Emulsion Ripening through Molecular Exchange at Droplet Contacts

Two coarsening mechanisms of emulsions are well established: droplet coalescence (fusion of two droplets) and Ostwald ripening (molecular exchange through the continuous phase). Here a third mechanism is identified, contact ripening, which operates through molecular exchange upon droplets collisions. A contrast manipulated small‐angle neutron scattering experiment was performed to isolate contact ripening from coalescence and Ostwald ripening. A kinetic study was conducted, using dynamic light scattering and monodisperse nanoemulsions, to obtain the exchange key parameters. Decreasing the concentration or adding ionic repulsions between droplets hinders contact ripening by decreasing the collision frequency. Using long surfactant chains and well‐hydrated heads inhibits contact ripening by hindering fluctuations in the film. Contact ripening can be controlled by these parameters, which is essential for both emulsion formulation and delivery of hydrophobic ingredients.     

Algebraic analysis of multigrid algorithms

We study the convergence rate of multilevel algorithms from an algebraic point of view. This requires a detailed analysis of the constant in the strengthened Cauchy–Schwarz inequality between the coarse‐grid space and a so‐called complementary space. This complementary space may be spanned by standard hierarchical basis functions, prewavelets or generalized prewavelets. Using generalized prewavelets, we are able to derive a constant in the strengthened Cauchy–Schwarz inequality which is less than 0.31 for the L² and H¹ bilinear form. This implies a convergence rate less than 0.15. So, we are able to prove fast multilevel convergence. Furthermore, we obtain robust estimations of the convergence rate for a large class of anisotropic ellipic equations, even for some that are not H¹‐elliptic. Copyright © 1999 John Wiley & Sons, Ltd.     

Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening

The classical Lifshitz–Slyozov–Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.     

Influence of the efficacy of interfacial modification on the coarsening of cocontinuous PS/HDPE blends during quiescent annealing

This article studies the quiescent annealing of three different cocontinuous polystyrene/high‐density polyethylene blends modified with two types of interfacial agents of widely different efficacies. Quantitative analysis of phase growth was obtained using mercury porosimetry. In a previous work, it was shown that one of these modifiers, a symmetrical diblock copolymer, has a high affinity for the interface and demonstrates virtually no micelle formation prior to saturation of the blend interface. The other modifier, a hydrogenated SEBS of 70/30 composition, forms micelles at elevated concentrations of modifier. In this study, it is shown that the cocontinuous phase size grows linearly without modifier, whereas the addition of both interfacial modifiers significantly suppresses the PE/PS phase coarsening and results in nonlinear phase growth behavior. The effect of the diblock copolymer on suppressing coarsening, however, is much more effective than that for the triblock case clearly supporting the tendency toward micelle formation for that latter modifier. In the case of unmodified PE/PS, the driving force for capillary pressure effects is so high that it is the capillary instability phenomena that dominate the coarsening and hence result in a linear growth of pore size with annealing time. When interfacial modifiers are added, the influence of reduced interfacial tension and lower pore size polydispersity significantly diminishes both capillary pressure effects and capillary instability phenomena. In that case, capillary pressure becomes the main rate determining step resulting in a nonlinear dependence of pore size with annealing time. It is shown that both the viscosity of the phases and the temperature of annealing can strongly influence coarsening behavior at low levels of interfacial modifier. Under all those conditions, however, nonlinear phase growth for the partially compatibilized system was maintained. These results clearly show that careful quantitative coarsening experiments using mercury porosimetry can be used as a tool to analyze the efficacy of interfacial modifiers for highly continuous or cocontinuous systems. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 711–721, 2006     

Monte Carlo simulation of antiphase boundaries and growth of antiphase domains in Al5Ti3 phase in Al-rich γ-TiAl intermetallics

Predominantly two kinds of antiphase boundaries (APBs) form in Al₅Ti₃, which is an Al-rich ordered derivative of the γ-TiAl (L1₀) phase. This phase can be viewed as a periodic arrangement of lean rhombs and squares on the Ti-rich (002) planes of the tetragonal L1₀. Energies of the two types of APBs were varied in a Monte Carlo simulation by suitably changing the pair interaction parameters. APBs of both types form boundaries of Al₅Ti₃ antiphase domains (APDs), which coarsen with time. An important observation in this regard is that mostly facetted APBs form at lower ageing temperatures, whereas curved APBs appear to form at relatively higher ageing temperatures. The findings of this work suggest that there exists a critical temperature, akin to the roughening transition temperature for crystals, that marks the transition from facetted to curved APBs.     

