Cahn Theory Research Articles

Wetting Effect Induced Depletion and Adsorption Layers: Diffuse Interface Perspective.

When a multi-component fluid contacts arigid solid substrate, the van der Waals interaction between fluids and substrate induces a depletion/adsorption layer depending on the intrinsic wettability of the system. In this study, we investigate the depletion/adsorption behaviors of A-B fluid system. We derive analytical expressions for the equilibrium layer thickness and the equilibrium composition distribution near the solid wall, based on the theories of de Gennes and Cahn. Our derivation is verified through phase-field simulations, wherein the substrate wettability, A-B interfacial tension, and temperature are systematically varied. Our findings underscore two pivotal mechanisms governing the equilibrium layer thickness. With an increase in the wall free energy, the substrate wettability dominates the layer formation, aligning with de Gennes' theory. When the interfacial tension increases, or temperature rises, the layer formation is determined by the A-B interactions, obeying Cahn's theory. Additionally, we extend our study to non-equilibrium systems where the initial composition deviates from the binodal line. Notably, macroscopic depletion/adsorption layers form on the substrate, which are significantly thicker than the equilibrium microscopic layers. This macroscopic layer formation is attributed to the interplay of phase separation and Ostwald ripening. We anticipate that the present finding could deepen our knowledge on the depletion/adsorption behaviors of immiscible fluids.

An a priori irreversible phase-field formulation for ductile fracture at finite strains based on the Allen–Cahn theory: a variational approach and FE-implementation

An a priori irreversible phase-field formulation for ductile fracture at finite strains based on the Allen–Cahn theory: a variational approach and FE-implementation

Effect of wall free energy formulation on the wetting phenomenon: Conservative Allen-Cahn model.

Cahn introduced the concept of wall energy to describe the interaction between two immiscible fluids and a solid wall [J. W. Cahn, J. Chem. Phys. 66, 3667-3672 (1977)]. This quintessential concept has been successfully applied to describe various wetting phenomena of a droplet in contact with a solid surface. The usually formulated wall free energy results in the so-called surface composition that is not equal to the bulk composition. This composition difference leads to a limited range of contact angles which can be achieved by the linear/high-order polynomial wall free energy. To address this issue and to improve the adaptability of the model, we symmetrically discuss the formulation of the wall free energy on the Young's contact angle via Allen-Cahn model. In our model, we modify the calculation of the fluid-solid interfacial tensions according to the Cahn's theory by considering the excess free energy contributed by the distorted composition profile induced by the surface effect. Additionally, we propose a semi-obstacle wall free energy which enforces the surface composition to be the bulk composition within the framework of bulk obstacle potential. By this way, the accuracy of the contact angle close to 0° and 180° is significantly improved in the phase-field simulations. We further reveal that the volume preservation term in the conservative Allen-Cahn model has a more significant impact on the wetting behavior on superhydrophobic surfaces than on hydrophilic surfaces, which is attributed to the curvature effect. Our findings provide alternative insights into wetting behavior on superhydrophilic and superhydrophobic surfaces.

A thermodynamically consistent diffuse interface model for the wetting phenomenon of miscible and immiscible ternary fluids

The wetting effect has attracted great scientific interest because of its natural significance as well as technical applications. Previous models mostly focus on one-component fluids or binary immiscible liquid mixtures. Modelling of the wetting phenomenon for multicomponent and multiphase fluids is a knotty issue. In this work, we present a thermodynamically consistent diffuse interface model to describe the wetting effect for ternary fluids, as an extension of Cahn's theory for binary fluids. In particular, we consider both immiscible and miscible ternary fluids. For miscible fluids, we validate the equilibrium contact angle and the thermodynamic pressure with Young's law and the Young–Laplace equation, respectively. Distinct flow patterns for dynamic wetting are presented when the surface tension and the viscous force dominate the wetting effect. For immiscible ternary fluids, we manipulate the wettability of two contact droplets deposited on a solid substrate according to three scenarios: (I) both droplets are hydrophilic; (II) a hydrophilic droplet in contact with a hydrophobic one; (III) both droplets are hydrophobic. The contact angles at each triple junction from the simulations are compared with Young's contact angle and Neumann's triangle rule. Simulations for the validation of our work are performed in two and three dimensions. In addition, we model the evaporation process of a ternary droplet and obtain the same power law as that of previous experiments. Our model allows one to relate the interfacial energies with surface composition, enabling the modelling of the coffee-ring phenomenon in further perspective.

