Surface Free Energy Density Research Articles

A novel constitutive model for surface elasticity at finite strains suitable across compressibility spectrum

The surface elasticity theory of Gurtin–Murdoch has proven to be remarkably successful in predicting the behavior of materials at the nano scale, which can be attributed to the fact that the surface-to-volume ratio increases as the problem dimension decreases. On the other hand, surface tension can deform soft elastic solids even at the macro scale resulting e.g. in elastocapillary instabilities in soft filaments reminiscent of Plateau–Rayleigh instabilities in fluids. Due to the increasing number of applications involving nanoscale structures and soft solids such as gels, the surface elasticity theory has experienced a prolific growth in the past two decades. Despite the large body of literature on the subject, the constitutive models of surface elasticity theory at large deformations are not suitable to capture the surface behavior from fully compressible to nearly incompressible elasticity, especially from a computational perspective. A physically meaningful and proper decomposition of the surface free energy density in terms of area-preserving and area-varying contributions remains yet to be established. We show that an immediate and intuitive generalization of the small-deformation surface constitutive models does not pass the simple extension test at large deformations and results in unphysical behavior at lower Poisson’s ratios. Thus, the first contribution of the manuscript is to introduce a novel decomposed surface free energy density that recovers surface elasticity across the compressibility spectrum. The second objective of this paper is to formulate an axisymmetric counterpart of the elastocapillary theory methodically derived from its three-dimensional format based on meaningful measures relevant to the proposed surface elasticity model. Various aspects of the problem are elucidated and discussed through numerical examples using the finite element method enhanced with surface elasticity.

Equation-of-state-dependent surface free-energy density for wettability in lattice Boltzmann method.

In thermodynamic theory, the liquid-vapor fluids can be described by a single multiphase equationof state and the surface wettability is usually characterized by the surface free-energy density. In this work, we propose an equation-of-state-dependent surface free-energy density for the wettability of the liquid-vapor fluids on a solid surface, which can lead to a simple closed-form analytical expression for the contact angle. Meanwhile, the thermodynamically derived equilibrium condition is equivalent to the geometric formulation of the contact angle. To numerically validate the present surface free-energy density, the mesoscopic multiphase lattice Boltzmann model with self-tuning equationof state, which is strictly consistent with thermodynamic theory, is employed, and the two-dimensional wetting condition treatment is extended to the three-dimensional situation with flat and curved surfaces. Two- and three-dimensional lattice Boltzmann simulations of static droplets on flat and curved surfaces are first performed, and the obtained contact angles agree well with the closed-form analytical expression. Then, the three-dimensional lattice Boltzmann simulation of a moving droplet on an inclined wall, which is vertically and sinusoidally oscillated, is carried out. The dynamic contact angles well satisfy the Cox-Voinov law. The droplet movement regimes are consistent with previous experiments and two-dimensional simulations. The dependence of the droplet overall velocity with respect to the dimensionless oscillation strength is also discussed in detail.

Light-induced Fredericks transition in the nematic liquid crystal cell with plasmonic nanoparticles at a cell bounding substrate.

The paper presents a theoretical description of the light-induced nematic liquid crystal reorientation in a cell with gold nanoparticles deposited on the surface of one of bounding substrates. It is shown that the surface plasmon resonance in the nanoparticles significantly affects the threshold of the director reorientation. The mathematical model of a surface free-energy density of nematic cell is given, which takes into account the influence of the local electric field on the near-surface nematic layer at the substrate with gold nanoparticles. The threshold intensity of a director orientation instability is calculated and its dependence on the wavelength of incident light and the degree of filling of the surface with gold nanoparticles is analyzed. Comparison of the theoretical calculations with experimental data confirms the full adequacy of the proposed theoretical model.

A continuum state variable theory to model the size-dependent surface energy of nanostructures

We propose a continuum-based state variable theory to quantify the excess surface free energy density throughout a nanostructure. The size-dependent effect exhibited by nanoplates and spherical nanoparticles i.e. the reduction of surface energy with reducing nanostructure size is well-captured by our continuum state variable theory. Our constitutive theory is also able to predict the reducing energetic difference between the surface and interior (bulk) portions of a nanostructure with decreasing nanostructure size.

