Anisotropic Surface Free Energy Research Articles

Effects of surface anisotropy on the surface morphological response of plasma-facing tungsten

Surface free energy has been hypothesized as the cause behind the experimental evidence (Parish et al., 2014) of a strong correlation between crystallographic orientation and surface morphology changes in helium ion-irradiated tungsten at the early stages of ‘fuzz’ growth in fusion plasma-facing tungsten. Here, this hypothesis is tested through self-consistent dynamical simulation based on an atomistically informed model of surface evolution in plasma-irradiated tungsten. Low-energy helium plasma irradiation of a tungsten surface, even when the ion energy is below the sputtering threshold, generates surface vacancy-adatom pairs in addition to adatoms from the flux of self-interstitial atoms to the surface resulting from the growth of high-pressure helium bubbles. Along with the stress-driven surface morphological instability as the primary driving force, preferential diffusion of these surface adatoms modulated by the associated Ehrlich–Schwoebel (ES) barriers, combined with the surface free energy anisotropy drive the anisotropic surface nanostructure growth. Simulation results for helium irradiation on W(100) show growth of pyramidal surface features, akin to experimental observations. This work confirms the hypothesis on the effect of tungsten surface crystallographic orientation on helium-ion-induced surface morphology changes, where surface mound formation is the outcome of a stress-driven surface morphological instability, while the anisotropic growth of the mounds is controlled by the ES-barrier-modulated anisotropic surface adatom diffusion and faceting of the mounds is controlled by the surface free energy anisotropy.

Impact of surface free energy on two-dimensional crystallization

Two-dimensional (2D) molecular crystals are emerging materials with fascinating optoelectronic properties. However, growing 2D organic single crystals is challenging because little is known about the factors that determine 2D crystallization. Herein, we investigated the relationship between 2D crystallization and the surface free energy of organic semiconductors. We proposed a term called surface free energy anisotropy factor (g) to measure the tendency of 2D crystallization. Taking two [1]benzothieno[3,2-b][1]benzothiophene derivatives as examples, we identified that the one with a larger g factor showed a higher tendency for 2D crystallization, irrespective of the external factors. This work provides a guideline for the rational design of organic semiconductors toward 2D crystallization.

Cu2+ induced self-assembly of urchin-like Co1−xCux into hollow microspheres toward wideband and thin microwave absorbers

Regulating the structure and ingredients of hollow structures can solve the problems of thickness, heaviness, and narrow broadband faced by metal microwave absorbers. Herein, we have developed for the first time a facile one-pot hydrothermal reduction approach to selectively preparing bimetallic Co1–xCux (x = 0, 0.012, 0.031, 0.062, 0.125, 0.250, 0.498) hollow microspheres (HMs) consisting of urchin-like superstructures. The selective adsorption of surfactants and prenucleated Cu seeds guide the growth of Co into 1D rods, which subsequently assembled into urchin-like Co1−xCux superstructures and further into HMs driven by the minimization of magnetic anisotropic energy or surface free energy. The combination of Co with Cu into Co1–xCux HMs can tune the structure, size, defect, components, conductivity, and magnetic properties. Moreover, urchin-like Co1−xCux HMs consisting of sparse rods with low Cu content (1%) and large diameter (5–10 µm) benefit the excellent absorption capacity by producing good impedance matching, multiple losses, and electrical/magnetic coupling effects. Interestingly, the Co0.990Cu0.010 HMs exhibited a stronger absorption (−37.94 dB), a wider bandwidth (9.68 GHz), a lower loading (30 wt%), and a smaller matching thickness (1.3 mm) compared with other absorbers. Our findings suggest that the Co1–xCux HMs will perform well as a superior absorber for EM wave absorption and shielding applications due to the adjustable saturation magnetization and outstanding absorption capacity.

