Adhesion toughness of multilayer graphene films

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1 Loughborough University Institutional eository Adhesion toughness of multilayer grahene films This item was submitted to Loughborough University's Institutional eository by the/an author. Citation: WOOD, J.D., HAVEY, C.. and WA, S., 7. Adhesion toughness of multilayer grahene films. ature Communiations, 8, 95. Additional Information: This is an Oen Aess Artile. It is ublished by ature ublishing rou under the Creative Commons Attribution 4. International Liene (CC Y). Full details of this liene are available at: htt://reativeommons.org/lienses/by/4./ etadata eord: htts://dsae.lboro.a.uk/34/777 Version: ublished ublisher: ature ublishing rou The Author(s) ights: This work is made available aording to the onditions of the Creative Commons Attribution 4. International (CC Y 4.) liene. Full details of this liene are available at: htt://reativeommons.org/lienses/ by/4./ lease ite the ublished version.

2 ATICLE DOI:.38/s w OE Adhesion toughness of multilayer grahene films Joseh D. Wood,, Christoher. Harvey & Simon Wang, Interfae adhesion toughness between multilayer grahene films and substrates is a major onern for their integration into funtional devies. esults from the irular blister test, however, dislay seemingly anomalous behaviour as adhesion toughness deends on number of grahene layers. Here we show that interlayer shearing and sliding near the blister rak ti, aused by the transition from membrane strething to ombined bending, strething and through-thikness shearing, dereases frature mode mixity II / I, leading to lower adhesion toughness. For silion oxide substrate and ressure loading, mode mixity dereases from 3% for monolayer films to 3% for multilayer films, ausing the adhesion toughness to derease from.44 J m to.365 J m. The mode I and II adhesion toughnesses are found to be I =.3 J m and II =.666 J m, resetively. With oint loading, mode mixity dereases from 74% for monolayer films to 6% for multilayer films, while the adhesion toughness dereases from.543 J m to.438 J m. Deartment of Aeronautial and Automotive Engineering, Loughborough University, Loughborough, Leiestershire LE 3TU, UK. Deartment of ehanial Engineering, Imerial College London, London SW7 AZ, UK. 3 Shool of ehanial and Equiment Engineering, Hebei University of Engineering, Handan 5638, China. Corresondene and requests for materials should be addressed to J.D.W. ( joseh.wood@imerial.a.uk) or to C..H. ( .m.harvey@lboro.a.uk) or to S.W. ( s.wang@lboro.a.uk) ATUE COUICATIOS 8: 95 DOI:.38/s w

3 ATICLE ATUE COUICATIOS DOI:.38/s w Koenig et al. suggested that one ossible ause for the large derease in adhesion toughness for multilayer grahene films in omarison to monolayer ones is the roughness of the substrate surfae. ultilayer grahene films may onform less well to the substrate than monolayer ones. Koenig et al. made roughness measurements on the to surfaes of grahene films and found a large dro in roughness from monolayer to two layer; however, they also found a large dro from two layer to three layer. This suggests that the roughness of the substrate surfae is unable to exlain the large derease in adhesion toughness. To investigate the effet of interfae roughness further, ao and Huang argued that the rough surfae of silion oxide auses grahene films to bend; hene, the total adhesion energy onsists of both van der Waals interation energy and a negative ontribution of bending strain energy. y assuming the substrate to have a sinusoidal rough surfae, they attemted to alulate the adhesion energy. They onluded that the large derease in adhesion toughness from monolayer to multilayer grahene films is due to the inrease in bending strain energy aused by the large inrease in the bending stiffness. Jiang and Zhu 3 measured the van der Waals interation energy between monolayer grahene films and silion oxide substrate using atomi fore mirosoy. Their measurements show, however, that the roughness inreases the interation energy. In ontrast, He et al. 4 studied the large derease in adhesion toughness from another ersetive. They roosed that the total adhesion energy onsists of both van der Waals interation energy and residual in-lane strain energy due to lattie mismath strain at the grahene film-silion oxide interfae. Their results show that the van der Waals interation energy remains nearly the same for grahene films with any number of layers, but that the residual in-lane strain energy and Young s modulus derease sharly from monolayer to multilayer grahene films. Koenig et al., however, reorted onvining exerimental results that show a onstant Young s modulus. This observation rovided a solid foundation for their subsequent adhesion toughness alulations using a ontinuum mehanis aroah. Koenig et al. also suggested ossible sliding between grahene layers in multilayer grahene films. The resent work follows Koenig et al. s ontinuum mehanis aroah but with onsideration for the interlayer shearing and sliding effet. Furthermore, the resent work onsiders the effet of shearing and sliding on the frature mode mixity. This is an imortant onsideration, sine interfae adhesion toughness is not a urely intrinsi material roerty, but instead also deends on the mode mixity. ote that the frature mode mixity and the interlayer shear and sliding effet are not onsidered anywhere in the urrent analytial mehanial models and we argue that this has aused onfusion when alulating adhesion toughness. Cao et al., did, however, reently reort studies on adhesion toughness between hotoresist films and oer substrates using blister tests and the finite element method. Two tyes of film are onsidered: One is ure hotoresist film and the other is ombined hotoresist film and a monolayer grahene. ode mixity is onsidered by using ohesive zone modelling. The resent work shows that adhesion toughness is mode mixity deendent, and that interlayer shearing and sliding near the blister rak ti, aused by the transition from membrane strething to ombined bending, strething and throughthikness shearing, dereases the mode mixity II / I, onsequently reduing the adhesion toughness. y onsidering the interlayer shearing and sliding effet, the mode I and mode II toughnesses are shown to be indeendent of the number of grahene layers. Aounting for the interlayer shearing and sliding effet on the frature mode mixity exlains the behaviour reorted in the literature, where adhesion toughness measurements seemingly deend on the film thikness (i.e., the number of grahene layers). One the mode I and mode II adhesion toughnesses have been found, the linear failure riterion an aurately determine the adhesion toughness under general loading onditions for real-world aliations of grahene filmsubstrate systems. esults Cirular blister test under a ressure load. Figure shows two tyes of irular blister test to determine the adhesion toughness of mono- and multilayer grahene films. The blister has a rak ti radius, the thikness of the monolayer grahene is t, n reresents the number of grahene layers and the Young s modulus of grahene is E. In Fig. a, the blister is under ressure loading. Aording to Jensen 3,4, the defletion δ at the entre of the blister in the membrane limit is =3 δ ¼ f ðνþ 4 ðþ net in whih is the ressure load and f(ν) is given by Stora kers 5 as f ðνþ ¼ : ð νþ =3 ðþ 7 ν The oeffiient of.9635 in Eq. () is introdued in the resent work to ahieve the benhmark value of f(/3) =.645 obtained by Jensen 3 sine Stora kers formula 5 f(ν) = [3( ν)/(7 ν)] /3 is aroximate. The bending moment er unit width, in-lane fore er unit width, and shear fore er unit width, at the blister rak ti 3,4 an be exressed in the following forms, a b nt δ nt δ Fig. Cirular blister tests to determine the adhesion toughness of mono- and multilayer grahene films. a A blister under a ressure load. b A blister under a oint load ATUE COUICATIOS 8: 95 DOI:.38/s w

