Calculation of Equilibrium Island Morphologies for Strained Epitaxial Systems

1996 ◽  
Vol 436 ◽  
Author(s):  
R. V. Kukta ◽  
L. B. Freund
Keyword(s):  
Free Energy ◽  
Stress Distribution ◽  
Aspect Ratio ◽  
Misfit Dislocation ◽  
Chemical Potential ◽  
Equilibrium Shape ◽  
Two Dimensional ◽  
Surface Chemical ◽  
Thick Substrate ◽  
Lower Height

AbstractStrained islands grown coherently on a relatively thick substrate with similar elastic properties are considered within the framework of continuum mechanics. The condition of uniform surface chemical potential is imposed to calculate two-dimensional equilibrium island shapes. The stress distribution in the equilibrium islands is shown to be highly nonhomogeneous. The effects of introducing a single misfit dislocation at the island-substrate interface are considered. It is found that there is critical island volume above which a dislocation decreases the total free energy of the system. The dislocation alters the stress distribution in the island, causing the island to relax via mass transport to an equilibrium shape with a lower height-to-width aspect ratio and a smaller surface chemical potential than the island prior to the introduction of the dislocation.

Equilibrium Shapes of Small Strained Islands

1995 ◽  
Vol 399 ◽  
Author(s):  
Brian J. Spencer ◽  
J. Tersoff
Keyword(s):  
Energy Density ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Chemical Potential ◽  
Equilibrium Shape ◽  
Two Dimensional ◽  
Asymptotic Results ◽  
Island Size ◽  
Equilibrium Shapes ◽  
Equilibrium Morphology

ABSTRACTWe calculate the equilibrium morphology of a strained layer, for the case where it wets the substrate (Stranski-Krastonow growth). Assuming isotropic surface energy and equal elastic constants in the film and substrate, we are able to calculate two-dimensional equilibrium shapes as a function of the island size and spacing. We present asymptotic results for the equilibrium shape of a thin island where the island height is much smaller than the island width. We also present numerical results of the full equations to describe the island shape when the islands are widely separated. From these solutions we are able to determine the chemical potential of the island as a function of island volume and the strain energy density along the surface of the island for small to medium-sized islands.

Observations on the Mechanics of Strained Epitaxial Island Growth

1995 ◽  
Vol 399 ◽  
Author(s):  
L. B. Freund ◽  
H. T. Johnson ◽  
R. V. Kukta
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Energy ◽  
Aspect Ratio ◽  
Misfit Dislocation ◽  
Strain Distribution ◽  
Strain Relaxation ◽  
Lattice Parameter ◽  
Island Growth ◽  
Numerical Finite Element

ABSTRACTAn epitaxial material island which has a lattice parameter differing by a small amount for that of its substrate is considered within the framework of continuum mechanics. The strain distribution in the island is determined for a range of aspect ratio, taking into account the compliance of the substrate. It is demonstrated that the total free energy of a strained island is minimum for some value of aspect ratio, and that this value depends on the volume of the island. To consider strain relaxation, the nucleation of a dislocation at the edge of a strained island is examined and the equilibrium aspect ratio of a dislocated island is computed. In particular, it is shown that an island can reduce its free energy by reducing its aspect ratio and, simultaneously, forming an interface misfit dislocation. The simulations are based on the numerical finite element method.

scholarly journals Analysis of Two-Layered Random Interfaces for Two Dimensional Widom-Rowlinson's Model

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Jun Wang ◽  
Jiguang Shao ◽  
Bingtuan Wang
Keyword(s):  
Free Energy ◽  
Chemical Potential ◽  
Probability Distributions ◽  
Random Phase ◽  
Polymer Chains ◽  
Two Dimensional ◽  
Cluster Expansions ◽  
Coexisting Phases ◽  
Random Interfaces

The statistical behaviors of two-layered random-phase interfaces in two-dimensional Widom-Rowlinson's model are investigated. The phase interfaces separate two coexisting phases of the lattice Widom-Rowlinson model; when the chemical potentialμof the model is large enough, the convergence of the probability distributions which describe the fluctuations of the phase interfaces is studied. In this paper, the backbones of interfaces are introduced in the model, and the corresponding polymer chains and cluster expansions are developed and analyzed for the polymer weights. And the existence of the free energy for two-layered random-phase interfaces of the two-dimensional Widom-Rowlinson model is given.

