 177 (2023) 104559 2 cons an s ess being applied o he sample) and elaxa ion (s ess diminishing wi h ime in he sample being de o med o a cons an le el) while ime a ying ( ypically ha monic de o ma ion o cing wi hin linea iscoelas ici y ange) in he case o oscilla o y measu emen . S anda d app oach o ma hema ical modelling o iscoelas ici y ypically in ol es he use o wo basic elemen s – an elas ic elemen ep esen ed by a sp ing and i s iscous coun e pa ep esen ed by a dashpo . These a e linked ei he in se ies (Maxwell model), in pa allel (Kel in-Voigh model) o mo e complica ed ne wo k, such as in Zene o Poyn ing-Thomson models (Oyen, 2014). Al hough on he quali a i e le el hese phenomenological models p o ide sa is ac o y esul s cap u ing he basic cha ac e o c eep (Kel in-Voigh ) o elaxa ion (Maxwell) es , hey a e a om being able o closely ollow cha ac e o eal ma e ial, whose s eep ini ial inc ease and subsequen ex emely slow le elling o a e well beyond capabili ies o a simple exponen ial unc ion ea u ed in he undamen al iscoelas ic models (Bon an i e al., 2020b; Macosko, 1996). E en hough hese sho comings can be deal wi h o a ce ain deg ee by including a numbe o basic elemen s (sp ings and dashpo s) in he model, he esul ing sys em s ill ails o i he expe imen al da a and also is o e complica ed due o la ge numbe o i ing pa ame e s ( wo o each elemen ) (Heymans and Bauwens, 1994). To o e come hese disad an ages a new g oup o ac ional isco- elas ic models (Heymans and Bauwens, 1994; Long e al., 2018; Main- a di, 2010; Schiessel e al., 1995; Xu and Jiang, 2017) based on he idea o non-in ege o de o de i a i e ha e been p oposed and p o en o be an e ec i e ool o desc ip ion o expe imen al iscoelas ic da a (c eep, s ess elaxa ion) o he mos di e se sys ems anging om shape memo y polyme s (Fang e al., 2015) o e gels (Fabe e al., 2017; Holde e al., 2018; Rosalina and Bha acha ya, 2002; Zhang e al., 2018) and ood (Mahiuddin e al., 2020) o biological (Bon an i e al., 2020a; Ca michael e al., 2015; C aiem e al., 2006; Li and Tian, 2021; Mahiuddin e al., 2020) and geological sys ems (Di Giuseppe e al., 2009; Chen and Ai, 2020; Wang, 2021). The key concep is an in o- duc ion o ano he elemen called a sp ing-po which exhibi s beha io be ween he sp ing and he dashpo and by design cap u es powe -law ma e ials (Bon an i e al., 2020b). The cu en wo k in es iga es a po en ial and easibili y o a iously concen a ed gela in o model ime dependen mechanical p ope ies o skin. The s udy is based on de ailed mechanical cha ac e iza ion ob- ained om iscoelas ic heological measu emen s as well as a comp ession es . Collec ed heological da a is i ed wi h sui able ac ional iscoelas ic models in o de o quan i y i s ime dependen beha io . 2. Expe imen al 2.1. Ma e ial 1. Po cine skin p epa a ion Po cine ea s we e clea ed and s o ed a −20 ◦C. P io o sample p epa a ion ea s we e de os ed na u ally in oom empe a u e and ho oughly cleaned unde unning wa e . Subsequen ly ci cula ca ilage- ee samples 25 mm in diame e we e cu ou . 