Effect Of Line Tension Research Articles

Properties of dislocation lines in crystals with strong atomic-scale disorder

We use discrete dislocation dynamics (DDD) to study the motion of a dislocation under strong stochastic forces that may cause bending and roughening of the dislocation line on scales that are comparable to the dislocation core radius. In such situations, which may be relevant in high entropy alloys (HEAs) exhibiting strong atomic scale disorder, standard scaling arguments based upon a line tension approximation may be no longer adequate and corrections to scaling need to be considered. We first study the wandering of the dislocation under thermal Langevin forces. This leads to a linear stochastic differential equation which can be exactly solved. From the Fourier modes of the thermalized dislocation line we can directly deduce the scale dependent effective line tension. We then use this information to investigate the wandering of a dislocation in a crystal with spatial, time-independent (‘quenched’) disorder. We establish the pinning length and show how this length can be used as a predictor of the flow stress. Implications for the determination of flow stresses in HEAs from molecular dynamics simulations are discussed.

The role of traction in membrane curvature generation.

Curvature of biological membranes can be generated by a variety of molecular mechanisms including protein scaffolding, compositional heterogeneity, and cytoskeletal forces. These mechanisms have the net effect of generating tractions (force per unit length) on the bilayer that are translated into distinct shapes of the membrane. Here, we demonstrate how the local shape of the membrane can be used to infer the traction acting locally on the membrane. We show that buds and tubes, two common membrane deformations studied in trafficking processes, have different traction distributions along the membrane and that these tractions are specific to the molecular mechanism used to generate these shapes. Furthermore, we show that the magnitude of an axial force applied to the membrane as well as that of an effective line tension can be calculated from these tractions. Finally, we consider the sensitivity of these quantities with respect to uncertainties in material properties and follow with a discussion on sources of uncertainty in membrane shape.

Numerical investigation of heterogeneous nucleation of water vapour on PM10 for particulate abatement

A heterogeneous nucleation model with inclusions of the line tension effect, the particle roughness effect, and the surface diffusion mechanism was presented. Effects of the particle roughness and the wetting agent on the heterogeneous nucleation behaviour were examined. The scaled nucleation barrier was analyzed and subsequently implications of the nucleation behaviour in the particulate abatement by vapour condensation were discussed. It was found that the effect of particle roughness on the nucleation behaviour was greatly affected by the line tension. There existed an optimal concentration of wetting agent at which the lowest nucleation barrier and critical saturation ratio could be obtained. The surface diffusion mechanism played an overwhelmingly important role in governing the embryo growth for hydrophilic particles with a diameter Dp > 0.1 µm and an embryo size smaller than the critical size, otherwise the contribution of direct vapour deposition mechanism could be significant. Based on the scaled nucleation barrier, three distinct nucleation regimes, i.e. athermal heterogeneous nucleation, thermal heterogeneous‐dominant nucleation, and homogeneous‐dominant nucleation, have been identified. When the contact angle was large, the wetting agent might need to be added to reduce the contact angle so as to reach the athermal heterogeneous and thermal heterogeneous‐dominant nucleation regimes, thus achieving efficient particulate abatement at low cost. The prediction results matched reasonably with the experimental data.

Morphology of Poly(styrene- co-butadiene) Random Copolymer Thin Films and Nanostructures on a Graphite Surface.

