Browsing by Author "Elder, K. R."

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  • Anomalous Fast Dynamics of Adsorbate Overlayers near an Incommensurate Structural Transition
    (2013) Granato, Enzo; Ying, S. C.; Elder, K. R.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We investigate the dynamics of a compressively strained adsorbed layer on a periodic substrate via a simple two-dimensional model that admits striped and hexagonal incommensurate phases. We show that the mass transport is superfast near the striped-hexagonal phase boundary and in the hexagonal phase. For an initial step profile separating a bare substrate region (or “hole”) from the rest of a striped incommensurate phase, the superfast domain wall dynamics leads to a bifurcation of the initial step profile into two interfaces or profiles propagating in opposite directions with a hexagonal phase in between. This yields a theoretical understanding of the recent experiments for the Pb/Si(111) system.
  • Dynamic scaling of out-of-plane fluctuations in freestanding graphene
    (2022-05-15) Granato, Enzo; Greb, Michelle; Elder, K. R.; Ying, S. C.; Ala-Nissila, T.
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study the dynamical behavior of the mean-square displacement of height fluctuations of freestanding graphene using a phase-field-crystal model introduced recently. We find that the dynamic scaling behavior obtained numerically at long times is well described by the scaling theory of polymerized membranes. The critical exponent characterizing the power-law increase with time depends only on the equilibrium roughening exponent ζ as α=ζ/(1+ζ). For sufficiently long times it crosses over to linear behavior for finite-size systems. The critical exponent α is in good agreement with the anomalous diffusion exponent observed experimentally in graphene, suggesting this is a property that could also be observable in other two-dimensional crystalline materials.
  • Dynamical transitions and sliding friction of the phase-field-crystal model with pinning
    (2010) Ramos, J. A. P.; Granato, E.; Ying, S. C.; Achim, C. V.; Elder, K. R.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal (PFC) model with pinning including both thermal fluctuations and inertial effects. The model provides a continuous description of adsorbed layers on a substrate under the action of an external driving force at finite temperatures, allowing for both elastic and plastic deformations. We derive general stochastic dynamical equations for the particle and momentum densities including both thermal fluctuations and inertial effects. The resulting coupled equations for the PFC model are studied numerically. At sufficiently low temperatures, we find that the velocity response of an initially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear response in the PFC model are similar to the results obtained previously with molecular dynamics simulations of particle models for adsorbed layers.
  • Dynamics of fluctuations and thermal buckling in graphene from a phase-field crystal model
    (2023-01-15) Granato, Enzo; Elder, K. R.; Ying, S. C.; Ala-Nissila, T.
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study the effects of thermal fluctuations and pinned boundaries in graphene membranes by using a phase-field crystal model with out-of-plane deformations. For sufficiently long times, the linear diffusive behavior of height fluctuations in systems with free boundaries becomes a saturation regime, while at intermediate times the behavior is still subdiffusive as observed experimentally. Under compression, we find mirror buckling fluctuations where the average height changes from above to below the pinned boundaries, with the average time between fluctuations diverging below a critical temperature corresponding to a thermally induced buckling transition. Near the transition, we find a nonlinear height response in agreement with recent renormalization-group calculations and observed in experiments on graphene membranes under an external transverse force with clamped boundaries.
  • Glassy phases and driven response of the phase-field-crystal model with random pinning
    (2011) Granato, E.; Ramos, J. A. P.; Achim, C. V.; Lehikoinen, J.; Ying, S. C.; Ala-Nissilä, Tapio; Elder, K. R.
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study the structural correlations and the nonlinear response to a driving force of a two-dimensional phase-field-crystal model with random pinning. The model provides an effective continuous description of lattice systems in the presence of disordered external pinning centers, allowing for both elastic and plastic deformations. We find that the phase-field crystal with disorder assumes an amorphous glassy ground state, with only short-ranged positional and orientational correlations, even in the limit of weak disorder. Under increasing driving force, the pinned amorphous-glass phase evolves into a moving plastic-flow phase and then, finally, a moving smectic phase. The transverse response of the moving smectic phase shows a vanishing transverse critical force for increasing system sizes.
  • Influence of dislocations in multilayer graphene stacks : A phase field crystal study
    (2024-10) Elder, K. R.; Huang, Zhi Feng; Ala-Nissila, T.
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    In this study, the influence of 5|7 dislocations in multilayer graphene stacks (up to six layers) is examined. The study is conducted using a recently developed phase-field crystal (PFC) model for multilayer systems incorporating out-of-plane deformations and parameterized to match to density functional theory calculations for graphene bilayers and other systems. The specific configuration considered consists of one monolayer containing four 5|7 dislocations (i.e., two dislocation dipoles) sandwiched between perfect graphene layers. This study reveals how the strain field from the dislocations in the defected layer leads to out-of-plane deformations, which in turn cause deformations of neighboring layers. Quantitative predictions are made for the defect-free energy of the multilayer stacks as compared to a defect-free system, which is shown to increase with the number of layers and system size. Furthermore, it is predicted that system defect energy saturates by roughly ten sheets in the stack, indicating the range of defect influence across the multilayer. Variations in stress field distribution and layer height profiles in different layers of the stack are also quantitatively identified.
