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Ultramicroscopy 103 (2005) 183–189

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Mapping the 3D-surface strain field of patterned tensilestainless steels using atomic force microscopy

Vincent Vignala,�, Eric Finotb, Roland Oltraa, Yvon Lacrouteb,Eric Bourillotb, Alain Dereuxb

aLRRS, UMR 5613 CNRS—Universite de Bourgogne, BP 47870, 21078 Dijon, FrancebLPUB, UMR 5027 CNRS—Universite de Bourgogne, BP 47870, 21078 Dijon, France

Received 24 May 2004; received in revised form 15 November 2004; accepted 23 November 2004

Abstract

The quantification of microstructural strains at the surface of materials is of major importance for understanding the

reactivity of solids. The present paper aims at demonstrating the potentialities of the atomic force microscopy (AFM)

for mapping the three-dimensional surface strain field on patterned tensile specimens. Electron beam (e-beam)

lithography has been used to deposit 16� 16 arrays of gold-squared pads. Monitoring the evolution of such a pattern

under applied strain allows to quantify the triaxial strains both at the micro-(plastic) domain and nanoscale (elastic)

domain vs. applied strain. The proposed method was applied to stainless steels after 4.5% plastic strain.

r 2005 Elsevier B.V. All rights reserved.

Keywords: AFM; Lithography; Strain; Stainless steel

1. Introduction

The quantification of strains at the scale ofmaterials microstructure is of major importancefor understanding the surface reactivity. Theexperimental techniques commonly used on me-tals, such as the neutron and X-ray diffraction(XRD) techniques [1,2] have limitations. Thedetermination of the strain by XRD is based on

e front matter r 2005 Elsevier B.V. All rights reserve

tramic.2004.11.021

ng author. Tel.: +33380 396 160;

6 132.

ss: vvignal@u-bourgogne.fr (V. Vignal).

the measurement of the angular shift of diffractionpeaks caused by a small homogeneous variation ofthe lattice plane in the sampling volume withrespect to the strain-free lattice plane spacing [3].This technique gives the average value of the strainin a phase calculated in a volume of a fewmicrometers thick (between 5 and 8 mm in the caseof stainless steels). In addition, applying thediffraction techniques to plastically strained speci-mens and highly anisotropic materials exhibitingnon-linear strain sin2c plots remains a difficulttask (where c is the angle between the normal tothe specimen surface and the normal to the

d.

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V. Vignal et al. / Ultramicroscopy 103 (2005) 183–189184

diffracting planes). In the former case, the straindistributions measured for several (h k l) planes arefitted to strain distributions issued from a least-square algorithm using the effective constantsobtained from the orientation distribution func-tions (ODFs) [4]. These techniques are notspatially resolved and do not allow to quantifythe stress gradients at the microscale. Suchmicrostresses, which can originate from differencesin elastic properties, plastic flow or thermalexpansion between phases, may affect the physi-co-chemical behaviour of materials [5].The micro-Raman spectroscopy, which is spa-

tially better resolved with a sampling volume ofapproximately 1mm3, could be used as an extenso-metry technique by assuming that any stress-induced change in the interatomic distance shouldchange the interatomic force constants and henceshift the atomic vibration wavenumber. However,this technique gives hydrostatic stress without anyfurther information on triaxial stress. The determi-nation of quantitative data from micro-Ramanremains the main problem as outlined by severalworkers [6]. Micro-Raman has been commonlyused on silicon [7] and some metallic substratescovered with a thermal oxide film, by usingmechanical models and calibration curves linkingthe Raman wavenumber shifts to the strainmeasured previously under straining conditions.This technique cannot be applied to metallicsurfaces.Atomic force microscopy (AFM) offers the

possibility to examine the three-dimensional (3D)surface morphology in order to evaluate in-planechanges (up to 150� 150 mm2) and to measure out-of-plane differences (up to 6 mm). AFM has beenused to study the surface topography of fatiguedstainless steels, especially in kinetic study ofgrowth where the shape of extrusions and intru-sions has been followed as a function of thenumber of loading cycles [8,9]. Description ofplastic deformation processes and strain gradientsaround grain boundaries has been proposed afteruniaxial tensile loading [10,11]. However, thesestudies remain qualitative and only a few para-meters, such as the surface roughness and the stepsheight, have been determined as a function ofapplied strain. A quantitative approach combining

