Finite Volume Fraction Research Articles

Ostwald Ripening Analysis Using Phase-Field Model

The Ostwald ripening phenomena is studied using a phase-field model. The phase-field parameters are obtained at a thin interface limit proposed by Kim et al. It is shown that small solid particles preferably melt out, and large particles grow and agglomerate due to the curvature undercooling. Calculation results are consistent with the LSW rule. The rate constant of average particle radius is estimated as 0.57 at 0.2 solid fraction. The change in shape of the invariant particle size distribution due to finite volume fraction agrees with experiment and a statistical model. The difference of the calculation results by Diepers et al. and by the present work is discussed. The change in the shape of the invariant size distribution depending on solid fraction is also shown. The shape becomes more wide and symmetry with increase of the solid fraction.

Unsteady laminar hydromagnetic fluid–particle flow and heat transfer in channels and circular pipes

Abstract The problem of unsteady laminar flow and heat transfer of a particulate suspension in an electrically conducting fluid through channels and circular pipes in the presence of a uniform transverse magnetic field is formulated using a two-phase continuum model. Two different applied pressure gradient (oscillating and ramp) cases are considered. The general governing equations of motions (which include such effects as particulate phase stresses, magnetic force, and finite particle-phase volume fraction) are non-dimensionalized and solved in closed form in terms of Fourier cosine and Bessel functions and the energy equations for both phases are solved numerically since they are non-linear and are difficult to solve analytically. Numerical solutions based on the finite-difference methodology are obtained and graphical results for the fluid-phase volumetric flow rate, the particle-phase volumetric flow rate, the fluid-phase skin-friction coefficient and the particle-phase skin-friction coefficient as well as the wall heat transfer for plane and axisymmetric flows are presented and discussed. In addition, these numerical results are validated by favorable comparisons with the closed-form solutions. A comprehensive parametric study is performed to show the effects of the Hartmann magnetic number, the particle loading, the viscosity ratio, and the temperature inverse Stokes number on the solutions.

Screening effects during late stage phase separation

Late stage phase separation is an important non-equilibrium thermodynamics process with a range of implications for applied systems such as thin film growth. A number of studies describe quantitatively Ostwald ripening, the dominant process under mass conservation. In these models the concept of a screening length occurs in different context, for three-dimensional systems to accommodate finite volume fractions of the minority phase, and for two-dimensional systems to circumvent the diverging steady state solution to the diffusion equation. We present a comprehensive review of this concept and show that the screening length in two- and three-dimensional systems has the same form if the average cluster radius and the average cluster cluster distance are used as parameters. In the experimental section, examples for two- and three-dimensional systems are given, showing that the screening length shields the interaction between clustering systems across a concentration step. This leads to sharper than expected interfaces between neighbouring ripening systems on a surface, and explains the resistance of sub-surface silicide clusters to dissolve into a uniform, buried film during annealing.

Criteria for crack deflection/penetration criteria for fiber-reinforced ceramic matrix composites

Deflection of a matrix crack at the fiber/matrix interface is the initial mechanism required for obtaining enhanced toughness in ceramic-matrix composites (CMCs). Here, energy release rates are calculated for matrix cracks that either deflect or penetrate at the interface of an axisymmetric composite as a function of elastic mismatch, fiber volume fraction, and length of the deflected or penetrated crack. The energy release rates for the competing fracture modes are calculated numerically by means of the axisymmetric damage model developed by Pagano, which utilizes Reissner’s variational principle and an assumed stress field to solve the appropriate boundary value problems. Crack deflection versus penetration is predicted by using an energy criterion analogous to that developed by He and Hutchinson. Results show that, for equal crack extensions in deflection and penetration, crack deflection is more difficult for finite crack extension and finite fiber volume fraction than in the He and Hutchinson limit of zero volume fraction and/or infinitesimal crack extension. Allowing for different crack extensions for the deflected and penetrating cracks is shown to have a small effect at larger volume fractions. Fracture-mode data on model composites with well-established constitutive properties show penetration into the fibers (brittle behavior), as predicted by the present criteria for crack extensions larger than 0·2% of the fiber radius and in contrast to the He and Hutchinson criterion, which predicts crack deflection. This result suggests that the latter criterion may overestimate the prospects for crack deflection in composites with realistic fiber volume fractions and high ratios of fiber to matrix elastic modulus.

