Base Radius Research Articles

Modeling the sharp focus of a radially polarized laser mode using a conical and a binary microaxicon

Based on the radial finite-difference time domain method we numerically show that the illumination of a conical glass microaxicon of base radius 7 μm and height 6 μm (numerical aperture NA=0.6) by a radially polarized annular R-TEM01 laser mode of wavelength λ=1 μm produces, 20 nm apart from the cone apex, a sharp focus of transverse diameter FWHM=0.30λ and longitudinal (axial) width FWHMz=0.08λ. The focal spot area at half-maximum intensity is HMA=0.071λ2. For comparison the focus diameter reported here is 1.7 times smaller than the diameter of the minimal diffraction spot (Airy disk, NA=1) of FWHM=0.51λ, with its area being 2.87 times smaller than that of the Airy disk HMA=0.204λ2. Also it is smaller than the diameter of a focal spot formed by a lens with NA=0.6 in an immersion medium n=1.5, FWHM=0.51λ/n NA=0.55λ, smaller than the diffraction limit in the medium (n=1.5, NA=1) FWHM=0.51λ/n=0.33λ and smaller than the focusing limit for the Bessel beam (NA=1)FWHM=0.36λ.

Computer generated hologram null compensator for optical metrology of a freeform surface

A computer-generated hologram (CGH) null compensator has been designed for optical metrology of a freeform surface (FFS) with positive base radius. The CGH is being employed in the freeform testing for the first time. The basic concept of the optical metrology and the testing procedure are explained explicitly. The design results of the CGH null compensator are presented in this article. Moreover, the design method and its results are discussed in detail.

Quantum limits to the electron field emission from tapered conductive sheets

A model has been constructed in order to study the effects of quantum confinement on the electron population on tapered conductive sheets. A two-dimensional rounded nanocone was considered for this study and the Schrödinger equation was solved analytically on the entire system. The average number of electrons on the tip region was shown to be strongly influenced by the geometrical parameters of the system, such as the tip and base radius and the overall length. Field emission from such structures was also investigated using a simple one-dimensional Wentzel–Kramers–Brillouin approximation. The limitations on the electron population imposed by the strong quantum confinement at the tip region will have important consequences on the field emission current obtained from these structures.

Dynamic Spreading of Polar Liquids on Offset Papers

The aim of this study was to investigate the spreading of sessile drops of polar probe liquids (water and ethylene glycol) on pigment coated offset papers. Furthermore, existing theoretical models for spreading were applied to evaluate the experimental results. The hydrodynamic model gave a better fit to the results at lower spreading rates, while the molecular-kinetic theory gave a good fit over a larger velocity range. Factors introduced to correct for the exponential dependency on time of the drop base radius and its contact angle can be interpreted as coupled processes. Differences in the spreading between the papers were found to correlate with the acid and base components of the surface energy, rather than with differences in surface roughness.

A Model for Fractal Dimension of Rough Surfaces

We report a model for the fractal dimension Ds of rough surfaces based on the fractal distribution of roughness elements on surfaces and the fractal character of surface profiles. The proposed model for the fractal dimension Ds is expressed as a function of the fractal dimensions D for conic roughness diameter/height and Dp for surface profile, maximum roughness base diameter λmax, the ratio β of conic roughness height to its base radius as well as the ratio λminλmax of the minimum to the maximal base diameter.

Post-Tanner stages of droplet spreading: the energy balance approach revisited

The spreading of a circular liquid drop on a solid substrate canbe described in terms of the time evolution of its base radiusR(t). In complete wetting, the quasistationary regime (far away from initialand final transients) typically obeys the so-called Tanner law, withR∼tαT,αT = 1/10. Late-time spreading may differ significantly from the Tanner law: in some cases the dropdoes not thin down to a molecular film and instead reaches an equilibrium pancake-likeshape; in other situations, as revealed by recent experiments with spontaneously spreadingnematic crystals, the growth of the base radius accelerates after the Tanner stage. Here wedemonstrate that these two seemingly conflicting trends can be reconciled within a suitablyrevisited energy balance approach, by taking into account the line tension contribution tothe driving force of spreading: a positive line tension is responsible for the formation ofpancake-like structures, whereas a negative line tension tends to lengthen the contactline and induces an accelerated spreading (a transition to a faster power law forR(t) than in the Tanner stage).

