Microscale Mapping of Structure and Stress in Barium Titanate.

Cross-correlation of electron backscatter diffraction (EBSD) patterns was used to generate rotation, strain, and stress maps of single-crystal tetragonal barium titanate (BaTiO3) containing isolated, small, sub-micrometer a domains separated from a c-domain matrix by 90° domain boundaries. Spatial resolution of about 30 nm was demonstrated over 5 µm maps, with rotation and strain resolutions of approximately 10-4. The magnitudes of surface strains and, especially, rotations peaked within and adjacent to isolated domains at values of approximately 10-2, i.e., the tetragonal distortion of BaTiO3. The conjugate stresses between a domains peaked at about 1 GPa, and principal stress analysis suggested that stable microcrack formation in the c domain was possible. The results clearly demonstrate the applicability of EBSD to advanced multilayer ceramic capacitor reliability and strongly support the coupling between the electrical performance and underlying mechanical state of BaTiO3-containing devices.

Predicting strength distributions of MEMS structures using flaw size and spatial density.

The populations of flaws in individual layers of microelectromechanical systems (MEMS) structures are determined and verified using a combination of specialized specimen geometry, recent probabilistic analysis, and topographic mapping. Strength distributions of notched and tensile bar specimens are analyzed assuming a single flaw population set by fabrication and common to both specimen geometries. Both the average spatial density of flaws and the flaw size distribution are determined and used to generate quantitative visualizations of specimens. Scanning probe-based topographic measurements are used to verify the flaw spacings determined from strength tests and support the idea that grain boundary grooves on sidewalls control MEMS failure. The findings here suggest that strength controlling features in MEMS devices increase in separation, i.e., become less spatially dense, and decrease in size, i.e., become less potent flaws, as processing proceeds up through the layer stack. The method demonstrated for flaw population determination is directly applicable to strength prediction for MEMS reliability and design.

Stress Measurements in Alumina by Optical Fluorescence: Revisited.

Stress measurements in single-crystal and polycrystalline alumina are revisited using a recently developed optical fluorescence energy shift method. The method simultaneously utilizes the R1 and R2 Cr-related ruby line shifts in alumina to determine two components of the stress tensor in crystallographic coordinates, independent of the intended or assumed stress state. Measurements from a range of experimental conditions, including high-pressure, shock, quasi-static, and bulk polycrystals containing thermal expansion anisotropy effects, are analyzed. In many cases, the new analysis suggests stress states and stress magnitudes significantly different from those inferred previously, particularly for shock experiments. An implication is that atomistic models relating stress state to fluorescence shift require significant refinement for use in materials-based residual stress distribution analyses. Conversely, the earliest measurements of fluorescence in polycrystalline alumina are shown to be consistent with recent detailed measurements of stress equilibrium and dispersion.

Determination of ceramic flaw populations from component strengths.

A procedure is outlined for determining the population of flaws in manufactured ceramics from strength measurements of sampled components. The broad applicability of the procedure is demonstrated in a quantitative manner, using strength measurements from a range of ceramic materials (eg, glass, glass-ceramic, single crystal, and polycrystal) with different flaw types (eg, bulk, surface, and edge). The deconvoluted flaw populations are mostly dominated by small flaws with extended large flaw tails and are all in domains of tens of micrometers. The procedure greatly extends the useful information to be gained by ceramics manufacturers and designers from strength distribution measurements and emphasizes the importance of identifying strength-limiting characteristics within a flaw population.

Review: Coefficients for Stress, Temperature, and Composition Effects in Fluorescence Measurements of Alumina.