Generalized Berreman's model of the elastic surface free energy of a nematic liquid crystal on a sawtoothed substrate

In this paper we present a generalization of Berreman's model for the elastic contribution to the surface free-energy density of a nematic liquid crystal in presence of a sawtooth substrate which favors homeotropic anchoring as a function of the wave number of the surface structure q, the tilt angle α, and the surface anchoring strength w. In addition to the previously reported nonanalytic contribution proportional to -q ln q, due to the nucleation of disclination lines at the wedge bottoms and apexes of the substrate, the next-to-leading contribution is proportional to q for a given substrate roughness, in agreement with Berreman's predictions. We characterize this term, finding that it has two contributions: the deviations of the nematic director field with respect to a reference field corresponding to the isolated disclination lines and their associated core free energies. Comparison with the results obtained from the Landau-de Gennes model shows that our model is quite accurate in the limit wL>1, when strong anchoring conditions are effectively achieved.

Boundary condition considerations in lattice Boltzmann formulations of wetting binary fluids

We propose a new lattice Boltzmann numerical scheme for binary-fluid surface interactions. The new scheme combines the existing binary free energy lattice Boltzmann method [Swift et al., Phys. Rev. E 54 (1996)] and a new wetting boundary condition for diffuse interface methods in order to eliminate spurious variations in the order parameter at solid surfaces. We use a cubic form for the surface free energy density and also take into account the contribution from free energy in the volume when discretizing the wetting boundary condition. This allows us to eliminate the spurious variation in the order parameter seen in previous implementations. With the new scheme a larger range of equilibrium contact angles are possible to reproduce and capillary intrusion can be simulated at higher accuracy at lower resolution.

Scaling of the elastic contribution to the surface free energy of a nematic liquid crystal on a sawtoothed substrate

We characterize the elastic contribution to the surface free energy of a nematic liquid crystal in the presence of a sawtooth substrate. Our findings are based on numerical minimization of the Landau-de Gennes model and analytical calculations on the Frank-Oseen theory. The nucleation of disclination lines (characterized by non-half-integer winding numbers) in the wedges and apexes of the substrate induces a leading order proportional to q ln q to the elastic contribution to the surface free-energy density, with q being the wave number associated with the substrate periodicity.

Specific surface free energy and etch pit density of synthesized quartz crystal

The specific surface free energy of growing faces of the synthetic quartz crystal, S, Z, +X, and −X faces was experimentally obtained using contact angle of water and formamide droplets on the crystal surfaces. The anisotropy of the specific surface free energy of the growing faces was explained by the difference of the steps on the faces, which originates from dislocations. In order to investigate the presence of the dislocation, the surface of the crystal was treated to form etch pits. The face with larger specific surface free energy has large density of etch pits, which suggests that the surface with high specific surface free energy has large number of steps originating from defects.

Pairwise Interactions between Linear Alkanes in Water Measured by AFM Force Spectroscopy

Pairwise interactions between n-alkanes from decane to octadecane in water have been studied by single-molecule force spectroscopy. The interacting molecules are covalently tethered to the glass substrate and to the probe of an atomic force microscope by water-soluble linkers to facilitate single-molecule detection. However, the measured distribution of rupture forces deviates significantly from the distribution predicted by theoretical models for rupture of individual bonds. To describe the statistics of rupture forces, an analytical model that considers near-simultaneous rupture of two bonds loaded by tethers with different lengths is introduced. The common most probable force analysis approach is used for comparison. In both data analyses, the possible systematic errors due to nonlinear elasticity of polymeric tethers and variations in the shape of the potential of mean force were considered. Experimental distributions of rupture forces are well-fit by the two-bond rupture model using a single set of kinetic parameters for different experiments, while the most probable force approach yields parameters that vary significantly for different samples. The measured activation energies for dissociation of alkanes are close to the free energies predicted by cavity models of hydrophobic interactions. The surface free-energy density is estimated to be approximately 21 kJ/(mol nm (2)) and is close to the upper limit of free energies used in the computer simulations of hydrophobic interactions in proteins. In contrast to the predictions of the cavity models, the measured activation energy does not increase monotonically with increase in alkane chain size. To explain this discrepancy and the measured distance to the transition-state barrier (approximately 0.6 nm), it is suggested that alkanes undergo conformational transition to the collapsed state upon dimerization. Change in the alkane conformation from extended to helical has been observed previously for binding of alkanes in water to hydrophobic synthetic receptors. Here, however, conformational change is suggested without geometrical constraints imposed by small cavitands. The proposed collapsed state of the alkane dimers has implications for the kinetics of self-assembly of surfactant micelles.