Effect of grain orientations on the thermal grain boundary grooving in a three‐dimensional setting

AbstractThe objective of this work is to study the effect of grain orientation on the thermal grooving by surface diffusion. Hackl et al. [1] have presented a finite element model for thermal grooving in three‐dimensions. This variational model involves surface energy, grain boundary energy, external and internal triple line energy. In this study, We use an orientation dependent surface energy. For {1 0 0} grain orientation in the normal direction, we have self‐similar groove profiles for increasing extent of anisotropy of the surface energy. For {1 1 0} and {1 1 1} orientations, there are formation of facets for critical anisotropic cases. These formations are due to so‐called missing orientations concerning the shape of an unconstrained crystal in equilibrium. The rate of grooving varies with change in the extent of surface free energy anisotropy. Flux along the triple line is also important in determining the groove root shape. Triple line energy and its mobility lead to deviate from a typical t1/4 scaling law. For all theses simulations, grain boundary energies are constant, satisfying Herring's relation. Comparisons are made for different values of mobilities for groove shape and its growth rate, using different grain orientations.

Faceted and dendritic morphology change in alloy solidification

We present methods and results for the simulation of faceted and dendritic crystal growth. Using a thermodynamically realistic isothermal alloy model for AlSi we demonstrate, in confirmation of experimental observations, a change in morphology from perfectly faceted hexagons at smaller undercooling to dendritic growth at larger undercoolings. We also demonstrate that there exists a cut off temperature which separates the two distinct morphologies, and indeed hybrid morphologies. These results suggest that the mechanism for morphology variation observed experimentally primarily lies in anisotropic surface free energy modelling, which we adopt in preference to kinetic anisotropy.

Extraction of effective solid-liquid interfacial free energies for full 3D solid crystallites from equilibrium MD simulations.

Molecular dynamics simulations of an embedded atom copper system in the isobaric-isenthalpic ensemble are used to study the effective solid-liquid interfacial free energy of quasi-spherical solid crystals within a liquid. This is within the larger context of molecular dynamics simulations of this system undergoing solidification, where single individually prepared crystallites of different sizes grow until they reach a thermodynamically stable final state. The resulting equilibrium shapes possess the full structural details expected for solids with weakly anisotropic surface free energies (in these cases, ∼5% radial flattening and rounded [111] octahedral faces). The simplifying assumption of sphericity and perfect isotropy leads to an effective interfacial free energy as appearing in the Gibbs-Thomson equation, which we determine to be ∼177 erg/cm2, roughly independent of crystal size for radii in the 50-250 Å range. This quantity may be used in atomistically informed models of solidification kinetics for this system.

Thermal Grooving by Surface Diffusion: a Review of Classical Thermo-Kinetics Approach

In polycrystalline materials wherever a grain boundary intersects a free surface and whenever the topographic variation associated with the atomic motion is favored by total free energy dissipation, the material surface grooves. In this review, we focused on the grain boundary grooving by surface diffusion which is an active mechanism at moderate temperatures and for grooves small in size. Starting with a description of the classical thermo-kinetics treatment of the process, we briefly reviewed Mullins’ very first modeling effort with a small slope assumption at the groove root and further considerations regarding finite slopes, different grain geometries, and anisotropic surface free energies. We concluded by giving examples of experimental observations in accord with theoretical calculations. Keywords: Thermal grooving; Grain boundary groove; Surface diffusion DOI: 10.17350/HJSE19030000042 Full Text: Review Article

Adaptive phase field modeling of morphological instability and facet formation during directional solidification of SiGe alloys

An adaptive phase field model is used to study the morphological instability and the facet formation during a directional solidification of SiGe alloys. Using highly anisotropic surface free energy and cusp kinetic functions, as well as a proper temperature gradient, the simulated critical growth velocity for various alloy concentrations is in good agreement with classical theories and experimental observations. The simulated facet formation and kinetics are also consistent with the experiments as well.