The numerical coefficients linearly relating the effects of stress (including pressure), temperature, and composition to shifts in the energies of the Cr-related fluorescence in alumina (Al2O3) are reviewed. The primary focus is the shift of the R1 and R2 "ruby" fluorescence lines under conditions typical for stress determination in polycrystalline Al2O3. No significant experimental difference in the R1 and R2 responses is observed for hydrostatic stress (or pressure) conditions (average shift coefficient of about 7.6 cm-1/GPa), changes in temperature (about 0.140 cm-1/K), or variations in composition (about 120 cm-1/mass fraction of Cr). There are significant differences in the R1 and R2 responses for nonhydrostatic stress conditions. In particular, for uniaxial stress along the a and c directions in the Al2O3 crystal, the R1 piezospectroscopic tensor coefficients (about 3.0 cm-1/GPa and 1.6 GPa cm-1/GPa, respectively) differ considerably, whereas the R2 coefficients (about 2.6 cm-1/GPa and 2.3 GPa cm-1/GPa, respectively) do not. Measurements of the piezospectroscopic tensor coefficients are shown to have interlaboratory relative consistency of about 4 % extending over 30 years, and are consistent with the scalar high-pressure measurements. Measurements of the temperature coefficients are shown to have interlaboratory relative consistency less than 1 % extending over 60 years. Fluorescence-based measurements of stress in polycrystalline Al2O3, although requiring temperature adjustment, are shown to have a relative uncertainty of about 2.5 %.

Determination of Residual Stress Distributions in Polycrystalline Alumina using Fluorescence Microscopy.

Maps of residual stress distributions arising from anisotropic thermal expansion effects in a polycrystalline alumina are generated using fluorescence microscopy. The shifts of both the R1 and R2 ruby fluorescence lines of Cr in alumina are used to create maps with sub-µm resolution of either the local mean and shear stresses or local crystallographic a- and c-stresses in the material, with approximately ± 1 MPa stress resolution. The use of single crystal control materials and explicit correction for temperature and composition effects on line shifts enabled determination of the absolute values and distributions of values of stresses. Temperature correction is shown to be critical in absolute stress determination. Experimental determinations of average stress parameters in the mapped structure are consistent with assumed equilibrium conditions and with integrated large-area measurements. Average crystallographic stresses of order hundreds of MPa are determined with characteristic distribution widths of tens of MPa. The stress distributions reflect contributions from individual clusters of stress in the structure; the cluster size is somewhat larger than the grain size. An example application of the use of stress maps is shown in the calculation of stress-intensity factors for fracture in the residual stress field.

Mapping Viscoelastic and Plastic Properties of Polymers and Polymer-Nanotube Composites using Instrumented Indentation.

An instrumented indentation method is developed for generating maps of time-dependent viscoelastic and time-independent plastic properties of polymeric materials. The method is based on a pyramidal indentation model consisting of two quadratic viscoelastic Kelvin-like elements and a quadratic plastic element in series. Closed-form solutions for indentation displacement under constant load and constant loading-rate are developed and used to determine and validate material properties. Model parameters are determined by point measurements on common monolithic polymers. Mapping is demonstrated on an epoxy-ceramic interface and on two composite materials consisting of epoxy matrices containing multi-wall carbon nanotubes. A fast viscoelastic deformation process in the epoxy was unaffected by the inclusion of the nanotubes, whereas a slow viscoelastic process was significantly impeded, as was the plastic deformation. Mapping revealed considerable spatial heterogeneity in the slow viscoelastic and plastic responses in the composites, particularly in the material with a greater fraction of nanotubes.

Quantitative Mapping of Stress Heterogeneity in Polycrystalline Alumina using Hyperspectral Fluorescence Microscopy.

The microstructurally-induced heterogeneous stress fields arising in a series of Cr-doped polycrystalline alumina materials are mapped with sub-micrometer sub-grain size resolution using fluorescence microscopy. Analysis of the hyperspectral data sets generated during imaging enabled both the amplitude and position of the characteristic Cr R1 fluorescence peak to be determined at every pixel in an image. The peak amplitude information was used to segment the images into individual grains and grain boundary regions. The peak position information, in conjunction with measurements on single-crystal controls, was used to quantify overall stress distributions in the materials and provide stress scales for maps. The combined information enabled spatial variations in the stress fields in crystallographic axes to be mapped and compared directly with microstructural features such as grains and grain boundaries. The mean c-axis stresses in these materials were approximately 200 MPa with stress distribution widths of about 70 MPa, both increasing with average grain size. Greatest variations in stress were observed at grain junctions; no trend in the stress for individual grains with grain size was observed.

