Parabolic Tip Research Articles

Loading effect of a metallic parabolic tip on S0 and A0 lamb waves

The loading effect induced by the contact between a parabolic duralumin tip and a free glass plate is investigated using Lamb waves and an optical heterodyne interferometric probe. The instrument detects 1-MHz impulse symmetric S₀ and antisymmetric A₀ Lamb wave trains launched in 1-mm-thick B270-type glass. Strain-optic modeling is carried out to explain optical measurement through the transparent medium and the loading effect of the tip. Three-wave optical interference modeling is also developed to explain the presence of fringes of equal thickness in C-scans of both modes propagating in a plate that has a 1.1-mrad wedge. Results show that through-glass probing inverts by a factor of -3.1 the signal that is normally returned by the interferometer at a free-air surface for the S₀ Lamb mode. Fringes of equal thickness reveal the spatial extension of the mechanical loading. Through-glass probing on A₀ produces about the same signal as in a free-air measurement configuration. This mode appears to be more appropriate for the evaluation of the loading effect of the tip. For this parabolic tip, we observe an A₀ attenuation of about 50% in the contact area.

Effect of Envelope Proteins on the Mechanical Properties of Influenza Virus

The envelope of the influenza virus undergoes extensive structural change during the viral life cycle. However, it is unknown how lipid and protein components of the viral envelope contribute to its mechanical properties. Using atomic force microscopy, here we show that the lipid envelope of spherical influenza virions is ∼10 times softer (∼0.05 nanonewton nm(-1)) than a viral protein-capsid coat and sustains deformations of one-third of the virion's diameter. Compared with phosphatidylcholine liposomes, it is twice as stiff, due to membrane-attached protein components. We found that virus indentation resulted in a biphasic force-indentation response. We propose that the first phase, including a stepwise reduction in stiffness at ∼10-nm indentation and ∼100 piconewtons of force, is due to mobilization of membrane proteins by the indenting atomic force microscope tip, consistent with the glycoprotein ectodomains protruding ∼13 nm from the bilayer surface. This phase was obliterated for bromelain-treated virions with the ectodomains removed. Following pH 5 treatment, virions were as soft as pure liposomes, consistent with reinforcing proteins detaching from the lipid bilayer. We propose that the soft, pH-dependent mechanical properties of the envelope are critical for the pH-regulated life cycle and support the persistence of the virus inside and outside the host.

A modified cellular automaton method for polydimensional modelling of dendritic growth and microsegregation in multicomponent alloys

Numerous numerical models for simulating solidification of metals on a microscopic scale have been proposed in the past, among them are most importantly the phase-field method and models based on cellular automata. Especially the models based on cellular automata (adopting the virtual front tracking (VFT) concept) published so far are often only suitable for the consideration of one alloying element. Since industrial alloys are usually constituted of multicomponent alloys, the possibility of applying cellular automata is rather limited. With the aim of enhancing this modelling technique, a new, modified VFT model, which allows for the treatment of several alloying elements, in the low Péclet number regime is presented. The model uses the physical fundamentals of solute and heat diffusion in two dimensions as a basis for determining the solidification progress. By a new and effective approach, based on a functional extrapolation of the concentration gradient, dendritic growth in multicomponent Fe-C-Si-Mn-P-S alloys could be studied. The model shows the typical behaviour of dendritic solidification, such as parabolic tip and secondary dendrite arm formation as well as selection of preferably aligned columnar dendrites. A validation of the model is performed by the evaluation of morphological parameters and comparing them to experimentally determined values. The results for free and constrained dendritic growth effectively demonstrate the capabilities of this new model. The model is especially attractive for bridging the gap between one-dimensional microsegregation models and multidimensional morphology models with regard to modelling the complex interrelations between segregation on a multidimensional level and morphology formation.

An Application of Conformal Mapping to Fracture Analysis of Composite Plate with Curved Crack

Fracture problem about anisotropic composite plate of curved crack is researched with the help of appropriate conformal mapping. The crack curve is transformed into a unit circle by the quoted conformal mapping function. Based on the stress boundary conditions, the special stress function is found and the expressions for the stress field and the stress intensity factor near the parabolic crack tip are obtained.

