Bridging between micro- and macroscales of materials by mesoscopic models

Bridging between micro- and macroscales of materialsby mesoscopic models

B.I. Lundqvist a,*, A. Bogicevic b, S. Dudiy a, P. Hyldgaard a, S. Ovesson a,C. Ruberto a, E. Schr€ooder a, G. Wahnstr€oom a

a Department of Applied Physics, Chalmers University of Technology and G€ooteborg University, S-412 96 G€ooteborg, Swedenb Scientific Research Laboratory, Ford Motor Company, Dearborn, MI 48121-2053, USA

Abstract

The importance of bridging length scales for materials is illustrated by three examples, nematic liquid crystals,

strength of materials, and epitaxial growth. Emphasis is on the microscopic scale, with first-principles calculations of

molecule–surface interaction, stacking-fault energies, interlayer interactions, diffusion barriers, and adsorbate–adsor-

bate interactions. Some pilot examples of using such information on the meso- and macroscales with models using

director fields, misfit densities of dislocation, and monomer and island densities are presented. The area is predicted to

have a great future. � 2002 Published by Elsevier Science B.V.

Keywords: Materials; Length scales; Liquid crystal; Strength of materials; Growth

1. Introduction

Materials theory, modelling, and simulations isan area in rapid development: In that processmultiscale modelling is a necessity. The gaps be-tween micro-, meso-, and macroscopic scales haveto be bridged, in order to fully implement theimmense potential of computational materialstheory. This is not only an inherent option andobligation of the field. Technologically, materialsbehavior thus modelled might imply reduced costsfor fabrication and increased manufacturing effi-ciencies.

One way to multiscale modelling is computa-tional design of hierarchically structured materials[1], a systems approach that integrates processing,structure, property, and performance relations.For, e.g., high-performance alloy steels, numericalimplementation of materials-science principles pro-vides a hierarchy of computational models thatdefine subsystem design parameters that are inte-grated, through computational thermodynamics, inthe comprehensive design ofmaterials as interactivesystems. Another approach uses mesoscale simu-lation to predict microstructure and microstructureevolution during thermodynamical processes. Suchan approach can be based on coupling parts by afinite-element-method deformation model at themicrostructural length scale with a Monte-Carlosimulation of the evolution of the deformationsubstructure during annealing.

Computational Materials Science 24 (2002) 1–13

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* Corresponding author. Tel.: +46-31-772-3198; fax: +46-31-

772-8426.

E-mail address: [email protected] (B.I. Lundqvist).

0927-0256/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S0927-0256 (02 )00158-1

mail to: [email protected]

This paper advocates the use of multiscalemodelling and simulation based on (i) generalmacroscopic descriptions, such as those for elasti-city, electromagnetism, hydrodynamics, and rheol-ogy, (ii) insightful modifications of the applicablefield theory on the mesoscopic scale, and (iii) mi-croscopic theory to provide key parameters ema-nating from atomic-scale behavior.As we are engaged in microscopic descriptions

of nature, bridging for us primarily means thelatter type of approach, starting with first-princi-ples calculations. Our materials consortium’s scopeto bridge the scales has so far only been filled withsome pilot studies. As an indication of what canbe done, three examples produced in our groupare briefly reviewed, concerning (i) liquid crystals,(ii) strength of materials, and (iii) crystal growth.In these cases the micro-, meso-, and macroscalescan be illustrated by molecule–surface interaction,director field, and optical properties in displays,stacking fault, misfit density of dislocation, andstrength, and diffusion barriers, island density, andcrystal growth, respectively.

2. Director configuration of nematic liquid crystal

The nematic liquid crystal (NLC) state of matteris the simplest state with an anisotropic electro-magnetic response. In NLC elongated molecules

have random positions but align themselves alongan average direction, giving the NLC directionalorder and positional isotropy, manifested in, e.g.,various biological molecules and membranes, sur-factants, and emulsifiers, and in extensive techno-logical applications in, e.g., optical displays. It isimportant to provide a mesoscopic NLC link be-tween micro- and macroscopic descriptions in asimple model.An analytical NLC link, describing the NLC

configuration as a function of anchoring condi-tions, has been derived for the NLC confinedbetween two planar plates [2]. Ultimately, a first-principles account of the molecule–surface inter-action is strived for, but the present account givesthe behavior as a function of the material con-stants and the molecular directions at the bound-aries (anchoring angles). The general analyticalsolution is derived in terms of Legendre third el-liptic functions of the variation in average molec-ular direction within the NLC (Fig. 1).The analytical solution for the molecular di-

rections has several important implications: (i) as itis given in established special functions, it is easilyavailable; (ii) the NLC system can be efficientlyand extensively analyzed; and (iii) the dependenceof the system behavior on materials and systemparameters can be obtained, exemplified by resultsfor the dependence of the optical transmission onparameters like the strength of the applied field,