2. Gela in Gela in om bo ine skin, ype B sui able o cell cul u e pu chased om Sigma Ald ich was used in his s udy. 2.2. Gela in samples p epa a ion Gela in samples we e p epa ed in a concen a ion se ies spanning om 10 o 50 w %. Co esponding amoun o gela in o p epa a ion o x [%] concen a ed gela in was calcula ed (Eq. (1)) and dissol ed in 10 mL o demine alized wa e a 60 ◦C o a leas 15 min while mixing. Subsequen ly he solu ion was degassed ( o emo e ai bubbles) in a acuum empe a u e chambe and injec ed (due o high iscosi y pou ing was ypically impossible) in o a cylind ic molds sized acco ding o used es ( heome y diame e =25 mm, hickness =2 mm; comp ession es diame e =20 mm, hickness =8 mm). Gela in was always le o cool down o lab empe a u e o 20 min p io o samples emo al and o he es ing. x=mgela ine mwa e (1) 2.3. Me hods 1. Ro a ional heome y All heological es s we e pe o med using an ad anced modula o a ional heome e Physica MCR502 (An on Paa , Aus ia) in e - connec ed wi h wa e -cooled Pel ie sys em P-PTD 200. A pa allel-pla e measu ing sys em wi h a diame e o 25 mm (PP25) was used, while he pla e gap was se o he sample hickness, i.e. 2.0 mm. All o he mea- su emen s we e pe o med a 0.2 ±0.02 N no mal o ce and a 25 ◦C. In equency sweep es s angula equency es ed anged om 0.1 o 100 ad s −1 wi h loga i hmic sampling o 10 p /decade. S ain applied o all in es iga ed samples du ing his measu emen was se o 0.5%, i.e. wi hin he iscoelas ic egion de e mined om ampli ude sweep be- o ehand. C eep es comp ised measu emen o s ain o 300 s a e exe ing cons an shea s ess o τ 0 =10 Pa wi h sampling adjus ed o he dynamics o he es . 2. Comp ession es Comp ession es s we e ca ied ou using a ensile es machine Tes ome ic M350 5CT (UK) wi h a 10 kg and a 1 kg load cell employed o gela in and po cine skin, espec i ely. Cylind ic samples 20 mm ac oss and 8 mm/1.5 mm (gela in/po cine skin) hick we e loaded a he cen e o wo me al pla es and comp essed a a cons an a e o 5 mm min −1 /1 mm min −1 (gela in/po cine skin) while co esponding o ce was eco ded. Measu emen s we e pe o med a oom empe a u e. 3. Scanning elec on mic oscopy (SEM) Mo phology o in es iga ed sys ems, i.e. gela in samples and po cine skin, was imaged using scanning elec on mic oscope No a NanoSEM 450 (FEI, Japan), p io o which he samples we e i s lyophilized o e nigh in o de o emo e wa e con en and hen spu e coa ed wi h a hin laye o gold. 4. F ac ional iscoelas ic models F ac ional iscoelas ic models wo k wi h a sp ing-po (Fig. 1), which Fig. 1. A undamen al ac ional iscoelas ic elemen – a sp ing-po ; he alue o α de e mines whe he i beha es mo e like an elas ic (sp ing) o a iscous (dashpo ) elemen . R. Mouˇ cka e al. Mechanics o Ma e ials 177 (2023) 104559 3 is a basic ac ional elemen ep esen ing con inuous ansi ion be ween a sp ing and a dashpo and has a ollowing go e ning equa ion (Bon an i e al., 2020b): σ ( ) = c α d α γ( ) d α ,(2) whe e σ [Pa] deno es ime dependen s ess, γ [−] is ime dependen s ain, α is he o de o Riemann-Lou ille ac ional de i a i e, 0 < α <1 (also some imes e e ed o as he ac ional exponen ) and c α [Pa s α ] is a “p ope y” o he sp ing-po . Subs i u ing sp ing-po in s anda d iscoelas ic models, gene alized ac ional models a e ob ained. In ou wo k gene alized ac ional Poyn ing-Thomson (FPT) model (Fig. 2) was used (Bon an i e al., 2020b; Xu and Jiang, 2017). The c eep compliancy o FPT model as a unc ion o ime is gi en as: J( ) = λ η 3Γ(1+λ)+ α η 1 E α −β,1+ α (− α −β η 1/ η 2),(3) whe e Γ is gamma unc ion and E p,q (z) is Mi ag-Le le unc ion Ep,q(z) = ∑ ∞ n=0 zn Γ(pn +q)(4) Complex modulus o he FPT