Contact Compression of Self-assembled Nano- and Micro-scale Pyramid Structures on Au (100) Surface

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1 Cntact Cmpressin f Self-assembled Nan- and Micr-scale Pyramid Structures n Au (100) Surface J. Wang, D. Ward, W.A. Curtin and K.-S. Kim Divisin f Engineering, Brwn University, Prvidence, Rhde Island ABSTRACT A prcess f self-assembly induced by electr-chemical etching was used t prduce nan and micrmeter scale pyramid-structures n (100) surfaces f gld. The pyramids grew in a selfsimilar fashin with the facets aligned in (114) plane. Using the unique characteristics f the selfsimilar pyramid structure, plastic cmpressin f the pyramids by a flat-surface platen was perfrmed t study length scale effects in the plastic defrmatin. A cntinuum limit analysis and a finite element simulatin as well as mlecular dynamics simulatins were carried ut t predict the defrmatin and lad-displacement behavir f the pyramid cmpressin. The limit analysis predicts that the lad f cmpressin is prprtinal t the square f the cntactcmpressin displacement. The cntinuum analysis prvides estimatin n the asympttic behavir f the elastic-plastic lad-deflectin respnse f the pyramid under cmpressin fr a large value f displacement. The three dimensinal mlecular dynamics simulatin was utilized t study the dislcatin activities during the early stage f the pyramid cmpressin. Experiments were als carried ut by pressing the pyramids with an atmically flat mica surface. The defrmatin f the cmpressed pyramid was measured using an Atmic Frce Micrscpe (AFM). The cntinuum analyses predict size independent values f the slpe change f the pyramid facets near the cntact edge, caused by plastic defrmatin. Hwever, atmistic simulatin predicts an ppsite value f the slpe change t the predictin f the cntinuum analyses. The AFM measurements f the slpe change shw size dependent transitin frm the predictin f the cntinuum analyses t that f the atmistic simulatins. The transitin data prvide an apparent characteristic length f the size dependence f plastic defrmatin in a small vlume. Mlecular dynamics shw that at very small length scale the size effect is strngly influenced by surface adhesin effects. INTRODUCTION Rapid advances in electrnics and structural materials are enabling the develpment f ever-smaller micr- and nan-electrmechanical (MEM and NEM) devices. When materials and structures are scaled dwn frm tens f micrn t a fractin f a micrn, metals display a strng sizedependence when defrmed nn-unifrmly int the plastic range due t the inhmgeneus plastic flw in crystalline slids. Cnventinal plasticity thery, which has n length scale, can n lnger sufficiently characterize the behavir f the materials and structures. This phenmenn has mtivated a large effrt in the mechanics and materials cmmunities t develp bth experiments and theries t investigate the material behavir at micrn and nanmeter scales. These include the recent established strain gradient plasticity thery [1] at micrmeter scale, discrete dislcatin plasticity thery [2] at tens f nanmeter scale, etc. Hwever, due t the difficulty in manufacturing sub-micrn structures systematically, experimental characterizatin f the length scale effect acrss micrn t nanmeter is challenging. At the nanmeter scale, much f the fabricatin prcess is cntrlled by the self-assembly f material. Thus the

2 develpment f a single experiment that can measure the size-dependence acrss several length scales is highly desirable. In this paper, a nvel electr-chemical etching apprach was adpted t prduce nan and micrmeter scale pyramid-structures n (100) surfaces f single crystal gld. The pyramids grew in a self-similar fashin with the facets aligned in (114) plane. Using the unique characteristics f the self-similar pyramid structure, cntact cmpressin experiments were perfrmed n the single crystal gld surface t study the length scale effect in the plastic defrmatin f the pyramids. A cntinuum limit analysis was used t characterize the self-similar defrmatin f the pyramid cmpressin, while mlecular dynamics simulatins were used t see the rigin f the scale dependence in plastic defrmatins at a very small length scale. SELF ASSEMBLY AND CONTACT COMPRESSION OF (114) GOLD PYRAMIDS The self assembly f fur sided square pyramids n Au (100) surface was recently discvered when a mechanically plished surface was immersed in a certain Au electrlyte. Even the eventual functin f the electrlyte was t remve the scratches frm mechanical plishing, it was fund that under center cnditins (current, temperature and etching time), the (11n) (n = 4 in mst cases) facets can be recnstructed during the etching and frm very well defined pyramid structures. The electrlyte cnsists f 25% hydrchlric acid, 25% ethylene glycl and 50% ethanl. Self-assembly f the pyramids was highly dependent n the etching parameters. The best pyramids were usually achieved at an electrical current f 2.5 ampers at an elevated temperature f 60 C fr duratin arund 2 minutes. Under this cnditin, cmplete pyramids f different sizes acrss nanmeter t micrmeter were btained. Under ther cnditins (varying current r etching time), either n pyramids were bserved r the pyramids became t big that the nearby pyramids start t calesce with each ther and the peaks gt etched away. Figure 1 shws an AFM scan f the self-assembled pyramids n tp f Au (100) surface grwn using the afrementined methd. Figure 2 shws the line prfiles acrss the ridge directin f 6 pyramids shwn in Figure 1. The line prfiles shw that the pyramids grew in a selfsimilar manner. Regardless f their individual sizes, the facets f mst f the pyramids are aligned in the (114) plane, althugh (113) and (115) rientatins were bserved in sme ther measurements. Measurements n different areas f the sample indicate that the size f these selfsimilar pyramids range frm tens and hundreds f nanmeter t a few micrns. Given the self-similar nature, these pyramids serve as perfect samples fr experimentally characterizing the size effect acrss micrn t nanmeter scales. Therefre, cntact cmpressin experiments were cnducted t see the length scale effect in the plastic defrmatin f these pyramid structures. T apply a cntact pressure lading, an Figure µm by 25 µm AFM scan f pyramid structures prduced frm electr-chemical plishing n a single crystal Au (100) surface. ) µm h t( Height (µm) H eig Distance (µm) Figure 2. Height prfile acrss the ridges f the pyramids.

3 atmically flat mica sheet was used t transfer the lad applied by an Instrn 4502 machine t the (100) Au sample surface. Different nminal pressure crrespnding t 1/6, 1/3, 1/2, 2/3 f the yielding lad f gld were applied. The initial and defrmed pictures fr a 25 µm by 25 µm area are shwn in Figure 3a and 3b. A line scan acrss the tw symmetric ridges f ne f the defrmed pyramids befre and after cmpressing is shwn in Figure 3c. By cmpressin, the peaks f the pyramids were flattened. Material was crushed dwnwards and sideways. The ridge angle became smaller after cmpressin, althugh the amunt f change was measured t be different fr different sizes f pyramids. 0.4 Height (µm) ) m h t(u H eig befre indentatin after indentatin Distance Distance (µm) (um) (a) (b) (c) Figure 3. Pyramid defrmatins: (a) AFM scan befre cmpressin; (b) AFM scan after defrmatin; (c) line scan alng the ridge f ne pyramid befre and after cmpressin. CONTINUUM MODELING In rder t mdel the defrmatin f a single pyramid under cntact pressure, cntinuum limit analysis was used t analyze the rigid-perfect-plastic punching f a single pyramid structure. Upper bund therem was adpted t predict the pressure lading at which the system starts t slip [3]. It was assumed that n frictin exist at the cntact surface and O N relative sliding was allwed between the C B punch and pyramid tp surface. L Fllwing the prcedure used by R.T. M N F Shield [4], the slip system was chsen t be D that shwn in Figure 4. The square area E LMNO in Figure 4a stands fr the tp surface f the pyramid, and is punched L dwn under a velcity f v under a unifrm M (a) pressure q. The tp surface is divided int N fur equal triangles by the diagnal LN and q MO. Due t symmetry, nly ne triangle v C D B E F B CMN is cnsidered. The dwnward C β mvement f the triangle is accmmdated α F by the flw in CDEFMN. The plyhedra D E γ M DCMN and EFMN are tetrahedral. Pints (b) (c) D, E are vertically belw line CF. MBDE and NBDE are tw symmetric sectins f Figure 4. Schematic f the pyramid slip system.

