254 research outputs found
Nonlinear effects for island coarsening and stabilization during strained film heteroepitaxy
Nonlinear evolution of three-dimensional strained islands or quantum dots in
heteroepitaxial thin films is studied via a continuum elasticity model and the
development of a nonlinear dynamic equation governing the film morphological
profile. All three regimes of island array evolution are identified and
examined, including a film instability regime at early stage, a nonlinear
coarsening regime at intermediate times, and the crossover to a saturated
asymptotic state, with detailed behavior depending on film-substrate misfit
strains but not qualitatively on finite system sizes. The phenomenon of island
stabilization and saturation, which corresponds to the formation of steady but
non-ordered arrays of strained quantum dots, occurs at later time for smaller
misfit strain. It is found to be controlled by the strength of film-substrate
wetting interaction which would constrain the valley-to-peak mass transport and
hence the growth of island height, and also determined by the effect of elastic
interaction between surface islands and the high-order strain energy of
individual islands at late evolution stage. The results are compared to
previous experimental and theoretical studies on quantum dots coarsening and
saturation.Comment: 19 pages, 12 figures; submitted to Phys. Rev.
Some remarks on stability for a phase-field model with memory
The phase field system with memory can be viewed as a phenomenological extension of the classical phase equations in which memory effects have been taken into account in both fields. Such memory effects could be important for example during phase transition in polymer melts in the proximity of the glass transition temperature where configurational degrees of freedom in the polymer melt constitute slowly relaxing "internal modes" which are di±cult to model explicitly. They should be relevant in particular to glass-liquid-glass transitions where re-entrance effects have been recently reported [27]. We note that in numerical studies based on sharp interface equations obtained from (PFM), grains have been seen to rotate as they shrink [35, 36]. While further modelling and numerical efforts are now being undertaken, the present manuscript is devoted to strengthening the analytical underpinnings of the model
Finite-time Singularities in Surface-Diffusion Instabilities are Cured by Plasticity
A free material surface which supports surface diffusion becomes unstable
when put under external non-hydrostatic stress. Since the chemical potential on
a stressed surface is larger inside an indentation, small shape fluctuations
develop because material preferentially diffuses out of indentations. When the
bulk of the material is purely elastic one expects this instability to run into
a finite-time cusp singularity. It is shown here that this singularity is cured
by plastic effects in the material, turning the singular solution to a regular
crack.Comment: 4 pages, 3 figure
Influence of uniaxial stress on the lamellar spacing of eutectics
Directional solidification of lamellar eutectic structures submitted to
uniaxial stress is investigated. In the spirit of an approximation first used
by Jackson and Hunt, we calculate the stress tensor for a two-dimensional
crystal with triangular surface, using a Fourier expansion of the Airy
function. crystal with triangular surface in contact with its melt, given that
a uniaxial external stress is applied. The effect of the resulting change in
chemical potential is introduced into the standard model for directional
solidification of a lamellar eutectic. This calculation is motivated by an
observation, made recently [I. Cantat, K. Kassner, C. Misbah, and H.
M\"uller-Krumbhaar, Phys. Rev. E, in press] that the thermal gradient produces
similar effects as a strong gravitational field in the case of dilute-alloy
solidification. Therefore, the coupling between the Grinfeld and the
Mullins-Sekerka instabilities becomes strong, as the critical wavelength of the
former instability gets reduced to a value close to that of the latter.
Analogously, in the case of eutectics, the characteristic length scale of the
Grinfeld instability should be reduced to a size not extremely far from typical
lamellar spacings. In a Jackson-Hunt like approach we average the undercooling,
including the stress term, over a pair of lamellae. Following Jackson and Hunt,
we assume the selected wavelength to be determined by the minimum undercooling
criterion and compute its shift due to the external stress. we realize the
shifting of the wavelength by the application of external stress. In addition,
we find that in general the volume fraction of the two solid phases is changed
by uniaxial stress. Implications for experiments on eutectics are discussed.Comment: 8 pages RevTex, 6 included ps-figures, accepted for Phys. Rev.
