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3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE I LECTURE 22: THEORETICAL ASPECTS OF NANOINDENTATION Outline : REVIEW LECTURE #21 : EXPERIMENTAL SINGLE MACROMOLECULE ELASTICITY........................ 2 NANOINDENTATION .............................................................................................................................3-7 Introduction ............................................................................................................................. 3 Indenter Geometries................................................................................................................ 4 Types of Deformation .............................................................................................................. 5 Oliver-Pharr Analysis : Geometric Set-Up .............................................................................. 6 Oliver-Pharr Analysis :Mathematical Formulation.................................................................... 7 APPENDIX ............................................................................................................................................8-13 Detailed Geometry of Indenters..........................................................................................8-10 Berkovich Contact Area ......................................................................................................... 11 Oliver-Pharr Citations ............................................................................................................ 12 Nanoindentor Instruments ..................................................................................................... 13 Objectives: To understand general theoretical formulations for reducing material properties from nanoindentation experiments Readings: Course Reader Documents 45 (one of the most cited papers in Materials Science)-46, Additional Historical Ref: Sneddon 1965 Int. J. Engng. 3, 47-57. 1 3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE SINGLE MOLECULE ELASTICITY OF TITIN (AFM) & DNA (OPTICAL TWEEZERS) - Structure and physiological role of Titin (Rief, et al. CHEMPHYSCHEM 2002, 3, 255-261)→sawtooth force profiles (Bustamante, et al. Science 1999, 271, 795) overstretching transition I. low stretched behaves like WLC (p ≈ 50 nm under physiological conditions, much larger than most polymers ~ 1nm, hence much smaller forces, need optical tweezers) II. intermediate stretches -some extensibility as apparent by finite slope beyond Lcontour (B-form) Force (pN) III. At 65 pN ~ 0.06 nN, reversible strain-induced conformational transition; chain "yields" and stretches out almost 2× its native B-form contour length at relatively constant force (plateau in force region) -All of hydrogen bonding and binding between 2 strands is still in tact, tilting of base pairs, tightened helix, reduction in diameter "overstretching transition" IV. entropic elasticity of S-form Distance (μm) V. can't see here - if you go to high enough stretches, separation between strains (mechanical "melting") Biological Relevance of Overstretching Transition? Ability to switch between different structures is critical to the processes of transcription, replication, condensaton, e.g. the base pairs are much more exposed in S-DNA than normal DNA, the transition may be biologically significant for accessing information contained in the DNA code 2 3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE INTRODUCTION TO NANOINDENTATION loading jump-to-contact 0 surface forces unloading adhesion contact forces 0 Tip-Sample Indentation Depth or Separation Distance, D (nm) -e.g. silicon or silicon nitride indenter probe on a cantilever force transducer -cantilever oriented at an angle to the surface (~11°) -indenter geometries, e.g. pyramidal (less well defined) -load range ~ nN-mN, smaller contact radii ~ 10s of nm Load, P Force, F (nN) Definition : Controlled compression and decompression of a Pmax probe tip into a sample surface while measuring force (load, indenter P) versus indentation displacement or depth, h (nm-scale) initial continuously surface → probe tip is relatively rigid compared to the sample → can measure mechanical properties (e.g. modulus, hardness) a hmax on areas nm-μm scale; e.g. thin films and small volume structures surface profile → called "nano" since the indentation depth is of nanometer scale, at Pmax however lateral contact areas and forces can be > nanoscale -multiaxial deformation AFM-based Indentation Instrumented or Depth-Sensing Indentation (DSI) Pmax loading unloading hmax Indentation Depth, h (Hysitron, Micromaterials, Appendix→extension of conventional hardness testing to smaller length scale) - diamond indenter - indenter oriented perpendicular to the surface - variable indenter geometries; Berkovich, cube corner, etc. - load range ~ μN-mN, larger contact radii ~ μm 3 3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE NANOINDENTATION : INDENTER GEOMETRIES AFM-Based Indentation a Side and back view Front view 56º Bottom view 120º ~18 μm Residual Berkovich Indent Impression Berkovich ~9 μm 33º 29º ~10 μm 33º 29º ~18 μm Instrumented Indentation 18º 18º b 33º ~9 μm End radius < 15 nm c 71º contact radius ~ 60 nm Nanomechanically mapped region 50 μm 2 μm Silicon tetrahedral probe tip indenter (k~ 56 N/m) 45º Cube Corner See Appendix for full geometric details a=Vickers, b= Berkovich, c= Knoop, d = conical, e=Rockwell, f=spherical 4 3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE NANOINDENTATION : TYPES OF DEFORMATION Elastoplastic or Inelastic hr= hf = residual / final depth Ue = elastic energy Ur = energy dissipated (elastoplastic / inelastic) Utotal = total work of deformation= Ue+Ur in.materials.drexel.edu/blogs/280_advanced_materials_lab/attachment/469.ashx Courtesy of Prof. Yury Gogotsi. Used with permission. 5 3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE OLIVER-PHARR ANALYSIS: GEOMETRIC SET-UP Linear Elastic, Isotropic, Continuum Contact Mechanics Theory (Oliver and Pharr, 1992 JMR, 7(6) 1564) : Geometry setup and definitions of geometric parameters : assumes "sink-in" hf Courtesy of George M. Pharr and Journal of Materials Research. Used with permission. P = applied load, Pmax= peak applied load h = indentation depth (at Pmax; h= hmax maximum depth) a = radius of contact circle hc= contact depth, vertical distance along which contact is made between sample and tip hs = displacement of the surface at the perimeter of contact From geometry : h = hc + hs A(hc) = contact (projected) area at hc ⎛ 1 - νi 2 ⎞ ⎛ 1 - ν2 ⎞ Er = reduced modulus = ⎜ +⎜ ⎟ ⎟ ⎝ E ⎠sample ⎝ Ei ⎠indenter (i.e. two springs in series) E = modulus ν = Poisson's ratio hf = residual final depth (indicates inelasticity; e.g. viscoelasticity, plasticity) dP S = contact ( initial unloading ) stiffness = ⎛⎜ ⎞⎟ ⎝ dh ⎠ Pmax (typically evalulated between 95% and 20% of Pmax ) -1 6 3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE OLIVER-PHARR ANALYSIS : MATHEMATICAL FORMULATION (Oliver and Pharr, 1992 JMR, 7(6) 1564) Er = π S → Sneddon Equation holds for any indenter geometry (1) 2 A(hc ) S is measured directly from the data (typically evalulated between 95% and 20% of Pmax ) hc = hmax − ε Pmax (2) S Tip Geometry flat-ended cylindrical punch paraboloid of revolution Cone ε 1 0.75 2(π-2)/π Indenter (Probe Tip) Area Function Calibration : A(hc ) = tip area function; representative of tip geometry, can be calibrated on sample of known modulus (e.g. fused quartz) by inverting Sneddon equation (1); 2 Courtesy of George M. Pharr and Journal of Materials Research. Used with permission. π⎛ S ⎞ A(hc ) = ⎜ ⎟ (3) 4 ⎝ Er ⎠ Carry out indentations at successively higher loads; at each Pmax calculate hc from (2) and A(hc ) from (3), these data are fit to a polynomial : A(hc ) = Co hc 2 + C1hc + C2 hc0.5 + C3 hc0.25 + C4 hc1/8 + C5 hc 1/16 Gives A(hc ) for every indentation depth, hc Co = 24.5; A(hc ) = 24.5hc 2 (Ideal Berkovich Geometry) (4) (see Appendix for Derivation), coefficients reflect indenter geometry Schematic courtesy of B. Bruet 7 3.052 Nanomechanics of Materials and Biomaterials Tuesday 05/08/07 Prof. C. Ortiz, MIT-DMSE Four appendices removed due to copyright restrictions. - Detailed geometry of indenters - Berkovich geometry calculation of contact area - Web screenshot: Oliver and Pharr JMR 1992 article is one of the most cited papers in Materials Science, has been cited >2975 times - Photos of nanoindentation instruments. 8