Phase-field study of three-dimensional steady-state growth shapes in directional solidification

We use a quantitative phase-field approach to study directional solidification in various three-dimensional geometries for realistic parameters of a transparent binary alloy. The geometries are designed to study the steady-state growth of spatially extended hexagonal arrays, linear arrays in thin samples, and axisymmetric shapes constrained in a tube. As a basis to address issues of dynamical pattern selection, the phase-field simulations are specifically geared to identify ranges of primary spacings for the formation of the classically observed "fingers" (deep cells) with blunt tips and "needles" with parabolic tips. Three distinct growth regimes are identified that include a low-velocity regime with only fingers forming, a second intermediate-velocity regime characterized by coexistence of fingers and needles that exist on separate branches of steady-state growth solutions for small and large spacings, respectively, and a third high-velocity regime where those two branches merge into a single one. Along the latter, the growth shape changes continuously from fingerlike to needlelike with increasing spacing. These regimes are strongly influenced by crystalline anisotropy with the third regime extending to lower velocity for larger anisotropy. Remarkably, however, steady-state shapes and tip undercoolings are only weakly dependent on the growth geometry. Those results are used to test existing theories of directional finger growth as well as to interpret the hysteretic nature of the cell-to-dendrite transition.

Weakly Nonlinear Theory of Dynamic Fracture

The common approach to crack dynamics, linear elastic fracture mechanics, assumes infinitesimal strains and predicts a r(-1/2) strain divergence at a crack tip. We extend this framework by deriving a weakly nonlinear fracture mechanics theory incorporating the leading nonlinear elastic corrections that must occur at high strains. This yields strain contributions "more divergent" than r(-1/2) at a finite distance from the tip and logarithmic corrections to the parabolic crack tip opening displacement. In addition, a dynamic length scale, associated with the nonlinear elastic zone, emerges naturally. The theory provides excellent agreement with recent near-tip measurements that cannot be described in the linear elastic fracture mechanics framework.

Effect of Shape of the Tip in Determining Interphase Properties in Fiber Reinforced Plastic Composites Using Nanoindentation

Fiber reinforced polymer composites are two component material systems in which fibers are embedded in a polymer matrix. Such a system inherently has an interface where the two components meet. Adjacent to the interface extending beyond the fiber surface is the “interphase region.” Properties within the interphase vary due to variations in the chemistry. The study of mechanical property variations with changing chemistry will help in better understanding and tailoring of the composite properties. The present work concentrates on the investigation of nanomechanical properties within the interphase of a glass fiber embedded in polyester matrix system. The glass fibers were coated with two types of silanes to produce a strong and a weak bond at the fiber-matrix interface. Nanoindentation techniques coupled with atomic force microscopy imaging capabilities have been used for this investigation. Two different tips were employed for indenting, one being a Berkovich diamond tip supplied by Hysitron, Inc., Minneapolis, MN and another being a parabolic tungsten tip, which was made in the laboratory. Indentations were performed within the interphase region, also in the bulk matrix, and on the glass fiber. The variation in mechanical properties such as modulus, stiffness, hardness, and penetration depth were obtained within the interphase by indenting at the fiber surface outward. Variations of the elastic modulus in the interphase region and its relation to the chemistry are presented. The results obtained using two different tip shapes have been compared. Phase imaging was performed using tapping mode atomic force microscopy to qualitatively identify the presence of an interphase near the glass fiber-polyester interface. These experiments show that when no coupling agent is used the interphase thickness is less than 0.1 μm, and its exact determination is limited by the spatial resolution of the tips employed and the process of indentation. Phase imaging results with composite samples made of coated glass fibers corroborate the results obtained from nanoindentation experiments.

The approximate ABCD matrix for a parabolic lens of revolution and its application in calculating the coupling efficiency

Using ray optics, we present an explicit formulation of the ABCD matrix for reflection and refraction of light beams at a parabolic interface separating media of different refractive indices under paraxial approximation. Based on the formulated ABCD matrix for refraction by a parabolic lens tip, we present a simple theoretical investigation of the coupling efficiency between a laser diode and a single-mode fiber with a parabolic lens formed on the fiber tip. The results show that this technique is effective and will be of benefit to designing suitable microlenses applying the laser-coupling technique.

