Roughening Transition Temperature Research Articles

The Electron–Phonon Interaction at Vicinal Metal Surfaces Measured with Helium Atom Scattering

Recently, it was demonstrated that inelastic helium atom scattering from conducting surfaces provides a direct measurement of the surface electron–phonon coupling constant (mass enhancement factor λ) via the temperature or the incident wave vector dependence of the Debye–Waller exponent. Here, previous published as well as unpublished helium atom scattering diffraction data from the vicinal surfaces of copper (Cu(11α), with α = 3, 5, 7) and aluminum (Al(221) and Al(332)) were analyzed to determine λ. The results suggested an enhancement with respect to the corresponding data for the low-index surfaces (111) and (001) above the roughening transition temperature. The specific role of steps compared to that of terraces is briefly discussed.

Influence of roughening transition on magnetic ordering.

In the literature of magnetic phase transitions, in addition to a critical point, the existence of another special point has been discussed. This is related to the broadening of the interface between two different ordering phases and is referred to as the point of roughening transition. While the equilibrium properties associated with this transition are well understood, the influence of this on nonequilibrium dynamics still needs to be investigated. In this paper we present comprehensive results, from Monte Carlo simulations, on coarsening dynamics in a system, over a wide range of temperature, in space dimension d=3, for which there exists a roughening transition at a nonzero temperature T_{R}. An advanced analysis of the simulation data, on structure, growth, and aging, shows that the onset of unexpected glasslike slow dynamics in this system, that has received attention in recent times, for quenches to zero temperature, actually occurs at this transition point. This implies that the structure and aging depend upon the final temperature, when the latter lies between 0 and T_{R}. This is a very interesting exception to universality in coarsening dynamics. The results also demonstrate an important structure-dynamics connection in the phase-ordering dynamics. We compare the key results with those from d=2, for which there exists no nonzero roughening transition temperature. The absence of the above-mentioned anomalous features in the latter dimension places our conjecture on the role of the roughening transition on a firmer footing.

Transmission of crystallization waves across the edge between the rough and faceted crystalline surfaces at the superfluid-solid He4 interface

The wavelike processes of crystallisation and melting or crystallisation waves are well-known to exist at the crystal 4He surface in its rough state. Below the roughening transition temperature the crystal surface experiences the transition to the smooth faceted state and the crystallisation waves represent the propagation of a train of crystalline steps at the velocity depending on the crystal step height. Here we analyse the transmission and reflection of crystallisation waves propagating across the crystal edge separating the crystal surface in the rough and faceted states.

Diffusion of interface and heat conduction in the three-dimensional Ising model.

We investigate the relationship between a diffusive motion of an interface, heat conduction, and the roughening transition in the three-dimensional Ising model. We numerically compute the thermal conductivity and the diffusion constant and find that the diffusion constant shows a crossover in its temperature dependence. The crossover temperature is equal to the roughening transition temperature in equilibrium and deviates from it when heat flows in the system. From these results, we discuss the possibility that heat conduction causes a shift of the roughening transition temperature.

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study analytically an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a continuum evolution law expressed by a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the nonlinear gradient flow of a convex, singular surface free energy with a certain exponential mobility in homoepitaxy. This evolution law is motivated both by an atomistic broken-bond model and a mesoscale model for crystal steps. By constructing an explicit solution to this PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the continuum law via a regularized surface free energy.

Overview: Understanding nucleation phenomena from simulations of lattice gas models.

Monte Carlo simulations of homogeneous and heterogeneous nucleation in Ising/lattice gas models are reviewed with an emphasis on the general insight gained on the mechanisms by which metastable states decay. Attention is paid to the proper distinction of particles that belong to a cluster (droplet), that may trigger a nucleation event, from particles in its environment, a problem crucial near the critical point. Well below the critical point, the lattice structure causes an anisotropy of the interface tension, and hence nonspherical droplet shapes result, making the treatment nontrivial even within the conventional classical theory of homogeneous nucleation. For temperatures below the roughening transition temperature facetted crystals rather than spherical droplets result. The possibility to find nucleation barriers from a thermodynamic analysis avoiding a cluster identification on the particle level is discussed, as well as the question of curvature corrections to the interfacial tension. For the interpretation of heterogeneous nucleation at planar walls, knowledge of contact angles and line tensions is desirable, and methods to extract these quantities from simulations will be mentioned. Finally, also the problem of nucleation near the stability limit of metastable states and the significance of the spinodal curve will be discussed, in the light of simulations of Ising models with medium range interactions.