Fig. 1. The NLC director field. (a) The director angle h versus the vertical position z. At fixed reduced magnetic potential

m ¼ ðlal0=K3Þ1=2Hd ¼ 2 or reduced voltage m ¼ ðeae0=K3Þ1=2Ed ¼ 2 the value of the elastic anisotropy runs through j ¼ðK3 � K1Þ=K3 ¼ 0, 0.25, 0.5, 0.75, 1 (top to bottom curves) where K1 is the splay elastic constant and K3 the bend elastic constant. The

inset illustrates the system of parallel plates encompassing the NLC, in this case with top anchoring at p=8 and bottom anchoring at

p=2. The director field is represented by small sticks. (b) The director angle hðzÞ at fixed elastic anisotropy j ¼ 0:25 for applied fields

m ¼ 0 (top curve), 2, 4, and 6 (bottom curve).

2 B.I. Lundqvist et al. / Computational Materials Science 24 (2002) 1–13

the elastic coefficients, and the anchoring angles(Fig. 2).Macroscopically, the NLC may be described

by a continuous vector field nðrÞ, a director field,representing the local average direction of themolecules at position r. For symmetry reasons,nðzÞ ¼ ðsin hðzÞ; 0; cos hðzÞÞ, where h is the anglebetween the director and the z-axis (Fig. 1). Fromthe Helmholtz free energy and the Euler–Lagrangeequations a relation between the angle h and thez-position is then obtained. The bulk splay andbend elastic coefficients of the NLC are accountedfor.The analytical expression for the director angle

of a NLC confined between parallel plates, withgeneral non-twist anchoring conditions, providesan explicit link between atomic-scale properties atthe plate interface to macroscopic bulk behavior,thereby bridging the gap between micro- andmacroscales.

3. Strength of materials

Mechanical properties of materials depend hi-erarchically on phenomena with length scales fromatomic up to macroscopic ones. Extended defects,such as dislocations and grain boundaries, are keyactors for large-scale behavior of solids, like plas-ticity, fracture, and the brittle–ductile transition.Theory is challenged to connect experimentally ob-served macroscopic properties to microscopic onesof extended defects, whose atomic-scale proper-ties today successively get accessible for studieswith electron-structure theory. At large distancescontinuum elasticity theory describes dislocationproperties and dislocation interactions in a usefulway. At the other extreme, an atomistic descrip-tion is required for the material specific, discreteatomic-core structure of the dislocation. Peierlsproposed early a hybrid model [3], where somedetails of the discrete dislocation core are incor-porated in an essentially continuum framework.The analytic solution of this Peierls–Nabarro (PN)model [4] gives a meaningful estimate of the latticeresistance to dislocation motion.

3.1. Dislocations

Dislocations are key concepts for the under-standing of mechanical properties of crystallinesolids. The PN model [3,4] provides a conceptualframework for dislocation structure and energet-ics. It is essentially a continuum treatment, but thedislocation core, the region of inelastic displace-ment, is given an approximate atomistic descrip-tion. The forces in the core, where the atomic-scalediscreteness really counts, are currently approxi-mated with the generalized stacking-fault (GSF)surface [5,6], which is the interplanar potentialenergy for sliding one half of a crystal over theother half.Mechanical properties are not only affected by

dislocations. Grain-boundary sliding is anotherpossible mechanism. Also here analogous meso-scopic descriptions exist.The bridging is here first exemplified for ele-

mentary semiconductors and metals. In fcc metalsthe experimentally observed dislocations have pla-nar cores and should therefore be favorable cases

Fig. 2. Theoretical predictions of the intensity of polarized light

transmitted through (a)–(c) a thin NLC film on top of an in-

homogeneous substrate or (d) a wetting droplet. (a) No applied

field. (b) External field applied perpendicular to the film. (c)

Polarization turned 45� with respect to (a). (d) Droplet on a

self-assembled monolayer (SAM) of organic molecules. The

intensity profile agrees very well with experiments and simula-

tions.