Numerical simulation of dynamic sand dunes

The physics of granular materials is interesting from many points of view because they exhibit a wealth of phenomena that have both fluid and solid aspects [C.S. Campbell, Annu. Rev. Fluid. Mech. 22 (1990) 57, H.M. Jaeger, S.R. Nagel, R.P. Behringer, Phys. Today 494 (1996) 32]. Recently a difficult pattern was observed if sand falls in the space between two plates and passes an obstacle [Y. Amarouchene, J.F. Boudet, H. Kellay, Phys. Rev. Lett. 86 (2001) 4286]. The interesting behaviour occurs on top of the obstacle where a dynamic dune with a parabolic tip is formed. Inside this parabola, a triangular region of non- or very slow flowing sand is observed. Using factor analysis it is possible to extract latent parameters from a dynamic process. Applying a three factor model we can clearly identify the inner triangle (1st factor) and the outer parabolic pattern (3rd factor). The second factor we interpret as shock wave. Most interactions between particles take place in a relatively small region. We show that the pattern formation process depends on the restitution coefficients (particle–particle and particle–obstacle) and also on the particle size. These findings cannot be observed if standard velocity profiles are used to analyse the data. Our findings show, that most interactions take place in a relatively small area correlating with the particle size. If the interactions between different particles and particle–obstacle are elastic the formation of a non-flowing triangular region is more difficult as if inelastic collisions are used. The factor curves also clearly show that a pattern formation process has to be finished, before the next pattern can be formed.

Theoretical comparison of the band broadening in nonretained electrically and pressure-driven flows through an ordered chromatographic pillar packing.

Using a well-validated computational fluid dynamics simulation method, based on a multi-ion transport model, a detailed analysis of the differences in band broadening between pressure-driven (PD) and electrically driven (ED) flows through perfectly ordered, identical chromatographic pillar packings has been made. It was found that, although the eddy-diffusion band-broadening contributions were nearly completely absent in the considered structure, the ED flow still yields much smaller plate heights than the PD flow. This difference could be fully attributed to the different ways in which the ED and PD velocity profiles reshape when passing through a tortuous pore structure with undulating cross section. Whereas in the PD case the parabolic tip of the band front is continually squeezed and extended each time it passes a pore constriction, the ED flow displays some kind of band front restoring mechanism, with which the fluid elements of the band front are (at least partly) laterally re-aligned after each pore constriction passage. This could be clearly visualized from a series of step-by-step images of the progression of a sharply "injected" species band moving through the packing under ED and PD conditions.

Growth of solutal dendrites: A cellular automaton model and its quantitative capabilities

A model based on the cellular automaton (CA) technique for the simulation of dendritic growth controlled by solutal effects in the low Peclet number regime was developed. The model does not use an analytical solution to determine the velocity of the solid-liquid (SL) interface as is common in other models, but solves the solute conservation equation subjected to the boundary conditions at the interface. Using this approach, the model does not need to use the concept of marginal stability and stability parameter to uniquely define the steady-state velocity and radius of the dendrite tip. The model indeed contains an expression for the stability parameter, but the process determines its value. The model proposes a solution for the artificial anisotropy in growth kinetics valid at zero and 45° introduced in calculations by the square cells and trapping rules used in previous CA formulations. It also introduces a solution for the calculation of local curvature, which eliminates mesh dependency of calculations. The model is able to reproduce qualitatively most of the dendritic features observed experimentally, such as secondary and tertiary branching, parabolic tip, arms generation, selection and coarsening, etc. Computation results are validated in two ways. First, the simulated secondary dendrite arm spacing (SDAS) is compared with literature values. Then, the predictions of the classic Lipton-Glicksman-Kurz (LGK) theory for steady-state tip velocity are compared with simulated values as a function of melt undercooling. Both comparisons are found to be in very good agreement.

