Effect Of Line Tension Research Articles

Higher-order phase transitions with line-tension effect

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincare, Anal. non lineaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal. 144 (1998) 1–46] for a first-order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies where u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when .

Monte Carlo Test of the Classical Theory for Heterogeneous Nucleation Barriers

Flat walls facilitate the condensation of a supersaturated vapor: classical theory of heterogeneous nucleation predicts that the free energy barrier DeltaF(het)*, which needs to be overcome for the formation of sphere-cap-shaped nucleation seeds, is smaller than the barrier DeltaF(hom)* for spherical droplets in the bulk by a factor 0 < f(theta) < 1, which only depends on the contact angle theta. In this Letter, we compute both DeltaF(hom)* and DeltaF(het)* from Monte Carlo simulations and test the theory for the lattice gas model (for which theta can be readily controlled). Even though the theory is only based on macroscopic arguments, it is shown to hold for experimentally relevant nanoscopic nucleation seeds (20 < or = DeltaF(hom)*/k(B)T < or = 200) if (independently estimated) line tension effects are considered.

Brittle fracture: thermodynamic refinement of the Griffith problem

The purpose of the work is (a) a rigorous formulation of the brittle fracture problem from the thermodynamic point of view and (b) widening the theory by accounting for not only surface tension, but also the line tension of a crack. A proper thermodynamic formulation of the Griffith problem is attained by introducing generalized Gibbs energy as a thermodynamic potential, both the 2d and 3d cases being included. Thermodynamic line tension is introduced as a new characteristic of a crack originating from changing surface tension near the crack tip, which can be of certain significance for nanocracks. The criterion of rupture is reformulated in the context of line tension. As an example, a detailed calculation of the surface and line tensions of a crack is performed for a molecular solid with dispersion forces. The effect of line tension on the ultimate strength is shown to be twice as much in the 3d case as compared with the 2d case.

On the stability of liquid droplets with line tension

Prompted by recent works on line tension effects upon the stability of liquid droplets laid on a rigid substrate, we prove existence and stability of equilibrium configurations when the substrate is either planar or spherical. For positive line tension our argument involves only set symmetrization, while for negative line tension a stability criterion is proposed which makes variational problems well-posed in the framework of sets with finite perimeter. Several applications illustrate the role of the criterion in providing a natural cutoff length that selects destabilizing modes.

Phase Separation of Palmitic Acid and Perfluorooctadecanoic Acid in Mixed Langmuir−Blodgett Monolayer Films

Deposition of mixtures of palmitic acid (C15H31COOH) and perfluorooctadecanoic acid (C17F35COOH) onto solid substrates gives rise to irregularly shaped, phase-separated domains under a variety of deposition conditions. The morphology and chemical composition of these phase-separated domains have been investigated using a combination of surface pressure-area isotherms, atomic force microscopy, X-ray photoemission electron microscopy, and confocal fluorescence microscopy imaging. While domain morphology and composition in 2D phase-separated mixed monolayer systems can typically be rationalized in terms of an interplay between line tension and dipole-dipole repulsion effects, it was found that for this system additional kinetic factors, including domain growth rates and the rate of dissolution of the fatty acid component into the aqueous subphase, also play a major role in controlling film properties. The potential importance of these effects for the controlled patterning of solid substrates is discussed.

Gamma convergence of an energy functional related to the fractional Laplacian

We prove a Γ-convergence result for an energy functional related to some fractional powers of the Laplacian operator, (−Δ) s for 1/2 < s < 1, with two singular perturbations, that leads to a two-phase problem. The case (−Δ)1/2 was considered by Alberti–Bouchitté–Seppecher in relation to a model in capillarity with line tension effect. However, the proof in our setting requires some new ingredients such as the Caffarelli–Silvestre extension for the fractional Laplacian and new trace inequalities for weighted Sobolev spaces.