model ele an o dynamic measu e- men o samples in oscilla ions is de ined as ollows: G∗( ω ) = η 3(i ω )λ[ η 1(i ω ) α + η 2(i ω )β] η 3(i ω )λ+ η 1(i ω ) α + η 2(i ω )β(5) 3. Resul s and discussion 3.1. Mo phology Gela in samples ha e a he uni o m po ous s uc u e wi h po e size dec easing wi h gela in concen a ion (Fig. 3 a-c). This in u n leads o dense sys em which is consequen ly also e lec ed in be e mechanical p ope ies in e ms o highe shea G and comp ession E moduli. On he con a y o simple po ous s uc u e o gela in, skin sample’s inne s uc u e is much iche wi h complex hie a chical s uc u e, om which only indi idual ibe s a e dis inguishable in he SEM (Fig. 4 b), which is ul ima ely esponsible o mo e g adua e onse o de o ma ion in he c eep es (Fig. 5) as well as s eepe inc ease o elas ic pa o complex shea modulus (Fig. 10) hen obse ed in gela in samples (Fig. 11). 3.2. Rheology 1. C eep es Typical c eep es o po cine skin sample is shown in Fig. 5. Ins an aneous elas ic esponse o skin’s compliance o s ep shea s ess is ollowed by g adually slowing inc ease o J wi h ime, which wi hin he in es iga ed ime ame showed no signs o eaching equilib ium compliance condi ions. Al hough a he e y beginning o he es a b ie in e al (0–0.01 s) ma ked wi h oscilla ions, whe e he skin beha es as an unde damped oscilla o (so called c eep- inging (Ewold and McKinley, 2007; Goudoulas and Ge mann, 2016)), can be obse ed (Fig. 5 igh ), hese mani es a ions a e signi ican ly smalle compa ed o gela in. Po cine skin, owing o i s ex emely complex hie a chical s uc u e comp ising iple helical collagen molecules (1.5 nm in diame e ) on he lowes le el assembled in pa allel in o ib ils (50–500 nm hick), mul iple o which o m ibe s and whole issues (Gau ie i e al., 2012), hus exhibi s g adual c eep o compliance wi h ime when subjec ed o cons an s ess. On he con a y, gela in samples (Fig. 6) exhibi p onounced c eep- inging in he i s phase o he c eep as gela in oge he wi h heom- e e o ms an unde damped mechanical oscilla o (Goudoulas and Ge mann, 2016). The oscilla ions end o diminish soone wi h gela in’s concen a ion (Fig. 6 igh ) as he damping o he sys em inc eases due o gela in’s changes in s uc u e, i.e. highe c osslinking densi y and smalle po e size (Fig. 3). Apa om ha inc easing gela in concen- a ion also conside ably a ec s alue o immedia e compliance, which dec eases in non-linea ashion, as well as he shape o he c eep cu es, which la en ou . All o hese mani es a ions clea ly ma k ansi ion o gela in om iscous-like o solid-like ma e ial wi h inc easing concen- a ion, which om ma e ial poin o iew s ems mos ly om highe c osslinking densi y and pa ially om dec easing po es’ size. In o de o be able o quan i y he cha ac e o he measu ed c eep da a in de ail and co espondingly ela e i o changes in he s uc u e o he in es iga ed sys em, da a was app oxima ed by he gene alized FPT model (Fig. 2). E en hough excep ional well ag eemen be ween FPT model and da a was ob ained, e y symme ical esul s in e ms o he exponen s o he pa allel elemen s o FPT model and ze o alue o he hi d one in se ies (Table 1) hin ed a he ac ha possibly e en simple ac ional model could su ice. T ying he e y basic sp ing-po model did no yield sa is ac o y i , howe e wo sp ing-po s in se ies i.e. gene alized ac ional Maxwell model (FMM - Fig. 7) did. C eep compliance o FMM has a ollowing o m: J( ) = α η 1Γ(1+ α )+ β η 2Γ(1+β).