Phase separation and the kinetics of phase coarsening in commercial impact polypropylene copolymers

The phase segregation and subsequent minor phase coarsening of a commercial impact polypropylene copolymer was studied. The major components of the impact polypropylene copolymer studied were 82.4 wt % polypropylene homopolymer and 17.6 wt % ethylene-propylene rubber (EPR). The system was artificially manipulated to ensure homogeneity by precipitation from solution with a nonsolvent. This ensured that the initial system did not exhibit large-scale phase segregation. The homogeneous initial system was subjected to storage in the melt at 193°C for a series of times. The two-phase morphology of commercial impact polypropylenes was generated in the melt state by storage in the melt for various periods of time from 5 s to 1 h. Small nuclei of particles appeared at short time and increased in volume with increasing time in the melt state. The coarsening of the minor phase EPR component was shown to follow the theoretically predicted d ～ t^1/3 and N ～ t^?1 (where d = diameter, N = number of particles, and t = time in the melt) relationships to a close approximation in accord with Ostwald ripening theory. At short times these relationships were not obeyed. The indication was that the long-time coarsening regime was not entered until several minutes elapsed in the melt state. The particle size distribution was initially quite narrow and exhibited a trend of broadening at longer times of coarsening. This may be due to a shift from the short-time regime to the long-time coarsening regime. The initial polymer, which was precipitated from solution, was shown not to have undergone large-scale phase segregation in that it exhibited a one-phase morphology (i.e., no particles with > 0.1 μm diameter) as determined by ¹²⁹Xe NMR spectroscopy. The precipitated blend produced incipient particle nuclei (> 0.1 μm diameter) after a very short time (5 s) in the melt state. © 1994 John Wiley & Sons, Inc.     

Optimal multigrid algorithms for calculating thermodynamic limits

Beyond eliminating the critical slowing down, multigrid algorithms can also eliminate the need to produce many independent fine-grid configurations for averaging out their statistical deviations, by averaging over the many samples produced in coarse grids during the multigrid cycle. Thermodynamic limits can be calculated to accuracy in justO(^-2) computer operations. Examples described in detail and with results of numerical tests are the calculation of the susceptibility, the -susceptibility, and the average energy in Gaussian models, and also the determination of the susceptibility and the critical temperature in a two-dimensional Ising spin model. Extension to more advanced models is outlined.     

The Stability of Several Triply Periodic Surfaces

The stability of six triply periodic surfaces of constant mean curvature (CMC) is investigated. The relative energy and curvature values of the surfaces comprising the P (Pm3¯m), I-WP (Im3¯m), and G (I4₁32) families are numerically calculated with K. Brakke's Surface Evolver. Regions where the I-WP surface can exist metastable to a complementary I-WP surface are found. This type of metastability is also found in the F-RD surface. Bifurcation points marking the stability limits of the P, I-WP, and G families are also calculated with Evolver. Modes of instability which may occur in the six CMC families are classified. Bifurcations in the P, G, I-WP, C(P), D, and F-RD families are attributed to fundamental instabilities. Lattices of spheres (LOS) are possible extremal surfaces at the bifurcations. It is determined that both the CMC surfaces and the LOS configurations are unstable to coarsening. Because the variation in curvature is lowest for the G family, it is the most robust of the six families to coarsening when the surfaces are otherwise equivalent.     

Simulations of multiphase flows with multiple length scales using moving mesh interface tracking with adaptive meshing

In multiphase flows, the length scales of thin regions, such as thin films between nearly touching drops and thin threads formed during the interface pinch-off, are usually several orders of magnitude smaller than the size of the drops. In this paper, a number of extra length criteria for adaptive meshes are developed and implemented in the moving mesh interface tracking method to solve these multiple-length-scale problems with high fidelity. A nominal length scale based on the solutions of Laplace’s equations with the unit normal vectors of surfaces as the boundary conditions is proposed for the adaptive mesh refinement in the thin regions. For almost flat interfaces/boundaries which are near to the thin regions, the averaged length of the interior edges sharing the two nodes with the boundary edge is introduced for the mesh adaptation. The averaged length of the interfacial edges is used for the interior elements near the interfaces but outside of the thin regions. For the interior mesh away from the interfaces/boundaries, different averaged length scales based on the initial mesh are employed for the adaptive mesh refining and coarsening. Numerous cases are simulated to demonstrate the capability of the proposed schemes in handling multiple length scales, which include the relaxation and necking of an elongated droplet, droplet–droplet head-on approaching, droplet-wall interactions, and a droplet pair in a shear flow. The smallest length resolved for the thin regions is three orders of magnitude smaller than the largest characteristic length of the problem.