Phase field modeling of Widmanstatten plate formation in Zr–2.5Nb material

Growth of Widmanstatten side-plates from pre-nucleated α in a matrix of β is a diffusion controlled process. The phase-field approach is formulated for Widmanstatten side plate formation through β-Zr (B.C.C)→α-Zr (HCP) transformation in binary substitutional Zr–2.5Nb alloy. Using Gibbs energy functional and diffusional mobility, microstructural evolution of parallel side-plates from pre-nucleated α phase has been simulated. The highly anisotropic nature of Widmanstatten plate and the presence of thin interface at the parent-product interface make the task challenging for FEM analysis. Based on this model, an estimate on the range of temperature over which Widmanstatten parallel side-plates can form is estimated and the extent of anisotropy of surface free energy on the morphology of the plates is analyzed.

Non-universal equilibrium crystal shape results from sticky steps

The anisotropic surface free energy, Andreev surface free energy and equilibrium crystal shape (ECS) z = z(x,y) are calculated numerically using a transfer matrix approach with the density matrix renormalization group (DMRG) method. The adopted surface model is a restricted solid-on-solid (RSOS) model with ‘sticky’ steps, i.e. steps with a point-contact-type attraction between them (p-RSOS model). By analyzing the results, we obtain a first-order shape transition on the ECS profile around the (111) facet; and on the curved surface near the (001) facet edge, we obtain shape exponents having values different from those of the universal Gruber–Mullins–Pokrovsky–Talapov (GMPT) class. In order to elucidate the origin of the non-universal shape exponents, we calculate the slope dependence of the mean step height of ‘step droplets’ (bound states of steps) ⟨n(p)⟩ using the Monte Carlo method, where p = (∂z/∂x,∂z/∂y) and ⟨⋅⟩ represents the thermal average. Using the result of the |p| dependence of ⟨n(p)⟩, we derive a |p|-expanded expression for the non-universal surface free energy feff(p), which contains quadratic terms with respect to |p|. The first-order shape transition and the non-universal shape exponents obtained by the DMRG calculations are reproduced thermodynamically from the non-universal surface free energy feff(p).

Generic role of the anisotropic surface free energy on the morphological evolution in a strained-heteroepitaxial solid droplet on a rigid substrate

A systematic study based on self-consistent dynamical simulations is presented for the spontaneous evolution of an isolated thin solid droplet on a rigid substrate, which is driven by the surface drift diffusion induced by the anisotropic capillary forces (surface stiffness) and mismatch stresses. In this work, we studied the effect of surface free energy anisotropies [weak and strong (anomalous)] on the development kinetics of the “Stranski–Krastanow” island type morphologies. The anisotropic surface free energy and the surface stiffness were treated with well accepted trigonometric functions. Although, various tilt angles and anisotropy constants were considered during simulations, the main emphasis was given on the effect of rotational symmetries associated with the surface Helmholtz free energy topography in two-dimensional space. Our computer simulations revealed the formation of an extremely thin wetting layer during the development of the bell-shaped Stranski-Krastanow island through the mass accumulation at the central region of the droplet via surface drift-diffusion. For weak anisotropy constant levels, instead of singlet islanding, we observed formation of doublet islanding, separated by a shallow wetting layer, for a set of specific tilt angles, ϕ=90° and ϕ=45°, respectively, for the twofold and fourfold rotational symmetry axis. No such formation has been detected for the sixfold symmetry. In the strong (anomalous) anisotropy constant domain, we demonstrated the existence of two distinct morphological modes: (i) the complete stability of the initial Cosine-shaped droplet just above a certain anisotropy constant threshold level by spontaneous slight readjustments of the base and the height of the cluster; (ii) the Frank-van der Merwe mode of thin film formation for very large values of the anisotropy constant by the spreading and coalescence of the droplets over the substrate surface. During the course of the simulations, we continuously tracked both the morphology (i.e., the peak height, the extension of the wetting layer beyond the domain boundaries, and the triple junction contact angle) and the energetics (the global Helmholtz free energy changes associated with the total strain and surface energy variations) of the system.