Mechanisms and kinetics of the migration of grain boundaries containing extended defects

A full understanding of the basic processes of grain boundary migration is a fundamental prerequisite for predictive models of microstructural evolution in polycrystalline materials in processing and in service. In a detailed study of the kinetics of a $[111] \mathrm{\ensuremath{\Sigma}}7$ symmetric tilt boundary, we have previously shown that defect-free, flat grain boundaries, below their roughening temperature, can be strictly immobile in the experimental limit. Here we present the results of molecular dynamics simulations of grain boundaries containing a variety of ``defects.'' These simulations show that the presence of some of these defects restores the mobility of flat boundaries, even well below the roughening transition temperature. These defects fundamentally alter the mesoscale mechanism of grain boundary migration from one involving homogeneous nucleation to a heterogenous process. At the atomistic level, the crystal lattice reorients via coordinated shuffling of groups of atoms. In the case of flat boundaries, these shuffles must accumulate to form critically stable nuclei, but in the case of boundaries containing defects the shuffling of a small number of atoms at the defects can be sufficient, fundamentally altering the mechanism and kinetics of migration.

Anisotropic interfacial tension, contact angles, and line tensions: a graphics-processing-unit-based Monte Carlo study of the Ising model.

As a generic example for crystals where the crystal-fluid interface tension depends on the orientation of the interface relative to the crystal lattice axes, the nearest-neighbor Ising model on the simple cubic lattice is studied over a wide temperature range, both above and below the roughening transition temperature. Using a thin-film geometry L(x)×L(y)×L(z) with periodic boundary conditions along the z axis and two free L(x)×L(y) surfaces at which opposing surface fields ±H(1) act, under conditions of partial wetting, a single planar interface inclined under a contact angle θ<π/2 relative to the yz plane is stabilized. In the y direction, a generalization of the antiperiodic boundary condition is used that maintains the translational invariance in the y direction despite the inhomogeneity of the magnetization distribution in this system. This geometry allows a simultaneous study of the angle-dependent interface tension, the contact angle, and the line tension (which depends on the contact angle, and on temperature). All these quantities are extracted from suitable thermodynamic integration procedures. In order to keep finite-size effects as well as statistical errors small enough, rather large lattice sizes (of the order of 46 million sites) were found to be necessary, and the availability of very efficient code implementation of graphics processing units was crucial for the feasibility of this study.

The effect of synthetic driving force on the atomic mechanisms associated with grain boundary motion below the interface roughening temperature

The mechanisms associated with grain boundary motion induced by synthetic, crystal-orientation-dependent driving forces are investigated for a large-angle [001] Ni symmetric tilt grain boundary. The application of non-physical forces by this method brings legitimate concern that it could lead to non-physical results. This concern is especially relevant below the interface roughening transition temperature where there is a substantial drop in grain boundary mobility and large driving force dependence. Using slip-vector analysis and examining continuum metrics for microrotation and strain, this work shows that the application of synthetic-driving forces does not alter the fundamental mechanisms leading to grain boundary motion. Results in this work are compared directly to shear driven simulations which reveal that the path and deformation history of grain boundary motion is unbiased by the non-physical nature of the driving force acting on the boundary. Nudged elastic band calculations show that the transition path for grain boundary motion is independent of the driving force magnitude and the energy barriers for motion are not appreciably altered by the application of the synthetic driving force.

Domain walls in helical magnets: Elasticity and pinning

Recently completely new types of domain walls (DWs) have been discovered in helical magnets, consisting generically of a regular array of pairs of magnetic vortex lines (Li F. et al., Phys. Rev. Lett., 108 (2012) 107203). Only for special orientations DWs are free of vortices. In this paper we calculate their elastic and pinning properties, using the pitch angle θ as a small parameter. In particular we show that vortex-free DWs exhibit long-range elasticity which makes them very stiff and suppresses their pinning by impurities. Their roughening transition temperature is of the order of the Néel temperature. DWs including vortices (either by orientation or due to step formation above their roughening transition) show short-range elasticity and strong pinning by impurities. These results apply both to centrosymmetric as well as to non-centrosymmetric systems. The application to chiral liquid crystals is briefly discussed.