Growth of solutal dendrites—A cellular automaton model

The stochastic cellular automaton (CA) model, originally proposed by Dilthey et al. (1997) was further refined and applied to the simulation of dendritic growth controlled by solutal effects in the low Péclet number regime. The use of square cells in the division of the computational domain introduces an artificial anisotropy when growth is simulated by capture of the closest neighboring cells. This artificial anisotropy becomes significant when the preferential growth direction is not aligned with the axis of the cell. The proposed model offers a solution for the artificial anisotropy using a new set of capturing rules. It also recommends a stability parameter whose value must converge to the stability constant, σ*, to enable the CA model to predict steady-state dendritic growth kinetics. The model reproduces qualitatively most of the dendritic features observed experimentally, such as secondary and tertiary branching, parabolic tip, arms generation and selection. Validation was performed by comparing the simulated secondary dendrite arm spacing with literature values and then by comparing the predictions of the classic Lipton-GIicksman-Kurz theory for steady state tip velocity, with simulated values as function of melt undercooling. Both comparisons were found to be in good agreement. However, results are still mesh dependent.

Numerical simulation of Zn coating solidification

A numerical model, which simulates nucleation and growth of Zn grains, has been developed in order to describe quantitatively the solidification of Zn coatings during the hot-dipping process. The inputs of the model are the nucleation distribution, which has been measured by electron backscattered diffraction (EBSD), and the dendritic growth kinetics, calculated with an analytical model of a parabolic dendrite tip modified to account for the interactions with the coating interfaces. The model predicts the shapes of the grain envelopes as a function of the grain orientation and the texture induced by growth. Three types of grain envelopes have been evidenced, depending on the angle between the c-axis and the normal to the coating plane. Moreover, it has been shown that growth reinforces the already existing {00.1} nucleation texture, in good agreement with experimental data. The model also predicts the cooling curve, including recalescence, and the grain size. Thus, it is used to describe the effects of Pb additions on solidification. In particular, it has been shown that Pb increases the nucleation undercooling and strongly decreases the density of active nuclei, thus resulting in a much larger grain size.

A dislocation mechanism of friction between a nanoprobe and solid surface

A dislocation mechanism of friction between an atomic-force microscope (AFM) probe and an atomically smooth solid surface is put forward. In this mechanism, the contact region is represented by an edge dislocation. The triboacoustic emission measured with an AFM shows the dislocation nature of friction. The friction force is calculated for a parabolic tip.

Phase-field simulations of dendritic crystal growth in a forced flow.

Convective effects on free dendritic crystal growth into a supercooled melt in two dimensions are investigated using the phase-field method. The phase-field model incorporates both melt convection and thermal noise. A multigrid method is used to solve the conservation equations for flow. To fully resolve the diffuse interface region and the interactions of dendritic growth with flow, both the phase-field and flow equations are solved on a highly refined grid where up to 2.1 million control volumes are employed. A multiple time-step algorithm is developed that uses a large time step for the flow-field calculations while reserving a fine time step for the phase-field evolution. The operating state (velocity and shape) of a dendrite tip in a uniform axial flow is found to be in quantitative agreement with the prediction of the Oseen-Ivantsov transport theory if a tip radius based on a parabolic fit is used. Furthermore, using this parabolic tip radius, the ratio of the selection parameters without and with flow is shown to be close to unity, which is in agreement with linearized solvability theory for the ranges of the parameters considered. Dendritic sidebranching in a forced flow is also quantitatively studied. Compared to a dendrite growing at the same supercooling in a diffusive environment, convection is found to increase the amplitude and frequency of the sidebranches. The phase-field results for the scaled sidebranch amplitude and wavelength variations with distance from the tip are compared to linear Wentzel-Kramers-Brillouin theory. It is also shown that the asymmetric sidebranch growth on the upstream and downstream sides of a dendrite arm growing at an angle with respect to the flow can be explained by the differences in the mean shapes of the two sides of the arm.

Structure formation in diffusional growth and dewetting

The morphology diagram of possible structures in two-dimensional diffusional growth is given in the parameter space of undercooling Δ versus anisotropy of surface tension ϵ. The building block of the dendritic structure is a dendrite with a parabolic tip, and the basic element of the seaweed structure is a doublon. The transition between these structures shows a jump in the growth velocity. We show the analogy of diffusional growth with dewetting patterns of a fluid film on a substrate. We also describe the structures and velocities of fractal dendrites and doublons destroyed by noise. The extension of these results to three-dimensional growth is briefly discussed.

