and microscale pyramid structures - Semantic Scholar

Acta Materialia 54 (2006) 3973–3982 www.actamat-journals.com

Size effect in contact compression of nano- and microscale pyramid structures Junlan Wang a

a,*

, Jie Lian a, Julia R. Greer b, William D. Nix b, Kyung-Suk Kim

c

Department of Mechanical Engineering, University of California, 900 University Avenue, Riverside, CA 92521, USA b Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA c Division of Engineering, Brown University, Providence, RI 02912, USA Received 21 December 2005; received in revised form 24 April 2006; accepted 24 April 2006 Available online 10 July 2006

Abstract An electrochemical etching approach was developed to fabricate self-similar nano- and microscale pyramid structures on singlecrystal gold (1 0 0) surfaces. Using their unique self-similar characteristics, pyramids of (1 1 4) facets were compressed to study the length scale effects in the contact pressure and plastic deformation. At first, many pyramids were compressed simultaneously with a flat mica sheet to measure the ridge angle changes of the deformed pyramids with respect to the sizes of the flattened area. The ridge angle changes were scattered between approximately 2 and 13 for compression displacements of 50–350 nm, in contrast to the perfect plasticity prediction of 4.7. Then, individual pyramids isolated with a focused ion beam were compressed with a flat tip nanoindenter for displacements of approximately 10–100 nm to obtain the relationship between the contact pressure and the compression depth. The plastic deformation-adjusted contact pressure evaluated by taking into account the initial 6–14 nm roundness offset of the pyramids is characterized by an initial increase up to approximately 2.5 GPa for a shallow compression depth within 10 nm followed by a gradual decay to approximately 450 MPa at a compression depth of 100 nm. This pressure seems to be still decaying towards an asymptotic value predicted by a continuum limit analysis. Given the size and self-similar nature of the pyramids, various mechanisms could possibly contribute to the observed scale dependence. The current study provides valuable experimental evidence for size-dependent material behavior at small length scales. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Size effect; PDA contact pressure; Single-crystal plasticity; Pyramid compression

1. Introduction Rapid advances in microelectronics and structural/functional materials have enabled the development of eversmaller micro- and nano-electromechanical systems (MEMS and NEMS). When materials and structures are scaled down to the sub-micrometer range, many of their properties, including mechanical, thermal, optical, electronic and magnetic properties, become different compared to their bulk counterparts, i.e., they are size dependent. In particular, the size-dependent mechanical properties of

*

Corresponding author. Tel.: +1 951 827 6429; fax: +1 951 827 2899. E-mail address: [email protected] (J. Wang).

crystalline materials have been demonstrated in a number of experiments. These include the depth-dependent indentation hardness [1–3], higher mechanical strength of nanocrystalline materials due to grain size hardening [4], higher flow stress in thin metal wires in tension and torsion test [5] and the size-dependent mechanical behavior of small volumes in general [6–8]. Conventional plasticity theory, which has no intrinsic length scale, can no longer sufficiently characterize the behavior of materials and structures at these small scales. In order to explain the observed size-dependent mechanical properties of metals, several theories have been developed and proposed recently. While the indentation size effect has been successfully explained by a strain gradient plasticity theory [5,9] based on the concept of geometrically

