Shear Wave Ultrasound Inspection of Flaws in Silicon Wafers Using Focused Transducers.

Silicon parts can contain micrometer-sized vertical cracks that are challenging to detect. Inspection using high-frequency focused ultrasound has shown promise for detecting defects of this size and geometry. However, implementing focused ultrasound to inspect anisotropic media can prove challenging, given the directional dependence of wave propagation and subsequent focusing behavior. In this work, back surface-breaking defects at various orientations within silicon wafers (0°, 15°, and 45° relative to the [010] crystallographic axis) are experimentally inspected in an immersion tank setup. Using 100 MHz unfocused and focused shear waves, the impact of medium anisotropy on focusing and defect detection is evaluated. The scattering amplitude and defect detection sensitivity results demonstrate orientation-dependent patterns that strongly rely on the use of focused transducers. The defects along the 45° orientation reveal two-lobe scattering patterns with maximum amplitudes less than half that of the defects in the 0° orientation, which in contrast show a one-lobe scattering pattern. The experimental results are further explored using finite element (FE) modeling and ray tracing to visualize the impact of focusing on wave propagation within the silicon. Ray tracing results show that the focused beam profiles for the 45° and 0° orientations form a butterfly wing and elliptical focusing profile, respectively, which correspond directly to experimentally found scattering patterns from defects. Additionally, the FE scattering results from unfocused transducers reveal single lobe scattering for both 0° and 45° orientations, proving the varying scattering patterns to be driven by the anisotropic focusing behavior.

Focal depth localization for highly focused transducers in isotropic materials.

Focusing equations aim to define the point in a solid at which a transducer beam will reach a minimum cross section. The most commonly used focusing equation relies on a small angle assumption that inherently excludes sharply focused transducers with significant curvature. In this article, a revised focusing equation is proposed for normal and oblique incidence through a fluid-solid interface. The closed-form equation is derived using ray tracing approaches similar to the conventional expression but circumvents the paraxial approximation, extending the applicability to sharply focused probes. Both conventional and modified focusing equations are compared through normal and oblique incidence ray diagrams, and the proximity to the computationally derived geometric focus is explored. The proposed modification to the focusing equation generally results in a closer approximation to the geometric focus, a smaller beam cross section, and a greater time convergence when compared to the conventional focusing equation.

Pressure influence on elastic wave attenuation in polycrystalline materials.

Traditionally, the acoustoelastic effect refers to the influence of stress in a solid on an elastic wave's phase velocity. Since the phase velocity can be represented by the real part of the complex wave number, a natural question arises regarding the effect of stress on the imaginary part or dissipation of the wave. In this article, the influence of pressure on the elastic wave's attenuation in polycrystalline materials is modeled. The constitutive behavior of an initially stressed solid is coupled into Weaver's scattering-based attenuation model [J. Mech. Phys. Solids 38, 55-86 (1990)]. As a result, the pressure-dependent longitudinal and shear wave attenuation coefficients are unveiled. As the traditional stress-free attenuation coefficients depend on the degree of single-crystal elastic anisotropy, it is shown that the pressure influence on attenuation depends on the anisotropy of the single-crystal's third-order or nonlinear elastic constants. Analysis of the model indicates linkages between pressure derivatives of velocity and attenuation to the material's linear and nonlinear elastic anisotropy, crystal structure, and type of atomic bonding.

Generalized ultrasonic scattering model for arbitrary transducer configurations.

Ultrasonic scattering in polycrystalline media is directly tied to microstructural features. As a result, modeling efforts of scattering from microstructure have been abundant. The inclusion of beam modeling for the ultrasonic transducers greatly simplified the ability to perform quantitative, fully calibrated experiments. In this article, a theoretical scattering model is generalized to allow for arbitrary source and receiver configurations, while accounting for beam behavior through the total propagation path. This extension elucidates the importance and potential of out-of-plane scattering modes in the context of microstructure characterization. The scattering coefficient is explicitly written for the case of statistical isotropy and ellipsoidal grain elongation, with a direct path toward expansion for increased microstructural complexity. Materials with crystallites of any symmetry can be studied with the present model; the numerical results focus on aluminum, titanium, and iron. The amplitude of the scattering response is seen to vary across materials, and to have varying sensitivity to grain elongation and orientation depending on the transducer configuration selected. The model provides a pathway to experimental characterization of microstructure with optimized sensitivity to parameters of interest.

Ultrasonic backscatter from elongated grains using line focused ultrasound.

Ultrasonic backscattering from polycrystalline materials with elongated grains is investigated. A normal incident line-focus transducer is employed such that refracted longitudinal and transverse waves are focused within the polycrystal and scatter at grain boundaries back to the transducer. A ray-based scattering model is developed to explain the dependence of the statistics of scattering measurements on grain elongation. The spatial variance of measured scattered signals from Al alloy (7475-T7) is compared to the model. This work promotes the ultrasonic backscatter technique for monitoring grain elongation of metals using one transducer with access to a single sample face.

Ultrasonic harmonic generation from materials with up to cubic nonlinearity.