Morphological growth criterion for faceted to non-faceted transition in pure materials

A criterion to explain the morphological growth transition from faceted to non-faceted (f-nf) morphology in the case of directional solidification is presented for pure materials. The criterion is based on the number of atomic layers at the solid–liquid interface and allows a foreknowledge of the solidification growth velocity at which f-nf transition becomes significant. The basis for the transition criterion hinges on the Cahn theory of interface structure and the theory of liquid–solid transition by the maximum entropy production rate (MEPR) principle. The criterion is tested with a number of pure materials and compared with experimental measurements for salol. The criterion fits quite well with available experimental data.

Modeling and Simulation of Nucleation and Growth Transformations with Nucleation on Interfaces of Kelvin Polihedra Network

Nucleation is a phenomenon associated to the start of the new phase, from a primary phase, named matrix. Growth is the increase in size of this new phase over time. In metallic materials, the nucleation may take place on the grain boundaries of the primary phase. A network of Kelvin polyhedra was used in this paper to represent the grains. A computer simulation was performed in which nucleation took places at the faces, edges and vertices of this polyhedral network. The Causal cone method was employed in the simulations. The results of the present computational simulations were compared with the classical Johnson Mehl-Avrami-Kolmogorov (JMAK) as well as with Cahn model for nucleation at the grain boundaries. JMAK theory considers nuclei to be uniform randomly located within the matrix. Cahn analytical model specifies that nucleation takes place on random planes. For a small number of nuclei, the simulations approached the JMAK model whereas as the number of nuclei increased the simulation results agree with Cahn's theory. Reasons for this are fully discussed.

A Space‐Charge Treatment of the Increased Concentration of Reactive Species at the Surface of a Ceria Solid Solution

AbstractA space‐charge theory applicable to concentrated solid solutions (Poisson–Cahn theory) was applied to describe quantitatively as a function of temperature and oxygen partial pressure published data obtained by in situ X‐ray photoelectron spectroscopy (XPS) for the concentration of Ce3+ (the reactive species) at the surface of the oxide catalyst Ce0.8Sm0.2O1.9. In contrast to previous theoretical treatments, these calculations clearly indicate that the surface is positively charged and compensated by an attendant negative space‐charge zone. The high space‐charge potential that develops at the surface (>0.8 V) is demonstrated to be hardly detectable by XPS measurements because of the short extent of the space‐charge layer. This approach emphasizes the need to take into account defect interactions and to allow deviations from local charge neutrality when considering the surfaces of oxide catalysts.

A variational approach to surface cation segregation in mixed conducting perovskites

A variational, phase-field methodology describing the segregation of dopant cations to the free surfaces of mixed conducting perovskite oxides is introduced. Based on the Cahn–Hilliard formalism for solute segregation in alloys and the Cahn theory of wetting of a solution at a solid surface, the model qualitatively predicts the experimentally observed behavior of solute segregation in oxides, which shows segregation layers on the order of tens of nanometers. Critical in this description are the interactions among defects and the gradient energy – two related concepts that can be expected to become influential at relatively high dopant concentrations. The analysis predicts that a first-order phase transition – a surface-mediated spinodal – may occur depending on the strength of dopant interactions and the affinity of dopant cations and oxygen vacancies for the free surface.

Enhancement of Faceted Growth of Primary Al<sub>3</sub>Ni Phase in High Magnetic Field

Faceted growth of primary Al3Ni phase in the hypereutectic Al-Ni alloy in a high magnetic field was investigated. It was found that faceted growth of primary Al3Ni phase was enhanced in the presence of the magnetic field. However, the fibrous to granular transition of Al-Al3Ni eutectics occurred. The undercooling of primary and eutectic phases during solidification was measured using differential thermal analysis. It was showed that the undercooling of primary phase was hardly changed but that of eutectics markedly increased in the magnetic field. According to Cahn theory of crystal growth, the critical driving force was used to satisfactorily explain the morphology transition in the magnetic field.