Nanoscale elasticity measurement with in situ tip shape estimation in atomic force microscopy

For a quantitative evaluation of nanoscale elasticity, atomic force microscopy, and related methods measure the contact stiffness (or force gradient) between the tip and sample surface. In these methods the key parameter is the contact radius, since the contact stiffness is changed not only by the elasticity of the sample but also by the contact radius. However, the contact radius is very uncertain and it makes the precision of measurements questionable. In this work, we propose a novel in situ method to estimate the tip shape and the contact radius at the nanoscale contact of the tip and sample. Because the measured resonance frequency sometimes does not depend so sensitively on the contact force as expected from the parabolic tip model, we introduced a more general model of an axial symmetric body and derived an equation for the contact stiffness. Then, the parameters in the model are unambiguously determined from a contact force dependence of the cantilever resonance frequency. We verified that this method is able to provide an accurate prediction of the cantilever thickness, the tip shape, and the effective elasticity of soft and rigid samples.

Formation of needle shaped twin domains in cordierite: A computer simulation study

The formation of needle shaped domains in cordierite is simulated using a simplified two-dimensional computer model with hexagonal symmetry and nonlocal (strain-induced) atomic interactions. Domain walls form along elastically soft directions as expected and sometimes needle domains appear during annealing. During annealing needles are then consumed by the coarsening process or retract into the domains to which they are attached. Needles have rounded, parabolic tips symptomatic of cordierite’s comparatively low anisotropy energy. Needle splitting is not observed but a novel wetting mechanism by which auxiliary domains form at the needle tip is observed. These domains form such as to make a section of the needle tip a strain allowed domain wall and so reduce the bending energies of the domain walls in this region. At higher temperatures, needle domain walls become only poorly orientated along soft directions and needle domains become patchy with blunt tips and increased “wetting” at the needle tips.

Needle twins and right-angled twins in minerals; comparison between experiment and theory

Transformation twinning in minerals forms isolated twin walls, intersecting twin walls with corner junctions, and wedge-shaped twins as elements of hierarchical patterns. When cut perpendicular to the twin walls, the twins have characteristic shapes, right-angled and needle-shaped wall traces, which can be observed by transmission electron microscopy or by optical microscopy. Theoretical geometries of wall shapes recently derived for strain-related systems should hold for most displacive and order-disorder type phase transitions: (1) right-angled twins show curved junctions; (2) needle-shaped twins contain flat wall segments near the needle tip if the elastic behavior of the mineral is dominated by its anisotropy; (3) additional bending forces and pinning effects lead to curved walls near the junction that make the needle tip appear more blunt. Experimental studies confirmed that these features occur in a wide range of materials. Bent right-angled twins were analyzed in Gd 2 (MoO 4 ) 3 . Linear needle tips were found in WO 3 , [N(CH 3 ) 4 ] 2 .ZnBr 4 CrAl, BiVO 4 , GdBa 2 Cu 3 O 7 , and PbZrO 3 . Parabolic tips occur in K 2 Ba(NO 2 ) 4 , and GeTe whereas exponential curvatures appear in BaTiO 3 , KSCN, Pb 3 (PO 4 ) 2 , CaTiO 3 , alkali feldspars, YBa 2 Cu 3 O 7 ?, and MnAl. The size and shape of the twin microstructure relates to its formation during the phase transition and the subsequent annealing history. The mobility of the twin walls after formation depends not only on the thermal activation but also on the structure of the wall, which may be pinned to impurities on a favorable structural site. Depinning energies are often large compared with thermal energies for diffusion. This leads to kinetic time scales for twin coarsening that are comparable to geological time scales. Therefore, transformation twins that exhibit needle domains not only indicate that the mineral underwent a structural phase transition but also contain information about its subsequent geological history.

Morphology diagram of possible structures in diffusional growth

A theory for the morphology diagram of possible structures in two-dimensional diffusional growth is presented. The main control parameters are undercooling Δ, anisotropy of surface tension ε and the strength of noise. Basic patterns are dendrites and seaweed. The building block of the dendritic structure is a dendrite with parabolic tip, and the basic element of the seaweed structure is a doublon. The transition between these structures shows a jump in the growth velocity. The possible extension of these results to three-dimensional growth is shortly discussed.