Nonreactive Spreading at High-Temperature Revisited for Metal Systems via Molecular Dynamics

The spreading for Cu and Ag droplets on top of a rigid solid surface modeling Mo is herewith considered via molecular dynamics. The dynamics of the base radius and the contact angle are recorded and fitted using the molecular-kinetic theory. A method is described to determine for liquid metals at the microscopic level the parameters appearing in this theory. These microscopic parameters are calculated directly in the simulations and compared to the fitted values. The agreement between the fitted values and the calculated ones shows that the dissipation of energy within the drop is caused primarily by the friction of liquid atoms over the substrate. This validation supports the understanding of the mechanisms controlling the spreading of liquid metals which, up to now, were based on experimental data and fitting procedures.

119Sn NMR spectroscopic and structural properties of transition metal complexes with terminal stannylene ligands

Abstract The coordination effect of transition metal TMLn (L = co-ligand) on stannylenes R2SnBn (B = base, n = 0–2) has been examined analysing 119Sn NMR data and X-ray molecular structures from the literature up to 2008. Coordination deshielding Δδ (119Sn) and chemical shift δ (119Sn) of BnR2Sn–TM(CO)n complexes, linearly correlate with δ (119Sn) of R2Sn. Slope, intercept and typical regions are interpreted through different ratios of σ/π-back bonding and impact of TM, R and Bn. Hybridisation changes explain dependencies of 1JSn–TM coupling constants on L, R, Bn and TM gyromagnetic ratios γTM. The Sn–TM bond lengths follow a parabolic profile along the TM period, fine-tuned by R, B and Ln. Electronic modifications on carbonyl complexes classify R2Sn as poor π-acceptors, while shortening of Sn–R and widening of angles R–Sn–R correspond to increased s-character of the Sn–TM bond. A “coordinative Lewis base radius” rcoord(SnR2) of 1.18 A is proposed.

BIDIMENSIONAL FINITE ELEMENT ANALYSIS OF SPUR GEAR: STUDY OF THE MESH STIFFNESS AND STRESS AT THE LEVEL OF THE TOOTH FOOT

The purpose of this work is to determine the spur gear mesh stiffness and the stress state at the level of the tooth foot. This mesh stiffness is derived from the calculation of the normal tooth displacements: local displacement where the load is applied, tooth bending displacement and body displacement [15]. The contribution of this work consists in, basing on previous works, developing optimal finite elements model in time calculation and results precision. This model permits the calculation of time varying mesh stiffness and the evaluation of stress state at the tooth foot. For these reasons a specific Fortran program was developed. It permit firstly, to obtain the gear geometric parameters (base radii, outside diameter,…) and to generate the data base of the finite element meshing of a tooth or a gear. This program is interfaced with the COSMOS/M finite element software to predict the stress and strain state and calculate the mesh stiffness of a gear system. It is noted that the mesh stiffness is periodic and its period is equal to the mesh period.

Infiltration time and imprint shape of a sessile droplet imbibing porous medium

The infiltration of a sessile droplet into a homogeneous porous medium for a constant droplet base radius case is solved numerically, where the porous medium is represented as a capillary network consisting of pores and throats. For a homogeneous medium, the network is built of the spherical pores of constant radius, and the cylindrical throats of constant radius and height. Having such defined network, the droplet imbibes porous medium in a single-phase flow for which the free interface in porous medium is smooth, and the liquid phase permeability and the capillary pressure are constant. Using the numerical solution we carry out the parametric study in which: (i) liquid viscosity and surface tension, (ii) droplet volume and base radius, and (iii) porous medium porosity and permeability are varied. The droplet infiltration time, and the imprint shape that is given with two spheroid half-axes are calculated. The dimensionless analysis is utilized to correlate the droplet infiltration parameters from which master curves for the droplet infiltration time and the droplet imprint shape are obtained. Using the infiltration time correlation, both numerical and experimental results show a linear behavior.