AFM with the electron back-scatter diffraction(EBSD) has shown that both the surface rough-ness and the misorientation of grains increaselinearly with increasing applied strain [12–15].AFM has also been used for determining themechanical characteristics of nanoscale structures(nanobeams of single-crystal silicon and SiO2)under bending [16].The numerical simulation [17,18] has appeared

as a powerful method to calculate the surfacestrain and stress fields from a complex micro-structure at a scale that cannot be reached fromthe experimental methods described above.The present paper aims at demonstrating the

potentialities of the AFM for mapping the 3Dsurface strain field on patterned tensile specimens.Electron beam (e-beam) lithography has been usedto deposit 16� 16 arrays of gold-squared pads.Monitoring the evolution of such a pattern withapplied strain allows to quantify the triaxial strainsboth at the micro-(plastic) domain and nanoscale(elastic) domain vs. applied strain. Similar ap-proaches have already been developed with scan-ning electron microscopy (SEM) [19–21].However, using AFM instead of SEM has fouradvantages: (i) the experiments are carried inambient conditions without any vaccuum require-ments, (ii) the AFM imaging provides a lateralresolution which allows to calculate the strain fieldaround particles at the nanoscale, (iii) both in-plane and out-of-plane strains can be assessedsimultaneously and (iv) the imaging is not limitedto the plastic domain as in the case of SEM (above2–3% plastic strain).

2. Experimental

2.1. Tensile specimens

Experiments were performed on a duplexstainless steel (UNS S32550, chemical composi-tion: C: 0.015wt%; Ni: 6.02%; Cr: 24.9%; Mn:1.13%; Cu: 1.65%; P: 0.014%; S: 0.001%;Si: 0.335%; Mo: 3.74% and N: 0.248%). Ahomogeneisation treatment at 1300 1C for 1 h,followed by a slow cooling down to 1080 1C(formation of the austenite) and a water quenchingwas performed to obtain a mean grain size of

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V. Vignal et al. / Ultramicroscopy 103 (2005) 183–189 185

about 50 mm and a volume fraction of austeniteand ferrite of 50:50. Tensile specimens having across-section of 2� 4mm2 and a gauge length of16mm were used. The yield and ultimate strengthof tensile specimens at 25 1C is 660 and 862MPa,respectively. Tensile specimens were mechanicallypolished using emeri papers and smoothed usingdiamond pastes (down to 1 mm).

2.2. Electron beam lithography

The e-beam lithography was then used to makethe arrays of circular shaped gold pads. Thedesired pattern was deposited on an area of1� 1mm2. The diameter of pads was about300 nm and the distance between centroids oftwo neighbouring pads was 4 mm. First, a 100 nmthick layer of 50 kD molecular weight of poly-methylmethacrylate (4% in content of polymer inchlorobenzene) was spin coated onto the tensilespecimens. They were then baked for 3 h at 80 1Cbefore being patterned by an e-beam. The e-beamdiameters of 1 nm or less were used to irradiate,and thereby chemically modify the PMMA layer.A conventional Raith Elphy plus pattern genera-tor, in connection with a JEOL 840 scanningelectron microscope equipped with a thermal fieldemitter and having a nominal resolution of 5 nm,was used to expose the PMMA layer. The beamenergy was 50 keV and the beam current was set at0.4 nA. Two exposure fields were used to matchthe resolution requirements of the structures. Thestructures were developed in MIBK for 1min,rinsed in deionised water and blown dry with purenitrogen. Cr (10 nm) and Au (20 nm) wereevaporated in a thermal evaporator (e-gun) andthe remaining resist was removed by lift-off inacetone. All the surface was cleaned with hotacetone, rinsed with isopropyl alcohol and driedon a hot plate at the temperature of 105 1C.

2.3. Mechanical tests and surface observations at

the microscale

Tensile specimens were then subjected to unixialtensile loading in air at 4.5% plastic strain appliedalong the X1-axis using a home-made tensilemicrostage. Surface observations were performed

after unloading using a D3100 AFM equippedwith a Nanoscope IIIa controller and a J typescanner (xy-scan range �150 mm). Images wereacquired in contact mode and at room tempera-ture. Unsharpened ‘D’ type silicon nitride canti-levers (Microlevers; nominal end-radius �50 nm;Thermomicroscopes (Sunnyvale, CA, USA) with anominal force constant of 0.1N/m) were used.

