Macroscopic Angle Research Articles

Scaling behavior of kinetic orientational distributions for dilute nematic polymers in weak shear

Analytical descriptions of shear-aligned nematic monodomains have a long history across continuum, mesoscopic and mean-field kinetic models. Their value lies in the prediction of explicit scaling properties of the orientational distribution and of normal and shear stresses, with respect to molecular and flow parameters. At the coarsest scale of continuum Leslie–Ericksen theory [Arch. Rational Mech. Anal. 4 (1960) 231; Arch. Rational Mech. Anal. 28 (1968) 265], an explicit macroscopic alignment angle formula always exists in terms of bulk Miesowicz viscosities; the theory applies only to nematic (highly concentrated) liquids with low molecular weight and in the weak flow limit [Phys. Soc. Jpn. 52 (1983) 3486; Phys. Soc. Jpn. 53 (1984) 1031; J. Chem. Phys. 116 (2002) 9120; A hydrodynamic theory for solutions of non-homogeneous nematic liquid crystalline polymers with density variations, in: Proceedings of ASME International Mechanical Engineering Congress, IMECE2002-32189, N.O., LA, 17-22 November 2002]. At mesoscales, orientation tensor models [J. Chem. Phys. 103 (1995) 3108; The Physics of Liquid Crystals, Oxford University Press, Oxford, 1993; Thermodynamics of Flowing Systems with Internal Microstructure, Oxford Science Publications, Oxford, 1994; J. Chem. Phys. 103 (1995) 807; The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, 1998] apply at all concentrations and shear rates; explicit “Leslie angle” formulas exist only in the weak shear limit [Macromolecules 23 (1990) 4446; J. Rheol. 37 (1993) 413; J. Rheol. 37 (1993) 289; J. Rheol. 41 (1997) 943; Phys. A 267 (1999) 294; J. Non-Newtonian Fluid Mech. 90 (2000) 283; Rheol. Acta 42 (2003) 20; J. Rheol. 47 (2003) 105], inheriting hysteresis in alignment properties from the quiescent isotropic–nematic transition. Since the fundamental results of Onsager [Ann. N.Y. Acad. Sci. 51 (1949) 627] for quiescent nematic equilibria, exact probability distribution functions (PDFs) of kinetic theory have proven elusive for flow-driven nematic states. In their stead, flow-aligned and unsteady PDF formulas, elegant though implicit, have been derived by Kuzuu–Doi [Phys. Soc. Jpn. 52 (1983) 3486; Phys. Soc. Jpn. 53 (1984) 1031] and Semenov [Sov. Phys. JETP 66 (1983) 321] by weak flow asymptotic analysis, and by Marrucci and Maffettone [Macromolecules 22 (1989) 4076] by restriction to two dimensions. A simpler problem concerns the dilute concentration regime where the quiescent equilibrium is isotropic, non-degenerate, and stable, which is the focus of this paper. Using weak-flow asymptotics, we explicitly construct, and establish stability of, stationary, shear-perturbed PDFs for dilute concentrations; our formulas prove unsteady tumbling of perturbed isotropic states cannot occur. Exact scaling properties are predicted, including explicit Leslie alignment angle and degree-of-alignment (birefringence) formulas, as well as normal and shear stresses, in terms of molecular parameters and normalized shear rate for dilute nematic polymers. Our formulas apply except in a neighborhood of the isotropic instability transition, which See et al. [J. Chem. Phys. 92 (1990) 792] analyzed by a singular perturbation analysis. We then show how to bridge our method with that of See et al. [J. Chem. Phys. 92 (1990) 792], thereby completing the existence, construction, and stability of shear-perturbed isotropic equilibria for all concentrations. We verify the formulas both by numerical simulations and by comparison with mesoscopic model predictions [Rheol. Acta 42 (2003) 20; J. Rheol. 47 (2003) 105; J. Chem. Phys. 92 (1990) 792].

Triaxial Compression of Sand Reinforced with Fibers

Results from drained triaxial compression tests on specimens of fiber-reinforced sand are reported. It is evident that the addition of a small amount of synthetic fibers increases the failure stress of the composite. This effect, however, is associated with a drop in initial stiffness and an increase in strain to failure. Steel fibers did not reduce initial stiffness of the composite. The increase in failure stress can be as much as 70% at a fiber concentration of 2% (by volume) and an aspect ratio of 85. The reinforcement benefit increases with an increase in fiber concentration and aspect ratio, but it also depends on the relative size of the grains and fiber length. A larger reinforcement effect in terms of the peak shear stress was found in fine sand, compared to coarse sand, when the fiber concentration was small (0.5%). This trend was reversed for a larger fiber concentration (1.5%). A model for prediction of the failure stress in triaxial compression was developed. The failure envelope has two segments: a linear part associated with fiber slip, and a nonlinear one related to yielding of the fiber material. The analysis indicates that yielding of fibers occurs well beyond the stress range encountered in practice. The concept of a macroscopic internal friction angle was introduced to describe the failure criterion of a fiber-reinforced sand. This concept is a straightforward way to include fiber reinforcement in stability analyses of earth structures.