We studied the morphology of poly(styrene- co-butadiene) random copolymers on a graphite surface. Polymer solutions were spin coated onto graphite, at various concentrations and molecular weights. The polymer films and nanostructures were imaged using atomic force microscopy. Above the overlap concentration, thin films formed. However, total wetting did not occur, despite the polymers being well above their Tg. Instead, dewetting was observed, suggesting the films were in a state of metastable equilibrium. At lower concentrations, the polymers formed networks, nanoislands, and nanoribbons. Ordered nanopatterns were observed on the surface; the polymers orientated themselves due to π-π stacking interactions reflecting the crystalline structure of the graphite. At the lowest concentration, this ordering was very pronounced. At higher concentrations, it was less defined but still statistically significant. Higher degrees of ordering were observed with poly(styrene- co-butadiene) than polystyrene and polybutadiene homopolymers as the copolymer's aromatic rings are distributed along a flexible chain, which maximizes π-π stacking. At the two lowest concentrations, the size of the nanoislands and nanoribbons remained similar with varying molecular weight. However, at higher concentrations, the polymer network features were largest at the lowest molecular weight, indicating that in this case, a large proportion of shorter chains stay on top of the adsorbed ones. The contact angles of the polymer nanostructures remained mostly constant with size, which is due to the strong polymer/graphite adhesion dominating over line tension and entropic effects.

Size-Dependent Submerging of Nanoparticles in Polymer Melts: Effect of Line Tension.

Adhesion of nanoparticles to polymer films plays a key role in various polymer technologies. Here we report experiments that reveal how silica nanoparticles adhere to a viscoelastic PMMA film above the glass transition temperature. The polymer was swollen with CO2, closely matching the conditions of nanoparticle-nucleated polymer foaming. It is found that the degree by which the particles sink into the viscoelastic substrate is strongly size dependent and can even lead to complete engulfment for particles of diameter below 12 nm. These findings are explained quantitatively by a thermodynamic analysis, combining elasticity, capillary adhesion, and line tension. We argue that line tension, here proposed for the first time in elastic media, is responsible for the nanoparticle engulfment.

The Role of Traction in Membrane Curvature Generation

Curvature in biological membranes can be generated by a variety of different molecular mechanisms such as protein scaffolding, lipid or protein asymmetry, cytoskeletal forces, etc. These mechanisms have the net effect of generating stresses on the bilayer that are translated into distinct final shapes of the membrane. We propose reversing this input-output relationship by using the shape of a curved membrane to infer physical quantities like the magnitude of the applied forces acting on the bilayer. To do this, we calculate the normal and tangential tractions along the membrane using the known material properties of the membrane along with its shape. These tractions are a quantitative measure of the response of the membrane to external forces or sources of spontaneous curvature. We demonstrate the utility of this approach first by showing that the magnitude of applied force can be inferred from the shape of the membrane alone in both simulations and experiments of membrane tubulation. Next, we show that membrane budding by local differences in spontaneous curvature is driven purely by the generation of traction in the radial direction and the emergence of an effective line tension at the boundary of these regions. Finally, we show that performing this calculation on images of phase-separated giant vesicles yields a line tension similar to experimentally determined values.

Finite-Size and Solvent Dependent Line Tension Effects for Nanoparticles at the Air-Liquid Surface.

The line tension for a nanoparticle (NP) at the air-liquid surface can be determined by examining the variation in NP solution surface tension with bulk NP concentration. In this publication the variation in line tension with liquid solvent is examined for the homologous series of liquids from n-decane through to n-octadecane. Finite-size line tension effects are also studied by examining the variation in line tension with NP size for NPs at the air-octadecane surface. Both the line tension variation with solvent and NP size can be qualitatively explained using an interface displacement model for the line tension.

Silica-Assisted Nucleation of Polymer Foam Cells with Nanoscopic Dimensions: Impact of Particle Size, Line Tension, and Surface Functionality