1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.04.030

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necessary dislocations for relatively large nanoindentation depths, e.g., 0.15–2 lm for copper, the mechanical properties of ultrathin films have been ascribed to grain size hardening and substrate confinement effects which can be modeled using discrete dislocation plasticity theory [10]. Very recently, Greer et al. observed the size-dependent flow stress of unconfined, uniaxially compressed single-crystal gold pillars [11] manufactured using the focused ion beam (FIB) milling technique developed by Uchic and co-workers [12]. As there are no severe strain gradients in the uniaxially compressed pillars, nor grain size or confinement effect, the increasing flow stress with decreasing pillar size has been explained using a concept of dislocation sourcecontrolled plasticity [11]. Although the FIB technique has proved to be powerful in producing samples in the submicrometer range, usually the fabrication of samples across micro- to nanometer sizes is both time-consuming and costly. Thus the development of a single experiment that can measure the size-dependent material behavior across several length scales is highly desirable. In this work, an electrochemical etching approach was developed to produce nano- and micrometer scale pyramid structures on (1 0 0) surfaces of single-crystal gold. The pyramids grew in a self-similar fashion with the facets coinciding with (1 1 4) planes. Given the self-similar nature, these pyramids serve as ideal samples for experimentally investigating the size-dependent material behavior across nanoto micrometer scales. Compared to previous studies, one unique advantage of the current study is that many pyramids of different sizes, from 100 nm to 10 lm in base side length, yet still self-similar, can be produced in a single etching experiment; thus the sample preparation is convenient and of low cost. The current fabrication method also eliminated the concern of possible Ga+ ion contamination associated with FIB-based fabrication methods [11,13]. Using the self-similar nature of the pyramid structures, contact compression experiments were designed to investigate the length-scale effect in the contact pressure and plastic deformation of the compressed pyramids. Experimentally, both multiple pyramid compression using a flat mica sheet and single pyramid compression using a flat nanoindenter tip were performed. Theoretically, a continuum limit analysis was carried out to predict the asymptotic behavior of the compressed pyramid. The fabrication and contact compression of the pyramids as well as the theoretical analysis are described in the following sections.

lytically for 2 min to remove the scratches from mechanical polishing. The electrolyte consisted of hydrochloric acid (25%), ethylene glycol (25%) and ethanol (50%). The single-crystal gold sample served as the anode and a thin graphite sheet served as the cathode. Self-assembly of the pyramids was observed under controlled polishing conditions. The optimal condition was found to be a combination of electrical current of around 2.5 A and an elevated temperature of approximately 60 C. Under these conditions, pyramids of varying sizes across nanometer to micrometer length scales were obtained. Other conditions produced either no pyramids or aggregated pyramids with incomplete peaks. Fig. 1 shows atomic force microscopy (AFM) topography images of the self-organized pyramids of varying sizes

2. Fabrication of self-similar pyramid structures The fabrication of four-sided square pyramids on the (1 0 0) surface of single-crystal gold was recently discovered when a mechanically polished gold surface was electrochemically etched in a gold electrolyte. The (1 0 0) surface of single-crystal gold (99.99%) was first mechanically polished using 0.3 lm alumina powder. After rinsing with methanol and distilled water, the sample was blown dry using nitrogen. The gold sample was then polished electro-

Fig. 1. AFM topography images of the pyramid structures produced from electrochemical polishing on a single-crystal gold (1 0 0) surface: (a) pyramid of sub-micrometer size; (b) pyramid of micrometer size; (c) singlepyramid of around 10 lm in size.


compression tests provide information of the plastic geometric change of the pyramids, single-pyramid compression was used to obtain quantitative information such as the load vs. compression displacement on each individual pyramid.

0.6

0.5

3.1. Multiple-pyramid compression

Height ( m)

0.4

0.3

0.2

0.1

0

3975

-2

-1

0

1

2

Distance ( m) Fig. 2. Height profile across the ridges of the pyramids measured using AFM.

[001] 0 0 1/4

100 O

010

010

[010]

10 0

[100]

Multiple-pyramid compression was accomplished by pressing an atomically flat mica sheet against a 1 mm diameter circular area on a (1 0 0) surface of gold using an Instron 4502 tensile/compression machine and a custom designed loading punch of 1 mm diameter. Different nominal pressures (load divided by the loading area) of 20, 40, 60 and 80 MPa were applied during the multiple-pyramid compression experiments. A line scan across the two symmetric ridges of one of the deformed pyramids before and after compressing is shown in Fig. 4. By compression, the peaks of the pyramids were flattened. Material was crushed downwards and sideways. The ridge angle – the angle made by the sloping ridge between two adjacent (1 1 4) planes relative to the (0 0 1) plane – became smaller after compression while the exact amount of change was measured to be dependent on the size of the pyramid. This ridge angle change (initial angle minus the final angle, which was obtained by measuring the global ridge slope change using AFM tomography image) vs. the final contact area is plotted in Fig. 5 for pyramids of different sizes. The experimental data show that the change of ridge angle remains constant, with some scatter, for contact areas larger than approximately 0.7 lm2, while the angle change decreases for contact areas less than 0.2 lm2. The large scatter in Fig. 5 is due to experimental errors associated with the

Fig. 3. Schematic of the pyramid orientation. The pyramid facets are along the (1 1 4) direction. 0.4