3. Results and discussion

3.1. Method for mapping the 3D-surface strain field

The resolution of scans was adapted to the grainsize. Relatively large areas (100� 100 mm2) werefirst scanned to identify regions of interest and todetermine changes in the misorientation betweenneighbouring grains. Smaller areas (50� 50 mm2)that correspond nearly to the grain size were thenscanned to improve the lateral resolution of thepads spacing within grains. At this resolution, thepixel size is 0.097 mm for a 8-bit image (512� 512pixels). Measurements at this scale allows tocompare the strain states of different grains.The average distance between centroids of pads

was calculated along the X1- and X2-axis at themicroscale by measuring the distance between twopads which are 10 pads apart on section profiles.The section profiles were smoothed using a low-pass frequency filter (f41Hz) and the standarddeviation was obtained by repeating the analysiswith 20 AFM images. The in-plane strain compo-nents were evaluated from the average distancesusing the following equations:

�11L1 � L1;0

L1;0; �22

L2 � L2;0

L2;0; (1)

where L1 and L2 are the average distance betweencentroids of pads along the X1- and X2-axis afterdeformation, respectively, L1,0 and L2,0 are theaverage distances before deformation. The off-diagonal components �12 of the strain tensor wascalculated as follows:

�12 ¼ tan Da; (2)

where Da is the angle shift between the twodirections after deformation. It should be

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0 10 20 30 40 50

0

100

200

300

4.29

66

8.98

38

13.4

757

18.0

6525

22.7

5245

27.5

373

32.3

2215

37.1

07

41.9

895

46.6

767

Hei

ght (

nm)

Distance (µm)

(b)

X2

[µm]

[µm

]

X1

(c)

Height from average (nm)

Fre

quen

cy [%

]

(a)

Fig. 1. (a) AFM image of a pattern on the unstrained specimen,

(b) typical smoothed section profile along the X1-axis with the

position of peaks (in mm) and (c) surface height distribution

determined between the arrays of pads on the image shown in

(a).

V. Vignal et al. / Ultramicroscopy 103 (2005) 183–189186

mentioned that a was measured directly fromAFM images. The out-of-plane strain componentwas deduced assuming that no changes in volumeoccurs with strain. As dilatation is the sum of thediagonal terms of the strain tensor, one has

�33 ¼ �ð�11 þ �22Þ: (3)

Out-of-plane shear stresses can be reasonnablyconsidered as negligible so that

�13 ¼ �23 ¼ 0: (4)

After appyling a low-pass frequency filter to theAFM images obtained on the strained surface, theerror made on the determination of the position ofpads was estimated to 1 pixel in both directions.Therefore, the error made on the measurement ofthe distance between 2 pads which are 10 padsapart was 2 pixels, corresponding to a relativeerror in percentage of the exact value of strain ofabout 7(2� 0.097)/(10�L0) ¼ 70.6% (where L0

is the the average distance between centroids ofpads before deformation, Eqs. (5)–(7)).The height distributions of the surface were

evaluated by measuring the height difference along50 scan lines of 50 mm in length between pads andfrom these data histograms were made. It wasfound that these height distributions follow aGaussian bell-shaped distribution where the full-width half-maximum (FWHM) was considered asrepresentative of the surface roughness at themicroscale.

3.2. Mapping the 3D-surface strain field after

unloading

The image analysis described above was appliedto three patterns on the unstrained specimen, asshown in Fig. 1(a). The parameters characterizingthese patterns were determined from profile sec-tions (Fig. 1(b)) as explained above. The followingvalues were obtained:

Pattern 1 : L1;0 ¼ 3:66� 0:02mm;

L2;0 ¼ 4:77� 0:02mm and a ¼ 90:7� 0:1; ð5Þ

Pattern 2 : L1;0 ¼ 3:64� 0:02mm;

L2;0 ¼ 4:77� 0:02mm and a ¼ 90:75� 0:05;ð6Þ

Pattern 3 : L1;0 ¼ 3:63� 0:02mm;

L2;0 ¼ 4:71� 0:02mm and a ¼ 90:8� 0:05: ð7Þ

These results were considered as the initial valuesfor calculating in-plane and out-of-plane strain

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-10

-5

0

5ε33 (%)

Out

-of-

plan

e st

rain

(b)

-2.8%

+4.13%

-1.3%

0 4 8 12 16 20-5

0

5

10 ε11 (%)ε22 (%)

In-p

lane

str

ains

Grain Label

0 4 8 12 16 20Grain Label

(a)

Fig. 2. Evolution of the diagonal terms of the strain tensor

determined at the microscale within different grains after 4.5%

plastic strain: (a) in-plane strains and (b) out-of-plane strain.