4 right circular cnes with MN as the axis. Fig.4b and 4c are the plane and vertical sectins thrugh CF. The stream lines f the flw are alng CDEF. Assuming BCD = β, the stream velcity = v sin β. The dwnward mtin f the lwer three triangles f MNCD is accmmdated in the same way while the remainder material being at rest. Energy is dissipated in the discntinuity surfaces between the material at rest and the material mving in vlume CDEFMN and als in the cnical regins MBDE and NDBE where the plastic strain rate is nt zer. Other parameters f the slip kinematics, the angles α and γ, are shwn in Figure 4c. By using the upper bund and slip-line therem [5], the cntact pressure q was fund t be a functin f the flw stress k and the angles f the pyramid and the slip line system, α, β, and γ, 2 { ( )} q( α, β; γ) k ( α + β γ) + 1+ sin β α + β γ + ctα + ctβ, with the minimum value crrespnds t the upper bund. Fr a given value f initial angle γ, ne can slve fr α and β by minimizing q. Fr γ = π /12 (15 ) which is clse t experimentally bserved angle, α = 47, β = 35 and q = k. Thus frm Figure 4c, the cntact area between the punch and pyramid will expand with a velcity v CB = v ct β = 1. 46v as the punch mves dwn with a speed f v. Assuming that changes in β is negligible, the gemetric relatins in Figure 4c further prvide the relatins between the cntact area A, punching lad P and the punch displacement δ as A= A ( δ δ ) 0 0, , P = q A= 5.24k A ( δ δ ) where A 0 and δ 0 are the initial cntact area and the punch displacement, respectively. Thus, cntinuum limit analysis predicts that the defrmatin f the pyramid abides a parablic relatin between the nrmal lading and displacement. The macrscpic behavir f the plastic defrmatin f the pyramid was als analyzed with finite element methds; hwever, due t space limit the results f FEM will be reprted else where. The cntinuum analysis prvides self similar shape changes f the pyramid defrmatin, and thus cnstant ridge-angle change caused by plastic bulging effect is expected fr large cmpressin displacement. At a small displacement f cmpressin sme size effects are expected t be bserved. ATOMISTIC SIMULATIONS Atmistic simulatins were perfrmed t catch the early perid f the defrmatin f the pyramid under cmpressin. In rder t cmpare with the experimental results, Au pyramids f (114) facet rientatins were cnsidered fr the mlecular dynamics simulatin f the defrmatin at 300K. The punch was created ut f Ir atms. The Suttn Chen ptential [6], a Finnis-Sinclair many bdy ptential, that includes lng range van der Waals interactins in the frm f a pair ptential was used t mdel the interactins between the punch and pyramids. The Ir atms were held fixed t simulate a rigid punch. A bi-material ally cmbinatin rule was used t determine the interactin between the punch and pyramid [7].