3. Pyramid compression experiments Experimental investigation of the compression behavior of these pyramids was performed in two ways: multipleand single-pyramid compression. While multiple-pyramid

0.35 0.3

Before compression

0.25

Height (μm)

grown on a (1 0 0) surface of gold using the aforementioned method. Measurements on different areas of the sample indicate that these pyramids grew in a self-similar manner and their sizes range from tens and hundreds of nanometers to a few micrometers in terms of their base side length. Fig. 2 shows the line profiles across the ridge direction of the six pyramids shown in Fig. 1(b). The aspect ratios of the pyramids shown in Fig. 2 provide evidence that these pyramids are all self-similar. Based on the full crystal orientation of the gold sample and the height to base length ratio of the pyramids, the majority of the pyramid facets were determined to be parallel to (1 1 4) planes, although (1 1 3) and (1 1 5) plane orientations were occasionally observed. Fig. 3 shows a schematic of the crystal orientation of a typical gold pyramid.

0.2 0.15 0.1

After compression

0.05 0

-3

-2

-1

0

1

2

3

Distance (µm) Fig. 4. AFM line scan along the ridge of one pyramid before and after compression.

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J. Wang et al. / Acta Materialia 54 (2006) 3973–3982 14

Angle change (degree)

12

10

8

6

4

2

0

0

1

2

3

4

5

Contact area (μm2) Fig. 5. Ridge angle reduction (initial subtracting final) vs. contact area from multiple-pyramid compression measured using AFM.

ridge angle measurement due to the imperfect pyramid shape and the bulging effect at the foothills of the pyramid after compression. However, it is unambiguous that all the ridge angles become smaller after compression, while the analysis of plane strain perfect plasticity predicts that the angle should increase by 4.71. 3.2. Single-pyramid compression 3.2.1. Experimental method In order to obtain quantitative load–displacement information for individual pyramids, pyramids of varying sizes were compressed using the dynamic contact module of a MTS Nanoindenter XP with a flat punch indenter tip. The flat punch tip was custom fabricated from a standard Berkovich indenter using a FIB to machine off the diamond tip, resulting in the projected area of an equilateral triangle with a 9-lm inscribed circle diameter. Prior to carrying out the single-pyramid compression tests, the pyramids of interest were isolated from the neighboring features utilizing the FIB technique. This procedure was critical to ensure that during the subsequent compression testing the indenter was in contact only with the pyramid of interest. In this approach, the surrounding areas were etched away resulting in the creation of an isolated pyramid on a pedestal as shown in Fig. 6. The depth of the crater as well as the height of the pedestal was approximately 3 lm while the outer diameter of the crater was 34 lm. These values of the depth and outer diameter of the crater proved to be sufficient to ensure the indenter contact with the pyramid only and not with the surrounding edges. The indenter system was thermally buffered from its surroundings to within 1 C; however, small temperature

fluctuations cause some of the machine components to expand and contract, and this thermal drift was corrected by monitoring the rate of displacement in the final 100 s of the hold period. Load–displacement data were collected in the continuous stiffness measurement (CSM) mode of the instrument. The experimental procedure involved locating the pyramid of choice under the topview 150· optical microscope, then calibrating the indenter to microscope distance to within a fraction of a micrometer on the surface of the sample away from the pyramids, and finally moving the calibrated flat indenter tip to the position directly above the pyramid. Thermal drift stabilization was then followed by compressing the pyramid at a constant nominal displacement rate of 12 nm/s. During the initial segment of the test, the instrument located the sample surface and then moved to the specified location and started the initial approach segment, decreasing the approach velocity to 5 nm/s when the indenter was less than 6 lm above the surface. Once the surface of the pyramid was detected, parameters such as the force on the pyramid, harmonic contact stiffness and the compressive displacement of the top of the pyramid from the point of contact were continuously measured and recorded. 3.2.2. Analysis of plastic deformation-adjusted contact pressure With the recorded load information, contact pressure can be obtained if the contact area is known. However, due to the characteristic geometry of the pyramid, the contact area will be continuously evolving during the compression process and it is not directly measurable from experiment. Based on the observed side bulging of the pyramid as shown in Fig. 4, a simple model analysis is developed to obtain the relation between the evolving effective contact area and the compression displacement by assuming plastic volume conservation. As shown in Fig. 7, the pyramid is assumed to have a constant base area, which is reasonable for the shallow compression depth of the small-scale pyramids. Once the top of the pyramid is flattened, the deformed volume is assumed to be accommodated by the rest of the pyramid. In the following derivation, ltop is half of the diagonal length of the contact area at currently measured displacement of u, l is half of the diagonal length of the deformed pyramid at any arbitrary vertical position z measured from the base of the pyramid, L is half of the diagonal length of the base area of the pyramid and H is the total height of the pyramid. For an originally perfect (1 1 4) pyramid, L = 4H. Thus the area of an arbitrary section of a distance z from the base is AðzÞ ¼ 2l2 :