Near-theoretical fracture strengths in native and oxidized silicon nanowires.

In this letter, fracture strengths σ f of native and oxidized silicon nanowires (SiNWs) were determined via atomic force microscopy bending experiments and nonlinear finite element analysis. In the native SiNWs, σ f in the Si was comparable to the theoretical strength of Si〈111〉, ≈22 GPa. In the oxidized SiNWs, σ f in the SiO2 was comparable to the theoretical strength of SiO2, ≈6 to 12 GPa. The results indicate a change in the failure mechanism between native SiNWs, in which fracture originated via inter-atomic bond breaking or atomic-scale defects in the Si, and oxidized SiNWs, in which fracture initiated from surface roughness or nano-scale defects in the SiO2.

Stochastic behavior of nanoscale dielectric wall buckling.

The random buckling patterns of nanoscale dielectric walls are analyzed using a nonlinear multi-scale stochastic method that combines experimental measurements with simulations. The dielectric walls, approximately 200 nm tall and 20 nm wide, consist of compliant, low dielectric constant (low-k) fins capped with stiff, compressively stressed TiN lines that provide the driving force for buckling. The deflections of the buckled lines exhibit sinusoidal pseudoperiodicity with amplitude fluctuation and phase decorrelation arising from stochastic variations in wall geometry, properties, and stress state at length scales shorter than the characteristic deflection wavelength of about 1000 nm. The buckling patterns are analyzed and modeled at two length scales: a longer scale (up to 5000 nm) that treats randomness as a longer-scale measurable quantity, and a shorter-scale (down to 20 nm) that treats buckling as a deterministic phenomenon. Statistical simulation is used to join the two length scales. Through this approach, the buckling model is validated and material properties and stress states are inferred. In particular, the stress state of TiN lines in three different systems is determined, along with the elastic moduli of low-k fins and the amplitudes of the small-scale random fluctuations in wall properties-all in the as-processed state. The important case of stochastic effects giving rise to buckling in a deterministically sub-critical buckling state is demonstrated. The nonlinear multiscale stochastic analysis provides guidance for design of low-k structures with acceptable buckling behavior and serves as a template for how randomness that is common to nanoscale phenomena might be measured and analyzed in other contexts.

In situ observations of Berkovich indentation induced phase transitions in crystalline silicon films.

The pressure induced phase transitions of crystalline Si films were studied in situ under a Berkovich probe using a Raman spectroscopy-enhanced instrumented indentation technique. The observations suggested strain and time as important parameters in the nucleation and growth of high-pressure phases and, in contrast to earlier reports, indicate that pressure release is not a precondition for transformation to high pressure phases.

Assessing strain mapping by electron backscatter diffraction and confocal Raman microscopy using wedge-indented Si.

The accuracy of electron backscatter diffraction (EBSD) and confocal Raman microscopy (CRM) for small-scale strain mapping are assessed using the multi-axial strain field surrounding a wedge indentation in Si as a test vehicle. The strain field is modeled using finite element analysis (FEA) that is adapted to the near-indentation surface profile measured by atomic force microscopy (AFM). The assessment consists of (1) direct experimental comparisons of strain and deformation and (2) comparisons in which the modeled strain field is used as an intermediate step. Direct experimental methods (1) consist of comparisons of surface elevation and gradient measured by AFM and EBSD and of Raman shifts measured and predicted by CRM and EBSD, respectively. Comparisons that utilize the combined FEA-AFM model (2) consist of predictions of distortion, strain, and rotation for comparison with EBSD measurements and predictions of Raman shift for comparison with CRM measurements. For both EBSD and CRM, convolution of measurements in depth-varying strain fields is considered. The interconnected comparisons suggest that EBSD was able to provide an accurate assessment of the wedge indentation deformation field to within the precision of the measurements, approximately 2×10(-4) in strain. CRM was similarly precise, but was limited in accuracy to several times this value.

Mechanical measurements of heterogeneity and length scale effects in PEG-based hydrogels.

