Some Aspects of Surface Diffusion Martin Burger Institut für Numerische und Angewandte Mathematik,...

Some Aspects of Surface Diffusion

Martin Burger

Institut für Numerische und Angewandte Mathematik, Center for Nonlinear Science CeNoS

Westfälische Willhelms-Universität Münster


Erlangen, February 2007 2

Outline Introduction: Motivation, Applications of Surface Diffusion

Strong anisotropies: Including strong anisotropies, curvature regularization, equilibria, dynamics, numerical simulation

Adatom diffusion: Change from 4th order to 2nd order system, change of equilibria, numerical simulation

Chemotaxis: limiting behaviour of packed cell densities



Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn)

Christian Schmeiser, Yasmin Dolak-Struss (Universität Wien)

Collaborations



Surface diffusion processes appear in various materials science applications, in particular in the (self-assembled) growth of nanostructures

Schematic description: particles are deposited on a surface and become adsorbed (adatoms). They diffuse around the surface and can be bound to the surface. Vice versa, unbinding and desorption happens.

Introduction



Various fundamental surface growth mechanisms can determine the dynamics, most important:

- Attachment / Detachment of atoms to / from surfaces

- Diffusion of adatoms on surfaces

Growth Mechanisms



Other effects influencing dynamics:- Anisotropy

- Bulk diffusion of atoms (phase separation)

- Exchange of atoms between surface and bulk

- Elastic Relaxation in the bulk

- Surface Stresses

Growth Mechanisms



Other effects influencing dynamics:

- Deposition of atoms on surfaces

- Effects induced by electromagnetic forces (Electromigration)

Growth Mechanisms



Isotropic Surface Diffusion Simple model for surface diffusion in the isotropic case:

Normal motion of the surface by minus surfaceLaplacian of mean curvature

Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity (ask Harald Garcke)



Level Set / Graph Formulation Level set function or graph parametrization u of surface determined from

-

(graph)

(level set)

@tu = ¡ div(P r · )

· = div( r uQ)

P =Q(I r uQ

r uQ)

Q=p1+ jr uj2

Q = jr uj



Level Set Formulation We have to deal with fourth-order equation, no maximum principle

No global level set formulation

Efficient computations and proofs still widely open (One of the „major mathematical challenges in materials science“, Jean Taylor, AMS, 2002 / Robert Kohn, SIAM, 2002)



Applications: Nanostructures SiGe/Si Quantum Dots

Bauer et. al. 99



Applications: Nanostructures SiGe/Si Quantum Dots



Applications: NanostructuresInAs/GaAs Quantum Dots



Applications: Nano / Micro Electromigration of voids in electrical circuits

Nix et. Al. 92



Applications: Macro Formation of Basalt Columns:

Giant‘s Causeway

Panska Skala (Northern Ireland) (Czech Republic)

See: http://physics.peter-kohlert.de/grinfeld.htmld



The energy of the system is composed of various terms:

Total Energy =

(Anisotropic) Surface Energy +

(Anisotropic) Elastic Energy +

Compositional Energy +

.....

We start with first term only

Energy



Surface energy is given by

Standard model for surface free energy

Surface Energy



Chemical potential is the change of energy when adding / removing single atoms

In a continuum model, the chemical potential can be represented as a surface gradient of the energy (obtained as the variation of total energy with respect to the surface)

For surfaces represented by a graph, the chemical potential is the functional derivative of the energy

Chemical Potential



Surface diffusion appears in many important applications - in particular in material and nano science

Growth of a surface with velocity

Surface Diffusion



F ... Deposition flux Ds .. Diffusion coefficient... Atomic volume ... Surface density k ... Boltzmann constant T ... Temperature n ... Unit outer normal ... Chemical potential =

energy variation

Surface Diffusion



Isotropic / Weakly Anisotropic:- Existence results Elliott-Garcke 1996

- Numerical simulation Bänsch-Morin-Nocchetto 2003, Deckelnick-Dziuk-Elliott 2004

Anisotropic: - Studies of equilibrium structures, Gurtin 1993, Spencer 2003, Cecil-Osher 2004

- Numerical simulation of asymptotic model (obtained from long-wave expansion), Golovin-Davies-Nepomnyaschy 2002 / 2003

Analysis and Simulation



In several situations, the surface free energy (respectively its one-homogeneous extension) is not convex. Nonconvex energies can result from different reasons:

- Special materials with strong anisotropy: Gjostein 1963, Cahn-Hoffmann1974

- Strained Vicinal Surfaces: Shenoy-Freund 2003

Surface Energy



Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004)

Surface Energy



In order to regularize problem (and possibly since higher order terms become important in atomistic homogenization), curvature regularization has beeen proposed by several authors (DiCarlo-Gurtin-Podio-Guidugli 1993, Gurtin-Jabbour 2002, Tersoff, Spencer, Rastelli, Von Kähnel 2003)

Curvature Regularization



Cubic anisotropy

surface energy becomes non-convex for

> 1/3

- Faceting of the surface

- Microstructure possible without curvature term

- Equilibria are local energy minimizers only

Anisotropic Surface energy°0(n) = 1+²

Pn4j



We obtain

Energy variation corresponds to fourth-order term (due to curvature variation)

Chemical Potential



Derivative

with matrix

Curvature Term



SD can be obtained as the limit ( →0) of minimization

subject to

Minimizing Movement: SD



Level set version:

subject to

Minimizing Movement: SD



Basic idea: Semi-implicit time discretization +

Splitting into two / three second-order equations +

Finite element discretization in space

Natural variables for splitting:

Height u, Mean Curvature , Chemical potential (denoted u, v, w in the following)

Numerical Solution



Based on variational principle, minimizing movement

subject to

Time Discretization



Quadratic approximation of the convex terms in the energy, linear approximation of the non-convex terms around u(t)

Rewrite local variational problem as minimization over u, v, and w With constraints defining v and w

KKT condition yields indefinite linear system,

Lagrangian variables are multiples of v and w

Time Discretization



Discretization of the variational problem in space by piecewise linear finite elements

and P(u) are piecewise constant on the triangularization, all integrals needed for stiffness matrix and right-hand side can be computed exactly

Spatial Discretization



After few manipulations we obtain indefinite linear system for the nodal values

A stiffness matrix from diffusion coefficient 1/Q B stiffness matrix from diffusion coefficient P/Q M mass matrix for identity, C mass matrix for 1/Q

Iterative solution by multigrid-precond. GMRES

Discrete Problem



SD = 3.5, = 0.02,



SD = 1.5, = 0.02,



Faceting Graph Simulation: mb JCP 04, Level Set Simulation: mb-Hausser-Stöcker-Voigt 06

Adaptive FE grid around zero level set



Faceting Anisotropic mean curvature flow



Faceting of Thin Films Anisotropic

Mean Curvature

Anisotropic Surface Diffusion

mb 04, mb-Hausser-

Stöcker-Voigt-05



Faceting of Bulk Crystals Anisotropic surface diffusion



Standard surface diffusion models have some strange aspects, in particular for nanostructures / epitaxy: - No kinetic effects

- Problems with topology change (atoms can only hop on single surface, not on a second one, even for small distances)- They do not correspond to the atomistic picture

Modelling Aspects



Standard Description (e.g. Pimpinelli-Villain):

- (Free) Adatoms hop on surfaces

- Coupled with attachment detachment kinetics

for the surface atoms on a crystal lattice

Atomistic Models on (Nano-)Surfaces



From Caflisch et. Al. 1999

Atomistic Models on (Nano-)Surfaces



Need two equations for two coupled processes

Need diffusion equation for adatoms

Modelling



Explicit model for surface diffusion including adatoms Fried-Gurtin 2004, mb 2006 Adatom density , chemical potential , normal velocity V, tangential velocity v, mean curvature , bulk density

Kinetic coefficient b, diffusion coefficient L, deposition term r

Modelling



Surface free energy is a function of the adatom density Chemical potential is the free energy variation

Surface energy:

Surface Free Energy



Relation to standard surface diffusion: convergence as the cost of free adatoms (in the surface free energy tends to infinity)

Modelling



Equilibrium shapes minimize the surface energy

at constant mass

Equilibrium Shapes



Equilibrium films: minimum at vanishing adatom density, flat surface. Same as without adatoms.

Equilibrium crystals: Wulff shape with vanishing adatom density is NEVER an equilibrium ! Isotropic equilibrium has nonzero adatom density and smaller radius than Wulff shape

Equilibrium Shapes



Model free energy

Parameter measures the cost of free adatoms

Equilibrium Crystals (Isotropic)



Equilibrium radius

Equilibrium Crystals (Isotropic)



Different regimes for surface energy:

- Convex for small adatom densities and shapes close to equilibrium

- Nonconvex for large adatom densities and shapes far away from equilibrium. The surface energy is consequently not lower semicontinuous

Surface Energy



Numerical Simulation



Numerical Simulation Flat initial shape, nonhomogeneous deposition



Numerical Simulation - Surfaces



Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity

Sensitivity function for quorum sensing derived by Painter and Hillen 2003 from microscopic model



Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity: Plateau formation



Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity: Plateau motion



Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity

Asymptotics at hyperbolic time-scale



Chemotaxis Limit is a nonlinear, nonlocal conservation law: we need entropy solutions

Entropy inequality



Chemotaxis Stationary solutions

These are entropy solutions iff



Chemotaxis Asymptotics for large time by time rescaling

Look for limiting solutions



Chemotaxis Asymptotic expansion in interfacial layer (as for Cahn-Hilliard)

Note: entropy condition



Chemotaxis We obtain a surface diffusion law with diffusivity

and chemical potential

Corresponding energy functional

D = ¡ 2@n S

¹ = ¡ S2 = ¡ S[ ]

2



Chemotaxis Flow is volume conserving

Flow has energy dissipation property



Chemotaxis Stability of stationary solutions can be studied based on second (shape) variations on the energy functional

Stability condition for normal perturbation

Instability without entropy condition ! Otherwise high-frequency stability, possible low-frequency instability



Chemotaxis Instability



Chemotaxis Surface Diffusion



Chemotaxis Surface Diffusion, 3D



Download and Contact Papers and Talks: Anisotropy: mb, JCP 2005

mb-Hausser-Stöcker-Voigt JCP 2007

Adatoms: mb, Comm. Math. Sci. 2006

Chemotaxis: mb-DiFrancesco-DolakStruss, SIMA 2007 mb-DolakStruss-Schmeiser, Preprint, 2006

www.math.uni-muenster.de/u/burger

e-mail: [email protected]

Some Aspects of Surface Diffusion Martin Burger Institut für Numerische und Angewandte Mathematik,...

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