(6) Also, wi h one exponen in he Maxwell model being p ac ically equal o ze o (β =0.002) means ha one o he sp ing-po s basically unc ions as a s anda d sp ing. Indeed sp ing-po in se ies wi h a sp ing which accoun s o immedia e pa o he compliance is su icien o cap u ing he cha ac e o he po cine skin sample in c eep, which is consis en wi h esul s om i ing using mo e complex FPT model whe e also he elemen in se ies unc ioned as a simple sp ing (λ =0). The e o e, he special case o ac ional Maxwell model con aining a sp ing-po and a sp ing in se ies (Fig. 7) was employed o all u he expe imen al c eep da a app oxima ion. Al hough wi hou igo ous physical meaning, o α <0.5 elas ici y plays a leading ole and he close α app oaches ze o he mo e Hooke’s elas ici y p e ails (Di Paola and Zingales, 2012). Ano he in e p e a ion ela es he o de o ac ional de i a i e α o he Debo ah numbe (Me zne e al., 1966) as ollows: Fig. 2. Gene alized ac ional Poyn ing-Thomson iscoelas ic model. R. Mouˇ cka e al. Mechanics o Ma e ials 177 (2023) 104559 4 Fig. 3. SEM o a iously concen a ed gela in (a–c: 10%, 30%, 50%). Fig. 4. SEM o po cine skin (a: side iew o he whole sample, b: de ail o indi idual ibe s). Fig. 5. C eep es o po cine skin in linea (le ) and loga i hmic ( igh ) ime scale o highligh c eep- inging a he ini ial phase o he es ; expe imen al da a (symbols connec ed wi h dashed line) i ed by he gene alized ac ional Maxwell iscoelas ic model ( ull line). R. Mouˇ cka e al. Mechanics o Ma e ials 177 (2023) 104559 5 De =1 α (7) wi h la ge De o solid-like ma e ials while small De ma k liquid. Thus, po cine skin wi h α =0.33 signi ies a he elas ic- ype o beha io despi e any clea signs o he c eep cu e eaching pla eau in he in es iga ed ime ame. Indeed, especially in he case o na u al ma- e ials, he inal ex emely long phase o he c eep can o en be mis- in e p e ed as iscous low due o limi ed (always ini e) du a ion o he es . Fo gela in samples FMM iden i ies ins an aneous elas ic pa o compliance as well as in he case o skin sample yielding β =0 o all in es iga ed concen a ions (Table 2). Sp ing cons an k (= η 2 ) in hus simpli ied FMM model is seen o inc ease non-linea ly wi h gela in concen a ion (Fig. 9) ma king g adual inc ease o he samples!’ elas ici y. Al hough ansi ion o gela in om mo e iscous-like sys em o a low concen a ions o mo e elas ic-like ma e ial as he loading app oaches 50% is appa en om he la ening o he measu ed c eep dependence, he usage o he FMM enables one o cap u e his ans- o ma ion e en quan i a i ely wi h α pa ame e , which g adually de- c eases om α 10% =0.83 o α 50% =0.27 (Table 2, Fig. 9) u ning he sp ing-po in he Maxwell model om dash-po -like mo e in o a sp ing (Fig. 1). This inding ag ees wi h physical-mechanical expec a ions o he sys em ising om la ge numbe o in e molecula in e ac ions o mo e concen a ed gela in leading ul ima ely o highe c osslinking densi y (Fig. 8). Mechanical spec a, i.e. G*( ω ), we e de e mined om subjec ing samples o a sinusoidal shea de o ma ion γ( ) = γ0ei ω o ampli ude γ 0 = 0.5% (wi hin linea iscoelas ici y egion) and ob aining co esponding complex s ess esponse. σ ( ) = G∗γ0ei ω = |G|eiθγ0ei ω = (G′+iG′′)γ0ei ω ,(8) whe e G*( ω ) is equency dependen complex