4 λ bη ATUE COUICATIOS DOI:.38/s w ATICLE a easurement number easurement number ρ easurement number d (J m ) II I Linear failure riterion ρ e (a) layer.5. layers , 4 and 5 layers δ (μm) f (μm) layers 4layers 5layers.5 layer layers δ (μm) onolayer grahene Two-layer grahene Three-layer grahene Four-layer grahene Five-layer grahene Theory Fig. Delaminating grahene films under a ressure load. a lots showing alulated values of the interlayer shearing and sliding arameter λ (a), the ratio η = S / J (b) and the frature mode mixity ρ = II / I () based on the measured values of and δ, and the material roerties of monolayer grahene. d lot showing adhesion toughness vs. the frature mode mixity ρ. e, f lots showing the measured and theoretial relationshis between the ressure load (e) and the blister radius (f) vs. the defletion at the entre of the blister δ resetively (Sulementary Fig. ): ¼ nt netδ = ð3þ 4 3 ð ν ÞφðνÞf ðνþ ¼ netδ = φðνþ f ðνþ ¼ in whih the oisson s ratio ν-deendent arameter φ(ν) is :78 þ :636ν φðνþ ¼ ð Þ=3 ð6þ 6 ½ ð ν ÞŠ =3 At this stage, the effet of interlayer shearing and sliding on the frature mode mixity an be introdued. An introdution to mixed-mode artition theory is given in Sulementary ote. This theory is then develoed and extended for the thin film blister test in Sulementary ote. The mode I and II energy release rates (Es) are obtained as 6 I ¼ :67 δ 8 II ¼ :3773 δ 8 ð:7578 :49ν þ λþ φðνþf ðνþ ð:4 þ :358νÞ φðνþf ðνþ and the mode mixity ratio ρ = II / I as :4 þ :358ν ρ ¼ :659 ð9þ :7578 :49ν þ λ The λ arameter in Eqs. (7) and (9) reresents the interlayer ð4þ ð5þ ð7þ ð8þ shearing and sliding effet at the blister rak ti, whih is given as λ ¼ λsðnþ ðþ y using Eq. () and Sulementary Eq. (5) in onjuntion with mixed-mode artition theory 6, the arameter λ in Eq. () an have the following alternative exressions: λ ¼ ζðνþ net where =3 δ δ =4 ¼ ζðνþ ¼ ζðνþ ðþ f ðνþ f ðνþnet ζðνþ ¼ 3:44 ν = φ ðþ In the ase of monolayer grahene films, the shear fore in Eq. (5) makes no ontribution to the E in the membrane limit beause there is no interlayer shearing and sliding. In the ase of multilayer grahene films, interlayer shearing and sliding ours near the blister rak ti, aused by the transition from membrane strething to ombined bending, strething and through-thikness shearing. Consequently, interlayer shearing and sliding ativates the shear fore in Eq. (5). Its ation is introdued through the λ arameter in onjuntion with the interlayer shearing and sliding fator S(n), whih is assumed to take the following form: Sn ð Þ ¼ e n ð3þ A more thorough and detailed exlanation for the origin of λ is given in Sulementary ote. The total E is simly the sum of the mode I E I in Eq. (7) and the mode II E II in Eq. (8). The mode mixitydeendent adhesion toughness an now be determined by using the mode I and mode II adhesion toughnesses and a linear failure riterion in whih = ( + ρ)/(/ I + ρ/ II ). ote that I and II are intrinsi interfae material roerties but is not. One major aim of the resent study is to determine ATUE COUICATIOS 8: 95 DOI:.38/s w 3