  • Interface pinning in spontaneous imbibition
    (2001) Dubé, M.; Majaniemi, S.; Rost, M.; Alava, M. J.; Elder, K. R.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    Evaporation and gravity induced pinning in spontaneous imbibition are examined within a phase field formalism. Evaporation is introduced via a nonconserving term and gravity through a convective term that constrains the influx of liquid. Their effects are described by dimensionless coupling constants ε and g, respectively. From liquid conservation, the early time behavior of the average interface position follows H(t)∼t1/2 until a crossover time t*(g,ε). After that the pinning height Hp(g,ε) is approached exponentially in time, in accordance with mean field theory. The statistical roughness of the interface is described by an exponent χ≃1.25 at all stages of the rise, but the dynamic length scale controlling roughness crosses over from ξ×∼H1/2 to a time independent pinning length scale ξp(ε,g).
  • Mass transport of adsorbates near a discontinuous structural phase transition
    (2016-12-09) Granato, E.; Ying, S. C.; Elder, K. R.; Ala-Nissilä, Tapio
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study the mass transport dynamics of an adsorbed layer near a discontinuous incommensurate striped-honeycomb phase transition via numerical simulations of a coarse-grained model focusing on the motion of domain walls rather than individual atoms. Following an initial step profile created in the incommensurate striped phase, an intermediate hexagonal incommensurate phase nucleates and grows, leading to a bifurcation into two sharp profiles propagating in opposite directions as opposed to broad profiles induced by atomic diffusive motion. Our results are in agreement with recent numerical simulations of a microscopic model as well as experimental observations for the Pb/Si(111) adsorbate system.
  • Modeling buckling and topological defects in stacked two-dimensional layers of graphene and hexagonal boron nitride
    (2021-03-12) Elder, K. R.; Achim, C. V.; Heinonen, V.; Granato, E.; Ying, S. C.; Ala-Nissila, T.
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    In this paper, a two-dimensional phase field crystal model of graphene and hexagonal boron nitride (hBN) is extended to include out-of-plane deformations in stacked multilayer systems. As proof of principle, the model is shown analytically to reduce to standard models of flexible sheets in the small deformation limit. Applications to strained sheets, dislocation dipoles, and grain boundaries are used to validate the behavior of a single flexible graphene layer. For multilayer systems, parameters are obtained to match existing theoretical density functional theory calculations for graphene/graphene, hBN/hBN, and graphene/hBN bilayers. More precisely, it is shown that the parameters can be chosen to closely match the stacking energies and layer spacing calculated by Zhou et al. [Phys. Rev. B 92, 155438 (2015)PRBMDO1098-012110.1103/PhysRevB.92.155438]. Further validation of the model is presented in a study of rotated graphene bilayers and stacking boundaries. The flexibility of the model is illustrated by simulations that highlight the impact of complex microstructures in one layer on the other layer in a graphene/graphene bilayer.
  • Modeling self-organization of thin strained metallic overlayers from atomic to micron scales
    (2013) Elder, K. R.; Rossi, G.; Kanerva, P.; Sanches, F.; Ying, S-C.; Granato, E.; Achim, C. V.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    A computational study of the self-organization of heteroepitaxial ultrathin metal films is presented. By means of a continuum complex field model, the relationship of the equilibrium surface patterns of the film to the adsorbate-substrate adhesion energy, as well as to the mismatch between the adsorbate and the substrate bulk lattice parameters, are obtained in both the tensile and the compressive regimes. Our approach captures pattern periodicities over large length scales, up to several hundreds of nm, retaining atomistic resolution. Thus, the results can be directly compared with experimental data, in particular for systems such as Cu/Ru(0001) and Ag/Cu(111). Three nontrivial, stable superstructures for the overlayer, namely, stripe, honeycomb, and triangular, are identified that closely resemble those observed experimentally. Simulations in nonequilibrium conditions are performed as well to identify metastable structural configurations and the dynamics of ordering of the overlayer.
