A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Multiple Dynamic Processes Contribute to the Complex Steady Shear Behavior of Cross-Linked Supramolecular Networks of Semidilute Entangled Polymer Solutions.

Molecular theories of shear thickening and shear thinning in associative polymer networks are typically united in that they involve a single kinetic parameter that describes the network -- a relaxation time that is related to the lifetime of the associative bonds. Here we report the steady-shear behavior of two structurally identical metallo-supramolecular polymer networks, for which single-relaxation parameter models break down in dramatic fashion. The networks are formed by the addition of reversible cross-linkers to semidilute entangled solutions of PVP in DMSO, and they differ only in the lifetime of the reversible cross-links. Shear thickening is observed for cross-linkers that have a slower dissociation rate (17 s(-1)), while shear thinning is observed for samples that have a faster dissociation rate (ca. 1400 s(-1)). The difference in the steady shear behavior of the unentangled vs. entangled regime reveals an unexpected, additional competing relaxation, ascribed to topological disentanglement in the semidilute entangled regime that contributes to the rheological properties.

Mechanism of Shear Thickening in Reversibly Cross-linked Supramolecular Polymer Networks.

We report here the nonlinear rheological properties of metallo-supramolecular networks formed by the reversible cross-linking of semi-dilute unentangled solutions of poly(4-vinylpyridine) (PVP) in dimethyl sulfoxide (DMSO). The reversible cross-linkers are bis-Pd(II) or bis-Pt(II) complexes that coordinate to the pyridine functional groups on the PVP. Under steady shear, shear thickening is observed above a critical shear rate, and that critical shear rate is experimentally correlated with the lifetime of the metal-ligand bond. The onset and magnitude of the shear thickening depend on the amount of cross-linkers added. In contrast to the behavior observed in most transient networks, the time scale of network relaxation is found to increase during shear thickening. The primary mechanism of shear thickening is ascribed to the shear-induced transformation of intrachain cross-linking to interchain cross-linking, rather than nonlinear high tension along polymer chains that are stretched beyond the Gaussian range.

Simulation of transverse and longitudinal magnetic ripple structures induced by surface anisotropy

Micromagnetic ripple structures on the surfaces of thick specimens of ultra-soft magnetic material having strong surface anisotropy Ks favouring out-of-surface magnetization have been calculated. These ripples have wavelengths of the order of 0.1 μm and extend to a depth ∼ √A/Ms, where A is the exchange constant and Ms is the saturation magnetization. The wave-vectors of the ripple structures are either transverse or parallel to the bulk magnetization. Both structures have lower energy than the one-dimensional structure discussed by O'Handley and Woods, and they exhibit stronger normal magnetization. The transverse structure requires a surface anisotropy Ks ≥ 0.80K0, where is that required for the one-dimensional structure. The threshold for longitudinal ripples is 0.84K0. It is suggested that the transverse structure probably constitutes the ground state. The magnitudes of Ks and A should be obtainable from measurements of the ripple wavelength and amplitude, and Ms.

Shear strength analysis of ball grid array (BGA) solder interfaces

Ball shear test is the most common test method used to assess the reliability of bond strength for ball grid array (BGA) packages. In this work, a combined experimental and numerical study was carried out to realize of BGA solder interface strength. Solder mask defined bond pads on the BGA substrate were used for BGA ball bonding. Different bond pad metallizations and solder alloys were used. Solid state aging at 150degC up to 1000 h has been carried out to change the interfacial microstructure. Cross-sectional studies of the solder-to-bond pad interfaces was conducted by scanning electron microscopy (SEM) equipped with an energy dispersive X-ray (EDX) analyzer to investigate the interfacial reaction phenomena. Ball shear tests have been carried out to obtain the mechanical strength of the solder joints and to correlate shear behaviour with the interfacial reaction products. An attempt has been taken to realize experimental findings by Finite Element Analysis (FEA). It was found that intermetallic compound (IMC) formation at the solder interface plays an important role in the BGA solder bond strength. By changing the morphology and the microchemistry of IMCs, the fracture propagation path could be changed and hence, reliability could be improved

Shear bond strengths of glass-ionomer cements to sound and to prepared carious dentine

The aim of this study was determine whether bonding of glass-ionomer cements to non-carious dentine differed from that to carious dentine. Five commercial cements were used, namely Fuji IX GP, Fuji IX capsulated, Fuji IX Fast capsulated (all GC, Japan), Ketac-Molar and Ketac-Molar Aplicap (both 3M-ESPE, Germany). Following conditioning of the substrate with 10% poly (acrylic acid) for 10 s, sets of 10 samples of the cements were bonded to prepared teeth that had been removed for orthodontic reasons. The teeth used had either sound dentine or sclerotic dentine. Shear bond strengths were determined following 24 h storage. For the auto-mixed cements, shear bond strength to sound dentine was found not to differ statistically from shear bond strength to sclerotic dentine whereas for hand-mixed cements, shear bond to sound dentine was found to be higher than to carious dentine (to at least p < 0.05). This shows that the chemical effects arising from interactions of glass-ionomer cements with the mineral phase of the tooth are the most important in developing strong bonds, at least in the shorter term.

Effect of machine dependence on wall failure property measurements, Jenike versus an annular shear tester

Annular, ring or torsional shear testers are commonly used in bulk solids handling research for the purpose of powder characterisation or equipment design. This paper reports from a DEFRA sponsored project which aims to develop an industrial powder flow-ability tester, (based on the annular shear tester) that is economic to buy and quick and easy to use in trained but unskilled hands. This paper compares the wall failure loci measured with an annular shear cell with measurements obtained using the accepted standard wall friction tester, the Jenike shear cell. These wall failure loci have been measured for several bulk solids which range from fine cohesive powders to free-flowing granular materials, on a stainless steel 304 2B wall surface.