B.I. Lundqvist et al. / Computational Materials Science 24 (2002) 1–13 3

for the PN model. However, in practice mechani-cal strength of close-packed metals is not deter-mined by the intrinsic resistance to dislocationmotion but depends on extrinsic obstacles, such assolute atoms, grain boundaries, and precipitates,which block the motion of the dislocations. Instronger covalent, ionic, and intermetallic crystalsthe plastic properties directly depend on nucle-ation and mobility of dislocations. The intrinsicmobility is also an important factor for creep be-havior. All these additions of course affect the PNdescription.Generalizations of the PN model and first-

principles calculations on precipitates and inter-faces are also touched upon.

3.2. Dislocation in Si

The nature of dislocations in silicon is an ob-ject for frequent studies. In Ref. [7] the interac-tion between two partial 90� edge dislocations isstudied with atomic-scale simulations using theeffective-medium tight-binding method. A largeseparation between the two dislocations (up to 30�AA, comparable to experimental values) is achievedwith a solution of the tight-binding Hamiltonianthat scales linearly with the number of atoms. Thepartial edge dislocation is found to be very ac-curately described by the PN dislocation model,using the calculated GSF restoring forces, as re-flected both in the interaction energy and in thedisplacement field. An asymmetric core recon-struction provides fourfold coordination, makingSi behave elastically down to atomic distances [7].

3.3. Dislocations in metals

The GSF surface plays an important role inproposed models for the brittle–ductile transition,and they can also be used for calibration of modelpotentials for large-scale simulations and as inputto quasicontinuum models [8].Accurate input to the PN model can be pro-

vided by the density-functional theory (DFT). Inthis theory the complex many-electron problem isreplaced by a simpler one, where a functional ofthe electron density is minimized. Methodologicaladvancement of DFT together with the improved

performance of computers and numerical methodsmake it realistic to address the present types ofquestions from first principles.The DFT has been used to calculate GSF curves

for both unrelaxed and relaxed Pd and Al alongmain crystal directions ([1 2 1] and [1 1 0]) [8], in-cluding dislocation profiles (Fig. 3), and barri-ers and stresses for dislocation motion. The GSFcurves are applied in the PN model to Shockleypartials and edge dislocations. The results for Pei-erls stresses compare well with experimental values.Two dissociated partial dislocations are shown tobe strongly elastically coupled in their motion.These interactions have effects on the Peierls bar-rier [8].

3.4. Peierls–Nabarro model and generalizations

The PN model provides a conceptual frame-work, and with atomic forces derived from the GSFsurface, dislocation properties can be determinedquantitatively. For planar dislocations the PNmodel gives a continuum solution for the disloca-tion fbðxÞ, from which a misfit energy can be com-puted and thus also energy barriers and stressesfor dislocation motion. A dislocation introducedinto a lattice generates stresses at the interface/glide plane, which are calculated according to

Fig. 3. GSF curve for Pd sliding in the [1 2 1] direction on the

(1 1 1) plane [8], calculated for unrelaxed lattice and for various

degrees of relaxations described in Ref. [8], where also the Ising-

model-type calculation (ANNNI) is described.


elasticity theory. These elastic stresses are bal-anced by atomic forces acting on either side ofthe glide plane due to the misfit of atomic planes.This is expressed in an integrodifferential equationfor the misfit density per unit area, or dislocationprofile, qbðxÞ ¼ dfbðxÞ=dx, [3,4,9]

Kb

2p

Z 1

�1qbðx0Þ

1

x� x0dx0 ¼ FbðfbðxÞÞ: ð1Þ

The constant Kb determines the magnitude of theelastic stresses caused by introducing the disloca-tion and is possible to determine by solving theelastic equations for the anisotropic case [9,10].The dislocation is viewed as a distribution of in-finitesimal Burgers vectors qbðxÞdx, with the misfitdensity normalized as

R1�1 qbðxÞdx ¼ b, the length

of the appropriate Burgers vector. The restoringatomic forces on the right-hand side of Eq. (1) maybe approximated by the appropriate GSF curve[5,6], from