5 The size f the system cnsists f 9736 atms crrespnding t 9 atmic layers f atms with an initial cntact area cnsisting f 9 atms by 9 atms. The bttm layer was held fixed creating an unnatural bundary cnditin but the system was large enugh t capture the imprtant features that ccur during the initial stages f the cntact cmpressin. Figure 5 shws the lad-displacement curve fr this simulatin. It can be seen that as the punch mves clse t the pyramid the surface atms jump t the punch. At rm temperature, the atms have enugh velcity that helps t vercme the energy barrier hlding the atms t the pyramid. This reactin causes the pyramid relcate the vlume frm the lwer part f the pyramid t the tp changing the ver all gemetry in a manner that is the inverse f that seen in macrscpic experiments. The defrmed cnfiguratin is shwn in Figure 6, in which each plus sign represents the prjectin f an atm n a (110) plane. The image is taken frm the increment step at which the first layer f the punch is at the initial psitin fr the tp layer f the pyramid. Due t an attractin between punch and pyramid, the tp layer f the pyramid atms mve alng the surface f the pyramid twards the punch. Since the atms are accepting defrmatin and mving t psitins that are in a lattice structure, the stress n the system is very lw at any given time. The atms are free t mve and are free t accmmdate the migratin f ther atms t previusly ccupied sites. As a result, an initially (114) pyramids finally appraches a structure mre clsely representing that f the (113) plane. The ridge angle change is increased fr an amunt f scale. SIZE EFFECT Lad (nn) Displacement (Angs) 4.2, which is ppsite t what was bserved in experiment f larger length Figure 5. Lad vs. displacement curve. Figure 6. 2D prjectin f a defrmed cnfiguratin f (114) pyramid n (110) plane. In rder t see the size effect in the plastic defrmatin f the pyramids under cntact pressure, the ridge angle change (initial angle subtract the final angle) f different sizes f pyramids vs. the cntact areas are pltted in Figure 7 fr bth experiments and atmistic simulatins. The experimental data shw that the change f ridge angle remains cnstant, with sme scattering, with respect t the change f cntact area fr the cntact defrmatin with cntact area larger than Ridge angle change 15 g e n 10 h a g lec 5 n a 0 Experiment MD simulatin R idge Cntact area (µm 2 ) Cntact area (µm 2 ) Figure 7. Angle change vs. cntact area frm experiment and atmistic simulatins.

6 apprximately 0.7 µm 2, while the angle change decreases with decreasing cntact area at very small cntact area less than 0.2 µm 2. Amazingly, the nly available atmistic simulatin result cnverges t the trend f experimental bservatin. DISCUSSIONS AND CONCLUSIONS A nvel electr-chemical etching technique was used t prduce self-rganized pyramid structures n a single crystal gld (100) surfaces. The facets f the pyramid have a preferred rientatin f (114) plane. Due t the self-similar nature and their size distributin, these pyramids serve as unique samples fr investigating the size effect in the plastic defrmatin f the single crystals. Cntact cmpressin experiments as well as cntinuum limit analysis and mlecular dynamics simulatins were cnducted t study the defrmatin f the pyramids. The limit analysis predicts a parablic relatin between the lad and displacement. It prvides an estimate n the asympttic behavir f the elastic-plastic lad-deflectin respnse f the pyramid under cmpressin fr a large value f deflectin. The three dimensinal mlecular dynamics simulatin reveals the very small vlume defrmatin prcess during the early stage f the pyramid cmpressin. The cntinuum analyses predict size independent values f the slpe change f the pyramid facets near the cntact edge, caused by plastic defrmatin. Hwever, atmistic simulatin predicts ppsite value f the slpe change t the predictin f the cntinuum analyses. The AFM measurements f the slpe change shw size dependent transitin frm the predictins f the cntinuum analyses t that f the atmistic simulatins. The transitin data prvide an apparent characteristic length f the size dependence f plastic defrmatin in a small vlume. The simulatin shws that the deviatin frm a cntinuum behavir at a very small length scale is clearly due t interface interactin and surface effects, while the transitin at a larger scale may be due t dislcatin mechanisms. ACKNOWLEDGEMENTS This wrk was supprted by Brwn/General Mtrs Cllabrative Research Labratry. Valuable discussins with Dr. Y. Qi and Dr. Y.T. Cheng are gratefully acknwledged. REFERENCES 1. Hutchinsn, J.W., Int. J. Slids Structures, 37, (2000) 2. Bitterncurt, E., Needleman, A., Gurtin, M.E., and Van der Giessen, E., JMPS, 51, (2003) 3. Jhnsn, K.L., Cambridge University Press (1984) 4. Shield, R.T., Drucker, D.C., Technical reprt, A Divisin f Applied Mathematics, Brwn University (1975) 5. Rice, J.R., Technical reprt, AT (11-1)-3084, n.13. Divisin f Engineering, Brwn University (1972) 6. Suttn, A.P., Chen, J., Phil. Mag. Lett., 61 (3), 139 (1990) 7. Rafii-Tabar, H., Suttn, A.P., Phil. Mag. Lett., 63 (4), 217 (1991)