dynamic shea modulus, eal pa o which (G′) de ines s o age (elas ic) modulus in phase wi h Fig. 6. C eep es o gela in in linea (le ) and loga i hmic ( igh ) ime scale o highligh c eep- inging a he ini ial phase o he es ; expe imen al da a (connec ed symbols) i ed by he gene alized ac ional Poyn ing-Thomson iscoelas ic model (line); legend: gela in concen a ion. Table 1 Fi ing pa ame e s o he gene alized ac ional Poyn ing-Thomson model. sample heo- mode α β λ η 1 η 2 η 3 po cine skin c eep 0.333 0.333 0.000 90.4 36.1 253.6 po cine skin oscilla o y 1.000 0.115 0.354 16.2 6580 36 207 gela ine 10% oscilla o y 0.004 0.270 1.000 1444 70 495 790 gela ine 20% oscilla o y 0.000 0.261 0.698 3552 243 459 447 gela ine 30% oscilla o y 0.000 0.324 0.282 5257 267 73 674 gela ine 40% oscilla o y 0.000 0.361 0.256 8397 276 85 709 gela ine 50% oscilla o y 0.001 0.421 0.243 12 501 303 89 703 Fig. 7. Gene alized ac ional Maxwell model (le ) and i s special case when β =0 ( igh ). Table 2 Fi ing pa ame e s o he ac ional Maxwell model employed o he c eep es . Sample α β η 1 η 2 (=k) po cine skin 0.333 0.002 1272 2514 gela ine 10% 0.825 0.000 6574 1149 gela ine 15% 0.594 0.000 5670 1255 gela ine 25% 0.498 0.000 8399 2151 gela ine 30% 0.405 0.000 9687 3158 gela ine 50% 0.265 0.000 40 807 18 042 Fig. 8. Changes in in e ac ions wi h gela ine concen a ion. R. Mouˇ cka e al. Mechanics o Ma e ials 177 (2023) 104559 6 exci a ion de o ma ion and imagina y pa (G′′) gi es loss modulus o iscous pa esponsible o ene gy dissipa ion. Al hough he c eep es da a is well i ed wi h i , he ac ional Maxwell model ailed o p o ide sa is ac o y esul s o equency measu emen o complex shea modulus and he e o e FPT model (Eq. (5)) was employed wi h much be e esul s e en hough bo h compo- nen s (G′and G′′) o he complex shea modulus we e app oxima ed wi h he same se o pa ame e s. Ex ac ed pa ame e s a e gi en in Table 1. E en hough he e y same se o pa ame e s o a conc e e model de e mined om c eep da a i ing should wo k e en o oscilla o y da a, in eali y his is un o una ely no he case and e en hough he model is capable o i ing dynamic heological da a he pa ame e s a e qui e di e en . This is a leas pa ially due o nume ous e ec s con- nec ed wi h sample geome y impe ec ions, loading and no mal o ce applica ion p io o measu emen . Mo eo e , he measu emen egion is always es ic ed ei he in ime ( o c eep) o o cing angula equency ( o oscilla ions) he esul o which being misma ch be ween i ing pa ame e s in each me hod. This anspi es e en in he case o complex modulus G* i ing, whe e basically wo cu es need o be i ed simul aneously. Ne e heless, a leas quali a i ely he ac ional model cap u es i s cha ac e . Spec a o bo h, po cine skin (Fig. 10) and gela in (Fig. 11), sys ems exhibi highe alues o s o age modulus o e loss modulus wi h an δ= G′′/G′o abou 0.3 and 0.05 o skin and gela in, espec i ely. Pa icu- la ly in case o gela in, such low an δ alues indica e well-de eloped ne wo k, in which elas ici y domina es o e iscous low, o which hey a e ypical (Van den Bulcke e al., 2000). S o age and loss moduli o po cine skin bo h inc ease g adually wi h angula equency which is cha ac e is ic o complex ma e ial comp ising s uc u al elemen s, e.g. ibe s, ib ils and collagen Fig. 9. Fi ing pa ame e s o ac ional Maxwell model o gela in (symbols) and po cine skin (line). 