Optimization of a Trapezoidal Profile Annular Fin

The article presented here is an analysis that leads to the optimization for a design of an annular trapezoidal fin using a new approach to a two-dimensional analytical method. A comparison between the new analytical approach and the generally used method for a rectangular annular fin is made to show the exactness of the new method of approach. For an annular trapezoidal fin, the heat loss, fin tip radius, and fin base height are optimized as a function of the ratio of convection characteristic numbers, fin shape factor, dimensionless fin volume, dimensionless fin base radius, and convection characteristic number.

Self-aligned hemispherical formation of microlenses from colloidal droplets on heterogeneous surfaces

This paper proposes a drop-on-demand (DOD) theory and scheme for constructing hemispherical refractive microlenses onto underlying heterogeneous (laterally structured) surfaces that consist of hydrophilic s-domains and hydrophobic p-domains. In theory, the drops would self-align themselves into the s-domains by repelling the p-domains due to surface tension, precisely determining the placement though disobeying the Young–Laplace equation. Using a droplet generator (inkjet printhead), in our experiments, evaporative polyurethane (PU) drops well fitted their footprints (base radii) onto the s-domains with a radius of 100 µm surrounded by the p-domains of Teflon, where the photoresist AZ4620 was used for lifting off the corresponding domains of Teflon. As a result, plano-convex shapes with spherical curvatures were fabricated in an array (spacing L ∼ 100 µm) with base (footprint) radius (Rb) ∼ 95 µm and curvature radius (Rc) ∼ 122 µm. Thus, both the theoretical and experimental results agreed well in a hemispherical shape (deviation < 5%) with the contact angle θ ∼ 51°. Further, such an angle could be tunable with Δθ over 20° simply by varying drop volume. In addition, one interesting and significant finding, for two (and more) overflow drops (Rb > Rs + L/2 = 150 µm, in excess of volume), indicates that more complex shapes than spherical ones (such as dumbbell and cross-like) can also be constructed by bridging the drops. Compared to those previous methods using photolithographic techniques, the present method is potentially appropriate for the varying radius and complex placement of array patterns.

A criterion for minimum wave drag of a body

Determination of the form of a body ensuring minimum drag under various isoperimetric conditions is a fundamental problem of aerodynamics. Very few results have been obtained for this problem in a rigorous formulation, as well as for supersonic motion of a body with a given fineness ratio. It is known for the plane problem that a wedge is a body ensuring the minimum wave drag for the zero coefficient of pressureperturbation reflection ( K = 0) from the shock attached to the wedge corner [1] or for the critical mach of the shock-layer gas [2]. A moving wedge with an attached shock is an almost optimal body under other conditions [2, 3]. The wedge ensures the minimum drag within the framework of the linear theory ( τ � 1) and the Newtonian theory ( τ ≤ 1) [4]. Here, τ = 1/( j λ ) and λ = L /( jR ), L , and R are the fineness ratio, length, and the base size of the body, respectively, in the plane case j = 1. According to the Newtonian theory for the axisymmetric problem, where j = 2 and R is the radius of the body base, the wage drag is minimum for slender (high-fineness) bodies ( τ � 1) with power-shaped longitudinal contours y = R ( x / L ) m , m = 3/4, where ( x , y ) are the Cartesian coordinates with the origin at the body nose (Fig. 1). Power-shaped bodies are used in practice. The most comprehensive information on the aerodynamic parameters of such bodies is presented in [5]. This is one of very few classes of bodies for which the aerodynamic parameters have been studied both theoretically and experimentally in a wide range of geometric parameters λ and m and Mach numbers M. The wave drag of low-fineness bodies is optimal for blunt-nosed shapes [4]. In the present work, the exact expression for the derivative of the wave-drag coefficient C x is derived using a specific global shape variation of a plane or axisymmetric body with a given fineness ratio. This derivative determines how C x changes if a body of the considered class is replaced by a λ -equivalent body with a tail skirt. It is shown that a positive derivative is a necessary, but not sufficient condition of the minimum wave drag of the initial body. Application of this criterion to the available data on the aerodynamic parameters of power-shaped axisymmetric bodies shows that the wave drag can be reduced if the comparison-body shape is obtained by not only weak, but even strong variation of the initial-body longitudinal contour.