Charge-induced instability and macroscopic quantum nucleation phenomena at a crystalline4He facet

The existence of a charge-induced instability is well known for the ${}^{4}$He crystal surface in the rough state. Much less is known about the charge-induced instability at the ${}^{4}$He crystal surface in the smooth well-faceted state below the roughening transition temperature. To meet the lack, we examine here the latter case. As long as the electric field normal to the crystal facet is below the critical value the same as for the rough surface, the crystal faceting remains absolutely stable. Above the critical field, unlike the absolutely unstable state of the rough surface, the crystal facet crosses over to the metastable state separated from the crushed state with a potential barrier proportional to the square of the linear facet step energy. The onset and development of the instability at the charged crystal facet has much in common with the nucleation kinetics of first-order phase transitions. Depending on the temperature, the electric breaking strength is determined either by thermal activation at high temperatures or by quantum tunneling at sufficiently low temperatures.

Grain Growth Stagnation Caused by the Grain Boundary Roughening Transition

Molecular dynamics simulations of bicrystals show that grain boundaries undergo a thermal roughening transition, and the grain boundary mobility increases abruptly when the boundary roughens. The roughening transition temperature varies widely from boundary to boundary, ranging from less than 0.4 to more than 0.9 of the melting temperature. Thus, at typical annealing temperatures we expect polycrystals to contain both smooth (slow) and rough (fast) boundaries, with the fraction of each type varying with temperature.

Interface-roughening phase diagram of the three-dimensional Ising model for all interaction anisotropies from hard-spin mean-field theory

The roughening phase diagram of the d=3 Ising model with uniaxially anisotropic interactions is calculated for the entire range of anisotropy, from decoupled planes to the isotropic model to the solid-on-solid model, using hard-spin mean-field theory. The phase diagram contains the line of ordering phase transitions and, at lower temperatures, the line of roughening phase transitions, where the interface between ordered domains roughens. Upon increasing the anisotropy, roughening transition temperatures settle after the isotropic case, whereas the ordering transition temperature increases to infinity. The calculation is repeated for the d=2 Ising model for the full range of anisotropy, yielding no roughening transition.

Monte Carlo simulation of antiphase boundaries and growth of antiphase domains in Al5Ti3 phase in Al-rich γ-TiAl intermetallics

Predominantly two kinds of antiphase boundaries (APBs) form in Al5Ti3, which is an Al-rich ordered derivative of the γ-TiAl (L10) phase. This phase can be viewed as a periodic arrangement of lean rhombs and squares on the Ti-rich (002) planes of the tetragonal L10. Energies of the two types of APBs were varied in a Monte Carlo simulation by suitably changing the pair interaction parameters. APBs of both types form boundaries of Al5Ti3 antiphase domains (APDs), which coarsen with time. An important observation in this regard is that mostly facetted APBs form at lower ageing temperatures, whereas curved APBs appear to form at relatively higher ageing temperatures. The findings of this work suggest that there exists a critical temperature, akin to the roughening transition temperature for crystals, that marks the transition from facetted to curved APBs.

Spectrum of facet crystallization waves at smooth 4He crystal surfaces

Wavelike crystallization and melting processes or crystallization waves are widely known to exist at 4He crystal surfaces in the rough state. Much less is known about crystallization waves for 4He crystal surfaces in the smooth, well-faceted state below the roughening transition temperature. Here we analyze the spectrum of facet crystallization waves and its dependence on wavelength, perturbation amplitude, and the number of possible facet steps distributed over a wavelength. All the distinctive features of facet crystallization waves compared to conventional waves at a rough surface result from a nonanalytic cusplike behavior in the angular dependence of the surface tension of smooth crystal facets.

Instability of a Crystal 4He Facet in the Field of Gravity

We analyze here the analog of the Rayleigh instability in the field of gravity for the superfluid–crystal 4He interface provided that the heavier 4He crystal phase occupies the space over the lighter superfluid phase. We find that the conditions and onset of the gravitational instability are different in kind above and below the roughening transition temperature when the crystal 4He surface is in the rough or in the smooth faceted state, respectively. In the rough state of the surface the gravitational instability is fully analogous to the classical case of the fluid–fluid interface. In contrast, in the case of the crystal faceted surface the onset of the gravitational instability is associated with surmounting some potential barrier. The potential barrier results from nonzero magnitude of the facet step energy. The size and the tilting angle of the crystal facet are also important parameters for developing the instability. The initial stage of the instability can be described as a generation of crystallization waves at the superfluid–crystal interface. In particular, we discuss the experiments (Demaria et al. in J. Low Temp. Phys. 89:385, 1992; Tsymbalenko in Fiz. Nizk. Temp. 21:162, 1995; Tsymbalenko in Low Temp. Phys. 21:120, 1995) which may concern the gravitational instability of the superfluid-crystal 4He interface.