Colloidal-probe spherical indentation load-relaxation experiments with a probe radius of 3 µm are conducted on poly(ethylene glycol) (PEG) hydrogel materials to quantify their steady-state mechanical properties and time-dependent transport properties via a single experiment. PEG-based hydrogels are shown to be heterogeneous in both morphology and mechanical stiffness at this scale; a linear-harmonic interpolation of hyperelastic Mooney-Rivlin and Boussinesq flat-punch indentation models was used to describe the steady-state response of the hydrogels and determine upper and lower bounds for indentation moduli. Analysis of the transient load-relaxation response during displacement-controlled hold periods provides a means of extracting two time constants τ1 and τ2, where τ1 and τ2 are assigned to the viscoelastic and poroelastic properties, respectively. Large τ2 values at small indentation depths provide evidence of a non-equilibrium state characterized by a phenomenon that restricts poroelastic fluid flow through the material; for larger indentations, the variability in τ2 values decreases and pore sizes estimated from τ2via indentation approach those measured via macroscopic swelling experiments. The contact probe methodology developed here provides a means of assessing hydrogel heterogeneity, including time-dependent mechanical and transport properties, and has potential implications in hydrogel biomedical and engineering applications.

Multi-Scale Effects in the Strength of Ceramics.

Multiple length-scale effects are demonstrated in indentation-strength measurements of a range of ceramic materials under inert and reactive conditions. Meso-scale effects associated with flaw disruption by lateral cracking at large indentation loads are shown to increase strengths above the ideal indentation response. Micro-scale effects associated with toughening by microstructural restraints at small indentation loads are shown to decrease strengths below the ideal response. A combined meso-micro-scale analysis is developed that describes ceramic inert strength behaviors over the complete indentation flaw size range. Nano-scale effects associated with chemical equilibria and crack velocity thresholds are shown to lead to invariant minimum strengths at slow applied stressing rates under reactive conditions. A combined meso-micro-nano-scale analysis is developed that describes the full range of reactive and inert strength behaviors as a function of indentation load and applied stressing rate. Applications of the multi-scale analysis are demonstrated for materials design, materials selection, toughness determination, crack velocity determination, bond-rupture parameter determination, and prediction of reactive strengths. The measurements and analysis provide strong support for the existence of sharp crack tips in ceramics such that the nano-scale mechanisms of discrete bond rupture are separate from the larger scale crack driving force mechanics characterized by continuum-based stress-intensity factors.

Accurate spring constant calibration for very stiff atomic force microscopy cantilevers.

There are many atomic force microscopy (AFM) applications that rely on quantifying the force between the AFM cantilever tip and the sample. The AFM does not explicitly measure force, however, so in such cases knowledge of the cantilever stiffness is required. In most cases, the forces of interest are very small, thus compliant cantilevers are used. A number of methods have been developed that are well suited to measuring low stiffness values. However, in some cases a cantilever with much greater stiffness is required. Thus, a direct, traceable method for calibrating very stiff (approximately 200 N/m) cantilevers is presented here. The method uses an instrumented and calibrated nanoindenter to determine the stiffness of a reference cantilever. This reference cantilever is then used to measure the stiffness of a number of AFM test cantilevers. This method is shown to have much smaller uncertainty than previously proposed methods. An example application to fracture testing of nanoscale silicon beam specimens is included.

Surface-engineered nanomaterials as X-ray absorbing adjuvant agents for Auger-mediated chemo-radiation.

We report a prototype approach to formulate gold nanoparticle-based X-ray absorbing agents through surface-engineering of a cisplatin pharmacophore with modified polyacrylate. The resulting agents exhibit both chemo-therapeutic potency to cancer cells and Auger-mediated secondary electron emission, showing great potential to improve the therapeutic efficacy of chemo-radiation.

Nanoscale mapping of contact stiffness and damping by contact resonance atomic force microscopy.

