Diffuse Interface Theory Research Articles

CO2 bubble nucleation in polystyrene: Experimental and modeling studies

AbstractPolymer foams are used extensively in a variety of applications. A firm understanding of bubble nucleation is vital to predict foam properties based on process conditions. However, a number of theoretical and experimental challenges have thus far limited progress in this area. We propose the use of a scaling theory to connect nucleation behavior to well understood bulk phase behavior of polystyrene‐CO2 systems, which can be predicted by equations of state, such as the Sanchez–Lacombe equation of state. Scaling theory of nucleation asserts that when the reversible work of critical nucleus formation is properly normalized and plotted against the normalized degree of supersaturation, the resulting scaling curve is insensitive to temperature and the materials being used. Once the form of the scaling function is known, it can be used to predict the nucleation barrier knowing only the initial foaming conditions and calculating only bulk thermodynamic values. Using an extension of diffuse interface theory, we determined the slope of the scaling curve near saturation. This initial slope allows us to constrain the scaling function for better predictions of the reversible work. We also performed a series of experiments to help verify the accuracy of the scaling theory. The scaled free energy barriers determined from our experiments are consistent with the scaling function so constructed, and our theoretical results qualitatively agree with those found previously. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012

Consistent pseudopotential interactions in lattice Boltzmann models

We outline a systematic procedure on how to construct and derive interactions for nonideal lattice fluids. Using a mesoscopic approach based on exact lattice theories, we prove the consistency of the resulting diffuse interface theory with continuum thermodynamics. Translated in the framework of the so called "Shan-Chen" model for nonideal lattice fluids, this paper shows how to adjust the associated pseudopotentials to reproduce a free energy model based on a square gradient theory of equilibrium interfaces.

Lattice Boltzmann simulations of bubble formation in a microfluidic T-junction

A lattice Boltzmann equation method based on the Cahn-Hilliard diffuse interface theory is developed to investigate the bubble formation process in a microchannel with T-junction mixing geometry. The bubble formation process has different regimes, namely, squeezing, dripping and jetting regimes, which correspond to the primary forces acting on the system. Transition from regime to regime is generally dictated by the capillary number Ca, volumetric flow ratio Q and viscosity ratio λ. A systematic analysis is performed to evaluate these effects. The computations are performed in the range of 10(-4)<Ca<1, 1<Q<20 and 10(-2)<λ<1, with the equilibrium contact angle varying from 30° to 150°.

Order in nanometer thick intergranular films at Au–sapphire interfaces

In recent years extensive studies on interfaces have shown that ∼1 nm thick intergranular films (IGF) exist at interfaces in different material systems, and that IGF can significantly affect the materials’ properties. However, there is great deal of uncertainty whether such films are amorphous or partially ordered. In this study specimens were prepared from Au particles that were equilibrated on sapphire substrates in the presence of anorthite glass, leading to the formation of 1.2 nm thick IGF at the Au–sapphire interfaces. Site-specific cross-section samples were characterized using quantitative high resolution transmission electron microscopy to study the atomistic structure of the films. Order was observed in the 1.2 nm thick films adjacent to the sapphire crystal in the form of “Ca cages”, experimentally demonstrating that ordering is an intrinsic part of IGF, as predicted from molecular dynamics and diffuse interface theory.

The effective energy in the Allen–Cahn model with deformation

The sharp interface limit of a diffuse interface theory of phase transitions is considered in static situations. The diffuse interface model is of the Allen–Cahn type with deformation, with a parameter \varepsilon measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics for \varepsilon\to0 of the equilibrium energy is determined. The interface energy is defined as the excess energy over the corresponding two phase state with a sharp interface without the interface energy. It is shown that to within the term of order {\rm o}(\varepsilon) the interface energy is equal to \sigma\varepsilon where the coefficient \sigma is given by a new formula that involves the mechanical contribution to the total energy. Also the corresponding equilibrium states are determined and shown to converge to a sharp interface state for \varepsilon\to0.