Nanoscale-faceting of metal surfaces induced by adsorbates

Using density functional theory and thermodynamic considerations, adsorbate-induced faceting of high-index metal surfaces such as Ir(210) and Re(112 1) has been studied. Focusing on these two systems we first discuss the adsorption behaviour of oxygen and nitrogen on the various surfaces relevant for the faceting, and afterwards use these energies to evaluate the stability of substrates and facets in the presence of oxygen and nitrogen. The faceting phase diagrams of Ir(210) and Re(112 1) show that both adsorbates enhance the anisotropy in surface free energy, finally causing nanofacets to become the thermodynamically favourable surface structure. We also generated analogous electrochemical phase diagrams for both surfaces in contact with an oxygen- or nitrogen-containing electrolyte and found that the same nanofacets should also become stable at positive electrode potentials. Thus, our calculations not only reproduce the experimentally observed surface faceting under UHV conditions, but also predict facet formation under electrochemical conditions.

Experimental Observation and Computer Simulations of 3D Triplet Structures in Diffusion Limited Growth of Xenon Crystals

Changes of growth morphologies are induced by a perturbation of the thermal diffusion field in the surrounding melt of a growing xenon crystal. Apart from the dendritic morphology, seaweed and doublon morphologies and for the first time transitions from dendritic to triplet structures (first predicted by T. Abel, E. Brener, and H. Müller-Krumbhaar [Phys. Rev. E 55, 7789 (1997)10.1103/PhysRevE.55.7789] were observed experimentally. With 3D phase-field simulations it was possible to reproduce the experimental procedure and to verify that triplet structures can grow in a stable way even in the presence of anisotropic surface free energy as found for experimental substances.

Surface motion of single crystals driven by anisotropic surface diffusion

In this paper we compute a mathematical model for the evolution of single crystal with a particular kind of anisotropic surface free energies that yields equilibrium crystal shapes close to octahedrons. Employing the developed model we exhibit the entire evolution path of single crystals to equilibrium with different levels of anisotropic surface free energy. We find that with a mildly anisotropic surface free energy, the crystal morphology is smooth in evolution and evolves to a unique equilibrium crystal shape. With a severely anisotropic surface free energy, edges, corners and faceting by hill-and-valley structures on crystal surface occur. The equilibrium crystal shape reached in computation approaches the analytic octahedron Wulff shape as the anisotropy in the surface free energy becomes more severe.

Nanoscale surface chemistry over faceted substrates: structure, reactivity and nanotemplates

Faceting is a form of self-assembly at the nanometre-scale on adsorbate-covered single-crystal surfaces, occurring when an initially planar surface converts to a "hill and valley" structure, exposing new crystal faces of nanometre-scale dimensions. Planar metal surfaces that are rough on the atomic scale, such as bcc W(111), fcc Ir(210) and hcp Re(1231), are morphologically unstable when covered by monolayer films of oxygen, or by certain other gases or metals, becoming "nanotextured" when heated to temperatures above approximately 700 K. Faceting is driven by surface thermodynamics (anisotropy of surface free energy) but controlled by kinetics (diffusion, nucleation). Surfaces can spontaneously rearrange to minimize their total surface energy (by developing facets), even if this involves an increase in surface area. In this critical review, we discuss the structural and electronic properties of such surfaces, and first principles calculations are compared with experimental observations. The utility of faceted surfaces in studies of structure sensitive reactions (e.g., CO oxidation, ammonia decomposition) and as templates for growth of metallic nanostructures is explored (122 references).

Facet Stability in Oxygen-Induced Nanofaceting of Re(123̅1)

The stability of the various facets in oxygen-induced faceting of Re(1231) has been studied by low-energy electron diffraction, scanning tunneling microcopy, and synchrotron-based high-resolution X-ray photoemission spectroscopy. When Re(1231) is annealed at 800-1200 K in oxygen (10(-7) Torr), the surface becomes completely covered with nanometer-scale facets, and its morphology depends on the substrate temperature and oxygen exposure. Especially, the (1121) facet competes with the (1011) facet in determining the surface morphology, and the stability of each facet relies on oxygen coverage. Using density functional theory, the O-Re binding energies on the facets for various oxygen concentrations are calculated to explain how the oxygen coverage affects the anisotropy of surface free energy, which in turn determines the morphology of the faceted surface.