ð1Þ

Assuming that l changes linearly with z: lðzÞ ¼ L kz;

ð2Þ

where k is the slope of the ridge of the deformed pyramid. The plastic volume conservation indicates


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Fig. 6. (a) 1.25 lm pyramid as is; (b) same pyramid post-FIB sample preparation; (c) 4.65 lm pyramid post-compression.

3 ðH uÞ 2 32 3 2 2 k 4H ðH uÞ k þ 16H ðH uÞ H ¼ 0: 6 3

z ltop

u

l ( z)

ð4Þ

H-u L

Fig. 7. Schematic of the cross-section of the pyramid along two symmetric ridges. Dashed line depicts the originally perfect (1 1 4) pyramid. Solid line depicts the deformed pyramid. u is the compressed displacement, H u is the height of the deformed pyramid.

1 Abase H ¼ 3

Z

H u 2

2l dz; 0

which yields

ð3Þ

Noting that l = 0 at z = H and solving for k, one obtains " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 6H 1 H þ 3u 1 kðu; H Þ ¼ : ð5Þ H u 3 H u Therefore

"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # H þ 3u ltop ðu; H Þ ¼ 2H 1 : H u

ð6Þ

And the contact area at displacement u from the apex is "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2 H þ 3u 2 2 Acontact ðu; H Þ ¼ 2ltop ¼ 8H ð7Þ 1 : H u

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Using the above contact area, the load information can be converted into contact pressure by dividing the load F over the contact area Acontact(u;H). Comparing with the nominal contact pressure by using the geometrically prospective contact area (i.e., 32u2), Eq. (7) reflects a first-order correction of plastic volume displacement, considering the scale effect of the pyramid size. Hence the calculated pressure is named plastic deformation-adjusted (PDA) contact pressure, i.e.,

8H 2

F hqffiffiffiffiffiffiffiffiffi H þ3u H u

1

ð8Þ

i2 :

0.2

3.2.3. Analysis of the pyramid roundness effect In the above PDA contact pressure calculation, u is assumed to be the displacement from the apex of a perfect (1 1 4) pyramid. The pyramids tested in the compression experiments, however, have rounded tops as demonstrated in Figs. 1 and 2; thus the apex of a real pyramid has an offset from that of the perfect one. In order to use Eq. (8) to calculate the PDA contact pressure, one has to take into account this offset distance. This was achieved by utilizing the harmonic stiffness data recorded during the compression experiment. Since all the pyramids in this experiment have similar apex geometry for a similar nominal contact area, once the pyramid apex is sufficiently flattened from the initial roundness, the local stiffness of the flattened contact can be considered specimen independent for a similar nominal contact area, and the local stiffness k Li ðuÞ for the ith pyramid can be expressed approximately as k Li ðuÞ

2

3

¼ Aðu þ d i Þ þ Bðu þ d i Þ þ Cðu þ d i Þ ;

i ¼ 1; 2; 3; . . .