Liquid phase demixing in ferroelectric/semiconducting polymer blends: An experimental and theoretical study

AbstractThis article describes a combined experimental and theoretical study on nanophase structure development as a result of liquid phase demixing in solution‐cast blends of the organic semiconductor poly(9,9′‐dioctyl fluorene) (PFO) and the ferroelectric polymer poly(vinylidene fluoride‐co‐trifluoroethylene) (P(VDF‐TrFE)). Blend layers (200 nm) are prepared by spin coating a 1:9 (w/w) PFO:P(VDF‐TrFE) blend solution in a common solvent on a poly(ethylenedioxy thiophene)/poly(styrene sulfonate) substrate. Owing to the pronounced incompatibility between the two polymers, a strong phase‐separated morphology is obtained, characterized by disk‐like nanodomains of PFO embedded in a P(VDF‐TrFE) matrix, as revealed by scanning electron microscopy. By varying the processing conditions, we find the average domain size and standard deviation to increase with spinning time. The considerable increase in domain size suggests the coarsening process not to be impeded by a steep rise in viscosity. This indicates solvent evaporation to be only moderate within the experimental time frame. The evolution of the observed phase morphology is modeled using ternary diffuse interface theory integrated with a modified Flory–Huggins (FH) treatment of the homogeneous (bulk) free energy of mixing, to account for significant molecular differences between the active blend components. Using calculated FH interaction parameters, the model confirms the phase separation to occur via spinodal decomposition of the blend solution during spin coating, as suggested by experimental observations. The simulated phase morphologies as well as the modeled trends in domain growth and standard deviation compare favorably with the experimental data. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 49: 1255–1262, 2011

Performance of some nucleation theories with a nonsharp droplet-vapor interface

Nucleation theories involving the concept of nonsharp boundary between the droplet and vapor are compared to recent molecular dynamics (MD) simulation data of Lennard-Jones vapors at temperatures above the triple point. The theories are diffuse interface theory (DIT), extended modified liquid drop-dynamical nucleation theory (EMLD-DNT), square gradient theory (SGT), and density functional theory (DFT). Particular attention is paid to thermodynamic consistency in the comparison: the applied theories either use or, with a proper parameter adjustment, result in the same values of equilibrium vapor pressure, bulk liquid density, and surface tension as the MD simulations. Realistic pressure-density correlations are also used. The best agreement between the simulated nucleation rates and calculations is obtained from DFT, SGT, and EMLD-DNT, all of which, in the studied temperature range, show deviations of less than one order of magnitude in the nucleation rate. DIT underestimates the nucleation rate by up to two orders of magnitude. DFT and SGT give the best estimate of the molecular content of the critical nuclei. Overall, at the vapor conditions of this study, all the investigated theories perform better than classical nucleation theory in predicting nucleation rates.

Determination of the Thickness and Composition Profiles for a Film of Binary Mixture on a Solid Substrate

We determine the steady-state structures that result from liquid-liquid demixing in a free surface film of binary liquid on a solid substrate. The considered model corresponds to the static limit of the diffuse interface theory describing the phase separation process for a binary liquid (model-H), when supplemented by boundary conditions at the free surface and taking the influence of the solid substrate into account. The resulting variational problem is numerically solved employing a Finite Element Method on an adaptive grid. The developed numerical scheme allows us to obtain the coupled steady-state film thickness profile and the concentration profile inside the film. As an example we determine steady state profiles for a reflection-symmetric two- dimensional droplet for various surface tensions of the film and various preferential attraction strength of one component to the substrate. We discuss the relation of the results of the present diffuse interface theory to the sharp interface limit and determine the effective interface tension of the diffuse interface by several means.

Crystal Nucleation in the Hard-Sphere System Revisited: A Critical Test of Theoretical Approaches

The hard-sphere system is the best known fluid that crystallizes: the solid-liquid interfacial free energy, the equations of state, and the height of the nucleation barrier are known accurately, offering a unique possibility for a quantitative validation of nucleation theories. A recent significant downward revision of the interfacial free energy from approximately 0.61kT/sigma(2) to (0.56 +/- 0.02)kT/sigma(2) [Davidchack, R.; Morris, J. R.; Laird, B. B. J. Chem. Phys. 2006, 125, 094710] necessitates a re-evaluation of theoretical approaches to crystal nucleation. This has been carried out for the droplet model of the classical nucleation theory (CNT), the self-consistent classical theory (SCCT), a phenomenological diffuse interface theory (DIT), and single- and two-field variants of the phase field theory that rely on either the usual double-well and interpolation functions (PFT/S1 and PFT/S2, respectively) or on a Ginzburg-Landau expanded free energy that reflects the crystal symmetries (PFT/GL1 and PFT/GL2). We find that the PFT/GL1, PFT/GL2, and DIT models predict fairly accurately the height of the nucleation barrier known from Monte Carlo simulations in the volume fraction range of 0.52 < varphi < 0.54, whereas the CNT, SCCT, PFT/S1, and PFT/S2 models underestimate it significantly.