FbðfbðxÞÞ ¼ � ocGSFðfbðxÞÞofb

; ð2Þ

where cGSF is obtained by calculating the energychange, when the system is sliding on the glideplane in the direction of the Burgers vector.A study of Shockley partials in Pd with both a

full-scale atomistic simulation of the structure andenergetics of the dislocation motion and the PNmodel shows the two approaches to result insubstantial differences [11]. In this case a reason-able improvement of the PN model is to solve theordinary PN equation, but with atomic forcesevaluated for a disregistry that corresponds toextrapolated positions of the atomic planes ac-cording to Eq. (1) but with a disregistry fbðx; yÞcalculated for atoms located at a finite distancey ¼ �d=2, above and below the glide plane, re-spectively [11]. Thus extrapolation of the resultsfrom the PN model to the real atomic positionsusing elasticity theory gives a better agreement[11]. A consistency could be obtained by modify-ing the original PN model and using the extrapo-lated disregistry positions in the expression for therestoring atomic forces. This modification im-proves the core structure considerably and wouldprobably have quantitative capabilities [11].

When critically examined, the classical Peierls–Nabarro theory of dislocations has a number ofinconsistencies [12]. A semidiscrete theory correctsthe inconsistencies of the classic theory and givesresults similar to actual atomistic calculations,both qualitatively and quantitatively, and providesa link between accurate nanometer-scale quan-tum mechanical calculations of the dislocationcore energetics to mesoscale continuum descrip-tions [12]. It would be interesting to combine thesemidiscrete PN model and the above modificationfor the simple-fcc-metal case to validate the PNmodel. A well-working PN model would be able togive trends for dislocation properties with inputfrom first-principles GSF surfaces and could be auseful tool for ‘screening’/correlating mechanicalproperties from a reliable atomistic input.

3.5. Strength of real materials

In the bridging interfaces are key mesoscalefeatures. Knowledge of interface structure and en-ergetics is important for modelling of both nucle-ation and growth in complex alloys and of theirstrength and other mechanical properties, such ashigh-temperature creep resistance.

Precipitates: In steels, non-shearable precipi-tates are obtained with carbide and nitride form-ers, such as Ti, V, Cr, and Nb. Precipitates withthe NaCl structure is one important class and areoften found as small thin discs (radius ’ 50–100�AA). The ‘flat’ part of the interface is semicoherent(that is, has a small misfit-dislocation density),whereas the ‘side’ of the disc has a large latticemismatch and is incoherent with the Fe matrix.The electronic and atomic structures of a model

system consisting of the semicoherent interfacebetween bcc-Fe and NaCl-VN have been calcu-lated with a pseudopotential implementation ofDFT [13]. A key result is the very small, and evennegative, value of the interface energy [13]. Thisdoes not only find its natural interpretation inelectron-structural terms but is also consistent withexperimental observations. In this way one pointsout key features of the electron structure thatmake the formation of VN precipitates in Fe fa-vorable, which in turn is important to make steeltough.


This kind of modelling may thus be used topredict long-time structural changes in steels inorder to improve high-temperature creep resis-tance. The thermodynamical modelling shouldalso include elastic energies due to volume differ-ences, which is often done separately. In contrast,a real interface has more degrees of freedom andmay, for instance, create misfit dislocations.

Interfaces: Highly relevant in industry, theproblem of metal–ceramic adhesion has attractedconsiderable materials-science attention, in par-ticular to rationalize the adhesion strength interms of the interface atomic and electronic struc-tures. While, e.g., most oxides are isolating ioniccrystals, the transition-metal carbides are charac-terized by covalent bonding mixed with metallicityand slight ionicity, which makes the mechanism ofthe metal–carbide adhesion very particular. Sin-tered hard metals, an extraordinarily successfulpowder-technological product, are produced bycompressing pellets of metals, like Co, and metalcarbides, e.g., WC and TiC.Recently, one such class, the interface between a