ð9Þ

D = 1.66 μm D = 1.90 μm D = 3.18 μm D = 6.57 μm 0.15

Load on pyramid (mN)

P PDA ¼

lengths of 1.66, 1.90, 3.18 and 6.57 lm, respectively. Fig. 9 shows the harmonic contact stiffness for the four pyramids together with the fitted values using the method described in Section 3.2.3. By fitting the experimental contact stiffness data using Eqs. (9) and (10), the 11 constants {A, B, C; di (i = 1, 2, 3, 4); k Si ði ¼ 1; 2; 3; 4Þ} for the above four pyramids were found to be {515.385958 N/[m(nm)], 1.81900471 N/[m(nm)2], 0.02103999 N/ 3 [m(nm) ]; 13.9060748, 6.4596270, 8.22404197, 6.43362278

0.05

0

where k Si is the stiffness of the substrate for specimen i. With this functional form of the compression stiffness, the constants (A, B, C; di; k Si ) could be best fitted to the harmonic stiffness data of the experiment. Once the offset displacement of the vertex point is obtained, the contact area at an indenter displacement u can be evaluated as Acontact(u + di; H) using Eq. (7) and the PDA contact pressure is calculated using Eq. (8) by replacing u with u + di.

60

90

120

36000

30000

24000

1.66 μm(exp) 1.66 μm(fit) 1.90 μm(exp) 1.90 μm(fit) 3.18 μm(exp) 3.18 μm(fit) 6.57 μm(exp) 6.57 μm(fit)

18000

12000

6000

0

3.2.4. Results of single-pyramid compression Pyramids of varying sizes were compressed using the method described in Section 3.2.1. Fig. 8 shows the load– displacement information of four pyramids with base side

30

Fig. 8. Load vs. displacement for four pyramids of different sizes. D is the length of the side of the square footprint of the pyramid.

Stiffness (N/m)

ð10Þ

0

Compression displacement (nm)

where u is the indenter displacement from the initial contact and di is the distance between the projected vertex point and the initial contact point of the ith pyramid. The coefficients A, B and C are constants to be determined from the experimental data. Then, the total stiffness ki(u) of the indentation can be expressed as 1 1 1 ¼ L þ S; k i ðuÞ k i ðuÞ k i

0.1

0

30

60

90

120

Compression displacement (nm) Fig. 9. Experimental and fitted harmonic stiffness vs. compression displacement.


nm; 3489185.66, 308159.457, 200087.699, 67766.625 N/m}. Once the offset displacement di (i = 1, 2, 3, 4) was obtained, the PDA contact pressure was calculated using Eq. (8) by replacing u with the offset displacement u + d. Fig. 10 shows the PDA contact pressure of the four pyramids as a function of the displacement from the projected pyramid vertex. Surprisingly, the contact pressure of all pyramids displays an initial increase for shallow compression with a maximum pressure of around 2.5 GPa reached at a compression depth of 10 nm (or 17 nm from the projected vertex) for the 1.90 lm pyramid. The rise in contact pressure may be due to the roundness of the apex of the pyramid causing the initial compression to be elastic. The contact pressure then follows an exponential decay for larger compression depths beyond 10 nm. The PDA contact pressure–displacement curves of different sizes fall on top of each other with increasing compression depth, which indicates the self-similar nature of the plastic deformation of the pyramids at the same contact depth once the pyramid apex is sufficiently flattened from the initial roundness. The variation of the PDA pressure between different sizes of the pyramids at the same compression depth for shallow compressions is probably due to the imperfect pyramid shape at the apex or other unknown reasons. Nevertheless, the PDA pressure vs. displacement data clearly show a scale dependence with the highest pressure demonstrated at shallow depths around 10 nm. For large compression depth beyond 30 nm from the projected vertex, the PDA contact pressure of all pyramids seems to converge approximately towards an asymptotic decaying trend curve with values of a few hundred megapascals. This asymptotic trend curve seems to be independent of the pyramid size.

3

PDA contact pressure (GPa)