Diffuse-interface model for smoothed particle hydrodynamics

Diffuse-interface theory provides a foundation for the modeling and simulation of microstructure evolution in a very wide range of materials, and for the tracking and capturing of dynamic interfaces between different materials on larger scales. Smoothed particle hydrodynamics (SPH) is also widely used to simulate fluids and solids that are subjected to large deformations and have complex dynamic boundaries and/or interfaces, but no explicit interface tracking or capturing is required, even when topological changes such as fragmentation and coalescence occur, because of its Lagrangian particle nature. Here we developed a SPH model for single-component two-phase fluids that is based on diffuse-interface theory. In the model, the interface has a finite thickness and a surface tension that depend on the coefficient k of the gradient contribution to the Helmholtz free energy functional and the density-dependent homogeneous free energy. In this model, there is no need to locate the surface (or interface) or to compute the curvature at and near the interface. One- and two-dimensional SPH simulations were used to validate the model.

Phase field theory of interfaces and crystal nucleation in a eutectic system of fcc structure: I. Transitions in the one-phase liquid region

The phase field theory (PFT) has been applied to predict equilibrium interfacial properties and nucleation barrier in the binary eutectic system Ag-Cu using double well and interpolation functions deduced from a Ginzburg-Landau expansion that considers fcc (face centered cubic) crystal symmetries. The temperature and composition dependent free energies of the liquid and solid phases are taken from CALculation of PHAse Diagrams-type calculations. The model parameters of PFT are fixed so as to recover an interface thickness of approximately 1 nm from molecular dynamics simulations and the interfacial free energies from the experimental dihedral angles available for the pure components. A nontrivial temperature and composition dependence for the equilibrium interfacial free energy is observed. Mapping the possible nucleation pathways, we find that the Ag and Cu rich critical fluctuations compete against each other in the neighborhood of the eutectic composition. The Tolman length is positive and shows a maximum as a function of undercooling. The PFT predictions for the critical undercooling are found to be consistent with experimental results. These results support the view that heterogeneous nucleation took place in the undercooling experiments available at present. We also present calculations using the classical droplet model [classical nucleation theory (CNT)] and a phenomenological diffuse interface theory (DIT). While the predictions of the CNT with a purely entropic interfacial free energy underestimate the critical undercooling, the DIT results appear to be in a reasonable agreement with the PFT predictions.

Diffuse-interface theory for structure formation and release behavior in controlled drug release systems

A common method of controlling drug release has been to incorporate the drug into a polymer matrix, thereby creating a diffusion barrier that slows the rate of drug release. It has been demonstrated that the internal microstructure of these drug–polymer composites can significantly impact the drug release rate. However, the effect of processing conditions during manufacture on the composite structure and the subsequent effects on release behavior are not well understood. We have developed a diffuse-interface theory for microstructure evolution that is based on interactions between drug, polymer and solvent species, all of which may be present in either crystalline or amorphous states. Because the theory can be applied to almost any specific combination of material species and over a wide range of environmental conditions, it can be used to elucidate and quantify the relationships between processing, microstructure and release response in controlled drug release systems. Calculations based on the theory have now demonstrated that, for a characteristic delivery system, variations in microstructure arising due to changes in either drug loading or processing time, i.e. evaporation rate, could have a significant impact on both the bulk release kinetics and the uniformity of release across the system. In fact, we observed that changes in process time alone can induce differences in bulk release of almost a factor of two and typical non-uniformities of ±30% during the initial periods of release. Because these substantial variations may have deleterious clinical ramifications, it is critical that both the system microstructure and the control of that microstructure are considered to ensure the device will be both safe and effective in clinical use.