transition metal and a transition-metal carbide,has been addressed in an ab initio theoreticalstudy of the representative Co/TiC(0 0 1) interface[14–17]. These first-principles computational ex-periments are based on DFT with GGA-PW91and consist of several steps: (i) performing total-energy calculations on a large number of hypo-thetical but realistic structures of the Co/TiCinterface, (ii) allowing the atoms to relax under thegiven, available external conditions, (iii) analyzingthe structural, bonding, and energetic results withrespect to trends and idealized electron-structuremodels, and (iv) deducing general results akin tothe available experimental experiences. They allowthe origin of the adhesion strength to be traceddown from the specifics of the interface atomic andelectronic structure. Especially interesting findingsconcern the interface Co–C bonds, which are un-expectedly strong and play a crucial role in theenergetics and the atomic structure of the Co/TiC(0 0 1) interface. At the same time the strengthof the Co–C bonds comes from intrinsic interfacialeffects that can be clearly identified in the elec-tronic structure. The effective number of thesebonds is controlled by an interplay between lattice

mismatch and the relaxation effects. The calculatedadhesion strength agrees reasonably well with re-sults of wetting experiments.Detailed analyses of local densities of states

(LDOS) show that there is a new interfacial effectat work: The interface Co–C bonds are muchstronger than the bulk ones. In bulk CoC theCo–C antibonding states are derived mainly fromthe Co-d states. The interface Co–C bonds have alarge concentration of the delocalized metallicstates of Co in the region of space and energy,where the Co–C antibonding states could be lo-cated. Thus the localization of the Co–C anti-bonding states inside a small space–energy regionis unfavorable due to the Pauli principle. Thus theinterface Co–C antibonding states should be muchmore delocalized than the bulk ones, losing theirantibonding character.First-principles interface calculations like these

should be possible to extend to the calculation ofatomic-scale parameters for mechanical proper-ties.

4. Crystal growth

Nucleation and growth of epitaxial films are ofgreat practical and fundamental interest. In sys-tems, where the overlayer wets the substrate, i.e. inlayer-by-layer or Frank-van der Merwe growth,the islands are only one atomic layer thick, withmorphologies ranging from ramified structuresat low temperatures to compact, polygonal shapesat higher temperatures. The great variety reflectsthe complex nature of the dynamical processesin such non-equilibrium systems. Experimentaltechniques, like scanning-tunnelling microscopy(STM), low-energy-electron microscopy, low-energy-electron diffraction, and grazing-incidenceX-ray scattering, allow studies of evolving adlayersfor length scales ranging from �AAngstr€ooms to mi-crons, and times ranging from picoseconds tominutes. This provides opportunities for develop-ing quantitative theories for thin-film growth,which ideally should span these length and timescales to connect first-principles electron-structurecalculations with thin-film morphology. This is at-tempted by first-principles calculations, kinetic


Monte-Carlo simulations (KMC), rate equations,and scaling theory.Models of epitaxial growth under typical mo-

lecular-beam-epitaxy conditions commonly assumethat atoms are deposited randomly onto an initiallyflat substrate at a constant rate of Fmonolayers persecond and diffuse freely on the crystalline sub-strate, until they encounter another atom, a groupof atoms or a defect, like a step. The population ofisolated adatoms increases initially linearly withtime, until small islands begin to nucleate. Duringthe ‘‘aggregation time’’, i.e. the time span when thenumber of islands increases due to increasing de-position but with this number still smaller than thenumber of monomers, the film growth can becharacterized by the dynamical evolution of theisland-size distribution, island shapes, and loca-tions.Our examples here primarily concern attempts

to understand the effects of elementary atomic-scale processes on the island shapes and distribu-tions, since these islands form the building blocksupon which all further growth proceeds. As themorphology of the first monolayer has a greatinfluence on the nucleation and growth of thesecond and subsequent layers, we also give anexample of interlayer couplings.

4.1. Mesoscale descriptions

The island-size distribution function hnsðhÞi,where the brackets represent an average over alarge area of the surface or an ensemble of systems,is defined as the number of islands per unit areacontaining s atoms at a coverage h ¼ Ft. One wayto describe its dynamical evolution is by a setof deterministic coupled rate equations. Therate coefficients that are inputs into these equa-tions should contain information about all of theatomic-scale processes, as well as long-range cor-relations between islands. The problem is a com-plex one and approximations have to be made.The rate equations describe the time evolutionduring early times up until the islands start tocoalesce, typically replacing time by coverageh ¼ Ft. Equations for the coverage derivative ofhnsðhÞi can be set up in terms of the adatom-dif-fusion constant D, flux F, and the capture numbers