5 ATICLE ATUE COUICATIOS DOI:.38/s w Table Average adhesion toughness of multilayer grahene films J (J m ) (J m ) ρ = II / I resent mehanial model Koenig et al. resent mehanial model Koenig et al. resent mehanial model Koenig et al. onolayer ultilayer values for I and II based on Koenig et al. s exerimental results. One these two roerties are known, the adhesion toughness under other loading onditions an be readily alulated. The total E, whih inludes the ontributions from the rak ti bending moment in Eq. (3), the in-lane fore in Eq. (4), and the rak ti shear fore in Eq. (5), an also be written in terms of the J omonent from Jensen s work 3,4, whih does not aount for the interlayer shearing and sliding effet, and the additional interlayer shearing and sliding omonent from the resent work s, as follows: ¼ J þ S ¼ J ð þ ηþ ð4þ Jensen s J omonent an be alulated as 3,4 J ¼ ζðνþ 4 4 =3 ¼ ζðνþ net δ 4 net f 4 ¼ ζðνþ δ ðνþ f ðνþ in whih the arameter ζ is ζðνþ ¼ ð νþφ þ 8φ ð5þ ð6þ The ratio η = S / J is λλþ:56 :858ν η ¼ ð Þ ð7þ :76 þ :835ν þ :543ν Koenig et al. found that Et = 347 m with E Ta. Taking oisson s ratio ν =.6 (following ref. ), then Eqs. (), (6), () and (6) give f(.6) =.697, φ(.6) =.399, ζð:6þ ¼ :89 and ζ(.6) =.45, resetively. Then, the essential equations above, namely Eqs. (5), (7), () and (9), beome, resetively J ¼ : =3 δ 4 ¼ :978nEt ¼ :657δ ð8þ net η ¼ :558λð:47 þ λþ ð9þ λ ¼ :89 =3 ¼ :738 δ δ =4 ¼ :75 ðþ net net :5 ρ ¼ ð:7349 þ λþ ðþ ote that Koenig et al. used J =.655 δ, whih is very lose to Eq. (8) in the resent work. Furthermore, by ombining either Eqs. (8) and (5), or Eqs. (9), () and II ( + /ρ) = J ( + η), then II ¼ :6986 J ðþ In the following, the ressure, the entral defletion δ and the radius of the multilayer grahene film blisters are taken from figures in Koenig et al. s Sulementary Information. The results are resented in Fig.. In Fig. a, the alulated values of λ, η and ρ, resetively, for monolayer and multilayer grahene films are lotted based on the measured values of and δ from Koenig et al.. In Fig. d, the alulated adhesion toughness is lotted vs. the frature mode mixity ρ. In Fig. e, f, omarisons are made between the measured values of, δ and, and the resent mehanial model for grahene films with different numbers of layers. ote that the Theory urve in Fig. e is obtained by substituting Eqs. (7) and (8) into the linear failure riterion and solving for ; then for Fig. f, use of Eq. () reasts the theory in terms of and δ. There is generally very good agreement between the resent mehanial model and the exerimental measurements. The numerial data for Fig. is also reorded in Sulementary Tables 5 for mono-, two-, three-, four- and five-layer grahene film blisters, resetively. To kee onsisteny with Koenig et al., results are alulated using the ressure and the entral defletion δ meaning that J =.657 δ and λ ¼ :75½δ= ðnetþš =4 from Eqs. (8) and () are the forms that used. For the urose of omleteness and omarison, results have also been alulated using the alternative exressions for λ in Eq. (), namely λ ¼ :89½ = ðnetþš =3 and λ ¼ :738δ=. The results are resented in Sulementary Tables 6 and Sulementary Tables 5, resetively. There is generally good agreement between the results when using the different exressions for λ. The values of the λ arameter, based on Koenig et al.'s measurements, are reorded in Sulementary Tables 5. There is a large derease from monolayer to two-layer grahene films and then only a small derease from two-layer to three-layer grahene films. For the three-, four- and five-layer grahene films, the values of the λ arameter are very lose to eah other. This shows the tyial interlayer shearing and sliding behaviour. The average adhesion toughnesses are =.44,.36,.389,.348 and.359 J m for the mono-, two-, three-, four- and fivelayer grahene film blisters, resetively, whih orresond to the following mode mixities ρ = II / I =.39,.4,.59,.63 and.7. There is a large derease in mode mixity for two-layer grahene film blisters in omarison to monolayer films, whih results in a large derease in the adhesion toughness. For higher numbers of grahene layers, the adhesion toughness does not hange signifiantly from the two-layer ase as there are no signifiant hanges in mode mixity. An overall average adhesion toughness for multilayer grahene films blisters is =.365 J m with ρ = II / I =.99. These results are shown in Table. ow the mode I and mode II adhesion toughnesses, I and II, are onsidered. He et al. 4 showed that the van der Waals interation energy remains nearly the same for grahene films with any number of layers at.66 J m. This suggests that I and II are the same for interfaes between monolayer grahene films and silion oxide substrates, and between multilayer grahene films and silion oxide substrates. As adhesion toughness is generally very small, a linear failure riterion an rovide an aurate reresentation of the frature mehanis in question 8. 4 ATUE COUICATIOS 8: 95 DOI:.38/s w