Pyroelectricity and thermodynamic theory of compositionally graded ferroelectric films

A thermodynamic theory of compositionally graded ferroelectric films (CGFF) is developed in this article. The Landau–Ginzburg thermodynamic expansion of the bulk and surface free energy density has been extended by terms of P.gradc type where P is polarization and c stands for the composition ratio of two different atoms. Both P and c are spatially dependent. Due to the additional terms in the free energy, additional polarization arises in the sample. The spatial distribution of polarization addressed in this article is influenced by the boundary conditions which are determined by the surface free energy density. However, the spatial distribution of polarization itself is not essential for the value of the pyroelectric coefficient since the latter depends on the value of spatially averaged polarization. In a CGFF, the surface contribution to a classical pyroelectric effect plays an essential role.

Atomistic Determination of Continuum Mechanical Properties of Ion-Bombarded Silicon

Highly disordered, ion-processed silicon is studied using a molecular dynamics simulation with empirical interatomic potentials. The surface free energy density, stress-strain relations, and continuum surface features of silicon, bombarded in the simulations to relatively high fluence by medium energy argon ions, are computed statistically by preparing multiple randomized ion-bombarded specimens. The surface-free energy per unit area for the ion-bombarded silicon is about 1.76J∕m2, much lower than the 2.35J∕m2 corresponding to a (001) unrelaxed, crystalline silicon surface. A stress-strain curve is obtained computationally by performing a constant strain test on the ion-bombarded specimens and by calculating stresses from the interatomic forces acting across different cross sections in the sample. The resulting tensile elastic modulus of the material, while slightly elevated due to the prominence of the free surface in the thin layer, is in good agreement with available experimental data. The surface is characterized using an interatomic potential-based C2 continuous sampling method.

Mind the Gap: The Shape of the Free Surface of a Rubber-Like Material in Proximity to a Rigid Contactor

Extending Andreussi and Gurtin's [1] pioneering work on the wrinkling of a free surface, we investigate the wrinkling stability of an incompressible elastic-like half-space whose surface is proximate to a contactor. Assuming a plane-strain deformation and accounting for both surface prestress and curvature dependence of the surface free-energy density, we impose balances of forces and torques both in the bulk and on the surface. From the resulting linearized bulk and superficial equations, we derive a quintic dimensionless dispersion relation and perform a parametric study to see when stable or unstable behavior of the free surface is manifested. When compared with the quadratic dispersion relation of Andreussi and Gurtin [1], we see here that the combined effects associated with surface prestress, curvature-dependence of the surface free-energy density, and the interactions between the surface and the proximate contactor always lead to an increased number of linearly stable wrinkled configurations.

Structure of random bidisperse foam

The Surface Evolver was used to compute the equilibrium microstructure of random soap foams with bidisperse cell-size distributions and to evaluate topological and geometric properties of the foams and individual cells. The simulations agree with the experimental data of Matzke and Nestler for the probability ρ ( F ) of finding cells with F faces and its dependence on the fraction of large cells. The simulations also agree with the theory for isotropic Plateau polyhedra (IPP), which describes the F-dependence of cell geometric properties, such as surface area, edge length, and mean curvature (diffusive growth rate); this is consistent with results for polydisperse foams. Cell surface areas are about 10% greater than spheres of equal volume, which leads to a simple but accurate relation for the surface free energy density of foams. The Aboav–Weaire law is not valid for bidisperse foams.

Nano-scale equilibrium crystal shapes

The finite size and interface effects on equilibrium crystal shape (ECS) have been investigated for the case of a surface free energy density including the free energy of step formation and inverse-square step–step interactions. Explicitly including the curvature of a crystallite leads to an extra boundary condition in the solution of the crystal shape, yielding a family of crystal shapes, governed by a constant of integration, which we call the shape parameter c. The total crystallite free energy, including interface energy, is minimized for c = 0, yielding in all cases the traditional Pakrovsky–Talapov shape. Solutions of the crystal shape for c ≠ 0 are presented and discussed in the context of meta-stable states due to the energy barrier for nucleation. Explicit scaled relationships for the ECS and meta-stable states in terms of the measurable step parameters and the interfacial energy are presented.