In this work, a new procedure is demonstrated to retrieve the conservative and dissipative contributions to contact resonance atomic force microscopy (CR-AFM) measurements from the contact resonance frequency and resonance amplitude. By simultaneously tracking the CR-AFM frequency and amplitude during contact AFM scanning, the contact stiffness and damping were mapped with nanoscale resolution on copper (Cu) interconnects and low-k dielectric materials. A detailed surface mechanical characterization of the two materials and their interfaces was performed in terms of elastic moduli and contact damping coefficients by considering the system dynamics and included contact mechanics. Using Cu as a reference material, the CR-AFM measurements on the patterned structures showed a significant increase in the elastic modulus of the low-k dielectric material compared with that of a blanket pristine film. Such an increase in the elastic modulus suggests an enhancement in the densification of low-k dielectric films during patterning. In addition, the subsurface response of the materials was investigated in load-dependent CR-AFM point measurements and in this way a depth dimension was added to the common CR-AFM surface characterization. With the new proposed measurement procedure and analysis, the present investigation provides new insights into characterization of surface and subsurface mechanical responses of nanoscale structures and the integrity of their interfaces.

Nanomechanical properties of polyethylene glycol brushes on gold substrates.

A necessary step in advancing the use of polyethylene glycol (PEG) surface coatings in critical biotechnological applications such as cancer treatments is to provide direct and reliable nanoscale property characterization. Measurements for such characterization are currently provided by scanning probe methods, which are capable of assessing heterogeneity of both surface coverage and properties with nanoscale spatial resolution. In particular, atomic force microscopy (AFM) can be used to detect and quantify the heterogeneity of surface coverage, whereas atomic force spectroscopy can be used to determine mechanical properties, thereby revealing possible heterogeneity of properties within coatings. In this work, AFM and force spectroscopy were used to characterize the morphology and mechanical properties of thiol-functionalized PEG surface coatings on flat gold substrates in aqueous PEG solution. Thiol-functionalized PEG offers a direct and simple method of attachment to gold substrates without intermediate anchoring layers and therefore can be exploited in developing PEG-functionalized gold nanoparticles. AFM was used to investigate the morphology of the PEG coatings as a function of molecular weight; the commonly observed coverage was in the form of sparse, brushlike islands. Similarly, force spectroscopy was utilized to study the mechanical properties of the PEG coatings in compression and tension as a function of molecular weight. A constitutive description of the mechanical properties of PEG brushes was achieved through a combinatorial analysis of the statistical responses acquired in both compression and tension tests. Such a statistical characterization provides a straightforward procedure to assess the nanoscale heterogeneity in the morphology and properties of PEG coverage.

Prototype cantilevers for quantitative lateral force microscopy.

Prototype cantilevers are presented that enable quantitative surface force measurements using contact-mode atomic force microscopy (AFM). The "hammerhead" cantilevers facilitate precise optical lever system calibrations for cantilever flexure and torsion, enabling quantifiable adhesion measurements and friction measurements by lateral force microscopy (LFM). Critically, a single hammerhead cantilever of known flexural stiffness and probe length dimension can be used to perform both a system calibration as well as surface force measurements in situ, which greatly increases force measurement precision and accuracy. During LFM calibration mode, a hammerhead cantilever allows an optical lever "torque sensitivity" to be generated for the quantification of LFM friction forces. Precise calibrations were performed on two different AFM instruments, in which torque sensitivity values were specified with sub-percent relative uncertainty. To examine the potential for accurate lateral force measurements using the prototype cantilevers, finite element analysis predicted measurement errors of a few percent or less, which could be reduced via refinement of calibration methodology or cantilever design. The cantilevers are compatible with commercial AFM instrumentation and can be used for other AFM techniques such as contact imaging and dynamic mode measurements.

Size measurement of nanoparticles using atomic force microscopy.

This chapter outlines procedures for sample preparation and the determination of nanoparticle size using atomic force microscopy (AFM). Several procedures for dispersing gold nanoparticles on various surfaces such that they are suitable for imaging and height measurement via intermittent contact mode, or tapping mode, AFM are first described. The methods for AFM calibration and operation to make such measurements are then discussed. Finally, the techniques for data analysis and reporting are provided. The nanoparticles cited are National Institute of Standards and Technology (NIST) Au nanoparticle Reference Materials RM 8011 (nominally 10 nm particles), RM 8012 (nominally 30 nm), and RM 8013 (nominally 60 nm).