6 ATUE COUICATIOS DOI:.38/s w ATICLE a (J m ) II I Linear failure riterion ρ b /(π ) (a) layer 8 6 layers 4 3, 4 and 5 layers δ (μm) (μm) layers 4layers layers δ (μm) 3layers layer Five-layer grahene Theoretial monolayer Theory Five-layer grahene Theory Fig. 3 Delaminating grahene films under a oint load. a lot showing adhesion toughness vs. the frature mode mixity ρ. b, lots showing the theoretial relationshis between the oint load (b) and the blister radius () vs. the defletion at the entre of the blister δ. ote that the average measured value of δ/ =.39 for five-layer grahene is also shown Let subsrits and + reresent values for monolayer and multilayer grahene films, resetively. Substituting the monolayer and multilayer results from Table into the linear failure riterion and solving simultaneously gives þ ρ ρ þ I ¼ ρ þ ρ þ ρþ þ ð þ ρ Þ ¼ :3 J m ð3þ and þ ρ II ¼ ρ þ ¼ :666 J m ð4þ þ ð þ ρ Þ þ ρ þ It is interesting to note that I =.3 J m is very lose to He et al. s 4 theoretial alulation of the van der Waals interation energy at.66 J m. In fat, the van der Waals interation energy is essentially the same in onet as the mode I adhesion toughness. The mode I adhesion toughness I an be determined using atomi fore mirosoy measurements 3 and JK model as I ¼ F adh 3π ti ¼ :98 J m ð5þ where F adh = 378 n is the van der Waals interation fore and ti = 45.4 nm is the radius of the miroshere ti used in the atomi fore mirosoy measurements. It is seen that the measured I =.98 J m is very lose to the resent value of I =.3 J m. In the following setion, the theory develoed above for the irular blister test under ressure loading and the determined values of I =.3 J m and II =.666 J m will be used to redit adhesion toughness under oint loading in order to examine the validity of the aroah. Cirular blister test under a oint load. A blister under a oint load (refs.,3 ) is shown in Fig. b. The mehanial model for it is very similar to the model develoed above for a ressure load. Some essential formulae are reorded here. Fitting a urve to the data in Jensen s 3 Fig. 5 gives φ(ν) as φðνþ ¼ :38ν 3 þ :3ν þ :48ν þ :4 The funtion f(ν) now beomes f ðνþ ¼ = ðφðνþþþφ ðνþ ν ð6þ ð7þ The ressure load an now simly be relaed everywhere with = π. y making this substitution in Eqs. (3) to(5), the mode I and II Es an be obtained as 6 I ¼ :67 δ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :557 ð ν Þφ 3 þ λ 8π φðνþf ðνþ ð8þ II ¼ :3773 δ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ :569 ð ν Þφ 3 ð9þ 8π φðνþf ðνþ In addition, Eqs. () and (5) beome, resetively =3 δ δ =4 λ ¼ ζðνþ ¼ ζðνþ ¼ ζðνþ π net f ðνþ π f ðνþnet ð3þ 4 =3 J ¼ ζðνþ π 4 4 net ¼ ζðνþ net δ 4 δ f 4 ¼ ζðνþ ðνþ π f ðνþ ð3þ Taking oisson s ratio ν =.6, then Eqs. (6), (7), () and (6) give φ(.6) =.4636, f(.6) =.497, ζð:6þ ¼ :33 and ζ (.6) =.3743, resetively. Equations (8) (3) then rodue the following: 4 =3 δ 4 J ¼ :3743 π 4 4 net ¼ :7446nEt ¼ :5 δ π ð3þ η ¼ :4485λð:3 þ λþ ð33þ =3 λ ¼ :33 ¼ :545 δ δ =4 ¼ :9 π net π net ð34þ :964 ρ ¼ ð:549 þ λþ ð35þ II ¼ :889 J ð36þ From Eq. (35), it an be seen that ρ = 7.47 for monolayer grahene under a oint load, whih is muh larger than for the ressure loading ondition at ρ =.39. The adhesion toughness for monolayer grahene under a oint load an be estimated using I =.3 J m, II =.666 J m and a linear failure riterion to be =.543 J m, whih is learly larger than for the ressure loading ase at =.44 J m. ATUE COUICATIOS 8: 95 DOI:.38/s w 5