Core–shell nanoparticles consisting of silica as core and surface-grafted poly(dimethylsiloxane) (PDMS) as shell with different diameters were prepared and used as heterogeneous nucleation agents to obtain CO2-blown poly(methyl methacrylate) (PMMA) nanocomposite foams. PDMS was selected as the shell material as it possesses a low surface energy and high CO2-philicity. The successful synthesis of core–shell nanoparticles was confirmed by Fourier transform infrared spectroscopy, thermogravimetric analysis, and transmission electron microscopy. The cell size and cell density of the PMMA micro- and nanocellular materials were determined by scanning electron microscopy. The cell nucleation efficiency using core–shell nanoparticles was significantly enhanced when compared to that of unmodified silica. The highest nucleation efficiency observed had a value of ∼0.5 for nanoparticles with a core diameter of 80 nm. The particle size dependence of cell nucleation efficiency is discussed taking into account line tension effects. Complete engulfment by the polymer matrix of particles with a core diameter below 40 nm at the cell wall interface was observed corresponding to line tension values of approximately 0.42 nN. This line tension significantly increases the energy barrier of heterogeneous nucleation and thus reduces the nucleation efficiency. The increase of the CO2 saturation pressure to 300 bar prior to batch foaming resulted in an increased line tension length. We observed a decrease of the heterogeneous nucleation efficiency for foaming after saturation with CO2 at 300 bar, which we attribute to homogenous nucleation becoming more favorable at the expense of heterogeneous nucleation in this case. Overall, it is shown that the contribution of line tension to the free energy barrier of heterogeneous foam cell nucleation must be considered to understand foaming of viscoelastic materials. This finding emphasizes the need for new strategies including the use of designer nucleating particles to enhance the foam cell nucleation efficiency.

Line tension effects on the wetting of nanostructures: an energy method

The superhydrophobicity and self-cleaning property of micro/nano-structured solid surfaces require a stable Cassie–Baxter (CB) wetting state at the liquid–solid interface. We present an energy method to investigate how the three-phase line tension affects the CB wetting state on nanostructured materials. For some nanostructures, the line tension may engender a distinct energy barrier, which restricts the position of the three-phase contact line and affects the stability of the CB wetting state. We ascertain the upper and lower limits of the critical pressure at the CB–Wenzel transition. Our results suggest that superhydrophobicity on nanostructures can be modulated by tailoring the line tension and harnessing the curvature effect. This study also provides new insights into the sinking phenomena observed in the nanoparticle-floating experiment.

Phase separation and folding in swelled nematoelastic films

We explore reshaping of nematoelastic films upon imbibing an isotropic solvent under conditions when isotropic and nematic phases coexist. The structure of the interphase boundary is computed taking into account the optimal nematic orientation governed by interaction of gradients of the nematic order parameter and solvent concentration. This structure determines the effective line tension of the boundary. We further compute equilibrium shapes of deformed thin sheets and cylindrical and spherical shells with the rectilinear or circular shape of the boundary between nematic and isotropic domains. A differential expansion or contraction near this boundary generates a folding pattern spreading out into the bulk of both phases. The hierarchical ordering of this pattern is most pronounced on a cylindrical shell.

Spreading law on a completely wettable spherical substrate: The energy balance approach.

The spreading of a cap-shaped spherical droplet on a completely wettable spherical substrate is studied. The nonequilibrium thermodynamic formulation is used to derive the thermodynamic driving force of spreading including the line-tension effect. Then the energy balance approach is adopted to derive the evolution equation of the spreading droplet. The time evolution of the contact angle θ of a droplet obeys a power law θ∼t^{-α} with the exponent α, which is different from that derived from Tanner's law on a flat substrate. Furthermore, the line tension must be positive to promote complete wetting on a spherical substrate, while it must be negative on a flat substrate.

Size-dependent contact angle and the wetting and drying transition of a droplet adsorbed onto a spherical substrate: Line-tension effect.

The size-dependent contact angle and the drying and wetting morphological transition are studied with respect to the volume change for a spherical cap-shaped droplet placed on a spherical substrate. The line-tension effect is included using the rigorous formula for the Helmholtz free energy in the droplet capillary model. A morphological drying transition from a cap-shaped to a spherical droplet occurs when the substrate is hydrophobic and the droplet volume is small, similar to the transition predicted on a flat substrate. In addition, a morphological wetting transition from a cap-shaped to a wrapped spherical droplet occurs for a hydrophilic substrate and a large droplet volume. The contact angle depends on the droplet size: it decreases as the droplet volume increases when the line tension is positive, whereas it increases when the line tension is negative. The spherical droplets and wrapped droplets are stable when the line tension is positive and large.