Heat and Fluid Flow Within an Anisotropic Porous Medium

A numerical experiment at a pore scale using a full set of Navier-Stokes and energy equations has been conducted to simulate laminar fluid flow and heat transfer through an anisotropic porous medium. A collection of square rods placed in an infinite two-dimensional space has been proposed as a numerical model of microscopic porous structure. The degree of anisotropy was varied by changing the transverse center-to-center distance with the longitudinal center-to-center distance being fixed. Extensive calculations were carried out for various sets of the macroscopic flow angle, Reynolds number and degree of anisotropy. The numerical results thus obtained were integrated over a space to determine the permeability tensor, Forchheimer tensor and directional interfacial heat transfer coefficient. It has been found that the principal axes of the permeability tensor (which controls the viscous drag in the low Reynolds number range) differ significantly from those of the Forchheimer tensor (which controls the form drag in the high Reynolds number range), The study also reveals that the variation of the directional interfacial heat transfer coefficient with respect to the macroscopic flow angle is analogous to that of the directional permeability. Simple subscale model equations for the permeability tensor, Forchheimer tensor and directional Nusselt number have been proposed for possible applications of VAT to investigate flow and heat transfer within complex heat and fluid flow equipment consisting of small scale elements.

Heterogeneous Mohr-Coulomb plastic material

We consider a heterogeneous medium which locally obeys a Mohr-Coulomb rigid-plastic behavior. We are interested in the macroscopic yield criterion based on an admissible stress formulation. We propose a two-dimensional analysis based on Airy potential formulation, using a first order perturbation. We first derive an upper bound for the macroscopic yield, and then construct explicitly an admissible stress field which reaches this bound. We finally show that within this framework, the macroscopic friction angle tends to the average of the local friction angle. Corrections to this value decay with the system size as [log(L)/L]1/2, for all (narrow) distributions of local friction angle.

Effective and microscopic contact angles in thin film dynamics

We introduce and analyse a class of quasi-self-similar solutions of the thin film equation to describe the dynamics of expanding liquid films on a solid surface. Using these solutions as intermediate asymptotics profiles, we obtain a quantitative expression for the shape of the film and a relation between the speed of the contact line and the macroscopic and microscopic contact angles.

Lattice model for the calculation of the angle of repose from microscopic grain properties

We study a simple lattice model for granular heap, which aims at calculating the macroscopic angle of repose from the microscopic grain properties. The model includes the effects of dissipation of the energy in the particle-particle collisions, and sticking of the particles to the pile. We obtain that, due to the discretization of the space, the angle of repose of the pile behaves as a complete devil’s staircase as a function of the model parameters. We present numerical and analytical considerations which characterize the properties of this staircase. @S1063-651X~98!09606-8#

Scanning thermal microscopy using batch fabricated thermocouple probes

We have developed scanning thermal microscopy probes for high resolution analysis of thermal properties in an atomic force microscope (AFM). Electron beam lithography and silicon micromachining have been used to batch fabricate Au/Pd thermocouples situated at the end of Si3N4 cantilevers. The cantilevers are patterned on the side of traditional style pyramidal AFM tips, giving a new shape of probe which is favorable for access to specimens containing significant topographic variation. Tip radius is approximately 50 nm and the probe has a macroscopic opening angle of 70°. The probes were scanned in the repulsive mode using a conventional AFM. Force feedback was optically employed to give topographic and thermal maps simultaneously by maintaining a constant force of approximately 5 nN. During initial scans using a photothermal test specimen, 80 nm period metal gratings were thermally resolved.

Ridging effects on wetting and spreading of liquids on solids

Analysis of the triple junction for a liquid on a solid has emphasized conditions pertinent for metal and ceramic systems. Complementary experiments are largely for liquid metals on Al2O3 and also liquid silicates on metals. Ridges can form at the triple line in response to the vertical component of force from surface tension. These are very small elastic distortions at lower temperatures. Where any local diffusion or solution-precipitation of the solid in the liquid can occur, the ridge can exceed atomic dimensions readily, and at high temperatures can become microns in size. If the liquid front stays attached, drag from these ridges can limit the spreading velocity of the liquid. Four time regimes of interest are identified. First, at short times or low temperatures, inelastic ridges do not form and motion can be controlled by viscous flow of the fluid. In regime II, a small ridge can be carried by the front, whilst the macroscopic angle can simultaneously approach that from the Young-Dupré relationship. Owing to the existence of this latter regime, a range of conditions exists wherein it is possible to satisfy the classical Young equation macroscopically, and a more complete equilibrium locally. Finally, at longer times more complex geometries will prevail while the system is evolving toward equilibrium, regime IV, which entails significantly deforming the solid.