Continuum approximation of the Peach–Koehler force on dislocations in a slip plane

We derive a continuum model for the Peach–Koehler force on dislocations in a slip plane. To represent the dislocations, we use the disregistry across the slip plane, whose gradient gives the density and direction of the dislocations. The continuum model is derived rigorously from the Peach–Koehler force on dislocations in a region that contains many dislocations. The resulting continuum model can be written as the variation of an elastic energy that consists of the contribution from the long-range elastic interaction of dislocations and a correction due to the line tension effect.

Facet evolution on supported nanostructures: Effect of finite height

The surface of a nanostructure relaxing on a substrate consists of a finite number of interacting steps and often involves the expansion of facets. Prior theoretical studies of facet evolution have focused on models with an infinite number of steps, which neglect edge effects caused by the presence of the substrate. By considering diffusion of adsorbed atoms (adatoms) on terraces and attachment-detachment of atoms at steps, we show that these edge or finite height effects play an important role in the structure's macroscopic evolution. We assume diffusion-limited kinetics for adatoms and a homoepitaxial substrate. Specifically, using data from step simulations and a continuum theory, we demonstrate a switch in the time behavior of geometric quantities associated with facets: the facet edge position in a straight-step system and the facet radius of an axisymmetric structure. Our analysis and numerical simulations focus on two corresponding model systems where steps repel each other through entropic and elastic dipolar interactions. The first model is a vicinal surface consisting of a finite number of straight steps; for an initially uniform step train, the slope of the surface evolves symmetrically about the centerline, i.e., the middle step when the number of steps is odd. The second model is an axisymmetric structure consisting of a finite number of circular steps; in this case, we include curvature effects which cause steps to collapse under the effect of line tension. In the first case, we show that the position of the facet edge, measured from the centerline, switches from O(t^1/4) behavior to O(t^1/5) (where t is time). In the second case, the facet radius switches from O(t^1/4) to O(t). For the axisymmetric case, we also predict analytically through a continuum shock wave theory how the individual collapse times are modified by the effects of finite height under the assumption that step interactions are weak compared to the step line tension.

Nonlinear effects of line tension in adhesion of small droplets

Three-phase line tensions may become crucial in the adhesion of micro-nano or small droplets on solid planes. In this paper we study for the first time the nonlinear effects in adhesion spanning the full range of physically possible parameters of surface tension, line tension, and droplet size. It is shown that the nonlinear adhesion solution spaces can be characterized into four regions. Within each region the adhesion behaves essentially the same. Especially, inside the characteristic regions with violent nonlinearities, the co-existence of multiple adhesion states for given materials is disclosed. Besides, two common fixed points in the solution space are revealed. These new results are consistent with numerical analysis and experimental observations reported in the literatures.

Calculating the surface tension between a flat solid and a liquid: a theoretical and computer simulation study of three topologically different methods

We discuss three topologically different methods for calculating the surface tension between a flat solid and a liquid from theoretical and computer simulation viewpoints. The first method, commonly used in experiments, measures the contact angle at which a static droplet of liquid rests on a solid surface. We present a new analysis algorithm for this method and explore the effects of line tension on the contact angle. The second method, commonly used computer simulations, uses the pressure tensor through the virial in a system where a thick, infinitely extended slab of liquid rests on a solid surface. The third method, which is original to this paper and is closest to the thermodynamic definition of surface tension, applies to a spherical solid in contact with liquid in which the flat solid is recovered by extrapolating the sphere radius to infinity. We find that the second and third methods agree with each other, while the first method systematically underestimates surface tension values.

Effect of the line tension at the vapor-liquid-solid boundary on the growth of silicon nanocrystals

We examine the thermodynamic aspects of the effect of the line tension at a bent vapor-liquid-solid boundary on quasi-one-dimensional growth of silicon nanowires. Experimental data demonstrate that there is a limiting size of solvent nanodrops below which they cannot exist in equilibrium and that there is a limiting nanowire diameter starting at which the increase in line tension with decreasing diameter prevents nanocrystal growth at a given supersaturation.