Free-Energy Barrier of Filling a Spherical Cavity in the Presence of Line Tension: Implication to the Energy Barrier between the Cassie and Wenzel States on a Superhydrophobic Surface with Spherical Cavities.

The free-energy barrier of filling a spherical cavity having an inner wall of various wettabilities is studied. The morphology and free energy of a lens-shaped droplet are determined from the minimum of the free energy. The effect of line tension on the free energy is also studied. Then, the equilibrium contact angle of the droplet is determined from the generalized Young's equation. By increasing the droplet volume within the spherical cavity, the droplet morphology changes from spherical with an equilibrium contact angle of 180° to a lens with a convex meniscus, where the morphological complete drying transition occurs. By further increasing the droplet volume, the meniscus changes from convex to concave. Then, the lens-shaped droplet with concave meniscus spreads over the whole inner wall, resulting in an equilibrium contact angle of 0° to leave a spherical bubble, where the morphological complete wetting transition occurs. Finally, the whole cavity is filled with liquid. The free energy shows a barrier from complete drying to complete wetting as a function of droplet volume, which corresponds to the energy barrier between the Cassie and Wenzel states of the superhydrophobic surface with spherical cavities. The free-energy maximum occurs when the meniscus of the droplet becomes flat, and it is given by an analytic formula. The effect of line tension is expressed by the scaled line tension, and this effect is largest at the free-energy maximum. The positive line tension increases the free-energy maximum, which thus increases the stability of the Cassie superhydrophobic state, whereas the negative line tension destabilizes the superhydrophobic state.

Bud formation of lipid membranes in response to the surface diffusion of transmembrane proteins and line tension

We study the formation of membrane budding in model lipid bilayers with the budding assumed to be driven by means of diffusion of trans-membrane proteins over a composite membrane surface. The theoretical model for the lipid membrane incorporates a modified Helfrich-type formulation as a special case. In addition, a spontaneous curvature is introduced into the model in order to accommodate the effect of the non-uniformly distributed proteins in the bending response of the membrane. Furthermore, we discuss the effects of line tension on the budding of the membrane, and the necessary adjustments to the boundary conditions. The resulting shape equation is solved numerically for the parametric representation of the surface, which has one to one correspondence to the membrane surface in consideration. Our numerical results successfully predict the vesicle formation phenomenon on a flat lipid membrane surface, since the present analysis is not restricted to the conventional Monge representation often adopted to the problems of this kind for the obvious computational simplicity, despite its limited capability to describe the deformed configuration of membranes. In addition, we show that line tension at the interface of the protein-concentrated domain makes a significant contribution to the bud formation of membranes.

Contact Angle for Spherical Nanodroplet in Cylindrical Cavity with Quadratic Curve Generatrix

<p class="1Body">Wetting of a spherical nanodroplet in smooth and homogeneous cylinder surface rotated by quadratic curve was studied by methods of thermodynamics. The solid-liquid-vapor system was separated into six parts using Gibbs method of dividing surface. The system free energy was calculated. A generalized Young equation for the equilibrium contact angle is proposed taking the line tension effects into consideration. On the basis of some assumptions, this generalized Young equation is the same as the classical Young’s equation.</p>

Line tension and morphology of a sessile droplet on a spherical substrate.