Ostwald ripening of liquid phase sintered Cu-Co dispersions at high volume fractions

An experimental study on the influence of a finite volume fraction of solid on Ostwald ripening of Cu-Co dispersions is presented. Cu-Co alloys with volume fractions of solid Co-particles in the range from 0.25 to 0.7 were annealed at 1473 K for times between 5 min and 70 h such that the Co particles coarsened within a liquid Cu(Co) matrix. The coarsening of the Co-particles was analyzed by various image analysis methods to yield statistically significant results. The experimental results on the coarsening rate and the particle size distribution as they vary with volume fraction are presented. The results show that there is not a definitive effect of the volume fraction on the rate of coarsening and that in all cases investigated the size distributions are broader than predicted by any theory of Ostwald ripening.

Numerical Simulation of Ostwald Ripening in Emulsions

Ostwald ripening at finite dispersed phase volumes was modeled successfully using linearized analytical solutions of the ripening equations and an explicit numerical routine. The numerical approach incorporated a number frequency distribution of drop radii rather than using a discrete number of drops. The effect of finite dispersed phase volume fraction was accounted for by using half the average separation distance between drops as the mass transfer boundary. The numerical predictions matched analytical predictions for infinitely dilute systems almost exactly and were in qualitative agreement with analytical predictions for infinitely concentrated systems. The numerical model was applied to the full range of dispersed phase concentrations and successfully predicted experimental cumulative frequency distributions. The growth rate, i.e. the change in the cube of the mean radius with time, was confirmed to be constant at any dispersed phase volume fraction. A simple expression was developed relating growth rate to dispersed phase volume fraction. Predicted growth rates at dispersed phase volume fractions less than 0.2 matched those found experimentally and by other numerical methods. Predicted growth rates at higher dispersed phase volume fractions agreed well with experimental data from the literature but were significantly higher than predictions from other numerical methods.

Unsteady flow of an electrically conducting dusty-gas in a channel due to an oscillating pressure gradient

Equations governing transient, laminar, hydromagnetic flow of a two-phase particulate suspension having a finite particulate volume fraction in a channel are developed. These equations are solved analytically in terms of cosine Fourier series subject to appropriate conditions by the separation of variables method. The flow of the suspension in the channel is assumed to take place due to the application of a time-dependent oscillating pressure gradient. The influence of the Hartmann number, particle loading, and the inverse Stokes number on the fluid and particle volume flow rates and the fluid-phase skin-friction coefficient is illustrated graphically. Comparisons with previously published work are made, and interpretation of the solutions is discussed.

Dynamics of a suspension of spheres and rigid polymers: Effect of geometrical mismatch

An effective medium approach together with a multiple scattering formalism is considered to study the steady-state dynamics of suspensions of spheres and polymer chains without excluded volume interactions. The polymer chains are assumed to obey rigid-body dynamics (without rotation) and are taken to be so long that Gaussian statistics is applicable. We have considered the dynamics of probe objects (either a sphere or a polymer) in a solution containing spheres and polymers. The significance of the dimensionless variables appearing when solving the effective medium equations and, in particular, the role of the geometrical parameter t=Rg/a (a is the radius of any sphere and Rg the radius of gyration of a polymer chain) are discussed. The translational diffusion coefficients of the moving probe sphere DS and of the center-of-mass of the probe polymer chain DP, and the shear viscosity of the suspensions have been derived. Both the friction coefficient of the probe sphere or that of the probe polymer chain and the shear viscosity diverge as ΦPOL→0.31 in the presence of polymers in the solution. Also, when polymers are diffusing in a suspension of fixed spheres an optimum range of ΦSP that maximizes the difference in the diffusion coefficients of polymer chains characterized by distinct t values has been noticed.