Light scattering behavior and the kinetics of pressure-induced phase separation in solutions of poly(ε-caprolactone) in acetone + CO2 binary fluid mixtures

The miscibility and the kinetics of pressure-induced phase separation in solutions of poly(ε-caprolactone) (PCL) in acetone + CO2 binary fluid mixtures have been studied at pressures up to 28 MPa and temperatures up to 410 K using a unique high pressure view-cell equipped with a dual set of pistons and dual set of sapphire windows. One set of the windows separated by 25.4 mm allows the assessment of the phase state and is used to monitor the transmitted light intensities. The second set of windows separated by 50 μm is used to monitor the scattered light intensities over a wide range of scattering vector q (from 0.35 to 4 μm−1) which allows the assessment of the mechanism of phase separation. Investigations have been carried out for a wide range of polymer concentrations, from 2.0 to 34.9 wt%, while holding the acetone-to-CO2 (wt:wt) ratio in each solution at a constant value of 2:1. The dual set of pistons that are employed which are synchronized and motorized create a churn-like action in the cell insuring effective mixing, even at the high polymer concentrations by translating the cell content across a magnetically-coupled rotating mixer impeller. The piston actions assure also that the solution is effectively introduced into the narrow gap between the scattering windows, and refreshed. The solutions at off-critical concentrations undergo pressure-induced phase separation via nucleation and growth mechanism which shows circular symmetric patterns in their light scattering patterns. For these solutions, the Debye–Bueche type scattering function was used to analyze the domain size of the new phase that forms and develops after a pressure quench. The phase separation in solutions at or near the critical polymer concentrations (9.0–15.0 wt%) proceeds via spinodal decomposition which is characterized by the formation and evolution of the spinodal ring patterns corresponding to a maximum in the angular variation of the scattered light intensities. The results in early stage of the spinodal decomposition were described by the linearized Cahn theory. The variation of the scattered light intensity maximum Im and its location in scattering vectors qm with time in the later stage of the spinodal decomposition obey power-law scaling according to Im ∼ tβ and qm ∼ t−α. The results for the 9.0 and 12.0 wt% solutions show that β/α changes its value from β/α > 3 to β/α ≈ 3 with time, indicating the progression of the spinodal decomposition from intermediate to late stage.

Molecular Dynamics Simulations and Kapitza Conductance Prediction of Si/Au Systems Using the New Full 2NN MEAM Si/Au Cross-Potential

Superlattices made by superposing dielectric and metal nanolayers are of great interest as their small size restricts the thermal energy carrier mean free path, decreasing the thermal conductivity and thereby increasing the thermoelectric figure of merit. It is, therefore, essential to predict their thermal conductivity. Potentials for Au and Si are discussed, and the potential of second nearest-neighbor modified embedded atom method (2NN MEAM) is chosen as being the best for simulating heat transfer in Si/Au systems. Full 2NN MEAM Si/Au cross-potential parameterization is developed, and the results are compared with ab initio calculations to test its ability to reproduce local density approximation (LDA) calculations. Volume-constant (NVT) molecular dynamics simulations are performed to deposit Au atoms on an Si substrate by physical vapor deposition, and the results of the intermixing zone are in good agreement with the Cahn and Hilliard theory. Nonequilibrium molecular dynamics simulations are performed for an average temperature of 300 K to determine the Kapitza conductance of Si/Au systems, and the obtained value of 158 MW/m 2 K is in good agreement with the results of Komarov et al. for Au deposited on isotopically pure Si- 28 and natural Si, with values ranging between 133 and 182 MW/m2 K.

Characterization of solid phases and study of transformation kinetics in m-chlorofluorobenzene by 35Cl nuclear quadrupole resonance

Polymorphism is of widespread occurrence in the world of molecular crystals. In this work we present experimental results showing the existence of four solid phases in m-chlorofluorobenzene. A glass structure is achieved by quenching the liquid phase at 77K. This glassy state crystallizes in a disordered phase at T∼143K, which in turn transforms to the high-temperature stable phase (phase I) at T∼153K. Depending on the thermal history of the sample, a different ordered phase (phase III) can be obtained. The disorder is attributed to a molecular orientational disorder. There is no evidence of molecular reorientation in any phase. A study of the disorder-order phase transformation kinetics, using nuclear quadrupole resonance, is presented. The results are analyzed following Cahn's theory. Nucleation seems to take place at grain boundaries. Growth rates for different temperatures have been determined.