2. Oscilla o y es Fig. 10. Mechanical spec um o a po cine skin; expe imen al da a i ed wi h FPT model (le ) and i ed FPT model ex apola ed in o wide equency domain ( igh ). Fig. 11. Mechanical spec a o gela in i ed wi h FPT model; legend: gela in concen a ion. R. Mouˇ cka e al. Mechanics o Ma e ials 177 (2023) 104559 7 molecules, wi h di e en elaxa ion imes. On he o he hand, gela in exhibi s pla eau o gen ly ising equency dependence o G′ o low (10% and 20%) o mo e concen a ed (30% and abo e) samples, espec i ely. An excep ionally good i o me- chanical spec a wi h FPT model (Fig. 11) enables one o use he i ing pa ame e s (Table 1) and p edic he beha io o he ma e ial ou side he measu ed equency domain. Thus, mas e cu es o a iously concen a ed gela in samples we e ob ained (Fig. 12), om which i can be seen ha he cha ac e o he gela in’s spec um conside ably changes be ween concen a ions 20 and 30%. A well-de ined G′′ peak a low equencies (10 −3 o 10 −2 ad s −1 ) obse ed o 10 and 20% is connec ed wi h elaxa ion o polyme chains’ disengagemen ( ep a- ion), and ma ks c oss-o e equency a which gela in u ns om liquid (G′′ >G′) in o elas ic s a e (G′>G′′). Fo mo e concen a ed gela in ( om 30% on) G′>G′′ holds always as he densi y o c osslinks is oo high o ma e ial o exhibi iscous low e en a e y low equencies. This is also e lec ed in he alues o FPT model pa ame e s wi h he sp ing-po in se ies unc ioning a i s as a dashpo (λ =1) wi h la ge iscosi y ( η ≈5 ×10 5 Pa s) o 10% gela in, while exhibi ing mo e elas ic- like beha io o 30, 40 and 50% gela in (λ ≈0.25). Ins an aneous elas ici y p esen in he sys em due o physical c oss- links in gela in’s chains is modelled by one o he sp ing-po s in pa allel ( α =0). I s “modulus” ( η 1 ) was seen o inc ease om 1400 Pa o 12 500 Pa e lec ing ise in he densi y o physical c osslinks in he sys em wi h gela in concen a ion (Fig. 13). A undamen al opological pa ame e cha ac e izing he polyme ne wo k in oduced by de Gennes is he en anglemen molecula weigh , M e , which is de ined as an a e age molecula weigh be ween opolog- ical cons ain s (Degennes, 1971). I is ypically in e ed om pla eau modulus G0 N o en ob ained as a alue o s o age modulus a equency o loss modulus minimum (G0 N=G′( ω )G′′→ min) (Liu e al., 2006). Rela- ionship be ween G0 N and M e is gi en as G0 N=4 5 ρ RT Me ,(9) whe e ρ [kg m −3 ] is densi y, R [J mol −1 K −1 ] mola gas cons an and T [K] he modynamic empe a u e. A ailabili y o wide ange mechanical spec a o gela in, owing o FPT model p edic ions, allows one o iden i y G′′ minima and subse- quen ly ex ac G0 N and calcula e M e . Polyme ne wo k can be desc ibed in e ms o densi y o i s c osslinks ρ x [mol m −3 ] which can be es ima ed as: ρ x=|G∗| RT (10) and hen used o he de e mina ion o mesh size: ξ= 6 πρ xNA 3 √,(11) whe e N A =6.022 ×10 23 [mol −1 ] is A ogad o’s cons an . En anglemen molecula weigh dec eases wi h gela in’s concen a- ion (Fig. 13) as in he mo e concen a ed gela in he e is highe inci- dence o in e pa icle in e ac ion leading o c ea ion o junc ion zones (Fig. 8). These a e egions o pa ially e o med iple-helices (o gela- ins’ polypep ide chains) s abilized by weak in e ac ions (H-bonds), which ac as physical c osslinks (Joly-Duhamel e al., 2002). Highe helix concen a ion consequen ly esul s in gela