2.5 1.66 μm 1.90 μm 3.18 μm 6.57 μm

2

1

0.5

0

30

60

90

4. Continuum limit analysis In order to understand the asymptotic behavior of the compressed pyramid at a much larger compression displacement, a continuum limit analysis was performed based on the rigid-perfectly plastic punching of a (1 1 4) pyramid structure. An upper bound theorem was adopted to predict the collapse pressure at which the system starts to slip. Frictionless contact was assumed and relative sliding was allowed between the punch and pyramid top surface. Following the procedure developed by Shield and Drucker [14], an admissible slip system was chosen to be that shown in Fig. 11. The square area LMNO in Fig. 11(a) stands for the top surface of the pyramid, and is punched down under a velocity of v under a uniform contact pressure Pc. The top surface is divided into four equal triangles by the diagonal LN and MO. Due to symmetry, only one triangle CMN is considered. The downward movement of the triangle is accommodated by the flow in CDEFMN. The polyhedra DCMN and EFMN are tetrahedral. Points D, E are vertically below line CF. MBDE and NBDE are two symmetric sections of right circular cones with MN as the axis. Fig. 11(b) and (c) shows the plane and vertical sections through CF. The stream lines of the flow are along CDEF. Assuming \BCD = b, the stream velocity is then v/ sin b. The downward motion of the lower three triangles of MNCD is accommodated in the same way while the remaining material is at rest. Energy is dissipated in the discontinuity surfaces between the material at rest and the material moving in volume CDEFMN and also in the conical regions MBDE and NDBE where the plastic strain rate is not zero. Other parameters of the slip kinematics, the angles a and c, are shown in Fig. 11(c). Using the upper bound [14] and slip-line theorem [15], the contact pressure Pc was found to be a function of the flow stress s0 and the angles of the pyramid and the slip line system, a, b, and c: P c ða; b; cÞ P s0 ða þ b cÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 þ sin2 bða þ b c þ cot a þ cot bÞ ; ð11Þ with the minimum value corresponding to the minimum upper bound of the collapse pressure of the admissible mechanism. For a given value of initial angle c, one can solve for a and b by minimizing Pc. For c = 15 which is close to experimentally observed angle, one obtains a ¼ 47 ; b ¼ 35 and P c ¼ 5:24s0 : ð12Þ

1.5

0

3979

120

Displacement from projected vertex (nm) Fig. 10. PDA contact pressure vs. displacement from projected vertex.

The limit pressure is about 12.6% higher than the plane strain perfect plastic punch pressure of (2 + 5p/6)s0 4.62s0 for this pyramid angle, showing a three-dimensional effect. If the initial flow stress s0 of bulk gold is taken to be about 20 MPa, then Pc = 105 MPa which seems to be the lower bound of the experimentally observed asymptotic value.

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(a)

N

O C

B M

L D

F E

O’

N’

B’

C’

L’

M’ N

(b)

B

C

D

E

F

M

(c)

Pc v B C F

D E

Fig. 11. Schematic of the pyramid slip system: (a) the slip-line field in the pressed pyramid; (b) top view of one-quarter of the slip system; (c) crosssectional view of one-quarter of the slip system along the CDEF plane with the punch included.

5. Discussion Both experimental investigations and the continuum limit analysis reveal important information about the plastic deformation of the pyramids under contact compression. Multiple-pyramid compression tests reveal that the ridge angle of the pyramids becomes smaller after compression, in contrast to the perfect plasticity prediction, and the amount of the ridge angle change is dependent on the size of the pyramid. Single-pyramid compression experiments yielded size-dependent PDA contact pressure vs. compression depth. Note that because the contact area is a continuously evolving parameter during compression and strongly dependent on the geometry and mechanical properties of the pyramids, the analysis of the contact area evolution is critical to the correct pressure interpretation. By using the conservation of plastic deformation volume and assuming the deformed volume is distributed along the entire height of the deformed pyramid, a simple depthand size-dependent contact area is obtained which is used

to calculate the PDA contact pressure. Another assumption in the PDA contact pressure calculation is that the pressure is uniformly distributed across the whole contact area. Although in reality the contact pressure might have spatial variations within the contact area, current smallscale plasticity theory is not sufficient to predict this variation. For all pyramids tested in this study (ranging from 1.66 to 6.57 lm in terms of base side length), a similar PDA contact pressure vs. displacement profile was observed. For compression displacements below 10 nm, the contact pressure increases dramatically with increasing displacement. The contact pressure then gradually decreases with increasing compression depth and eventually converges to an asymptotic decaying trend curve with values of a few hundred megapascals. The maximum contact pressure at compression depths around 10 nm is almost 5 times higher than that at larger depths around 100 nm. Variations in contact pressure with displacement provide a relative measure of the strength of the size-dependent compression