Optimizing Super-Hydrophobic Surfaces: Criteria for Comparison of Surface Topographies

Two useful criteria are employed for comparing the super-hydrophobicity of surface designs from a wetting point of view. The first is the apparent contact angle at the transition point between the homogeneous (Wenzel) wetting regime and the heterogeneous (Cassie–Baxter) wetting regime. The second criterion is the wetted area of the solid surface in the heterogeneous wetting regime. Four different types of model rough surfaces are compared using the above criteria, in order to demonstrate optimization considerations in the design of super-hydrophobic surfaces. These model surfaces consist of protrusions of cylinders, truncated cones, paraboloids, and hemispheres. It appears that protrusions of very small base radii are not desirable, since they must be very high in order to conform to the first criterion. For higher base radii it seems that rounded protrusions, such as the paraboloids, are advantageous. This is an interesting conclusion, since the protrusions of the Lotus leaf are of a similar geometry. Based on the case of hemispherical protrusions it also appears that more than one geometrical degree of freedom is necessary in order to optimize a super-hydrophobic surface.

Distribution of axial lava domes along a superfast overlapping spreading center, 27–32°S on the East Pacific Rise

Deep‐towed DSL‐120 bathymetric data are used to investigate the pattern of lava dome formation along a superfast spreading portion of the southern East Pacific Rise (EPR), including the overlapping limbs of a giant (120 × 120 km) propagator near 29°S. Along the 670 km of the axis surveyed, 1172 small domes were identified using a closed contour algorithm. Their abundance, defined by spatial density, is well correlated with the along‐axis relief of the ridge crest. Where the western and eastern limbs plunge toward the overlap zone, densities are high (3–6 km−2); however, where the axial depth profile is shallow and flat, densities are comparably low (0.4 km−2). Volcanic domes within the low abundance areas are characterized by lower ratios of height to basal radius (0.15 versus 0.22), smaller maximum heights (18 versus 40 m), and a larger relative percentage of small versus large mounds. The zone of high dome abundance encompasses the overlapping limbs of the rift and extends more than 100 km to the north and south beyond the overlap zone. Domes form dominantly during low effusion rate, point‐source eruptions, which suggests that discontinuous melt lenses underlie the ridge axis proximal to the overlapper. Conversely, fissure‐fed sheet flows dominate along the more distal segments, implying the presence of a more continuous axial magma lens. Throughout the survey area, dome abundance increases systematically near second‐order segment boundaries. Within the high abundance zone, some third‐order offsets also correlate with increased dome production, but local peaks in abundance are not tied exclusively to higher‐order ridge offsets. Where dome abundance is low, domes are clustered tightly near second‐order offsets and there is no increase in dome abundance near third‐order segment boundaries.

Microbubble or pendant drop control described by a general phase diagram

Controlling states and growths of a microscale bubble (or pendant drop) in a static liquid on a solid surface (or orifice) can be achieved by introducing general dimensionless phase diagrams provided in this work. Nowadays, microbubbles are often used to control transport phenomena in various micro- and nano-technologies. This work parametrically presents general three-dimensional phase diagrams of a microbubble on a solid surface by applying perturbation solutions with accuracy to the second power of Bond number of Young–Laplace equation in the literature. The phase diagrams are found to be divided into three regions, depending on if the microbubble surface contains the inflection point or neck. The general growth, departure and entrapment of a microbubble thus can be described by path lines on diagrams by adjusting two of three dimensionless parameters governing the apex and base radii, and contact angle to satisfy the desired requirement. The initial condition is Bond number, defined as the ratio of hydrostatic and capillary pressures at an initial or critical state. Validity of this model is confirmed by comparing with available theoretical data, agreed with experimental results in the literature.