V. Vignal et al. / Ultramicroscopy 103 (2005) 183–189 187

components of the strained specimens. The surfaceroughness was also evaluated from the methoddescribed previously and a value of about 5 nmwas obtained after mechanical polishing, as shownin Fig. 1(c).A high dispersion in the average distances was

observed on the strained specimens at the micro-scale, confirming that a heterogeneous strain fieldwas developed under straining conditions at thesurface of polycrystalline materials. The mainadvantage of combining the AFM and the surfacepaterning was to provide quantitative values of �ij

and to map variation in strains (comparisonbetween the average distances in different grains).A significant increase in the average distancebetween pads was systematically observed alongthe X1-axis (between 3.71 and 3.86 mm) whereas aslight decrease in the spacing was obtained alongthe X2-axis (between 4.59 and 4.75 mm). Thediagonal components of the strain tensor, �ii

(i ¼ 1; 2 or 3), were deduced using Eqs. (1) and(3) and Fig. 2 shows the dispersion of the resultsfor different grains. The strain in the loadingdirection, �11, was found to vary significantly as afunction of grains, from 2.2% to 6.2%, as shownin Fig. 2(a). The strain state of grains depends ontheir morphology, size and crystallographic orien-tation with respect to the loading direction. Theaverage value, of about 4.1%, was of the sameorder of magnitude as the global applied strain,indicating that the local measurements gaveaccurate data. On the other hand, the in-planestrain component in the direction perpendicular toloading, �22, was smaller and remained more orless constant over a large number of grains (ofabout �1.3%), as shown in Fig. 2(a). In poly-crystalline materials, the sweeling of grains in thisdirection was restricted by the constraints imposedby neighbouring grains. However, grains at thesurface of the specimen have another degree offreedom in deformation along the X3-axis, com-pared to interior grains, because of the presence offree surface. By result, dilatation was promoted inthis direction and the strain �33 was found tofluctuate widely between �6% and +0.23%around a mean value of �2.8%, as shown in Fig.2(b). This former value was much larger than thatdetermined along the X2-axis (of about 1.3%),

confirming that the deformation was enhancedalong the X3-axis.As it was already shown [12], the surface

roughness increased significantly with appliedstrain, from 5 nm in the absence of strain to valuesranging between 40 and 80 nm after 4.5% plasticstrain (Fig. 3). In addition, the surface roughnessincreased with �33 according to a linear law, asshown in Fig. 4. This suggests that the surfaceroughness could be interpreted as a measure ofout-of-plane surface displacements. Dispersion inthe results might result from the dependance of thesurface roughness on out-of-plane displacementsof underlying grains that is not considered in thecalculations of �33.Although some large variations in the angle

between the two directions were detected in a fewgrains (negative variation of �1.21 and positivevariation of +21 for grains 8 and 18 in Fig. 5,respectively), it can be seen that the variations inthe angle were close to 0.71 at the microscale after4.5% plastic strain, corresponding to an average

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Fig. 3. (a) AFM image of a pattern inside a grain after 4.5%

plastic strain and (b) surface height distribution determined

between the arrays of pads on the image shown in (a).

0 240

50

60

70

Out-of-plane strain, ε33 (%)

FW

HM

4 6 8

Fig. 4. Evolution of the surface roughness vs. �33 after 4.5%plastic strain (FWHM: full-width half-maximum of the

gaussian distribution of the height).

0 4 8 12 16 20-5

0

5

10

ε12 (%)

In-p

lane

str

ain

Grain Label

1%

Fig. 5. Evolution of the off-diagonal term of the strain tensor,

�12, determined at the microscale within different grains after

4.5% plastic strain.

V. Vignal et al. / Ultramicroscopy 103 (2005) 183–189188

value of �12 of about 1%. Therefore, the actualloading path of individual grains was not a puretension while a pure uniaxial tension was globally

applied to the specimen. These specific sheareffects inside grains could be related to theexistence of grains with different crystallographicorientations, leading to a complex strain state ofthe specimen.

3.3. Distorsion of the patterns after unloading

No distorsions in the patterns were observed onthe unstrained specimen, even at a large scale ofseveral micrometers. Misorientations around 3.51were observed between some neighbouring grainsafter 4.5% plastic strain, as shown in Fig. 6. Thehigh density of slip bands within these grainsindicates that such distorsions appear at highplastic strains.

4. Conclusions

The use of AFM on patterned tensile specimensopens a large field of investigation in studying themechanical behaviour of engineering surfaces atthe micro-(plastic) and nanoscale (elastic)domain. This method can be used for quantifyingthe changes observed in the mechanical behaviourof engineering materials after surface treatmentsor in the presence of coatings. The sameanalysis could be performed at the nanoscalein order to map the strain components within agrain using nanolithography patterns. Therefore,the major impact of the proposed method is to

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Fig. 6. AFM image showing a distorsion in the pattern after

4.5% plastic strain.

V. Vignal et al. / Ultramicroscopy 103 (2005) 183–189 189

provide quantitative information on triaxialstrains between and inside grains. In addition,combining AFM on patterned tensile specimenswith EBSD will allow to determine the misorienta-tion between highly strained grains and todetermine the misorientation angle leading tothese distorsions.

Acknowledgements

This work was supported by the GERMECAM (www.u-bourgogne.fr/REACTIVITE/GERMECAM).

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