7 ATICLE ATUE COUICATIOS DOI:.38/s w The adhesion toughness for multilayer grahene under oint loading an be estimated in a similar way as above for ressure loading but now using exerimental data from Zong et al. 5 in whih they used nanoartiles to reate a oint load on five-layer grahene membrane blisters. The blisters tyially ossessed a radius in the range 5 3 nm and entral defletion δ in the range 5 7 nm. They used the formula =.65nEt(δ/ ) 4 with E =.5 Ta and nt =.7 nm. ote that Zong et al. s 5 value for E is half of that used by Koenig et al., and that n 5. Zong et al. reorted the adhesion toughness as =.5 J m meaning that δ/ =.39. When using Koenig et al. s value of E =. Ta, then Eq. (3) gives J =.36 J m, and Eq. (4) gives the total measured adhesion toughness as =.438 J m. ow using ρ =.64 from Eq. (35), the linear failure riterion, and the mode I and mode II adhesion toughnesses, I =.3 J m and II =.666 J m, the redited value of is =.437 J m, whih is extremely lose to measured =.438 J m. It an be seen that the mode mixity lays a key role in determining the adhesion toughness and that the auray of I =.3 J m, II =.666 J m and the linear failure riterion is very good. Figure 3 shows the behaviour of delaminating grahene films under a oint load. Figure 3a follows the same style as Fig. d f; however, the measured data 5 is now only for films with five layers. In artiular, it is seen in Fig. 3 that the measured value of δ/ =.39 is very lose to the theoretial redition of δ/ =.98. Disussion In reent work, (following ref. ), oddeti et al. reorted further studies on the adhesion toughness between monolayer grahene and silion oxide substrates. The adhesion toughness was found to be =.4 J m, whih is signifiantly smaller than =.45 J m, reorted by Koenig et al.. oddeti et al. suggest that the differene arises from the differenes in interfae roerties suh as roughness and hemial reativity between the samles in ref. and the samles in refs.,. In line with this suggestion, the resent work suggests that the redution is aused by redution of the mode I and mode II adhesion toughnesses at the interfae, I and II, whih are now estimated. Taking the Young s modulus and oisson s ratio still as E = Ta and ν =.6, Eq. () gives the mode II E omonents at failure as II = =.68 J m. Then the mode I E omonent at failure is easily obtained as I =.7 Jm. If the ratio between I and II is taken to be the same as that in ref., i.e., II / I =.896, then I and II are then alulated to be I =.3 J m and II =.377 J m. Clearly they are signifiantly smaller than I =.3 J m and II =.666 J m for the samles in ref.. ore information on adhesion toughness of grahene an be found in the latest review aer 3. The methodology develoed above is also alied in the authors reent work (manusrit in review) to determine the mode I and mode II adhesion toughness of thin films by using blister tests. The analytial reditions agree very well with the exerimental results reorted by Cao et al.. It should be noted that a general methodology has been resented, and the substrate should not be restrited to silion oxide substrates. Furthermore, the adhesion energy ommonly used in the literature is generally different from the adhesion toughness unless the mode I adhesion toughness is equal to mode II adhesion toughness, whih is not generally the ase. It is the adhesion toughness that matters for the design of grahene filmsubstrate material systems. Data availability. The authors delare that the data suorting the findings of this study are available within the artile and its Sulementary Information file. eeived: June 7 Aeted: 7 ovember 7 eferenes. Koenig, S.., oddeti,.., Dunn,. L. & unh, J. S. Ultrastrong adhesion of grahene membranes. at. anotehnol. 6, ().. ao, W. & Huang,. Effet of surfae roughness on adhesion of grahene membranes. J. hys. D Al. hys. 44, 45 (). 3. Jiang, T. & Zhu, Y. easuring grahene adhesion using atomi fore mirosoy with a miroshere ti. anosale 7, 76 (5). 4. He, Y., Chen, W. F., Wu, W.., Ouyang,. & Yang,. W. Anomalous interfae adhesion of grahene membranes. Si. e. 3, 66 (3). 5. Zong, Z., Chen, C.-L., Dokmei,.. & Wan, K.-T. Diret measurement of grahene adhesion on silion surfae by interalation of nanoartiles. J. Al. hys. 7, 64 (). 6. Cao, Z. et al. A blister test for interfaial adhesion of large-sale transferred grahene. Carbon 69, 39 4 (4). 7. Yoon, T. et al. Diret measurement of adhesion energy of monolayer grahene as-grown on oer and its aliation to renewable transfer roess. ano Lett., (). 8. Yue, K., ao, W., Huang,. & Liehti, K.. Analytial methods for the mehanis of grahene bubbles. J. Al. hys., 835 (). 9. Wang,., ao, W., Cao, Z., Liehti, K.. & Huang,. umerial analysis of irular grahene bubbles. J. Al. eh. 8, 495 (3).. eorgiou, T. et al. rahene bubbles with ontrollable urvature. Al. hys. Lett. 99, 933 ().. Cao, Z., Tao, L., Akinwande, D., Huang,. & Liehti, K.. ixed-mode tration-searation relations between grahene and oer by blister tests. Int. J. Solids Strut. 84, (6).. Cao, Z., Tao, L., Akinwande, D., Huang,. & Liehti, K.. ixed-mode interations between grahene and substrates by blister tests. J. Al. eh. 8, 88 (5). 3. Jensen, H.. The blister test for interfae toughness measurement. Eng. Frat. eh. 4, (99). 4. Jensen, H.. Analysis of mode mixity in blister tests. Int. J. Frat. 94, (998). 5. Stora kers,. Variation riniles and bounds for the aroximate analysis of lane membranes under lateral ressure. J. Al. eh. 5, (983). 6. Wang, S. & Harvey, C.. ixed mode artition theories for one dimensional frature. Eng. Frat. eh. 79, (). 7. Harvey, C.. & Wang, S. ixed-mode artition theories for one-dimensional delamination in laminated omosite beams. Eng. Frat. eh. 96, (). 8. Harvey, C.., Elett,.. & Wang, S. Exerimental assessment of mixedmode artition theories for generally laminated omosite beams. Comos. Strut. 4, 8 (5). 9. Wood, J. D., Harvey, C.. & Wang, S. artition of mixed-mode fratures in D elasti orthotroi laminated. Comos. Strut. 49, (6).. Wang, S., Harvey, C.. & Wang,. oom temerature sallation of α-alumina films grown by oxidation. Eng. Frat. eh. 78, 4 45 (7).. oddeti,.. et al. ehanis of adhered, ressurized grahene blisters. J. Al. eh. 8, 499 (3).. oddeti,.. et al. rahene blisters with swithable shaes ontrolled by ressure and adhesion. ano Lett. 3, 66 6 (3). 3. Akinwande, D. et al. A review on mehanis and mehanial roerties of D materials rahene and beyond. Extrem. eh. Lett. 3, 4 77 (7). Aknowledgements This work was suorted by the UK Engineering and hysial Sienes esearh Counil (ESC) under grant referene E/958/. Author ontributions S.W. and C..H. oneived the ideas. All authors ontributed to the theory derivations. J.D.W. erformed the alulations and reared the manusrit draft. S.W. and C..H. heked all of J.D.W. s alulations, and reared the final draft of the manusrit. S.W. 6 ATUE COUICATIOS 8: 95 DOI:.38/s w

8 ATUE COUICATIOS DOI:.38/s w ATICLE and C..H. jointly suervised the whole rojet. All authors artiiated in disussions about the rojet. Additional information Sulementary Information aomanies this aer at htts://doi.org/.38/s w. Cometing interests: The authors delare no ometing finanial interests. erints and ermission information is available online at htt://ng.nature.om/ rerintsandermissions/ ublisher's note: Sringer ature remains neutral with regard to jurisditional laims in ublished mas and institutional affiliations. Oen Aess This artile is liensed under a Creative Commons Attribution 4. International Liense, whih ermits use, sharing, adatation, distribution and rerodution in any medium or format, as long as you give aroriate redit to the original author(s) and the soure, rovide a link to the Creative Commons liense, and indiate if hanges were made. The images or other third arty material in this artile are inluded in the artile s Creative Commonsliense, unless indiated otherwise in a redit line to the material. If material is not inluded in the artile screative Commons liense and your intended use is not ermitted by statutory regulation or exeeds the ermitted use, you will need to obtain ermission diretly from the oyright holder. To view a oy of this liense, visit htt://reativeommons.org/ lienses/by/4./. The Author(s) 7 ATUE COUICATIOS 8: 95 DOI:.38/s w 7

9 Sulementary Figures Sulementary Figure Double antilever beam. a, eometry and loading onditions. b, Details loal to the rak ti. Sulementary Figure lister test interfae rak. a, Thin layer on a thik substrate. b, Effetive rak ti fores and bending moments.