An investigation of water-gas transport processes in the gas-diffusion-layer of a PEM fuel cell by a multiphase multiple-relaxation-time lattice Boltzmann model

In order to study water-gas transport processes in the gas-diffusion-layer (GDL) of a proton exchange membrane (PEM) fuel cell system, a multiphase, multiple-relaxation-time lattice Boltzmann model is presented in this work. The model is based on the mean-field diffuse interface theory and can handle the multiphase flows with large density ratios and various viscosities. By using the standard bounce back boundary condition and an approximate average scheme for the non-slip and wetting boundary walls, respectively, detailed liquid-gas transportation in the GDL, in which exact boundary condition is difficult to be implemented, can be simulated. Unlike most of lattice Boltzmann methods based on the Bhatnagar–Gross–Krook collision operator, the present model shows a viscosity-independent velocity field, which is very important in simulating multiphase flows where various viscosities coexist. We validate our model by simulating a static droplet on a wetting wall and compare with theoretical predictions. Then, we simulate a water-gas flow in the GDL of a PEM fuel cell and investigate the saturation-dependent transport properties under different conditions. The results are shown to be qualitatively consistent with the previous numerical and theoretical works.

Molecular Dynamics Studies on the Kinetics of Phase Changes of Nanoparticles: Properties, Structures, and Crystal Nucleation of Copper Nanoparticle Cu453

AbstractMolecular dynamics (MD) computer simulations have been carried out to study the structures, properties and crystal nucleation of nanoparticles with 453 Cu atoms. Structure information was analyzed from the MD simulations, while properties of nanoparticles of Cu453, such as melting point, freezing temperature, heat capacity and molar volumes, have been estimated. The face center cubic (FCC) phase and icosahedron (Ih) phase were observed during the quenching process, and nucleation rates of crystallization to FCC crystal of Cu453 at temperatures of 650, 700, 750, and 800 K were analyzed. Both classical nucleation theory (CNT) and diffuse interface theory (DIT) were used to interpret our observed nucleation rates. The free energy and diffuse interface thickness between the liquid and the FCC crystal phases were estimated by the CNT and DIT respectively, and the results show that the DIT does not work properly to the system.

Interface-Stabilized Nanoscale Quasi-Liquid Films and Interfacial Prewetting and Premelting Transitions

AbstractEquilibrium-thickness, intergranular films (IGFs) have been observed in various ceramic materials. Recently, surficial amorphous films (SAFs) of similar character have also been found. Furthermore, a series of studies revealed the stabilization of disordered (quasi-liquid) IGFs and SAFs well below the bulk solidus or eutectic temperatures, wherein analogies to the phenomena of premelting and prewetting can be made. Accordingly, combined interfacial premelting and prewetting models have been developed using a diffuse-interface theory. This paper outlines the key results of two model experiments in support of the above theory: namely observation of quasi-liquid grain boundary films (metallic IGFs) in W-Ni and searching of a complete wetting transition for Bi2O3 on ZnO where SAFs become macroscopically thick. We propose that simple combined interfacial premelting and prewetting models apply to metallic IGFs, but only serve as a basis to understand IGFs and SAFs in ceramics where additional interactions, e.g. dispersion forces and space-charges, should be added separately and may result in more complex behaviors.

Crystal nucleation in hydrous rhyolite: Experimental data applied to classical theory

Feldspar nucleation rate data obtained by laboratory decompression of hydrous silicate melt are interpreted in view of the classical theory of nucleation (CNT) and a non-classical variation, the diffuse-interface theory (DIT). The nucleation rate data can be modeled by the CNT formalism only if the interfacial free energy (σ) is allowed to vary as a function of composition. The values thus obtained vary by a factor of four (0.024–0.100 J/m2) and decrease systematically over a sixfold increase in dissolved H2O content (0.8–4.8 wt%). This result is qualitatively consistent with the effects of dissolved H2O on the liquid-vapor interfacial free energy in haplogranite magma (Mangan and Sisson 2000) and the liquid-crystal interfacial free energy in the one-component Li-disilicate system (Davis et al. 1997). The DIT states that the interfacial region between the bulk solid and bulk melt has thermodynamic properties intermediate between these phases, and that σ is defined as the difference between the interfacial enthalpy ( H int) and interfacial entropy ( TS int). If the DIT model is correct, the nucleation rate data for feldspar may indicate that: (1) dissolved H2O content controls the spatial distribution of enthalpy and configurational entropy around incipient crystals, and (2) the spatial gradients of these potentials diverge during devolatilization. This study suggests that crystal nucleation studies may yield insights into the structure and thermodynamics of hydrous melts; likewise, experimental studies are important for refining a physical understanding of nucleation phenomena. Our results can be applied to quantitative numerical models of ascent-driven magma crystallization.