A unique state of solid matter: Stochastic spinodal modulations in the Au–50Ni transition above 600 K

Our observation of spinodal modulations in gold–50 at.% nickel (Au–50Ni) transformed at high temperatures (above 600 K) contradicts non-stochastic Cahn theory with its ∼500 K modulation suppression. These modulations are stochastic because simultaneous increase in amplitude and wavelength by diffusion cannot be synchronized. The present theory is framed as a second-order differential uphill/downhill diffusion process and has an increasing time-dependent wavenumber and amplitude favouring Hillert’s one-dimensional prior formulation within the stochastic association of wavelength and amplitude.

Dissipative phenomena and acoustic phonons in isothermal crystals: A density-functional theory study

We derive the nonlinear transport equations for isothermal crystalline solids using a density-functional theory based free energy and the Poisson bracket formalism. The model is comprised of three characteristic time scales: A fast one of sound waves and two slow ones of diffusive phenomena. It also combines long-range elastic features with a density field $\ensuremath{\rho}$ having an atomistic resolution. At the level of linearized hydrodynamics, our results are identical to transport equations obtained from the Mori projection operator treatment when the coupling of mass density and phonons becomes weak. At the linear level, we also show that Fick's law breaks down without dislocations in accordance with Cahn's theory of stress effects on diffusion in solids. In contrast, with no bilinear couplings between the density and the displacement fields, it is possible to use a nonlinear local theory to describe diffusion and elastic effects using just a single scalar field $\ensuremath{\rho}$, which is numerically efficient. We show how a variety of phenomenological phase field crystal models---which have recently demonstrated their power in materials science applications---can be derived, and how their parameters can be related to more microscopic ones.

Calculation of the surface pressure and the contact energy in the adsorption of alkanes on water

In an attempt to complete the theoretical description of the complex wetting of alkanes on aqueous substrates, the surface pressure in the adsorption of pentane, hexane, and heptane on water is calculated as a function of the bulk gas pressure using a two-dimensional van der Waals equation in conjunction with Hirasaki’s expression for the Henry’s law constant. The contact energy as defined within Cahn theory is subsequently deduced using Dobbs’ modified version of this theory and employed to calculate the wetting behavior of the above-mentioned alkanes on water. The experimentally observed sequence of two changes in the wetting state, namely a first-order transition between a microscopically thin and a mesoscopically thick film, followed by a continuous divergence of the film thickness, is found within the theory as well. While the critical wetting transition is brought about by long-range forces and can be described by Dzyaloshinskii–Lifshitz–Pitaevskii theory, the first-order thin–thick transition involves the short-range interactions between substrate and adsorbate. In the absence of a complete theory for the latter, they are accounted for in an effective way in the present approach, which relies only on the macroscopic bulk properties of the different media involved and which leads to satisfactory results for the transition temperatures.

Lattice Boltzmann simulation of liquid–gas flows through solid bodies in a square duct

The lattice Boltzmann method for two-phase immiscible fluids with large density differences proposed by Inamuro et al. [T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys. 198 (2004) 628–644] is applied to the problem of liquid–gas flows through solid bodies in a square duct. A wetting boundary condition is introduced so that partial wetting on solid surfaces is realized to agree with Cahn theory. Using this method, we investigate the characteristics of wettability in terms of dynamic contact angles between two fluids and a solid wall. Also, we carry out simulations of liquid–gas rising flows through solid bodies in a square duct. It is found from these simulations that the present method can be useful for the problems of liquid–gas flows through complicated geometries.

Phase Behavior and Morphology of Poly(Methyl Methacrylate)/Poly(α‐Methyl Styrene‐co‐Acrylonitrile) Blends Under Quiescent and Shear Flow

The kinetics of spinodal decomposition (SD) for the binary blend poly(methyl methacrylate), PMMA, and Poly(α‐methylstyrene‐co‐acrylonitrile), PαMSAN, with 31 wt% AN content (LCST‐type phase diagram) has been thoroughly studied using a time‐resolved light scattering technique. The early stage SD was dominated by a diffusion process and can be well described within the framework of the linearized Cahn‐Hilliard theory. The spinodal temperature could be evaluated from the analysis of the early stage SD based on the Cahn theory. In addition, viscoelastic properties of this system have been systematically investigated at temperatures below and above the LCST phase diagram. The linear viscoelastic properties of the blends were found to be greatly changed by phase separation in the two‐phase regime. This change in the linear viscoelastic properties attributed to an additional contribution of concentration fluctuations to the material functions at the phase separation temperatures. The phase diagram of the blends was also estimated rheologically through the dynamic temperature ramps of G′, G″ and η*. Furthermore, the phase behavior and morphology of this system has been studied under different shear rates using simple shear apparatus and transmission electron microscopy (TEM), respectively.