Ridging effects on wetting and spreading of liquids on solids

Analysis of the triple junction for a liquid on a solid has emphasized conditions pertinent for metal and ceramic systems. Complementary experiments are largely for liquid metals on Al2O3 and also liquid silicates on metals. Ridges can form at the triple line in response to the vertical component of force from surface tension. These are very small elastic distortions at lower temperatures. Where any local diffusion or solution-precipitation of the solid in the liquid can occur, the ridge can exceed atomic dimensions readily, and at high temperatures can become microns in size. If the liquid front stays attached, drag from these ridges can limit the spreading velocity of the liquid. Four time regimes of interest are identified. First, at short times or low temperatures, inelastic ridges do not form and motion can be controlled by viscous flow of the fluid. In regime II, a small ridge can be carried by the front, whilst the macroscopic angle can simultaneously approach that from the Young-Dupré relationship. Owing to the existence of this latter regime, a range of conditions exists wherein it is possible to satisfy the classical Young equation macroscopically, and a more complete equilibrium locally. Finally, at longer times more complex geometries will prevail while the system is evolving toward equilibrium, regime IV, which entails significantly deforming the solid.

Molecular-statistical description of the smectic-A–smectic-C phase transition

A simple molecular-statistical model of the smectic-A--smectic-C phase transition is proposed. Assuming a bilinear mean-field potential, the macroscopic tilt angle is calculated by Boltzmann statistics as a thermal average of the molecular tilt. The calculation leads to an equation of state for the smectic-C phase which is a self-consistent-field equation involving the Langevin function of a reduced tilt and a reduced temperature. This molecular-statistical approach of the smectic-C--smectic-A phase transition is in principle analogous to a ferromagnetic phase transition with spin number S\ensuremath{\rightarrow}\ensuremath{\infty}. An excellent description of experimental tilt angle data for the complete temperature range of the smectic-C phase is achieved with only one or two fit parameters, respectively. Differences between tilt angle data obtained by optical and by x-ray experiments originate from different moments of the tilt angle distribution function and from different interaction parameters for the mesogenic cores and the complete molecule.

Second harmonic generation in Langmuir-Blodgett films of an asymmetrically substituted metallophthalocyanine

The formation of Langmuir-Blodgett (LB) films of an asymmetrically substituted metallophthalocyanine, copper nitro-tri- tert-butyl-phthalocyanine, is reported. From the second-harmonic generation (SHG) measurement, macroscopic second-order susceptibility, molecular hyperpolarizability and tilt angle for the molecules on a substrate were determined to be 1.63 × 10 −5 esu, 2.11 × 10 −27 esu and 26.1°, respectively. The SHG behaviour both in Z-type deposition and in alternating deposition LB films were investigated. A nearly quadratic relation between SH intensity and layer numbers was observed up to seven layers.

Direct Numerical Simulation of Fluid Flow in an Anisotropic Porous Medium.

A direct numerical simulation has been carried out to investigate viscous flow in an anisotropic porous medium. A collection of square rods placed in an infinite two-dimensional space has been proposed as a numerical model of microscopic porous structure. The degree of anisotropy was varied by dislocating alternate rows of square rods. Extensive calculations were carried out for various sets of the macroscopic flow angle, Reynolds number, porosity and degree of anisotropy. The numerical results thus obtained were integrated over space to determine the directional permeability from a purely theoretical. The predicted directional permeability closely follows the formula proposed by Scheidegger.

On the meandering of steps on vicinal GaAs (100) surfaces

Using scanning tunneling microscopy, we determine the terrace-width distribution on molecular-beam epitaxially grown vicinal GaAs(100) surfaces. We find a pronounced peak in the frequency at 4 nm, independent of macroscopic tilt angle. To explain this preferred terrace width, which is found by us and other authors, we introduce a simple numerical model of step meandering. The model accounts well for the shape of the terrace-width distribution found experimentally. This suggests that the preferred terrace width of 4 nm can be explained as a direct consequence of the thermodynamics on the surface, without introducing any long-range step-step interactions.