  • Moiré patterns and inversion boundaries in graphene/hexagonal boron nitride bilayers
    (2023-02) Elder, K. R.; Huang, Zhi Feng; Ala-Nissila, Tapio
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    In this paper a systematic examination of graphene/hexagonal boron nitride (g/hBN) bilayers is presented, through a recently developed two-dimensional phase field crystal model that incorporates out-of-plane deformations. The system parameters are determined by closely matching the stacking energies and heights of g/hBN bilayers to those obtained from existing quantum-mechanical density functional theory calculations. Out-of-plane deformations are shown to reduce the energies of inversion domain boundaries in hBN, and the coupling between graphene and hBN layers leads to a bilayer defect configuration consisting of an inversion boundary in hBN and a domain wall in graphene. Simulations of twisted bilayers reveal the structure, energy, and elastic properties of the corresponding moiré patterns and show a crossover as the misorientation angle between the layers increases from a well-defined hexagonal network of domain boundaries and junctions to smeared-out patterns. The transition occurs when the thickness of domain walls approaches the size of the moiré patterns and coincides with the peaks in the average von Mises and volumetric stresses of the bilayer.
  • Nonlinear driven response of a phase-field crystal in a periodic pinning potential
    (2009) Achim, C. V.; Ramos, J. A. P.; Karttunen, M.; Elder, K. R.; Granato, E.; Ala-Nissilä, Tapio; Ying, S. C.
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study numerically the phase diagram and the response under a driving force of the phase field crystal model for pinned lattice systems introduced recently for both one- and two-dimensional systems. The model describes the lattice system as a continuous density field in the presence of a periodic pinning potential, allowing for both elastic and plastic deformations of the lattice. We first present results for phase diagrams of the model in the absence of a driving force. The nonlinear response to a driving force on an initially pinned commensurate phase is then studied via overdamped dynamic equations of motion for different values of mismatch and pinning strengths. For large pinning strength the driven depinning transitions are continuous, and the sliding velocity varies with the force from the threshold with power-law exponents in agreement with analytical predictions. Transverse depinning transitions in the moving state are also found in two dimensions. Surprisingly, for sufficiently weak pinning potential we find a discontinuous depinning transition with hysteresis even in one dimension under overdamped dynamics. We also characterize structural changes of the system in some detail close to the depinning transition.
  • Patterning of Heteroepitaxial Overlayers from Nano to Micron Scales
    (2012) Elder, K. R.; Rossi, G.; Kanerva, P.; Sanches, F.; Ying, S-C.; Granato, E.; Achim, C. V.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    Thin heteroepitaxial overlayers have been proposed as templates to generate stable, self-organized nanostructures at large length scales, with a variety of important technological applications. However, modeling strain-driven self-organization is a formidable challenge due to different length scales involved. In this Letter, we present a method for predicting the patterning of ultrathin films on micron length scales with atomic resolution. We make quantitative predictions for the type of superstructures (stripes, honeycomb, triangular) and length scale of pattern formation of two metal-metal systems, Cu on Ru(0001) and Cu on Pd(111). Our findings are in excellent agreement with previous experiments and call for future experimental investigations of such systems.
  • Phase diagram and commensurate-incommensurate transitions in the phase field crystal model with an external pinning potential
    (2006) Achim, C. V.; Karttunen, M.; Elder, K. R.; Granato, E.; Ala-Nissilä, Tapio; Ying, S. C.
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study the phase diagram and the commensurate-incommensurate transitions in a phase field model of a two-dimensional crystal lattice in the presence of an external pinning potential. The model allows for both elastic and plastic deformations and provides a continuum description of lattice systems, such as for adsorbed atomic layers or two-dimensional vortex lattices. Analytically, a mode expansion analysis is used to determine the ground states and the commensurate-incommensurate transitions in the model as a function of the strength of the pinning potential and the lattice mismatch parameter. Numerical minimization of the corresponding free energy shows reasonable agreement with the analytical predictions and provides details on the topological defects in the transition region. We find that for small mismatch the transition is of first order, and it remains so for the largest values of mismatch studied here. Our results are consistent with results of simulations for atomistic models of adsorbed overlayers.
  • Phase diagram of pinned lattices in the phase field crystal model
    (2008) Achim, C. V.; Karttunen, M.; Elder, K. R.; Granato, E.; Ala-Nissilä, Tapio; Ying, S. C.
     School of Science | A4 Artikkeli konferenssijulkaisussa
    We study the phase diagram and the commensurate-incommensurate phase transitions of a two-dimensional phase field crystal model for adsorbed layers. The model allows for both elastic and plastic deformations on atomic and diffusive time-scales, and provides a continuum description of lattice systems, such as adsorbed atomic layers or two-dimensional vortex lattices. Analytically, mode expansion analysis and numerical minimization of the free energy are used to determine the ground states as a function of the pinning potential and lattice mismatch parameter. The results show a rich phase diagram with several different types of commensurate and incommensurate phases.