A molecular dynamics study of the motion of a nanodroplet of pure liquid on a wetting gradient

The dynamic behavior of a nanodroplet of a pure liquid on a wetting gradient was studied using molecular dynamics simulation. The spontaneous motion of the droplet is induced by a force imbalance at the contact line. We considered a Lennard-Jones system as well as water on a self-assembled monolayer (SAM). The motion of the droplet for the Lennard-Jones case was found to be steady with a simple power law describing its center-of-mass position with time. The behavior of the water droplet was found to depend on the uniformity of the wetting gradient, which was composed of methyl- and hydroxyl-terminated alkanethiol chains on Au(111). When the gradient was nonuniform the droplet was found to become pinned at an intermediate position. However, a uniform gradient with the same overall strength was found to drive a droplet consisting of 2000 water molecules a distance of 25 nm or nearly ten times its initial base radius in tens of nanoseconds. A similar result was obtained for a droplet that was twice as large. Despite the many differences between the Lennard-Jones and water-SAM systems, the two show a similar overall behavior for the motion. Fair agreement was seen between the simulation results for the water droplet speed and the theoretical predictions. When the driving force was corrected for contact angle hysteresis, the agreement was seen to improve.

On evaporation rate and interfacial temperature of volatile sessile drops

A theoretical description of the evaporation of sessile drops is proposed. The linear dependency of the evaporation rate on droplet's base radius is described as well as the interfacial temperature. The analysis shows that most of evaporation is indeed concentrated near the contact line as demonstrated by experiments from literature. The evaporative cooling induced by evaporation results in a non-uniform temperature profile along the interface, being colder on the apex and hotter near the contact line. This result is counter-intuitive. It is the reason why the physical explanations are provided. Interfacial temperature gradient leads to Marangoni thermocpaillary driven flow within the drop.

Explicit integration of one problem in nonholonomic mechanics

where w is the angular velocity of the body, n is the normal at the contact point, a is the radius of the spherical base, and b is the radius of the spherical shell of the body (Fig. 1). Here and below, all vectors and tensors are set in a moving system of coordinates related to the principal axes of the body. The moment M at the contact point is related to the angular velocity w by the linear relation

Charge-Dependent Cavity Radii for an Accurate Dielectric Continuum Model of Solvation with Emphasis on Ions: Aqueous Solutes with Oxo, Hydroxo, Amino, Methyl, Chloro, Bromo, and Fluoro Functionalities

Dielectric continuum solvation models are widely used because they are a computationally efficacious way to simulate equilibrium properties of solutes. With advances that allow for molecular-shaped cavities, they have reached a high level of accuracy, in particular for neutral solutes. However, benchmark tests show that existing schemes for defining cavities are unable to consistently predict accurately the effects of solvation on ions, especially anions. This work involves the further development of a protocol put forth earlier for defining the cavities of aqueous solutes, with resulting advances that are most striking for anions. Molecular cavities are defined as interlocked spheres around atoms or groups of atoms in the solute, but the sphere radii are determined by simple empirically based expressions involving the effective atomic charges of the solute atoms (derived from molecular electrostatic potential) and base radii. Both of these terms are optimized for the different types of atoms or functional groups in a training set of neutral and charged solutes. Parameters in these expressions for radii were fitted by minimizing residuals between calculated and measured standard free energies of solvation (DeltaG(s)*), weighted by the uncertainty in the measured value. The calculations were performed using density functional theory with the B3LYP functional and the 6-311+G** basis set and the COnductor-like Screening MOdel (COSMO). The optimized radii definitions reproduce DeltaG(s)* of neutral solutes and singly charged ions in the training set to within experimental uncertainty and, more importantly, accurately predict DeltaG(s)* of compounds outside the training set, in particular anions (J. Phys. Chem. A 2003, 107, 5778). Inherent to this approach, the cavity definitions reflect the strength of specific solute-water interactions. We surmise that this feature underlies the success of the model, referred to as the CD-COSMO model for Charge-Dependent (also Camaioni-Dupuis) COSMO model. These findings offer encouragement that we can keep extending this scheme to other functional groups and obtain better accuracy in using continuum solvation models to predict equilibrium properties of aqueous ionic solutes. The approach is illustrated for a number of test cases, including the determination of acidities of an amine base, a study of the tautomerization equilibrium of a zwitterionic molecule (glycine), and calculating solvation energies of transition states toward a full characterization of reaction pathways in aqueous phase, here in S(N)2 exchange reactions. The calculated reaction barriers in aqueous solution are in excellent agreement with experimental values.