10 Sulementary Tables Sulementary Table Adhesion toughness of monolayer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.8a rou average a rou average a rou average a rou average Overall average Koenig et al

11 Sulementary Table Adhesion toughness of two-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a rou average a rou average Overall average Koenig et al Sulementary Table 3 Adhesion toughness of three-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a a rou average Overall average Koenig et al

12 Sulementary Table 4 Adhesion toughness of four-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a a Overall average Koenig et al Sulementary Table 5 Adhesion toughness of five-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a rou average a rou average Overall average Koenig et al

13 Sulementary Table 6 Alternative alulations for the adhesion toughness of monolayer grahene films with based on Koenig et al. s measurements 9 a μm μm. based on resent mehanial model S J II I - J m 3.8a rou average a rou average a rou average a rou average Overall average Koenig et al

14 Sulementary Table 7 Alternative alulations for the adhesion toughness of two-layer grahene films with based on Koenig et al. s measurements 9 a μm μm. based on resent mehanial model S J II I - J m 3.5a rou average a rou average a rou average Overall average Koenig et al Sulementary Table 8 Alternative alulations for the adhesion toughness of three-layer grahene films with based on Koenig et al. s measurements 9 a μm μm. based on resent mehanial model S J II I - J m 3.5a rou average a a rou average Overall average Koenig et al

15 Sulementary Table 9 Alternative alulations for the adhesion toughness of four-layer grahene films with based on Koenig et al. s measurements 9 a μm μm. based on resent mehanial model S J II I - J m 3.5a rou average a a Overall average Koenig et al Sulementary Table Alternative alulations for the adhesion toughness of five-layer grahene films with based on Koenig et al. s measurements 9 a μm μm. resent mehanial model based S J II I J m on 3.5a rou average a rou average a rou average Overall average Koenig et al

16 Sulementary Table Alternative alulations for the adhesion toughness of monolayer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.8a rou average a rou average a rou average a rou average Overall average Koenig et al

17 Sulementary Table Alternative alulations for the adhesion toughness of two-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a rou average a rou average Overall average Koenig et al Sulementary Table 3 Alternative alulations for the adhesion toughness of three-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a a rou average Overall average Koenig et al

18 Sulementary Table 4 Alternative alulations for the adhesion toughness of four-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a a Overall average Koenig et al Sulementary Table 5 Alternative alulations for the adhesion toughness of five-layer grahene films with based on. Koenig et al. s measurements 9 a μm μm based on resent mehanial model S J II I - J m 3.5a rou average a rou average a rou average Overall average Koenig et al. 9.39

19 Sulementary otes Sulementary ote. ixed-mode artition theory During the last deade or so, the authors and their olleagues have develoed an orthogonal ure mode artition methodology for artitioning mixed-mode D interfae fratures in layered omosite materials into their ure mode omonents. One examle of D interfae frature is the irular blister frature of thin films, whih onsists of only the mode I and II frature modes. D interfae frature an be readily reresented by a double antilever beam (DC) of unit width,, as shown in Sulementary Fig.. Its loading onditions onsist of ti bending moments er unit width, and, ti axial fores er unit width, and, and ti through-thikness shear fores er unit width, and. Sulementary Fig. b shows the internal loads at the rak ti and the sign onvention of the interfae normal stress n and shear stress s. Extensive analytial and numerial studies have been arried out to rove the validity of the methodology 3 and various indeendent exerimental test results have been used to assess the methodology 7. It is found to be sound and the develoment is lear and thorough. A detailed exlanation of the methodology 3 even just the asets that are losely-related to the resent work is not ossible here. Therefore, in order to fous on the resent work, only the most essential arts are given for hysial understanding in what follows. ased on the well-known virtual rak losure tehnique and linear elasti frature mehanis, the mode I and mode II energy release rates (Es) an be written as Fodo I lim () dshfsh II lim () in whih F o and d o reresent the rak ti oening fore er unit width and oening dislaement resetively; F sh and d sh reresent the rak ti interfae shearing fore er unit width and dislaement resetively; and reresents the rak extension length. Sulementary Eqs. () and () an be written in the following forms 7 based on D elastiity:

20 I I -D -D -D -D 3-D 3-D 4-D 4-D 5-D 5-D (3) II II -D -D -D -D 3-D 3-D 4-D 4-D 5-D y omaring Sulementary Eq. (3) with Sulementary Eq. (), it is seen that the terms in the first and seond brakets of Sulementary Eq. (3) orresond to F o and d o resetively, whih are linearly roortional to the rak ti loads. This is required by linear elasti frature mehanis. The rak ti loads onsist of the bending moments er unit width, and, the axial fores er unit width, and, and the through-thikness shear fores er unit width, and. The oeffiients I and II are onstants. The arameters i-d and i-d (with i,,3,4, 5 ) are indeendent of the rak ti loads and deendent on the DC material roerties, interfae roerties, frature loation, rak extension size, et. They are alled ure mode II modes for reasons best shown by examle: When the rak ti loading onditions are, D,,,,, or in a T vetor form T D 5-D with the suersrit T denoting transosition, the first braket in Sulementary Eq. (3), orresonding to F o, equals zero and therefore mode I E I ; hene, -D is alled a ure mode II mode due to zero rak ti oening fore. Similarly, (with i,3,4, 5 ) are also alled ure mode II modes due to zero rak ti oening fore. Using the equivalent exlanation, (with i,,3,4, 5 ) are alled ure mode II modes due to zero rak ti i-d oening dislaement. It is worth noting that i-d and i-d (with i,,3,4, 5 ) are different from eah other in the ase of bi-material interfaes beause the material mismath auses a hase differene between the variations of stress and dislaement 8,9,8, and they are also rak ti extension size-deendent 8,9,8. Similarly, y omaring Sulementary Eq. (4) with Sulementary Eq. (), it is seen that the terms in the first and seond brakets of Sulementary Eq. (4) orresond to d sh and F sh resetively. y using the same exlanation as above, (with i,,3,4, 5 ) are alled ure i-d i-d (4)