in’s be e mechanical p ope ies mani es ed by highe shea and comp ession (shown) la e on modulus. C. Comp ession es To expand he gela in’s mechanical beha io o he mac oscopic scale o la ge de o ma ions a comp ession es o po cine skin and gela in samples was ca ied ou . Con a y o he ensile es o he comp ession es i is a he di icul o p ecisely iden i y a which poin he b eakdown o he sample occu s. The e o e, only ini ial phase, i.e. loading (comp ession) o he gela in samples wi h p essu e, which has been in en ionally clipped a he s ain o 60%, alue a ainable o all samples, is p esen ed (Fig. 15). Po cine skin exhibi s con exly shaped comp ession es cu es (Fig. 14) wi h wo linea egions. One is loca ed a he beginning ( ε = 0–20%) o he es wi h comp ession modulus E 0 =1.75 kPa while he o he appea s a mo e subs an ial comp ession de o ma ions (subs an- ial sca e o he onse o he second phase be ween he ou measu ed po cine skin samples is caused by impe ec na u e o skin samples, whose op and bo om a e no pe ec ly pa allel) and has modulus o abou 30 kPa (E 1 ). Bo h alues app oxima ely co espond o skin’s shea modulus |G*| de e mined a he ends o in es iga ed shea a e ange in heological measu emen s. Comp ession es s o gela in samples show egion o linea elas ici y o all in es iga ed samples (10–50 w % gela in concen a ion) up o abou 25% o s ain abo e which he de i a i e d σ /d ε inc eases (Fig. 15). Comp ession modulus, E, o he indi idual samples was calcula ed om he linea egion, namely o ε ∈(5%;10%), as a alue o he de i a i e o he expe imen al enginee s ess-s ain cu es: E=d σ d ε (12) Ex ac ed modulus alues as well as he comp ession es cu es bo h show ha in e ms o comp ession de o ma ion e en lowes concen- a ed gela in in es iga ed (10%) has sligh ly be e mechanical p op- e ies han po cine skin wi h mo e concen a ed gela in samples clea ly exceeding comp ession modulus o po cine skin. Fig. 12. Ex apola ion o i ed da a wi h FPT model in o wide equency domain; legend: gela in concen a ion. R. Mouˇ cka e al. Mechanics o Ma e ials 177 (2023) 104559 8 E en hough gela in’s comp ession modulus, E, is abou one o de o magni ude smalle compa ed o he absolu e alue o complex shea modulus, |G*|, measu ed in oscilla ions a equency o 10 ad s −1 , i s concen a ion dependence has a qui e simila sligh ly con ex cha ac e (Fig. 16), which suppo s he idea o g adual ein o cemen o he gela in polyme ne wo k wi h c osslinks due o mo e equen gela in in- e ac ions as gela in concen a ion is inc eased. Despi e di e en de o ma ion mode (shea e sus comp ession) and magni ude (small e sus la ge) he undamen al e ec o gela in concen a ion on i s mechanical p ope ies is main ained. To comp ehensi ely assess iscoelas ic p ope ies o gela in as a possible su oga e o na u al skin i can be claimed ha al hough much simple in e ms o i s s uc u al complexi y, gela in o app op ia e concen a ion esponds o bo h s a ic (c eep) and dynamic exci a ion in many ega ds in a way much simila o a eal skin. This concen a ion o he used bo ine gela in was iden i ied as 40%, a which complex dy- namic modulus G* a ele an angula equencies (5–10 ad s −1 co - esponding o walking– unning (Hol e al., 2008)) as well as pa ame e s o FMM exhibi ed highes ag eemen . IV. Conclusions. Expe imen ally de e mined iscoelas ic p ope ies o po cine skin using o