Structure of Random Foam

The Surface Evolver was used to compute the equilibrium microstructure of dry soap foams with random structure and a wide range of cell-size distributions. Topological and geometric properties of foams and individual cells were evaluated. The theory for isotropic Plateau polyhedra describes the dependence of cell geometric properties on their volume and number of faces. The surface area of all cells is about 10% greater than a sphere of equal volume; this leads to a simple but accurate theory for the surface free energy density of foam. A novel parameter based on the surface-volume mean bubble radius R32 is used to characterize foam polydispersity. The foam energy, total cell edge length, and average number of faces per cell all decrease with increasing polydispersity. Pentagonal faces are the most common in monodisperse foam but quadrilaterals take over in highly polydisperse structures.

The Influence and Interpretation of Surface Parameters for Wetting Transitions in Ternary Mixtures

Starting from a microscopic Hamiltonian defined on a semi-infinite cubic lattice, and employing a mean-field approximation, the surface parameters relevant for wetting in confined ternary mixtures are derived. These are found in terms of the microscopic coupling constants, and yield a physical interpretation of their origins. In comparison with the standard expression for the surface free-energy density, several new terms arising from the derivation are identified. The influence of the surface parameters on a predicted unbinding transition in a mixture of oil, water, and amphiphile demonstrate that existing results are robust to the addition of the extra surface terms.

Computer Simulation of Transition Processes between Bistable States in Bistable Nematic Liquid Crystal Cell Using Breaking Phenomenon of Surface Anchoring

The transition process between bistable states in the bistable nematic (Bi-Nem) cell based on the anchoring breaking phenomenon has been strictly simulated by computer. In our simulation, the surface tilt angle dependence of the contribution of azimuthal anchoring to the surface free energy density has been taken into account. This leads to successful reproduction of the bistable operation of the Bi-Nem cells by the computer simulation. As a result, it has been clarified that in the transition process from the uniform state to the twisted state, the molecular orientation takes the 270 deg. twisted state at first and then changes toward to the 180 deg. twisted state. From our simulations, it is confirmed that the bistable transition occurs in limited regions of polar and azimuthal anchoring energy on the slave plate.

Surface energy and spatial dispersion in ferroelectrics: the influence on the opticaltransmission and reflection spectra in the paraelectric state

The description of surface and size effects in ferroelectrics is based on the combinedcontribution of surface free energy density and gradient dependent terms occurring in thevolume free energy density. The dynamics of the polarization waves and, consequently, theoptical properties in the infrared range, can be significantly affected in the case of smallenough soft mode damping, even in the paraelectric phase at temperatures above those ofthe ordered ferroelectric state. Reflection and transmission spectra for semi-infinite mediaand thin films are calculated within this model, using the orders of magnitude generallypublished in the literature for ferroelectric oxides as regards the values of the physicalparameters. It is shown that the tangential and the normal interface extrapolationlengths, as well as the bulk gradient coefficient related to the model studied,could be subjected to a spectroscopical evaluation. The frequencies of the sharpsingularities appearing in the transmission and reflection spectra are given with anexcellent precision by the frequencies of the surface and of the guided modescalculated in the electrostatic approximation for wavevector equal to zero, thusproviding a rather simple way of determining the pertinent physical quantities.

Microscopic origin and relevance of surface parameters for confined ternary mixtures

On the basis of a microscopic Hamiltonian on a semi-infinite cubic lattice, we derive the excess surface free energy for a confined ternary mixture of oil, water and amphiphile. Using the standard mean-field approximation for a spin-1 model we calculate the surface parameters in terms of the microscopic coupling constants which provides a physical interpretation of their origin. We further discuss in detail the corresponding surface free-energy density in the continuous Ginzburg–Landau theory and demonstrate that it takes the form of an expansion in powers of the surface order parameter and its local gradient. In comparison with the commonly accepted expression, new terms are found and their significance for the description of surface phenomena is described.

Alternative to the Shuttleworth formulation of solid surface stress

We examine two derivations of the Shuttleworth equation (which is a relation between surface stress $g,$ surface tension \ensuremath{\gamma}, and surface strain), and identify the flaws we perceive. Rectifying the perceived flaws leads not to the Shuttleworth equation but to $g\ensuremath{-}\ensuremath{\gamma}$ equivalence. We conclude that surface stress is merely the generalization of the concept of surface tension to an elastically anisotropic system; the surface free-energy density is one-half of the trace of the surface stress tensor, to lowest order. In our opinion, our conclusions lead to a more coherent and elegant form of surface thermodynamics which should prove useful in controlling and in understanding nanometer-scale fabrication.