Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin,

Transcription

1 Buckling mode of elatic thin film on elatic ubtrate Haixia Mei and Rui Huang * Department of Aeropace Engineering and Engineering Mechanic, Univerity of Texa, Autin, TX Jun Young Chung and Chritopher M. Stafford Polymer Diviion, National Intitute of Standard and Technology, Gaitherburg, MD Hong-Hui Yu Department of Mechanical Engineering, The City College of the City Univerity of New York, New York, NY Abtract Two buckling mode have been oberved in thin film: buckle-delamination and wrinkling. Thi Letter identifie the condition for electing the favored buckling mode for elatic film on elatic ubtrate. Tranition from one buckling mode to another i predicted a the tiffne ratio between the ubtrate and the film or the interfacial defect ize varie. The theoretical reult are demontrated experimentally by oberving the coexitence of both buckling mode and mode tranition in one film-ubtrate ytem. * Electronic mail: ruihuang@mail.utexa.edu 1

2 Thin film material are ued in a wide range of application including microelectronic, coating, and medical device. A frequent failure mechanim in thee material i buckling of the thin film, reulting in interfacial delamination and fracture [1,2]. Recently, undertanding of buckle pattern ha alo led to application in metrology [3-5], tretchable interconnect [6,7], and optical grating [8]. Previou tudie on thin film buckling have focued on one of the two buckling mode, buckle-delamination or wrinkling, a illutrated in Fig. 1. The former conider buckling of the film when it i partly delaminated from the ubtrate [9,10], while the latter aume no delamination a the ubtrate deform coherently with the film [11-13]. The characteritic of both buckling mode are often obervable, with localized pattern for buckledelamination (e.g., telephone cord bliter) and homogeneou pattern for wrinkling. One of the unanwered quetion i: what determine the election of either buckling mode for a given material ytem? Intuitively, buckle-delamination typically occur with relatively tiff ubtrate and wrinkling occur only when the ubtrate i very oft. However, a quantitative criterion i lacking. In thi Letter, we develop a theoretical undertanding of the buckling mode election for elatic thin film on elatic ubtrate. Thi i demontrated experimentally by oberving both the buckling mode and a mode tranition in a ingle film-ubtrate ytem with varying tre and interfacial defect. A a tre-driven intability, the tudy of buckling can be traced back to the claical Euler buckling of an elatic column. The fundamental buckling mode minimize the elatic train energy and depend on the type of contraint. For a thin elatic film bonded to a thick elatic ubtrate, the buckling intability i contrained by the ubtrate. Without delamination, buckling of the film (i.e., wrinkling) require coherent deformation of the ubtrate, which i poible only 2

3 when the ubtrate i relatively compliant. Baed on an energetic analyi [11-13], the critical tre for wrinkling i: 2 / 3 E f 3 4 E σ w =, (1) E f where E f and E are the plane-train moduli of the film and ubtrate, repectively. When the compreive tre in the film, σ > σ w, the film buckle pontaneouly, forming wrinkle throughout the film urface. A particular wrinkle wavelength i etablihed to minimize the total elatic energy in the film and the ubtrate [11-13]. The effect of ubtrate and film thicknee on the wrinkling tre have alo been tudied previouly [14,15]. For tiff ubtrate, buckling deformation of the film i highly contrained, leading to high critical tree for wrinkling. However, the ubtrate contraint may be locally mitigated by interfacial defect that lead to partial delamination of the film. In thi cae, the delaminated portion of the film buckle, which in turn drive growth of delamination through interfacial fracture [9,10]. The co-development of buckling and delamination lead to abundant bliter pattern uch a telephone-cord bliter. Compared to wrinkling, the buckle-delamination pattern are typically localized and enitive to interfacial defect [16,17]. Early tudie of buckle-delamination often aumed a fixed-end condition at the edge of delamination, which eentially neglected the effect of elatic deformation in the ubtrate. Under uch a condition, the critical tre for the onet of buckling i identical to that for a free-tanding heet [9], i.e., 2 h 2 B0 = π E f σ, (2) 12 b 3

4 B where h i the film thickne and b i the half width of the delamination. The buckling tre σ B0 i independent of the ubtrate propertie but depend on the relative ize of the interfacial delamination, b/h. Recent tudie [18-20] have hown that the buckling tre can be ignificantly lower than that predicted by Eq. (2) when the elatic deformation of the ubtrate i conidered, epecially for compliant ubtrate. Yu and Hutchinon [20] derived an implicit expreion σ σ B0 B tan π σ B σ B0 2 πh a = 12 12b b / h + a 11 a 22, (3) where a 11, a 22, and a 12 are determined numerically, either by direct finite element calculation or by olving an integral equation, a dimenionle pring contant at the edge of the buckled film, which depend on the ratio b/h and two Dundur parameter. The firt Dundur parameter, α, i related to the tiffne ratio a E = ( 1 α ) (1 + α). The econd Dundur parameter, β, E f typically play a le important role than α [20]. In thi tudy, we concentrate on the effect of α and take β = 0 by etting Poion ratio ν ν = 0. 5 in the numerical calculation. f = A comparion between the critical tree for wrinkling and buckle-delamination i preented in Fig. 2. The tree are normalized by the plane-train modulu of the film and plotted veru the ubtrate-film tiffne ratio. Uing the log-log cale, the critical wrinkling tre in Eq. (1) i a traight line with a lope 2/3. The critical tre for buckling, σ B, obtained from Eq. (3), decreae a the relative delamination ize b/h increae. For a contant delamination ize, the buckling tre increae a the tiffne ratio increae, but at a lower rate compared to the increae of the wrinkling tre. The interection of the two critical tree define a critical tiffne ratio, R c. When the relative ubtrate tiffne i greater (i.e., E / E > R ), the buckling tre i lower than the wrinkling tre, dictating that buckle- f c 4