a ional heome y (c eep and oscilla o y es ) we e modelled wi h gene alized ac ional Maxwell model ( wo sp ing-po s in se ies) in o de o imp o e upon accu acy and comp ehensibili y o he i o e ed by adi ional iscoelas ic models used. Fi ing e ealed ha e en a special case o ac ional Maxwell model comp ising a sp ing-po and a s anda d sp ing in se ies is capable o e y p ecise app oxima ion o he Fig. 13. E olu ion o gela in’s ne wo k cha ac e is ics (c osslinking densi y/black squa es/, mesh size/blue ci cles/and en anglemen molecula weigh /black c osses/) wi h concen a ion. (Fo in e p e a ion o he e e ences o colou in his igu e legend, he eade is e e ed o he Web e sion o his a icle.) Fig. 14. Comp ession es o po cine skin. Fig. 15. Comp ession es o gela in; legend: gela in concen a ion. Fig. 16. Comp ession and shea modulus e sus gela in concen a ion. R. Mouˇ cka e al. Mechanics o Ma e ials 177 (2023) 104559 9 c eep es da a. P og essi e dec ease o α exponen cap u es gela in ansi ion om iscous-like o mo e elas ic-like s a e wi h i s concen- a ion in he sys em. This inding is u he suppo ed by g adual in- c ease o sp ing cons an k in he model as well as bo h comp ession and shea moduli inc easing wi h gela in con en . On s uc u al le el his ansi ion o iscoelas ici y mo e owa ds elas ic pa o he spec um s ems om highe densi y o in e molecula in e ac ions be ween gela in molecules. Fo i ing esponse o dynamic exci a ion (mechanical spec a) o ma e ial a mo e complex ac ional model (Poyn ing–Thomson) was employed, which was able o i gela in excep ionally well wi h model’s pa ame e s con i ming an inc ease o i s elas ici y wi h gela in concen- a ion. This was also con i med by g adual dec ease o calcula ed en anglemen molecula weigh o gela in wi h concen a ion. Imp o emen o mechanical p ope ies, namely comp ession modulus, wi h concen a ion o gela in was obse ed e en a la ge scale de- o ma ions du ing comp ession es . E en hough gela in’s po ous in e nal s uc u e is much simple compa ed o hie a chical s uc u e o be ound in biological sys em o po cine skin, i ce ainly has he po en ial o model he skin in e ms o mechanical p ope ies as has been shown o small de o ma ions using heome y. Pe o med la ge scale de o ma ion comp ession es con- i ms his inding wi h e en low concen a ed gela in exhibi ing highe comp ession modulus han po cine skin. Thus, om he iewpoin o iscoelas ic p ope ies, gela in and i s modi ica ions (e.g. h ough c osslinking) can be seen as sui able ma e ial o u he es ing (e.g. la ge de o ma ion iscoelas ici y) in he ield o skin subs i u es. CRediT au ho s a emen Robe Mouˇ cka: Concep ualiza ion, Me hodology, Fo mal analysis, In es iga ion, W i ing – O iginal D a , Visualiza ion. Michal Sedlaˇ cík: Concep ualiza ion, In es iga ion, W i ing – O iginal D a , W i ing - Re iew & Edi ing, Supe ision, Funding acquisi ion. Zuzana P´ a íko ´ a: Fo mal analysis, W i ing - Re iew & Edi ing. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing inancial in e es s o pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape . Da a a ailabili y Da a will be made a ailable on eques . Acknowledgemen s The au ho s g a e ully acknowledge p ojec DKRVO [RP/CPS/2022/ 007] suppo ed by he Minis y o Educa ion, You h and Spo s o he Czech Republic. 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