  • Phase-field-crystal models and mechanical equilibrium
    (2014) Heinonen, V.; Achim, C. V.; Elder, K. R.; Buyuddagli, S.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    Phase-field-crystal (PFC) models constitute a field theoretical approach to solidification, melting, and related phenomena at atomic length and diffusive time scales. One of the advantages of these models is that they naturally contain elastic excitations associated with strain in crystalline bodies. However, instabilities that are diffusively driven towards equilibrium are often orders of magnitude slower than the dynamics of the elastic excitations, and are thus not included in the standard PFC model dynamics. We derive a method to isolate the time evolution of the elastic excitations from the diffusive dynamics in the PFC approach and set up a two-stage process, in which elastic excitations are equilibrated separately. This ensures mechanical equilibrium at all times. We show concrete examples demonstrating the necessity of the separation of the elastic and diffusive time scales. In the small-deformation limit this approach is shown to agree with the theory of linear elasticity.
  • Striped, honeycomb, and twisted moiré patterns in surface adsorption systems with highly degenerate commensurate ground states
    (2017-11-29) Elder, K. R.; Achim, C. V.; Granato, E.; Ying, S. C.; Ala-Nissila, T.
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    Atomistically thin adsorbate layers on surfaces with a lattice mismatch display complex spatial patterns and ordering due to strain-driven self-organization. In this work, a general formalism to model such ultrathin adsorption layers that properly takes into account the competition between strain and adhesion energy of the layers is presented. The model is based on the amplitude expansion of the two-dimensional phase field crystal (PFC) model, which retains atomistic length scales but allows relaxation of the layers at diffusive time scales. The specific systems considered here include cases where both the film and the adsorption potential can have either honeycomb (H) or triangular (T) symmetry. These systems include the so-called (1×1), (3×3)R30â, (2×2), (7×7)R19.1â, and other higher order states that can contain a multitude of degenerate commensurate ground states. The relevant phase diagrams for many combinations of the H and T systems are mapped out as a function of adhesion strength and misfit strain. The coarsening patterns in some of these systems is also examined. The predictions are in good agreement with existing experimental data for selected strained ultrathin adsorption layers.
  • Thermal buckling transition in graphene : Static and dynamical critical exponents
    (2025-01) Granato, Enzo; Elder, K. R.; Ying, S. C.; Ala-Nissila, T.
     A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study numerically the thermal buckling transition in graphene membranes under compressive strain and clamped boundaries employing an atomistic quasiharmonic model. The numerical simulations combine three different Monte Carlo methods, local moves, collective wave moves and parallel tempering. We determine the static and the dynamical critical exponents by finite-size scaling, and the nonlinear response to a transverse force near the transition. The correlation length exponent and the nonlinear response are in good agreement with recent renormalization-group calculations of elastic membranes. Despite the applied strain, we find a dynamical critical exponent and diffusion exponent of height fluctuations at the transition close to the value for freestanding graphene, z=2(1+ζ) and α=ζ/(ζ+1), where ζ is the static roughening exponent, as obtained in a recent study with a phase-field crystal model.
  • Thermal fluctuations and phase diagrams of the phase-field crystal model with pinning
    (2008) Ramos, J. A. P.; Granato, E.; Achim, C. V.; Ying, S. C.; Elder, K. R.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We study the influence of thermal fluctuations in the phase diagram of a recently introduced two-dimensional phase field crystal model with an external pinning potential. The model provides a continuum description of pinned lattice systems allowing for both elastic deformations and topological defects. We introduce a nonconserved version of the model and determine the ground-state phase diagram as a function of lattice mismatch and strength of the pinning potential. Monte Carlo simulations are used to determine the phase diagram as a function of temperature near commensurate phases. The results show a rich phase diagram with commensurate, incommensurate, and liquidlike phases with a topology strongly dependent on the type of ordered structure. A finite-size scaling analysis of the melting transition for the c(2×2) commensurate phase shows that the thermal correlation length exponent ν and specific heat behavior are consistent with the Ising universality class as expected from analytical arguments.
  • Thermodynamics of bcc metals in phase-field-crystal models
    (2009) Jaatinen, A.; Achim, C. V.; Elder, K. R.; Ala-Nissilä, Tapio
     School of Science | A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
    We examine the influence of different forms of the free-energy functionals used in the phase-field-crystal (PFC) model, and compare them with the second-order density-functional theory (DFT) of freezing, by using bcc iron as an example case. We show that there are large differences between the PFC and the DFT and it is difficult to obtain reasonable parameters for existing PFC models directly from the DFT. Therefore, we propose a way of expanding the correlation function in terms of gradients that allows us to incorporate the bulk modulus of the liquid as an additional parameter in the theory. We show that this functional reproduces reasonable values for both bulk and surface properties of bcc iron, and therefore it should be useful in modeling bcc materials. As a further demonstration, we also calculate the grain boundary energy as a function of misorientation for a symmetric tilt boundary close to the melting transition.