The Departure from Equilibrium of Turbulent Boundary Layers

1968 ◽  
Vol 19 (1) ◽  
pp. 1-19 ◽  
Author(s):  
H. McDonald
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Kinetic Energy ◽  
Stress Distribution ◽  
Boundary Layers ◽  
Turbulent Boundary Layers ◽  
Two Dimensional ◽  
Pressure Distributions ◽  
Momentum Integral ◽  
State Examination

SummaryRecently two authors, Nash and Goldberg, have suggested, intuitively, that the rate at which the shear stress distribution in an incompressible, two-dimensional, turbulent boundary layer would return to its equilibrium value is directly proportional to the extent of the departure from the equilibrium state. Examination of the behaviour of the integral properties of the boundary layer supports this hypothesis. In the present paper a relationship similar to the suggestion of Nash and Goldberg is derived from the local balance of the kinetic energy of the turbulence. Coupling this simple derived relationship to the boundary layer momentum and moment-of-momentum integral equations results in quite accurate predictions of the behaviour of non-equilibrium turbulent boundary layers in arbitrary adverse (given) pressure distributions.

scholarly journals A 4d $$ \mathcal{N} $$ = 1 Cardy Formula

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Joonho Kim ◽  
Seok Kim ◽  
Jaewon Song
Keyword(s):  
Black Holes ◽  
Free Energy ◽  
Asymptotic Behavior ◽  
High Temperature ◽  
Gauge Theory ◽  
Chemical Potential ◽  
Temperature Limit ◽  
Superconformal Index ◽  
High Temperature Limit ◽  
Cardy Formula

Abstract We study the asymptotic behavior of the (modified) superconformal index for 4d $$ \mathcal{N} $$ N = 1 gauge theory. By considering complexified chemical potential, we find that the ‘high-temperature limit’ of the index can be written in terms of the conformal anomalies 3c − 2a. We also find macroscopic entropy from our asymptotic free energy when the Hofman-Maldacena bound 1/2 < a/c < 3/2 for the interacting SCFT is satisfied. We study $$ \mathcal{N} $$ N = 1 theories that are dual to AdS5 × Yp,p and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

Anisotropic misfit dislocation nucleation in two-dimensional grown InAs/GaAs(001) heterostructures

1998 ◽  
Vol 73 (8) ◽  
pp. 1074-1076 ◽  
Author(s):  
Achim Trampert ◽  
Klaus H. Ploog ◽  
Eric Tournié
Keyword(s):  
Misfit Dislocation ◽  
Dislocation Nucleation ◽  
Two Dimensional

Stars and stripes. Nanoscale misfit dislocation patterns on surfaces

2002 ◽  
Vol 74 (9) ◽  
pp. 1663-1671 ◽  
Author(s):  
Raghani Pushpa ◽  
Shobhana Narasimhan
Keyword(s):  
Surface Structure ◽  
Misfit Dislocation ◽  
Domain Walls ◽  
Chemical Potential ◽  
Model System ◽  
Misfit Dislocations ◽  
Face Centered Cubic ◽  
Hexagonal Close Packed ◽  
Structure Changes ◽  
Face Centered

Close-packed metal surfaces and heteroepitaxial systems frequently display a structure consisting of regularly spaced misfit dislocations, with a network of domain walls separating face-centered cubic (fcc) and hexagonal close-packed (hcp) domains. These structures can serve as templates for growing regularly spaced arrays of nanoislands. We present a theoretical investigation of the factors controlling the size and shape of the domains, using Pt(111) as a model system. Upon varying the chemical potential, the surface structure changes from being unreconstructed to the honeycomb, wavy triangles, "bright stars", or Moiré patterns observed experimentally on Pt(111) and other systems. For the particular case of Pt(111), isotropically contracted star-like patterns are favored over uniaxially contracted stripes.

scholarly journals The adsorption of vapours on mercury. I. Non-polar substances

1946 ◽  
Vol 187 (1008) ◽  
pp. 53-73 ◽  
Keyword(s):  
Free Energy ◽  
Total Energy ◽  
Langmuir Equation ◽  
Phase Changes ◽  
Two Dimensional ◽  
Surface Phase ◽  
Standard State ◽  
Spreading Pressure ◽  
Benzene Toluene ◽  
Entropy Of Adsorption

Reversible results for the adsorption of benzene, toluene and n -heptane vapours on mercury have been obtained. The films were found to be gaseous and obeyed the Volmer eqution F ( A - b ) = kT , where F = spreading pressure, A =area per molecule and b = co-area. The possibility that the films might be immobile was considered and the Langmuir equation was applied but found unsatisfactory. A standard state for the surface phase was defined and the free energy, total energy and entropy of adsorption evaluated. The heat of adsorption was shown to increase with the amount on the surface. A number of phase changes were found to occur after the completion of monolayer adsorp­tion, the most striking being interpreted as the change over from ‘flat’ to ‘vertical’ adsorp­tion of the toluene molecules. Others were thought to be either two-dimensional condensation or adsorption of a second layer.

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