Effects of seed particle size and composition on heterogeneous nucleation of n-nonane

Heterogeneous nucleation of supersaturated n-nonane vapour on seed particles of different size and composition has been investigated using a fast expansion chamber. Monodisperse seed particle sizes were ranging from about 4 nm up to about 24 nm in diameter. By using different types of particle generators WO x , Ag and (NH 4) 2SO 4 particles were generated. For direct comparison between different particle compositions overlapping sizes have been generated for WO x and Ag at about 7 nm particle diameter as well as for Ag and (NH 4) 2SO 4 at about 15 nm. Nucleation temperature was kept constant at about 278 K. Experimental data were compared to Kelvin equation and Fletcher theory including the effect of line tension. It was found that heterogeneous nucleation of n-nonane seems to be independent of seed particle composition and starts well below the Kelvin curve. Good agreement was achieved with Fletcher theory including the effect of line tension.

Line Tension and Coalescence in Heterogeneous Membranes

Membrane inhomogeneity gives rise to a perturbation energy that may be manifested as line tension between regions of different compositions. Here we calculate the perturbation energy of a phase-separated fluid membrane composed of domains embedded in a continuous phase and relate it to the line tension in the system. We find that the effective line tension due to the thickness mismatch between the phases varies nonmonotonically as a function of the domain size and spacing when those decrease below about 20 nm. Although we find that the characteristic time scale for domain coalescence increases with domain size, the coalescence time for domains on the order of 25 to 50 nm may be as long as several days. The time scale for domain coalescence is found, quite surprisingly, to be maximal when the bending and area moduli of the domain and continuous phase are similar and to decrease when one phase is stiffer than the other. Domain stability is also found to vary as a function of spacing; as a result, small domains on the order of 5 nm become unstable and tend to coalesce when the domain-phase area fraction increases above 0.5, whereas larger domains on the order of 50 nm become unstable only when the area fraction increases above 0.8.

Effect of Line Tension on the Lateral Organization of Lipid Membranes

The principles of organization and functioning of cellular membranes are currently not well understood. The raft hypothesis suggests the existence of domains or rafts in cell membranes, which behave as protein and lipid platforms. They have a functional role in important cellular processes, like protein sorting or cell signaling, among others. Theoretical work suggests that the interfacial energy at the domain edge, also known as line tension, is a key parameter determining the distribution of domain sizes, but there is little evidence of how line tension affects membrane organization. We have investigated the effects of the line tension on the formation and stability of liquid ordered domains in model lipid bilayers with raft-like composition by means of time-lapse confocal microscopy coupled to atomic force microscopy. We varied the hydrophobic mismatch between the two phases, and consequently the line tension, by modifying the thickness of the disordered phase with phosphatidylcholines of different acyl chain length. The temperature of domain formation, the dynamics of domain growth, and the distribution of domain sizes depend strongly on the thickness difference between the domains and the surrounding membrane, which is related to line tension. When considering line tension calculated from a theoretical model, our results revealed a linear increase of the temperature of domain formation and domain growth rate with line tension. Domain budding was also shown to depend on height mismatch. Our experiments contribute significantly to our knowledge of the physical-chemical parameters that control membrane organization. Importantly, the general trends observed can be extended to cellular membranes.

MFM observations of domain wall creep and pinning effects in amorphous CoxSi1−x films with diluted arrays of antidots

Magnetic force microscopy (MFM) has been used to analyse the behaviour of domain walls in uniaxial amorphous CoxSi1−x films patterned with diluted arrays of antidots by electron beam lithography. The walls are found to be pinned by the antidot array when the antidot density is high enough along the easy axis. The expansion of reversed nuclei under the influence of the tip stray field has been observed in several consecutive MFM images of the same area, showing how the competition between line tension effects and pinning by the patterned holes governs the creep motion of the 180° walls across the array of antidots.

Wettability study of modified silicon dioxide surface using environmental scanning electron microscopy

The wettability analysis is often used to characterize a surface in micro and nanometer scale. At these small scales, effects of the contact line tension are also expected to play a significant role. Wettability effect is studied using environmental scanning electron microscopy on silicon dioxide surface modified by a low-energy electron irradiation method. Electron-induced wettability variation and patterning at micrometer scale on silicon dioxide substrate allow investigating the contact angle dependence on the water droplet line curvature and calculating values of the line tension of a three-phase system (solid-liquid-vapor) of about 10−9 J/m that is consistent with theoretical estimations. It is found that the sign of the line tension alters from positive for hydrophilic surface to negative for hydrophobic one.