Mobility and interactions of weakly nonwetting droplets

Lubrication equations based on diffuse interface theory are applied to description of slow motion and interaction of droplets on solid support. For the case of quasistationary motion of a weakly nonwetting fluid, we derive integral relations reducing the problem to computation of ratios of applicable “thermodynamic forces” to a dissipative integral characterizing an individual droplet. This allows us to describe the droplet motion without computing explicitly the distortion of their shape. Explicit computations are carried out for a droplet sliding on an inclined plane, and for an ensemble of droplets interacting through the precursor layer.

Size dependence of ordering in FePt nanoparticles

We have investigated the size effect of A1→L10 ordering of FePt nanoparticles in FePt–Al2O3 granular and FePt/SiO2 particulate films by transmission electron microscopy (TEM). The TEM results have shown convincingly that ordering does not progress when the particle size has a diameter of less than 4 nm. Calculation of the order parameter profile from the surface to the volume of the FePt nanoparticles based on diffuse-interface theory justified the experimentally observed size dependence of the ordering. The transition length from disorder to order depends on the interfacial energy, hence the critical particle size of ordering should vary depending on the type of matrix and substrate.

Molecular Dynamics Studies of the Kinetics of Phase Changes in Clusters IV: Crystal Nucleation from Molten (NaCl)256 and (NaCl)500 Clusters

AbstractMolecular dynamics computer simulation based on the Born‐Mayer‐Huggins potential function has been carried out to study the effects of cluster size and temperature on the nucleation rate of sodium chloride clusters in the temperature range of 580 K to 630 K. Clusters with 256 and 500 NaCI molecules have been studied and the results have been compared with those obtained from 108 molecule clusters. The melting point (MP) of the dusters were observed to increase with the size of the clusters and can be well described by a linear equation MP = 1107(37)‐1229(23)N−1/3(N is the number of molecules in the cluster). The nucleation rate was found to decrease with increasing the cluster size or temperature. Various nucleation theories have been used to interpret the nucleation rates obtained from this molecular dynamics simulation. It is possible to use a constant diffuse interface thickness to interpret the nucleation rate from the diffuse interface theory in the temperature range of this study. However, the interfacial free energy estimated from classical nucleation theory and diffuse interface theory increases too fast with increasing the temperature while that from Gran‐Gunton theory does not change with changing temperatures. The sizes of critical nuclei estimated from all the theories are smaller than those estimated from our simulations.

Molecular dynamics studies of the kinetics of phase changes in clusters III: structures, properties, and crystal nucleation of iron nanoparticle Fe 331

Molecular dynamics (MD) computer simulations have been carried out to study the structures, properties, and crystal nucleation of iron nanoparticles with 331 Fe atoms or with diameter around 2 nm. Structure information for the nanoparticles was analyzed from the MD simulations. Three crystalline phases and one amorphous phase were obtained by cooling the nanoparticles from their molten droplets at different cooling rates or with different lengths of cooling time periods. Molten droplets froze into three different solid phases and a solid–solid transition from a disordered body-centered cubic (BCC) phase to an ordered BCC phase were observed during the slow cooling and the quenching processes. Properties of nanoparticle Fe 331, such as melting point, freezing temperature, heat capacity, heat of fusion, heat of crystallization, molar volume, thermal expansion coefficient, and diffusion coefficient, have been estimated. Nucleation rates of crystallization to two solid phases for Fe 331 at temperatures of 750, 800, and 850 K are presented. Both classical nucleation theory and diffuse interface theory are used to interpret our observed nucleation results. The interfacial free energy and the diffuse interface thickness between the liquid phase and two different solid phases are estimated from these nucleation theories.