Compressible two-phase boundary-layer flow with finite particulate volume fraction

A continuum mathematical model governing compressible boundary-layer fluid-particle flow and heat transfer over a semi-infinite flat plate is developed. This model is based on separate balance laws of mass, linear momentum, and energy for both phases and allows for finite particle-phase volume fraction and viscous stresses. It is found that the volume fraction of suspended particles and, therefore, the particle-phase density is uniform in the boundary layer. Using this fact, the governing nonlinear partial differential equations are simplified and solved numerically by an iterative finite-difference method. Graphical results for flow and heat transfer properties are presented for various values of the particle-fluid viscosity ratio and the power index parameter for the viscosity-temperature relation.

Ostwald ripening of solid-liquid Pb-Sn dispersions

An experimental study on the influence of a finite volume fraction of solid on Ostwald ripening of Pb-Sn dispersions is presented. Dispersions of lead particles in a near-eutectic Pb-Sn liquid were stabilized against gravitationally induced sedimentation by the use of Lorentz forces. The Lorentz forces were induced by an electrical current within the samples and a magnetic field perpendicular to it. Dispersions from 0.3 to 0.71 vol fraction solid Pb could thus be kept free of any large-scale sedimentation for times of minutes to 15 hours. The results show that there is a definitive effect of the volume fraction on the rate of coarsening and that the experimentally measured kinetics are in close agreement with theoretical predictions. However, it is also clear that Lorentz forces accelerate the coarsening process whenever a solid skeleton of the dispersoids is present. It is unclear if this acceleration is also present in the experiments with a low volume fraction of solid.

Solutions for fluid-particle flow and heat transfer in a porous channel

Equations governing transient flow and heat transfer aspects of a particulate suspension with a constant finite volume fraction in a porous channel are developed. The steady-state problem is solved analytically and the transient problem is solved numerically by an iterative implicit finite difference scheme. The variations of the velocity and temperature profiles, volume flow rates, skin-friction coefficient, and the wall heat transfer coefficient with time and other physical parameters are illustrated graphically and discussed.

Control of Surfactant Level in Starve Fed Emulsion Polymerization: 3. Langmuir Site Adsorption Model for Competitive Adsorption of in Situ and Added Surfactants in the Presence of Anchored Groups

In emulsion polymerization processes it is important to control the surfactant level to avoid secondary nucleation (too much surfactant) and coagulation (too little). Persulfate initiators generate two types of surface active sulfate-containing species; anchored sulfate groups (10-20%) and mobile in situ surfactant (ISS; 50-70%). This paper examines the significance of the competition between these two species and an added surfactant, sodium dodecylbenzene sulfonate (SDBS). SDBS soap titrations were performed on five uncleaned methyl methacrylate/butyl acrylate latexes with particle sizes from 500 nm to 3.25 μm at volume fractions from 1 to 30%. Not surprisingly, traditional Langmuir adsorption analyses were unsatisfactory, even after correction for finite volume fraction of solids. We develop a Langmuir site adsorption model to describe the competitive adsorption of ISS and SDBS in the presence of anchored sites and fit the surface tension data from 84 data points to obtain three parameters: the adsorption equilibrium constants for SDBS ( K s = 760 ± 30 liter mol -1) and ISS ( K 1 = 50 ± 12 liter mol -1) and the limiting area per molecule (APM = 0.37 ± 0.01 nm 2). These parameters are quite consistent with literature values determined in noncompetitive situations. The anchored groups were most significant on smaller particles, and the mobile ISS adsorption was most significant on smaller particles at high volume fraction. The relatively weak attraction of the ISS to the particle surface is due to its low molecular weight (∼400 Da per sulfate group) and results in desorption upon latex dilution.

Elastic-plastic thermal stresses and deformation of short-fibre composites

A micromechanics model is proposed to analyse residual stresses and deformations that develop in short-fibre composites upon an applied uniform temperature change. The model is based on Eshelby's equivalent inclusion method and treats the interaction among fibres at finite volume fractions through the Mori-Tanaka mean field theory. The model treats the matrix as an elastic/plastic material while the fibre is elastic and is able to account for the effects of the composite microgeometry. To this end, the effects of misoriented short fibres, the orientation of which is described by a density distribution function, are considered. Numerical results obtained from the proposed model indicate that the misorientation of short fibres has a significant effect on both the stress and deformation behaviour of short-fibre composites.