Estimation of line tension and contact angle from heterogeneous nucleation experimental data

Using the classical nucleation theory corrected with line tension and experimental data of heterogeneous nucleation of n-nonane, n-propanol, and their mixture on silver particles of three different sizes, the authors were able to estimate the line tensions and the microscopic contact angles for the above mentioned systems. To do this they applied generalized Young's equation for the line tension and calculated the interfacial tensions using Li and Neumann's equation [Adv. Colloid Interface Sci. 39, 299 (1992)]. It has been found that, for both unary and binary systems, the line tension is negative and the resulting microscopic contact angle derived from experimental nucleation data is most of the time larger than the macroscopic one. This is in contrast to earlier studies where the influence of line tension has not been accounted for. The values of the three phase contact line tension obtained in this way are of the same order of magnitude as the estimations for other systems reported in literature. The line tension effect also decreases considerably the nucleation barrier.

Fatty-Acid Monolayers at the Nematic/Water Interface: Phases and Liquid-Crystal Alignment

The two-dimensional (2D) phases of fatty-acid monolayers (hexadecanoic, octadecanoic, eicosanoic, and docosanoic acids) have been studied at the interface of a nematic liquid crystal (LC) and water. When observed between crossed polarizers, the LC responds to monolayer structure owing to mesoscopic alignment of the LC by the adsorbed molecules. Similar to Langmuir monolayers at the air/water interface, the adsorbed monolayer at the nematic/water interface displays distinct thermodynamic phases. Observed are a 2D gas, isotropic liquid, and two condensed mesophases, each with a characteristic anchoring of the LC zenithal tilt and azimuth. By varying the monolayer temperature and surface concentration we observe reversible first-order phase transitions from vapor to liquid and from liquid to condensed. A temperature-dependent transition between two condensed phases appears to be a reversible swiveling transition in the tilt azimuth of the monolayer. Similar to monolayers at the air/water interface, the temperature of the gas/liquid/condensed triple-point temperature increased by about 10 degrees C for a two methylene group increase in chain length. However, the absolute value of the triple-point temperatures are depressed by about 40 degrees C compared to those of analogous monolayers at the air/water interface. We also observe a direct influence by the LC layer on the mesoscopic and macroscopic structure of the monolayer by analyzing the shapes and internal textures of gas domains in coexistence with a 2D liquid. An effective anisotropic line tension arises from elastic forces owing to deformation of the nematic director across phase boundaries. This results in the deformation of the domain from circular to elongated, with a distinct singularity. The LC elastic energy also gives rise to transition zones displaying mesoscopic realignment of the director tilt or azimuth between adjacent regions with a sudden change in anchoring.

Cu/ZnO and Cu/ZrO2interactions studied by contact angle measurement with TEM

A technique of contact angle measurement was applied to the nano-scale oxide-supported metal particles. For Cu supported on ZnO and ZrO2 the angles were found to increase and the work of adhesion to decrease with increasing particle size. Such a trend is interpreted as an effect of negative contact line tension of 2.1 x 10(-9) J m(-1) and 1.0 x 10(-9) J m(-1) in the Cu/ZnO and Cu/ZrO2 system, correspondingly. For the small-sized Cu particles the apparent work of adhesion on ZnO support is higher than that on ZrO2.

Sessile droplets on a curved substrate: effects of line tension

The equilibrium and stability of a sessile droplet laid on a spherical substrate in the presence of line tension are studied within a continuum approach. The destabilizing role played by the substrate's curvature is analysed in detail for both positive and negative line tensions. In the former case, several bifurcation diagrams exist which show, among other things, that stable equilibria can fail to exist when the droplet size, the substrate size and the ratio between the line and the surface tension are comparable to one another. When line tension is negative, the substrate's curvature enhances the destabilization of sessile droplets.