Mechanism of surface-tension reduction by electric-field application: Shape changes in single-crystal field emitters under thermal-field treatment

The shape change of the ⟨100⟩-oriented tungsten single-crystal field emitter tip under thermal-field (TF) treatment was systematically studied with increasing remolding field (i.e., reversely polarized field opposite to field-emission polarity) at a constant temperature. The initially hemispherical emitter tip builds up to polyhedral shapes with increasing remolding voltage. These shapes are formed by certain low-index crystal planes. On further increase of the voltage, an overremolding state is reached in which the emitter takes a rugged surface structure. A thermodynamic model is proposed to explain the buildup mechanism utilizing the reduction of the surface tension under the influence of a strong electric field. The theory assumes the emitter-extractor system to comprise an electric capacitor and includes the electrostatic-field energy stored around the emitter into the system under consideration. It is shown that the electrostatic-field energy term in the Gibbs free energy could be interpreted as an effective reduction of the surface tension. The initially moderate anisotropy of the intrinsic surface tension is increasingly enhanced by the electrostatic-field energy through the reduction mechanism. The facets on the emitter surface consequently develop in accordance with the Wulff construction of the equilibrium crystal shape (ECS). By application of an electric field, the surface free-energy anisotropy can be made large enough to induce complete faceting of the crystal even at an elevated temperature where the crystal shape can immediately follow the ECS.

Self-assembling of Ge dots on nanopatterns: Experimental investigation of their formation, evolution and control

Understanding and predicting quantum dots ordering is a central problem for many physical processes. Here, we investigate the mechanisms which govern the self-assembling of Ge dots on nanopatterned vicinal Si substrate. In a first part, we investigate the formation of nanopatterns by self-organization of growth instabilities which develop during epitaxial growth of Si and ${\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$ layers on Si substrates. Evolution laws of kinetic and thermodynamic growth instabilities as a function of temperature, deposited thickness, and strain are determined on vicinal (111) and (001). In a second part, we analyze the Ge dots ordering on full scale wafer patterns formed by self-organization of the growth instabilities. Depending on the nature of the latter ones, we show that Ge dots can align either in the valley of the instability undulations or on their top. We explain these results by the predominant effect of surface diffusion, surface free energy anisotropy, and strain energy relaxation, respectively.

The effect of kinetics in the surface evolution of thin crystalline films

The effect of kinetics on the faceting of crystal surfaces caused by strongly anisotropic surface free energies is shown by numerical simulations. We compare the evolution of a crystal towards its equilibrium shape by surface diffusion and surface diffusion with kinetics and observe a significant slow down of the dynamics. For quantitative studies of surface morphologies this effect is crucial, as kinetics are assumed to play a dominant role in the surface evolution of thin crystalline films.

The effect of imposing a corner condition on the evolution of two-dimensional crystal morphologies by surface diffusion with anisotropic surface free energies

Continuity of the chemical potential is required from the physics of the evolution of crystal morphology by surface diffusion and in equilibrium crystal shapes (ECS). With an anisotropic surface free energy, multiple ECS with corners have been reported and instability developed in computing evolution of crystal morphology. For a cornered ECS that minimizes the total surface free energy, a corner condition is needed. Hence, in addition to continuity of the chemical potential, we add the corner condition in the evolution of crystal morphology and ensure the condition is satisfied in the numerical computation of the evolving surfaces. The evolution is governed by a nonlinear 4th order time dependent partial differential equation. By enforcing the corner condition, we obtain better stability in the evolution, which leads to ECS having lower total surface free energies than those obtained previously without satisfying the corner condition. Our study shows that the corner condition plays an important role for the cases of critically and severely anisotropic surface free energies that yield surface corners and facets.