behavior. Both the single- and multiple-pyramid compression experiments indicate a strong size-dependent mechanical compression response. Considering the size and the geometric characteristics of the pyramids, there are at least four major mechanisms which could possibly contribute to the observed scale dependence. One is the initial roundness of the apex. The roundness of the apex can be represented by the offset distance, d, of the projected vertex point from the peak point of the round apex. When the round apex is flattened, the effect of the initial roundness on the volume displacement of the plastic deformation is a factor of (d/u)3. If it is flattened by twice d from the peak of the round apex the effect is only about 4%. The second source of the scale effect is the dislocation–starvation–plasticity boundary layer near the free surface [11]. For an annealed single-crystal gold, the average spacing between dislocations is 1–10 lm [16]. When the deformation volume is small compared to the average dislocation spacing, deformation is likely to be dislocation source-limited or dislocation-starved. Under such circumstances, the stress field under the indenter does not move and multiply dislocations. Pre-existing dislocations will not be activated by the compression as the probability of dislocations lying in the pyramid is negligible; dislocations must be nucleated for incipient plasticity. The third source is the indenter/specimen contact interface plasticity, which is described as nanoscale contact surface plasticity [17], and the fourth source is the large plastic strain gradient sector converging to the edge of the contact. It was also noted that due to the constant displacement rate control used in the single-pyramid compression, the resulting strain rate is varied during the pyramid compression. From a simple analysis, the strain rate e_ can be defined as the time rate of the height change of the _ divided by the total height H u deformed pyramid, u, _ (see Fig. 8), i.e., u=ðH uÞ. Given the range of height H of the four pyramids tested in this study and the range of displacement u, the strain rate change is significantly less than an order of magnitude both during the testing of each individual pyramid and between different pyramids. Thus the strain rate change is not likely to contribute to the observed scale-dependent contact pressure. To gain a better understanding of the possible mechanisms contributing to the observed size-dependent compression behavior of the pyramids, ongoing work is focusing on applying various plasticity theories with different hardening laws as well as atomistic simulations to investigate the size-dependent transition in the contact pressure observed in these experiments. 6. Conclusions A novel electrochemical etching technique was used to produce self-similar pyramid structures on single-crystal gold (1 0 0) surfaces. The facets of the pyramid have a preferred (1 1 4) plane orientation. Due to the self-similar nature and their size distribution, these pyramids serve as

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unique samples for investigating the size effect in the plastic deformation of single-crystal surface structures. Multipleand single-pyramid contact compression experiments as well as continuum limit analysis were conducted to study the deformation of the pyramids. The AFM measurements of the ridge slope change show a size-dependent transition of the plastic deformation. The transition data provide an apparent characteristic length of the size dependence of plastic deformation in a small volume. Single-pyramid compression experiments revealed a size-dependent PDA contact pressure vs. compression depth. While the PDA contact pressure increases to a maximum value of approximately 2.5 GPa for shallow compression depths below 10 nm, it gradually decays to an asymptotic trend curve for larger depths beyond 50 nm which further decays towards the predicted value of the continuum limit analysis. Different mechanisms could possibly contribute to the size-dependent behaviors observed in pyramid compression, including the initial roundness of the pyramid apex, the dislocation source limited plasticity, nanoscale contact surface plasticity and strain gradient hardening. The data obtained in this study provide valuable experimental evidence to further explore the size-dependent material behavior at small length scales as well as to test the validity of various plasticity theories. Compared to previous experiments regarding sizedependent properties, a unique advantage of the current study is the self-similar nature of the pyramids. An additional advantage is that the current study avoids the concern of possible Ga+ contamination associated with FIB-based fabrication methods. Although the FIB technique was still used to isolate the individual pyramids tested in this work, the diameter of the pedestal underneath the pyramid is much larger than the pyramid size and the Ga+ beam is never pointed over the top of this pedestal. The Ga+ ions strike the sides of the pedestal at very low angles of incidence; thus Ga+ ion implantation is believed not to be an issue in this study. Alternatively, the gold (1 0 0) surface could be FIB patterned first before fabricating the pyramids. This will completely resolve the Ga+ ion contamination issue. Acknowledgements J.W. and J.L. acknowledge the financial support from the University of California Regents and Academic Senate. W.D.N. and J.R.G. acknowledge financial support through a grant provided by the Department of Energy (DE-FG02-04-ER46163). K.S.K. and J.W. acknowledge the support from the Brown University/General Motors collaborative research laboratory. References [1] De Guzman MS, Neubauer G, Flinn PA, Nix WD. Mater Res Symp Proc 1993.

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