The effects of line tension on the morphology of a sessile droplet placed on top of a convex spherical substrate are studied. The morphology of the droplet is determined from the global minimum of the Helmholtz free energy. The contact angle between the droplet and the spherical substrate is expressed by the generalized Young's formula. When the line tension is positive and large, the contact angle jumps discontinuously to 180^{∘}, the circular contact line shrinks towards the top of the substrate, and the droplet detaches from the substrate, forming a spherical droplet if the substrate is hydrophobic (i.e., the Young's contact angle is large). This finding is consistent with that predicted by Widom [J. Phys. Chem. 99, 2803 (1995)JPCHAX0022-365410.1021/j100009a041]; the line tension induces a drying transition on a flat substrate. On the other hand, the contact angle jumps to 0^{∘}, the circular contact line shrinks towards the bottom of the substrate, and the droplet spreads over the substrate to form a wrapped spherical droplet if the substrate is hydrophilic (i.e., the Young's contact angle is small). Therefore, not only the drying transition of a cap-shaped to a detached spherical droplet but also the wetting transition of a cap-shaped to a wrapped spherical droplet could occur on a spherical substrate as the surface area of the substrate is finite. When the line tension is negative and its magnitude increases, the contact line asymptotically approaches the equator from either above or below. The droplet with a contact line that coincides with the equator is an isolated, singular solution of the first variational problem. In this instance, the contact line is pinned and cannot move as far as the line tension is smaller than the critical magnitude, where the wetting transition occurs.

Line tension and morphology of a droplet and a bubble attached to the inner wall of a spherical cavity.

The effects of line tension on the morphology of a lens-shaped droplet and bubble placed on the inner wall of a spherical cavity are studied. The contact angle between the lens-shaped droplet and the concave spherical substrate is expressed by the generalized Young's formula. The equator of the spherical substrate is found to play a crucial role. Neither a droplet with its contact line on the upper hemisphere of the substrate nor one with its contact line on the lower hemisphere can transform into each other continuously. On a hydrophobic substrate, the contact angle jumps discontinuously to 180(∘), and the droplet is detached from the substrate to form a spherical droplet when the line tension is positive and large. This is similar to the drying transition on a flat substrate. On the other hand, on a hydrophilic substrate, the contact angle jumps discontinuously to 0(∘) when the line tension is positive and large. Then, the droplet spreads over the whole inner wall to leave a spherical bubble. Therefore, not only the drying transition but also the wetting transition is induced by positive line tension on a concave spherical substrate. There also exist stable as well as metastable droplets, whose phase diagrams can be complex. When the line tension is negative and its magnitude increases, the contact line approaches the equator infinitesimally from either above or below. However, it cannot cross the equator of a spherical cavity continuously. The droplet with a contact line that coincides with the equator is a singular droplet. The contact line is pinned and cannot move, irrespective of the magnitude of the line tension.

The contact angle for a droplet on homogeneous and spherical concave surfaces

Wetting of droplets on homogeneous and spherical concave rough surfaces is investigated based on thermodynamics. In this study, neglecting the droplet gravity and the thickness of the precursor film of the liquid–vapor interface, the three-phase system is divided into six parts using Gibbs concept of dividing surface. The system Helmholtz free energy is established based on thermodynamics. Supposing the temperature and chemical potential to be constant, a new generalized Young’s equation of the equilibrium contact angle for a spherical droplet on a spherical concave rough surfaces is obtained including the line tension effects. Under certain conditions, this generalized Young’s equation is the same as the Rusanov’s equation.

Local well‐posedness for volume‐preserving mean curvature and Willmore flows with line tension

We show the short‐time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume‐preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non‐local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface.

Line-tension-induced scenario of heterogeneous nucleation on a spherical substrate and in a spherical cavity.

Line-tension-induced scenario of heterogeneous nucleation is studied for a lens-shaped nucleus with a finite contact angle nucleated on a spherical substrate and on the bottom of the wall of a spherical cavity. The effect of line tension on the free energy of a critical nucleus can be separated from the usual volume term. By comparing the free energy of a lens-shaped critical nucleus of a finite contact angle with that of a spherical nucleus, we find that a spherical nucleus may have a lower free energy than a lens-shaped nucleus when the line tension is positive and large, which is similar to the drying transition predicted by Widom [B. Widom, J. Phys. Chem. 99, 2803 (1995)]. Then, the homogeneous nucleation rather than the heterogeneous nucleation will be favorable. Similarly, the free energy of a lens-shaped nucleus becomes negative when the line tension is negative and large. Then, the barrier-less nucleation with no thermal activation called athermal nucleation will be realized.