rs, expressing the efficiency of an island of size s incompeting for the available monomers on thesurface. They contain terms for the rate at whichdiffusing monomers are added to an island of sizes� 1ðsÞmultiplied by the total density of islands ofthat size, processes that increase (decrease) thenumber of islands of size s, and terms that accountfor the direct capture of deposited atoms by is-lands of size s� 1 and s, respectively.In particular, the equation for the density of

monomers can be written [18]

dhn1idh

¼ 1� 2DF

� �r1hn1i2 �

DF

� �hn1i

�X1s¼2

rshnsi � j1hn1i �X1s¼1

jshnsi; ð3Þ

where the terms on the right-hand side accountfor the flux of atoms onto the surface, for the lossof monomers to dimer formation, for the loss toattachment to islands, for the loss of diffusingmonomers due to direct capture of the depositedflux, and for the loss of flux to the direct im-pingement onto existing islands and monomers,respectively. The factor of 2 in the second termdescribes the fact that when one dimer is formedtwo monomers are lost.Any information about island structure and

spatial correlation between islands has to be con-tained in the time-dependent capture numbers rs.As the average quantities hnsðhÞi contain incom-plete information about the correlations, the sys-tem of equations (3) has to be augmented withfurther information or assumptions, as is done innumerous approximations in the literature. Typi-cally, a mean-field assumption is made, for in-stance, as in the following model [18]: Consider acircular island of radius Rs that is embedded in anensemble-averaged system of islands and mono-mers. The local density of monomers, n1ðr; hÞ,responds to the presence of this island. In par-ticular, if adatoms attach irreversibly, the densityof monomers vanishes at the edge of the island,n1ðRs; hÞ ¼ 0. The simplest possible diffusion equa-tion describing the spatial variations of n1 is

on1oh

¼ DF

� �r2n1 þ |� D

F

� �n�2n1; ð4Þ


where | and n can be calculated from the self-consistency condition, n1ðr ! 1; hÞ ¼ n1ðhÞ, i.e.the presence of this island is not felt infinitely faraway. In this limit, Eq. (4) must reproduce Eq. (3),which leads to

| ¼ 1�X1s¼1

jshnsi ð5Þ

and

n�2 ¼ 2r1hn1i þX1s¼2

rshnsi þDF

� �j1; ð6Þ

where n is the average distance a monomer travelsbefore being captured by an island or anothermonomer and | is the fraction of the flux, whichlands on the bare substrate. In this approach themean-field assumption that at every point outsidethe island the local densities nsðr; hÞ taking on theiraverage values hnsðhÞi ðfor sP 2Þ is implicit.Despite ‘‘simplicity’’, Eq. (4) is difficult to solve,

due to its complicated time dependence ½n ¼ nðtÞ�and the growth of the island, i.e. the movingboundary. An approximate solution can be found,however, by assuming that the rate of adatomdiffusion is large compared to the growth rate ofthe island. It is sufficient then to fix the radius ofthe island and solve for the instantaneous con-centration of monomers. One cannot satisfy Eq.(4), as r ! 1, if the coverage (time) derivative ofthe more general nðr; hÞ is neglected. Instead,subtracting Eq. (3) from Eq. (4),

FD

on1oh

�� dhn1i

dh

�¼ r2n1 � n�2ðn1 � hn1iÞ: ð7Þ

Neglecting deviations of the coverage derivativefrom its average value makes the right-hand mem-ber of Eq. (7) approximately zero. The resultingHelmholtz equation has a straightforward solutionfor n1ðr; hÞ in terms of modified Bessel functionsfrom which one can readily obtain the capturenumbers rs.There are cases, where the continuum approach

produces an excellent agreement between themean-field rate equations and KMC simulations,given the same assumptions about the elementaryprocesses [18].