21 mode I modes due to zero rak ti shearing dislaement, and (withi,,3,4, 5) are alled ure mode I modes due to zero rak ti shearing fore. Again, i-d and i-d (with i,,3,4,5 ) are different from eah other in the ase of bi-material interfaes beause the material mismath auses a hase differene between variations of stress and dislaement 8,9,8 and they are also rak ti extension size deendent 8,9,8. In the ase of homogeneous interfaes, i-d and i-d (with i,,3,4, 5 ) are equal to eah other, and i-d and (with i,,3,4, 5 ) are also equal to eah other beause then there is no hase i-d differene between variations of stress and dislaement 8,9,8 and they are also indeendent of rak ti extension size 8,9,8. Then, Sulementary Eqs. (3) and (4) beome I I (5) -D -D 3-D 4-D 5-D II II (6) -D -D 3-D 4-D 5-D Furthermore, in the ase of isotroi materials, Sulementary Eqs. (5) and (6) redue to 7, where e e I I (7) -D -D 3-D 4-D. The ure modes, i-d and i-d (with i,,3, 4 ), in e II II (8) -D -D 3-D 4-D Sulementary Eqs. (7) and (8) were derived by the authors 7, using a owerful orthogonal ure mode methodology and have been thoroughly verified against numerial simulations (interested readers are advised to read refs. 7 and ). They are reorded below. -D -D -D h h i-d (9) () () 3

22 -D h 3 h if if () -D -D 3-D (3) -D -D -D -D -D -D 3-D (4) -D -D -D -D -D (5) 4-D -D 3-D (6) 4-D -D3-D The thikness ratio of the beams h h is denoted by ; also 6 5 e 3 3 and e with. For through-thikness shear fores at the rak ti, and (with ), the ure modes -D and -D are -D, -D, ex.9866atanh i where (4) are i log (7). The remaining arameters in Sulementary Eqs. (7), (8), (3) and -D I -D, -D -D II -D (8) -D -D 6 -D (9) -D b h E -D 6 -D () -D b h E -D -D b h E () -D -D b h E () -D where b is the unit width of the beam, and E E for lane stress or E E for lane strain, with E being the Young s modulus of the beam and being the oisson s ratio. The two thikness ratio -deendent orretion fators, namely the through-thikness shear 4

23 orretion fator and the ure-mode-ii E orretion fator using the following elegant exressions: ex.3993 i, an be alulated (3) -D C (4) -D F C F.79 ex (5) It is worth noting that in the absene of through-thikness shear fores, and, Sulementary Eqs. (7) and (8) are extremely lose to Suo & Huthinson s D artitions, ; that is, the ure modes in Sulementary Eqs. (9) () are nearly idential to Suo & Huthinson s ure modes, and are just resented in different forms. i Sulementary ote. Develoment of mixed-mode artition theory for thin film blister test In the following, Sulementary Eqs. (7) and (8) are extended to the ase of thin films in the blister test to determine the adhesion toughness, for examle, the adhesion toughness of multilayer grahene films 9. The substrate is treated as infinitely thik and the films as very thin, as shown in Sulementary Fig. a; therefore, the thikness ratio tends to infinity. The authors latest work on the mehanial behaviour of thin film sallation 4 7 shows that exellent agreement is ahieved with exerimental results 4 when the material mismath between a film and its substrate is negleted. Furthermore, in these studies 4,5 slightly worse agreement was found with exerimental results, when the mismath 8,9 was taken into aount. Therefore, the resent work also neglets the material mismath, and Sulementary Eqs. (7) and (8) beome where, I I (6) D 3 D II II (7) D 3 D and, whih are shown in Sulementary Fig. b, are the effetive rak ti bending moment, axial fore and shear fore resetively. The ure modes D and when, based on Suo & Huthinson,, are D 5

24 (8) D, D, h h where h is the film thikness. ote that the authors ure modes D and D in Sulementary Eqs. () and () also give very lose values to Sulementary Eq. (8); however, in the resent work, Sulementary Eq. (8) is used. The oeffiient arameters, I in Sulementary Eq. (6) and II in Sulementary Eq. (7), must be determined: In the absene of through-thikness shear fore, the total E is given by 6 h 3 Eb h (9) In the ase of ure mode D, that is, with D and, Sulementary Eqs. (6), (8) and (9) give 3 6 I.67 Eb h (3) Similarly, in the ase of ure mode D, that is, with D and, Sulementary Eqs. (7), (8) and (9) give 6 II.3773 Eb h (3) The ure modes, 3D in Sulementary Eq. (6) and 3D in Sulementary Eq. (7), must also be determined: y rearranging Sulementary Eqs. (3) and (4), they beome 3 -D 3-D (3) -D -D -D -D -D 3-D (33) -D -D -D -D ote that before Sulementary Eqs. (3) and (33) an be used in Sulementary Eqs. (6) and (7), whih are for thin films, they must be redued to the limit where. To do this, eah of the ratios D, D and -D - D -D in Sulementary Eqs. (3) and - D (33) must also be redued to this limit, and they are now eah onsidered in turn. The ratio in Sulementary Eqs. (3) and (33) when is determined D D first. In the general ase with rak ti moments by and only, the total E is given 6