Body-centered cubic phase in 3-arm star mesogens: a torsional tapping AFM and GISAXS study

The mode of liquid crystal (LC) self-assembly of asymmetric three-arm star oligobenzoate mesogens has been investigated by torsional tapping mode AFM imaging, and by bulk and grazing-incidence X-ray diffraction. It was confirmed that the cubic liquid crystal phase has Imm symmetry and established that it consists of spherical aggregates arranged on a body-centered lattice, rather than having a bicontinuous structure. Molecular simulation suggests that, in spite of their perceived rigidity, the oligobenzoate mesogens are folded, often sharply, within the supramolecular spheres, and that they act effectively as conical objects. The recently introduced torsional tapping AFM technique has allowed high resolution and contrast to be obtained from the soft mobile surface of the samples, showing the (110) plane of highly ordered supramolecular spheres. Notably the fine structure of the observed steps reveals that nearly isolated micelles still preserve their integrity. Finally, the equilibrium habits of the cubic LC droplets were shown by optical microscopy to be polyhedral with crystallographic facets, rather than spherical. This suggests that the air–LC interface is below the roughening transition temperature, which is attributed to the whole micelles rather than the isolated molecules acting as the interacting units.

Kinetic roughening of a Σ5 tilt grain boundary in SrTiO 3

The grain boundary kinetics of a SrTiO 3 bicrystal with a misorientation relationship of 36.8°/[1 0 0] (Σ5) is examined at elevated temperatures by in situ high-resolution transmission electron microscopy. At 973 K, the grain boundary migrates by the propagation of atomic steps with a height of half a unit cell (∼0.2 nm), but at 1173 K, it becomes rough and advances without the requirement of steps. All the temperatures examined are below the thermal roughening transition temperature of the grain boundary. These results give a clear indication of grain boundary kinetic roughening. We suggest that our present results may lead to the development of a unifying framework for previously suggested grain boundary migration mechanisms.

Continuum model for the long-range elastic interaction on stepped epitaxial surfaces in2+1dimensions

In heteroepitaxy, the mismatch of lattice constants between the crystal film and the substrate causes misfit strain and stress in the bulk of the film, driving the surface of the film to self-organize into various nanostructures. Below the roughening transition temperature, an epitaxial surface consists of facets and steps and changes its morphology by lateral motion of steps. In this paper, we present a $2+1$-dimensional continuum model for the long-range elastic interaction on stepped surface of a strained film. The continuum model is derived rigorously from the discrete model for the interaction between steps; thus it incorporates the discrete features of the stepped surfaces. Examples show that our continuum model is much more accurate as an approximation to the discrete model than the traditional continuum approximation. Moreover, in the linear instability of a planar surface, our continuum model gives the transition from step bunching instability to step undulation instability as the distance between adjacent steps increases, which agrees with the experimental observations and the results of discrete models and is missing using the traditional continuum approximation. Numerical simulations of the surface evolution using our model in the nonlinear regime show several different surface morphologies, including the morphology of step bunching which cannot be obtained using the traditional continuum approximation.

Derivation of a Continuum Model for the Long-Range Elastic Interaction on Stepped Epitaxial Surfaces in $2+1$ Dimensions

In heteroepitaxy, the mismatch of lattice constants in the crystal film and the substrate causes a misfit stress in the bulk of the film, driving the self-organization of the film surface into various nanostructures. Below roughening transition temperature, the epitaxial surface consists of terraces separated by atomic-height steps, and the misfit results in a long-range elastic interaction between surface steps. In this case, the surface morphology is determined by the motion of the steps, and the widely used continuum models for surface evolution above the roughening transition temperature do not apply directly. In this paper, we present a continuum model for this long-range elastic interaction on a stepped heteroepitaxial surface in $2+1$ dimensions. The continuum model is derived rigorously by taking the continuum limit from the discrete model for the interaction between steps, thus incorporating the discrete features of the stepped surface.