Interplay of internal stresses, electric stresses and surface diffusion in polymer films
We investigate two destabilization mechanisms for elastic polymer films and
put them into a general framework: first, instabilities due to in-plane stress
and second due to an externally applied electric field normal to the film's
free surface. As shown recently, polymer films are often stressed due to
out-of-equilibrium fabrication processes as e.g. spin coating. Via an
Asaro-Tiller-Grinfeld mechanism as known from solids, the system can decrease
its energy by undulating its surface by surface diffusion of polymers and
thereby relaxing stresses. On the other hand, application of an electric field
is widely used experimentally to structure thin films: when the electric
Maxwell surface stress overcomes surface tension and elastic restoring forces,
the system undulates with a wavelength determined by the film thickness. We
develop a theory taking into account both mechanisms simultaneously and discuss
their interplay and the effects of the boundary conditions both at the
substrate and the free surface.Comment: 14 pages, 7 figures, 1 tabl
Distributional fixed point equations for island nucleation in one dimension: a retrospective approach for capture zone scaling
The distributions of inter-island gaps and captures zones for islands
nucleated on a one-dimensional substrate during submonolayer deposition are
considered using a novel retrospective view. This provides an alternative
perspective on why scaling occurs in this continuously evolving system.
Distributional fixed point equations for the gaps are derived both with and
without a mean field approximation for nearest neighbour gap size correlation.
Solutions to the equations show that correct consideration of fragmentation
bias justifies the mean field approach which can be extended to provide
closed-from equations for the capture zones. Our results compare favourably to
Monte Carlo data for both point and extended islands using a range of critical
island size . We also find satisfactory agreement with theoretical
models based on more traditional fragmentation theory approaches.Comment: 9 pages, 7 figures and 1 tabl
Orientation dependence of the elastic instability on strained SiGe films
At low strain, SiGe films on Si substrates undergo a continuous
nucleationless morphological evolution known as the Asaro-Tiller-Grinfeld
instability. We demonstrate experimentally that this instability develops on
Si(001) but not on Si(111) even after long annealing. Using a continuum
description of this instability, we determine the origin of this difference.
When modeling surface diffusion in presence of wetting, elasticity and surface
energy anisotropy, we find a retardation of the instability on Si(111) due to a
strong dependence of the instability onset as function of the surface
stiffness. This retardation is at the origin of the inhibition of the
instability on experimental time scales even after long annealing.Comment: 3 pages, 4 figure
Asymptotics of capture zone distributions in a fragmentation-based model of submonolayer deposition
We consider the asymptotics of the distribution of the capture zones associated with the islands nucleated during submonolayer deposition onto a one-dimensional substrate. We use a convolution of the distribution of inter-island gaps, the asymptotics of which is known for a class of nucleation models, to derive the asymptotics for the capture zones. The results are in broad agreement with published Monte Carlo simulation data (O'Neill et al., 2012) [13]
Morphological instability, evolution, and scaling in strained epitaxial films: An amplitude equation analysis of the phase field crystal model
Morphological properties of strained epitaxial films are examined through a
mesoscopic approach developed to incorporate both the film crystalline
structure and standard continuum theory. Film surface profiles and properties,
such as surface energy, liquid-solid miscibility gap and interface thickness,
are determined as a function of misfit strains and film elastic modulus. We
analyze the stress-driven instability of film surface morphology that leads to
the formation of strained islands. We find a universal scaling relationship
between the island size and misfit strain which shows a crossover from the
well-known continuum elasticity result at the weak strain to a behavior
governed by a "perfect" lattice relaxation condition. The strain at which the
crossover occurs is shown to be a function of liquid-solid interfacial
thickness, and an asymmetry between tensile and compressive strains is
observed. The film instability is found to be accompanied by mode coupling of
the complex amplitudes of the surface morphological profile, a factor
associated with the crystalline nature of the strained film but absent in
conventional continuum theory.Comment: 16 pages, 10 figures; to be published in Phys. Rev.
Phase Field Modeling of Fracture and Stress Induced Phase Transitions
We present a continuum theory to describe elastically induced phase
transitions between coherent solid phases. In the limit of vanishing elastic
constants in one of the phases, the model can be used to describe fracture on
the basis of the late stage of the Asaro-Tiller-Grinfeld instability. Starting
from a sharp interface formulation we derive the elastic equations and the
dissipative interface kinetics. We develop a phase field model to simulate
these processes numerically; in the sharp interface limit, it reproduces the
desired equations of motion and boundary conditions. We perform large scale
simulations of fracture processes to eliminate finite-size effects and compare
the results to a recently developed sharp interface method. Details of the
numerical simulations are explained, and the generalization to multiphase
simulations is presented
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