Wetting transitions of ionic solutions.

Cahn's phenomenological theory of wetting of a solid substrate by a saturated vapor is generalized to the case where the substrate is charged and the wetting film contains counterions, with or without added salt. The electrostatic contribution to the grand potential associated with these ions is calculated within a nonlinear Poisson-Boltzmann theory. In the salt-free case, when the wetting film includes only counterions released by the substrate, the wetting transition is always first order, regardless of its nature in a neutral system. When salt is present, other wetting scenarios may arise, depending on the salt concentration and substrate surface charge. Over a restricted range of salt concentrations, a wetting scenario similar to that of prewetting, is predicted to occur along the liquid-vapor coexistence line. This scenario includes a discontinuous wetting transition between microscopic and mesoscopic film thicknesses, followed by a continuous divergence of the film thickness at higher temperatures.

Transition temperatures and contact angles in the sequential-wetting scenario of n-alkanes on (salt) water

Alkanes of medium chain length show an unusual wetting behavior on (salt) water, exhibiting a sequence of two changes in the wetting state. When deposited on the water surface at low temperatures, the liquid alkane forms discrete droplets that are interconnected only by a molecularly thin film. On increase of the temperature, there occurs a sudden jump of the film thickness and, at this first-order transition, a mesoscopically thick layer of liquid alkane is formed. Heating the system further leads to a divergence of the film thickness in a continuous manner. While the long-range forces between substrate and adsorbate are responsible for the critical transition, which occurs at the temperature at which the Hamaker constant changes sign, it is primarily the short-range components of the forces that bring about the first-order transition. We calculate the Hamaker constant of the system and show how, within a modified Cahn theory, accurate predictions of the first-order transition temperatures can be obtained for n-alkanes (pentane and hexane) on water and even on brine. Furthermore, we study the variation of the contact angle as a function of temperature.

Spinodal Decomposition in Binary Blend of Poly(methyl methacrylate)/Poly(α-methyl styrene-co-acrylonitrile): Analysis of Early and Late Stage Demixing

Summary: The phase separation kinetics of a poly(methyl methacrylate), PMMA, and poly(α-methylstyrene-co-acrylonitrile), PαMSAN, blend exhibiting a LCST-type phase diagram have been investigated as functions of temperature and demixing time for the near critical composition (PαMSAN/PMMA = 25:75) using a time-resolved light scattering technique. We found that the scattering data in the early stage spinodal decomposition (SD) can be well described by the linearized Cahn–Hilliard theory. Spinodal temperature Ts ∼ 171 °C was determined from Dapp versus T and qequation/tex2gif-stack-1.gif versus T based on the analysis of Cahn theory in the early stage SD, where Dapp is the apparent diffusion coefficient and qm is the scattering vector at the maximum scattering intensity. The value of Ts obtained from the analysis of light scattering data was in good agreement with the phase diagram obtained visually at the equilibrium condition. The LCST-type phase diagram of this blend was also calculated using the Flory–Huggen theory. The estimated interaction parameter used for the calculation of the phase diagram was found to be composition and temperature dependent. The coarsening behavior of the blend at the late stage SD was also studied by analyzing the magnitudes of qm and Im at various demixing times and temperatures based on the nonlinear statistical theories. Both of qm and Im obtained at different times and temperatures can be superposed and reduced to the respective master curves by horizontal shifting when they are plotted against log (t/aT) at a given reference temperature, where aT is the temperature-dependent shift factor. The shape of the phase separation domain structures as a function of time during the SD was also considered. The scaling structure function F(x,t;T) = qm(t;T)3I(q,t;T) versus q/qm was found to be time independent and falls onto a universal curve in the late stage SD as a result of the dynamical self-similarity accompanying with the phase separation process. Optical microscopy pictures for the structure development of spinodal decomposition for PαMSAN/PMMA = 25:75 blend at 180 °C for different demixing times.