紙のミクロとマクロな繊維配向性

Macroscopic fiber orientation angle is known to affect stack lean of paper (e.g. z-fold continuous form) and a lot of research have been devoted to control macroscopic fiber orientation. But little attention has been paid to microscopic fiber orientation. The objective of this work is to clarify the relationship between microscopic and macroscopic fiber orientation and to obtain any hint or clue in controlling fiber orientation. The distribution of fiber orientation angle for nine kinds of wood-free paper was evaluated by both laser diffusion technique and ultrasonic elasticity measurement. The main difference between these two method is magnitude of the area subjected for measurement ; the area of the former method is much smaller than same of the latter. Therefore, results of the former method were thought to represent microscopic fiber orientation and of the latter macroscopic fiber orientation. By comparing the fiber orientations obtained with these two methods, the relationship between microscopic and macroscopic fiber orientation could be found. The results are summarized as follows : (1) The correlation coefficient between microscopic and macroscopic fiber orientation angles for the same sample position varies paper to paper.(2) If the variation of fiber orientation angle for cross-machine direction is large, the correlation coefficient between microscopic and macroscopic fiber orientation angles becomes large.(3) The paper which has larger variation of microscopic fiber orientation angle along machine direction shows poor correlation between microscopic and macroscopic fiber orientation angles.(4) Sheet formation seems to affect uniformity of fiber orientation angle along machine direction.

Studies on curing mechanism of PE-PU oligomer/HEMA mixture

In review of former studies on curing mechanism of the mixture of oligomer/monomer studied with macroscopic angle, but not with micro- cosmic angle, radiation curable PE-PU oligomer had been synthesed and studies on curing mechanism of the mixture of oligomer/monomer in the level of molecules was introduced in this paper.

Observations on stress-force-fabric relationships in idealized granular materials

The paper presents a series of theoretical developments and numerical experiments directed at quantifying important features of the micromechanical behavior of granular media by introducing average characteristics of fabric anisotropy and certain statistical averages of contact forces. The introduced characteristics are explicitly linked to stress through a stress-force-fabric relationship. Numerical simulations are used to verify this relationship for plane assemblies and to trace the evolution of induced anisotropy in contact orientations and contact forces. It is shown that at large strain the degree of anisotropy in contact orientations achieves some limiting value and the macroscopic angle of friction at large strains can be expressed directly in terms of this value. Effects of the angle of interparticle friction and magnitude of interparticle stiffness on the shear capacity of numerical assemblies at large strain are investigated.

Wetting behavior of two-dimensional meniscus formed under a holizontal plate. (Effect of infinitesimal roughness on a solid surface).

The wetting behavior of a two-dimensional liquid meniscus fomred under a horizontal plate was investigated theoretically. In the previous report, we considered the wetting mechasism from a macroscopic viewpoint ; that is, we assumed that Young's equation could be applied to the apparent macroscopic wetting problem. In this report, in order to substantiate the assumption made in a previous report, the same problem was analyzed precisely from a microscopic viewpoint when there existed infinitesimal two-dimensional roughnesses on the horizontal plate. The thermodynamic energy change of the system with the meniscus moving along the roughend surface was calculated. When the apparent macroscopic angle between the meniscus and the plate was equal to the advancing or receding contact angle, the system became unstable. This result agreed with that of the previous report. The assumption that Young's equation could be used macroscopically was verified by this precise analysis.

Macroscopic boundary angles of wetting of sintered capillary structures of heat pipes

We measured the macroscopic boundary angles of wetting by water, acetone, and pentane of the capillary structures of antigravity heat pipes in the 293–363 °K temperature range.

Microscopic and macroscopic contact angles

A simple relation is derived between the microscopic and macroscopic angles of contact of a fluid interface with a solid by taking into account the interaction of the interfaces in the region of contact. It is also shown how the macroscopic angle can be related to the perturbed interfacial tensions. Both angles are a property of the three phases in contact, independent of geometry and weak gravitational fields. Finally, calculated profiles of axially symmetric fluid interfaces in the neighbourhood of a planar solid surface are compared with those that would occur in the absence of interaction between the interfaces.

Exsolution of Ca-clinopyroxene from orthopyroxene aided by deformation

Monoclinic calcium-poor shear-transformation lamellae and calcium-rich exsolution lamellae occur parallel to (100) in orthopyroxene. The formation of both structures from an orthopyroxene host involves a shear on (100) parallel to [001], with additional cation exchange in the exsolution case. The shear transformation involves a macroscopic simple shear angle of 13.3° (shear strain of 0.236) and produces a specific a-axis orientation with respect to the sense of shear; we have found that this orientation dominates in exsolution lamellae in kinked orthopyroxene, where the sense of shear is known. In undeformed orthopyroxene, there is generally no preferred sense of orientation of the monoclinic a axes. We advance a specific model for exsolution involving nucleation and growth by shear transformation combined with cation exchange, thus circumventing the classical nucleation barrier and permitting exsolution at lower solute supersaturations.