5 delamination occur firt a the compreive tre develop in the film. On the other hand, for more compliant ubtrate ( E / E < R ), the wrinkling tre i lower and the film wrinkle. f c Therefore, a tranition in the buckling mode i predicted quantitatively a the tiffne ratio between the ubtrate and the film varie. The critical value R c a a function of the delamination ize i plotted in Fig. 3. Thi plot repreent a buckling mode election map: if the tiffne ratio and the interfacial defect ize render a point below the Rc curve, wrinkling i energetically favored; otherwie, buckle-delamination i favored. It hould be noted that the preent dicuion i limited to the onet of the initial buckling mode only. We note that, for each delamination ize, the buckling tre determined from Eq. (3) ha two limit: for high ubtrate/film tiffne ratio, the buckling tre approache σ B0 a given by Eq. (2); for very low tiffne ratio, the buckling tre approache σ / 4. Thee limiting tree are plotted a horizontal dahed line in Fig. 2. While the upper limit correpond to the buckling of the delaminated film with fixed end a an approximation for a rigid ubtrate, the lower limit i identical to that for buckling of a free-tanding film of length 2b imply upported at both end. Therefore, a the relative ubtrate tiffne decreae, the effective contraint at the edge of delamination relaxe from eentially fixed end (contraint on both diplacement and rotation) to imple upport (no contraint on rotation). Thi i reaonable if one conider B0 buckling of the delaminated region only. However, a dicued above, when E / E < R, f c buckling i no longer limited to the delaminated region. Conequently, Eq. (3) hould be valid only for the regime where E / E > R. Another limit for Eq. (3) i et by the ue of the plate f c equation for the delaminated part, which roughly require b/h > 3. 5

6 To validate the buckling mode election criterion decribed above, we deigned an experiment that enabled obervation of both the buckling mode a well a a mode tranition. Here, a polytyrene (PS) film (h = 120 nm) wa bonded to a 1 mm thick polydimethyliloxane (PDMS) ubtrate. The material elatic moduli are etimated a E f ~ 4 GPa and E ~ 2 MPa, with a ratio E / E f = [3]. From Eq. (1), the critical tre for wrinkling i σ w ~ 13 MPa. By annealing the pecimen at 120 C, which i above the gla tranition temperature of PS (T g ~ 105 ºC), and then lowly cooling, a compreive tre develop in the PS film due to differential thermal expanion. Auming an initially mall defect at the interface (ay, b/h = 5), a illutrated by the vertical line in Fig. 2, the compreive tre in the PS film firt reache the wrinkling tre, at which point wrinkling occur pontaneouly over the film urface. A the temperature i further decreaed, the tre eventually reache the critical buckling tre for the initial defect ize, and buckle-delamination occur alongide with the exiting wrinkle. Figure 4 how a et of optical image of the PS film on the PDMS ubtrate, which demontrate the occurrence of wrinkling followed by buckle-delamination during the firt cooling cycle (top row). The inet Fourier pectrum how a well-defined wrinkle wavelength at 60 C. When the material ytem wa heated back to 120 C from the room temperature, the compreive tre in the film vanihed and the elatic film recovered with no obervable buckling or wrinkling. Next, the ytem wa ubjected to a econd cooling (Fig. 4, econd row). Compared to the firt cooling, the interfacial defect ize ha increaed due to the growth of buckle-delamination in the firt cycle. An etimate of the delamination width (Fig. 4, top row, 35 C) give b/h ~ 20 for the econd cycle, for which the buckling tre i lower than the wrinkling tre with E / = a hown in Fig. 2. Conequently, buckle-delamination occurred firt, at location imilar to thoe in the firt cycle. Further cooling led to growth of buckle-delamination and eventually wrinkling E f 6

7 of the bonded region a well. Different buckling and wrinkling pattern are oberved for the firt and econd cycle, which can be attributed to the different evolution path for mode tranition and interaction. In ummary, by comparing the critical tree for wrinkling and buckle-delamination, we develop a quantitative criterion for the election of initial buckling mode in elatic thin film. A buckling mode election map i contructed with repect to the tiffne ratio and the interfacial defect ize. The theoretical reult i demontrated by an experiment that oberved the coexitence of both buckling mode and a mode tranition. Acknowledgment HM and RH are grateful for the financial upport by National Science Foundation under Grant No Portion of thi work wa conducted at the National Intitute of Standard and Technology Combinatorial Method Center. Thi manucript i an official contribution of the National Intitute of Standard and Technology, not ubject to copyright in the United State. 7