25 Eb h (34) In the ase of ure mode D, that is, with D and, Sulementary Eqs. (7), (3) and (34) give D D D D Similarly, in the ase of ure mode D, that is, with D and, Sulementary Eqs. (8), (3) and (34) give D D D D When, Sulementary Eqs. (35) and (36) give -D -D.659 ote that the ure modes in Sulementary Eq. (8) (refs.,) are used in deriving Sulementary Eq. (37). When ure modes in Sulementary Eqs. () and (), derived by the authors 7, are used, the ratio beomes whih is very lose -D -D to. 659 in Sulementary Eq. (37). In the resent work, Sulementary Eq. (37) is used as it is believed to be more aurate. The quantities -, D -, D - and D - in Sulementary Eqs. (3) and (33) when D are determined next. When Then, Sulementary Eqs. (7) and () give, Sulementary Eq. (3) gives (35) (36) (37) -D b h E (38) In the ase of ure mode D, that is, with D and, Sulementary Eqs. (8), (3) and (37) give (39) -D 3 b h E In the ase of ure mode D, that is, with D and, Sulementary Eqs. (8), (3) and (37) give D 3 b h E (4) 7

26 When () gives, then C from Sulementary Eq. (5). Therefore, Sulementary Eq. F (4) -D ow, substituting Sulementary Eqs. (7) and (37) (4) into Sulementary Eqs. (3) and (33) gives 3-D and 3-D as ote that as 3-D.63 (4) 3-D, 3-D, h, the effetive rak ti through-thikness shear fore only ontributes to the mode I omonent of the E I. All terms in Sulementary Eqs. (6) and (7) have now been derived. Again, it is imortant to note that the through-thikness shear fore inreases the mode I E omonent, and that this effet itself inreases for thiker films, as shown by Sulementary Eqs. (6) and (4). Therefore, the thiker the film is, the lower the adhesion toughness. Sulementary Eqs. (6) and (7) have also been used 7 to study the adhesion toughness of thin hotoresist films under linear bending with small defletion. The films were exerimentally tested by Cao et al. 7 using the irular blister test. The exerimental results indeed show the redution of the adhesion toughness. ow, Sulementary Eqs. (6) and (7) are alied to study the adhesion toughness of multilayer grahene films, suh as those in Koenig et al. s 9 work. It is noted that films an generally be under bending, strething and shearing; however, when films are in the membrane limit (i.e. only under strething), they are referred to as membranes. In Koenig et al. s 9 work, multilayer grahene films are in this membrane-strething state, and the rak ti fores in Sulementary Eqs. (6) and (7) for a irular blister of radius under a ressure nt 4 nt 4 load are then given by 5,6 3 net nt net f net 3 f (43) 3 net net net (44) f f (45) 8

27 ote that a irular blister is now seifially studied; therefore b as the fores in Sulementary Eqs. (43) (45) are er unit width. The quantities n, E and t reresent the number of grahene layers, the Young s modulus, and the thikness of monolayer grahene, resetively. The entre defletion, and the oisson s ratio -deendent arameters, and, are given in Eqs. (), () and (6) in the main artile. ote that various exressions for f and f are reorted in literature 8 due to different aroximations being used in their derivations. Jensen s 5,6 total E results, however, are very lose to Henky s 9, as shown in the main artile. Furthermore, Jensen s 5,6 total E results for monolayer membranes agree very well with Wang & Tong s 3 values from finite element simulations. Therefore, based on these onsiderations, Jensen s 5,6 exressions for f and, along with Sulementary Eqs. (43) (45) are used in the resent work. Substituting Sulementary Eqs. (43) (45) into Sulementary Eqs. (6) and (7) gives where I 6.67 E 6.67 E II nt nt f net 4 net f E E nt nt f net 4 net f (46) (47) 9

28 nt 3 43 / (48) with and S( n) S( n) (49) S 3-D 3-D (5) n n e (5) The origin of the arameter is obvious, but the origin of the fator S n needs to be exlained. onolayer films are onsidered first. First, some ertinent observations: As mentioned earlier, Jensen s 5,6 total E results for monolayer membranes agree very well with Wang & Tong s 3 values from finite element simulations. Jensen, however, only inluded the ontributions from the bending moment and the axial fore, as given by Sulementary Eqs. (43) and (44) resetively. This indiates that the through-thikness shear fore, given by Sulementary Eq. (45), does not ontribute to E for monolayer membranes. Furthermore, in other work by the authors 7, two senarios are onsidered using the methodology develoed in the resent work: () linear bending of monolayer films at small defletion, inluding the through-thikness fore, and () membrane strething of monolayer films at large defletion without inluding the through-thikness fore. The analytial reditions for the adhesion toughness between hotoresist films and oer substrates are in exellent agreement with the exerimental results 7. The thiknesses of the hotoresist films are μm (for membrane strething at large defletion), and 3 μm and 6 μm (both for linear bending at small defletion). This indiates that the through-thikness shear fore has no effet on E for membrane strething of monolayer films at large defletion, while it does have effet on E for linear bending of monolayer films at small defletion. ow, the exlanation for these observations is given: In the ase of linear bending at small defletion, through-thikness shear strain is rodued by the through-thikness shear fore. They together result in through-thikness shear strain energy and ontribute to the E at rak ti. In the ase of membrane strething at large defletion, there is no throughthikness shear fore in the membrane blister resulting in no through-thikness strain. Although transition from membrane strething to ombined bending, strething and throughthikness shearing ours near the rak ti, the through-thikness shear strain energy at rak