The flow induced by a rotating disk in a particulate suspension

A typical finite volume fraction two phase flow model is used to investigate the influence of assumptions about the nature of particle phase viscosity and the structure of particle phase boundary conditions at a solid surface on solutions to the Von Karman problem for a particle/fluid suspension. Solutions are obtained numerically by an iterative finite difference method. Comparisons of these solutions with one another and with previously published work reveal that both the particle phase viscosity and solid surface boundary conditions influence the nature of predictions in important ways. Selected numerical results are presented graphically to illustrate these conclusions

Unsteady Flow of a Power-Law Dusty Fluid With Suction

Equations governing flow of a particulate suspension exhibiting finite volume fraction in non-Newtonian power-law fluids are developed and applied to the problem of unsteady flow past an infinite porous flat plate with suction. Numerical results for small volume fraction for the displacement thicknesses for both phases and the skin-friction coefficient for the fluid phase are obtained using an implicit finite difference scheme and presented graphically to elucidate interesting features of the solutions.

Crossover effects on the osmotic pressure and the diffusion coefficient in moderately concentrated polymer solutions

The effects of a finite volume fraction of polymer on the reduced osmotic pressure and the reduced diffusion coefficient in solutions, covering an extended concentration range, are analysed in the framework of a recent theoretical approach, which incorporates a renormalization group treatment of a modified interaction Hamiltonian. The resulting analytical functions are consistent with the large amount of experimental data accumulated from the literature. In the interpretation of the diffusion behaviour, over a large concentration interval, the concept of hydrodynamic screening plays a crucial role in obtaining a uniform picture. In the region of strongly overlapping chains a breakdown of the universal features of the reduced osmotic pressure and the reduced diffusion coefficient can be inferred from the theoretical relations.

Elastic moduli of composites with rigid sliding inclusions

We investigate the effect of interfaces on the elastic properties of composites with randomly distributed inclusions. Initially, the solution for a single isolated sliding inclusion is obtained and this result is used in two separate effective-medium theories, the self-consistent method and the differential scheme, to predict the elastic properties of composites containing a finite volume fraction of inclusions. For simplicity, the inclusions are assumed to be rigid. In the analysis, a parameter is introduced to describe the degree of sliding at the interface. Two limiting cases are perfect bonding and pure sliding at the inclusion—matrix interface. We show that the Poisson's ratio of the composite tends towards a universal value that is independent of the material parameters of the matrix, as the number of inclusions is increased. In contrast to the perfect-bonding case, both effective-medium theories give remarkably similar results in the pure-sliding limit.

Finite volume fraction effects on coarsening—II. Interface-limited growth

We study the asymptotic behavior of the interface-limited coarsening on the basis of the statistical theory recently developed by us. This theory takes into account the cooperative effects arising from statistical correlations among precipitates. Using the theory, we discuss the finite precipitate volume fraction effects on the form of the precipitate size distribution function and the coarsening rate. Then we find that such effects play important roles on these properties, even though the volume fraction is small.

Direct numerical simulation of bubble behavior in oscillatory flow

The motion of spheres (bubbles) in an incompressible liquid undergoing a small-amplitude oscillatory motion is calculated by a multipole expansion method. In the limit of small viscosity, the Stokes layer is confined to the vicinity of the surface of the bubbles, which therefore interact approximately only through the pressure field. The motion of the spheres is parametrized in terms of added mass, Basset, and drag forces; the coefficients of which are obtained from the simulation. To obtain results useful for the study of pressure wave propagation in bubbly liquids, several bubble configurations are studied for different (finite) volume fractions and the results then averaged. The effects of surface tension and bubble density are also considered. [Work supported by DOE.]

The surface energy of metastable Al 3Li precipitates from coarsening kinetics

Small angle X-ray scattering has been used to measure the coarsening kinetics of metastable Al 3Li precipitates in a dilute Al Li binary alloy. The results are compared to previous transmission electron microscopy studies of similar alloys. Lifschitz-Slyozov-Wagner coarsening theory was invoked as a means of obtaining the α/Al 3Li interfacial energy. However, corrections accounting for effects due to finite volume fractions and limited solubility of Li in the precipitate phase are included. The surface energy was found to be ∼0.005J/m 2, significantly less than previously reported. A possible explanation for the discrepancy is discussed.