8 Reference 1. J.W. Hutchinon, M.D. Thoule, E.G. Liniger, Acta Metall. Mater. 40, 295 (1992). 2. H. Yin, R. Huang, K.D. Hobart, J. Liang, Z. Suo, S.R. Shieh, T.S. Duffy, F.J. Kub, J.C. Sturm, J. Appl. Phy. 94, 6875 (2003). 3. C.M. Stafford, C. Harrion, K.L. Beer, A. Karim, E.J. Ami, M.R. VanLandingham, H.-C. Kim, W. Volken, R.D. Miller, E.E. Simonyi, Nat. Mater. 3, 545 (2004). 4. E.A Wilder, S. Guo, S. Lin-Gibon, M.J. Faolka, C.M. Stafford, Macromolecule 39, 4138 (2006). 5. C.M. Stafford, B.D. Vogt, C. Harrion, D. Julthongpiput, R. Huang, Macromolecule 39, 5095 (2006). 6. M. Watanabe, H. Shirai, T. Hirai, J. Appl. Phy. 92, 4631 (2002). 7. S.P. Lacour, S. Wagner, Z.Y. Huang, Z. Suo, Appl. Phy. Lett. 82, 2404 (2003). 8. C. Harrion, C.M. Stafford, W. Zhang, A. Karim, Appl. Phy. Lett. 85, 4016 (2004). 9. J.W. Hutchinon and Z. Suo, Adv. Appl. Mech. 29, 63 (1992). 10. M. Ortiz and G. Gioia, Adv. Appl. Mech. 33, 119 (1997). 11. T.W. Shield, K.S. Kim, R.T. Shield, J. Appl. Mech. 61, 231 (1994). 12. J. Groenewold, Phyica A298, 32 (2001). 13. Z.Y. Huang, W. Hong, Z. Suo, J. Mech. Phy. Solid 53, 2101 (2005). 14. R. Huang, J. Mech. Phy. Solid 53, 63 (2005). 15. R. Huang, C.M. Stafford, B.D. Vogt, J. Aeropace Engineering 20, 38 (2007). 16. M.-W. Moon, J.-W. Chung, K.-R. Lee, K.H. Oh, R. Wang, A.G. Evan, Acta Mater. 50, 1219 (2002). 17. C. Coupeau, Thin Solid Film 406, 190 (2002). 8

9 18. B. Cotterell and Z. Chen, Int. J. Fracture 104, 169 (2000). 19. G. Parry, J. Colin, C. Coupeau, F. Foucher, A. Cimetiere, J. Grilhe, Acta Mater. 53, 441 (2005). 20. H.H. Yu and J.W. Hutchinon, Int. J. Fracture 113, 39 (2002). 9

10 Lit of Figure Fig. 1: (Color online) Schematic of two buckling mode in thin film. The critical tree, σ w and σ B, are given by Eq. (1) and (3), repectively. Fig. 2: (Color online) Comparion between the critical tree for wrinkling and buckledelamination. The open ymbol are numerical reult from Eq. (3) for variou b/h ratio. The dahed line indicate the limiting tree for buckle-delamination. The vertical line repreent the PS/PDMS ytem with E / = E f Fig. 3: The critical tiffne ratio a a function of the relative delamination ize. Below the line, wrinkling i favored; above the line, buckle-delamination i favored. The dahed portion how the qualitative trend beyond the limit of the plate theory ued in deriving Eq. (3). Fig. 4: Optical micrograph of a 120 nm PS thin film on a PDMS ubtrate cooled from 120 C to room temperature, where wrinkling i the favored buckling mode during the 1 t cooling cycle (top row) and buckle-delamination i favored in the 2 nd cycle. The Fourier pectrum at 60 C (inet) in the firt cycle how a well-defined wrinkle wavelength. 10

11 h σ Film Subtrate σ > σ w σ > σ B 2b Wrinkling Buckle-delamination Fig. 1: (Color online) Schematic of two buckling mode in thin film. The critical tree, σ w and σ B, are given by Eq. (1) and (3), repectively. 11

12 Fig. 2: (Color online) Comparion between the critical tree for wrinkling and buckledelamination. The open ymbol are numerical reult from Eq. (3) for variou b/h ratio. The dahed line indicate the limiting tree for buckle-delamination. The vertical line repreent the PS/PDMS ytem with E / = E f 12

13 Fig. 3: The critical tiffne ratio a a function of the relative delamination ize. Below the line, wrinkling i favored; above the line, buckle-delamination i favored. The dahed portion how the qualitative trend beyond the limit of the plate theory ued in deriving Eq. (3). 13

14 Fig. 4: Optical micrograph of a 120 nm PS thin film on a PDMS ubtrate cooled from 120 C to room temperature, where wrinkling i the favored buckling mode during the 1 t cooling cycle (top row) and buckle-delamination i favored in the 2 nd cycle. The Fourier pectrum at 60 C (inet) in the firt cycle how a well-defined wrinkle wavelength. 14