This letter considers the combined effects of quadratic and cubic nonlinearity on plane wave propagation in generally anisotropic elastic solids. Displacement solutions are derived that represent the fundamental, second-, and third-harmonic waves. In arriving at the solutions, the quadratic and cubic nonlinearity parameters for generally anisotropic materials are defined. The effects of quadratic and cubic nonlinearity are shown to influence the amplitude and phase of the fundamental wave. In addition, the phase of the third-harmonic depends on a simple combination of the quadratic and cubic nonlinearity parameters. Nonlinearity parameters are given explicitly for materials having isotropic and cubic symmetry. Lastly, acoustic nonlinearity surfaces are introduced, which illustrate the nonlinearity parameters as a function of various propagation directions in anisotropic materials.

Diffuse ultrasonic backscatter using a multi-Gaussian beam model.

Diffuse ultrasonic backscatter is widely used to evaluate microstructural parameters of heterogeneous materials. Recent singly scattered response (SSR) models utilize a single-Gaussian beam (SGB) assumption which is expected to have limitations. Following a similar formalism, a model is presented using a multi-Gaussian beam (MGB) assumption to characterize the transducer beam for longitudinal-to-longitudinal scattering at normal incidence through an interface with arbitrary curvature. First, the Wigner transform of the transducer field is defined using conjugate double-layer MGB expressions. The theoretical analysis shows that ten groups of Gaussian beams are sufficient for convergence. Compared with the SGB-SSR curve, the shape of MGB-SSR curve is positive skewed. Differences between the MGB-SSR model and the SGB-SSR model are quantified and shown to be complex functions of frequency, sample curvature, transducer parameters, and focal depth in the material. Finally, both models are used to fit experimental spatial variance data from a 304 stainless steel pipe with planar, convex, and concave surfaces. The results show that the MGB-SSR has some characteristics suggesting a better fit to the experiments. However, both models result in grain size estimates within the uncertainty of the optical microscopy suggesting that the SGB is sufficient for normal incidence pulse-echo measurements.

Ultrasonic attenuation of polycrystalline materials with a distribution of grain sizes.

Elastic wave scattering at grain boundaries in polycrystalline media can be quantified to determine microstructural properties. The amplitude drop observed for coherent wave propagation (attenuation) as well as diffuse-field scattering events have been extensively studied. In all cases, the scattering shows a clear dependence on grain size, grain shape, and microstructural texture. Models used to quantify scattering experiments are often developed assuming dependence on a single spatial length scale, usually, mean grain diameter. However, several microscopy studies suggest that most metals have a log normal distribution of grain sizes. In this study, grain size distribution is discussed within the context of previous attenuation models valid for arbitrary crystallite symmetries. Results are presented for titanium using a range of distribution means and widths assuming equiaxed grains and no preferred crystallographic orientation. The longitudinal and shear attenuations are shown to vary with respect to the frequency dependence for varying distribution widths even when the volumetric mean grain size is held constant. Furthermore, the results suggest that grain size estimates based on attenuation can have large errors if the distribution is neglected. This work is anticipated to play an important role in microstructural characterization research associated with ultrasonic scattering.

Evaluating grain size in polycrystals with rough surfaces by corrected ultrasonic attenuation.

Surface roughness of a sample has a great effect on the calculated grain size when measurements are based on ultrasonic attenuation. Combining modified transmission and reflection coefficients at the rough interface with a Multi-Gaussian beam model of the transducer, a comprehensive correction scheme for the attenuation coefficient is developed. An approximate inverse model of the calculated attenuation, based on Weaver's diffuse scattering theory, is established to evaluate grain size in polycrystals. The experimental results showed that for samples with varying surface roughness and matching microstructures, the fluctuation of evaluated average grain size was ±1.17µm. For polished samples with different microstructures, the relative errors to optical microscopy were no more than ±3.61%. The presented method provides an effective nondestructive tool for evaluating the grain size in metals with rough surfaces.

Mode-converted ultrasonic scattering in polycrystals with elongated grains.

Elastic wave scattering is used to study polycrystalline media for a wide range of applications. Received signals, which include scattering from the randomly oriented grains comprising the polycrystal, contain information from which useful microstructural parameters may often be inferred. Recently, a mode-converted diffuse ultrasonic scattering model was developed for evaluating the scattered response of a transverse wave from an incident longitudinal wave in a polycrystalline medium containing equiaxed single-phase grains with cubic elastic symmetry. In this article, that theoretical mode-converted scattering model is modified to account for grain elongation within the sample. The model shows the dependence on scattering angle relative to the grain axis orientation. Experimental measurements were performed on a sample of 7475-T7351 aluminum using a pitch-catch transducer configuration. The results show that the mode-converted scattering can be used to determine the dimensions of the elongated grains. The average grain shape determined from the experimental measurements is compared with dimensions extracted from electron backscatter diffraction, an electron imaging technique. The results suggest that mode-converted diffuse ultrasonic scattering has the potential to quantify detailed information about grain microstructure.