4.2. First-principles example of interlayer influences

In nucleation and growth of epitaxial films themorphology of the first monolayer could havegreat influence on the nucleation and growth of thesecond and subsequent layers, etc. The importanceof interlayer coupling is illustrated in a very recentwork on Al2O3 nucleation on TiC(1 1 1) [20],crucial in the deposition of j-Al2O3 throughchemical-vapor deposition for the production ofwear-resistant coatings on cemented-carbide cut-ting tools. It is found that structures characterizedby the Al atoms in sites similar to the j-Al2O3

structure are more favored than structures havinghexagonal Al networks similar to a-Al2O3. Thisindicated preferred growth of j-Al2O3 on a perfectTiC(1 1 1) surface is understood in terms of at-tractive bonding between neighboring Al atomschemisorbed on the TiC(1 1 1) surface on top of adense O(1� 1) layer (Fig. 4). This originates fromthe dominance of the 2pxy orbitals of O in the O–Tibonds in the first layer of chemisorbed O onTiC(1 1 1). Therefore, at sufficiently high Al cov-erage, the system finds it favorable to use thedangling O2pz bonds to bind to the next layer ofchemisorbed Al atoms, provided that the anti-

Fig. 4. LDOS profiles describing attractive bonding between

neighboring Al atoms chemisorbed on the TiC(1 1 1) surface on

top of a dense Oð1� 1Þ layer [20].


bonding Al–O orbitals that result from this findsimilar states in neighboring Al adatoms to formstabilizing bonds with.

4.3. First-principles account of elementary growthprocesses

Epitaxial growth offers an exotic variety ofsurface morphologies that emanate from a handfulof elementary atomic diffusion processes. The firststep towards a detailed understanding of this isto map out all relevant atomic processes, terracediffusion, corner-crossing, kink-breaking and soon, which has been done on Al(1 1 1) (Fig. 5) bydensity-functional calculations [19]. A descriptionwith a lattice-gas model, where atoms are allowedto occupy certain grid positions only, for Al(1 1 1)the fcc and hcp sites, is adequate in epitaxy andmany other situations within a broad temperaturerange. The diffusion of atoms adsorbed on thismesh can be described by transition-state theory(TST). Experimental techniques, such as field-ionmicroscopy and STM, help to ascertain that nodiffusion processes are left out.

The next step is to determine the rates for allprocesses. For each process i, one has to find theactivation energy Ei and the corresponding pre-factor m0i , yielding a collection of rates mi given bythe Arrhenius law (TST)

mi ¼ m0i e�Ei=kBT ; ð8Þ

where kB is the Boltzmann constant and T is thesubstrate temperature.This information can be used in KMC simula-

tions of growth, but the relative importance ofvarious growth modes can be estimated (Fig. 6).From Eq. (8) an activation temperature Ti can bedefined for each process i, below which the partic-ular event is frozen out. Defining the process tobe activated when it happens at a rate C, the on-set temperature is determined by Ti ¼ ðEi=kBÞ=ðln mi=CÞ. The value of C depends on the experi-mental growth rate, but is typically of the order of

Fig. 5. Elementary diffusion processes at atom islands for an

fcc(1 1 1) surface [19,22]. Each process is characterized by a

letter (T for terrace, E for Edge, K for kink, and C for corner)

and a subscript that indicates the number of in-layer nearest

neighbors before and after the jump. The processes can take

place on both A steps, with a 1 0 0 microfacet, and B steps, with

a 1 1 1 microfacet.

Fig. 6. Temperature scale indicating the onset of elementary

diffusion processes at Al islands on Al(1 1 1) [19]. The activation

temperatures are calculated from Eq. (8), as described in the

text. The notations are described in Fig. 5.


1/s. As the temperature is increased, more andmore of these processes become activated (Fig. 6).Already around 17 K, terrace diffusion (T0!0)becomes activated. The growth occurs by a hit-and-stick mechanism, where atoms diffuse on the ter-races and irreversibly attach to existing islands. Byslightly raising the substrate temperature, a tran-sition occurs from fractal to dendritic growth withramified fractal patterns growing in three distinctdirections, as observed experimentally for severaldifferent metal-on-metal systems. The asymmetrybetween A and B steps (Fig. 5) for several ele-mentary diffusion processes is striking (Fig. 6).Dimer diffusion has also been studied for Al2 on

Al(1 1 1) [21] for adatom pairs on an open metalsurface [23].The next step is to make KMC simulations with

energy barriers obtained from such density-func-tional calculations. For instance, the microscopicorigin of compact triangular islands on close-packed surfaces has been identified in this way[22]: corner-diffusion anisotropy controls the shapeof compact islands at intermediate temperatures.The correlation between the orientation of den-drites grown at low temperatures and triangularcompact islands grown at higher temperatures(Fig. 7) is rationalized, and an explanation for whyin some systems dendrites grow fat before turningcompact is also provided in this way [22].

4.4. Adsorbate interactions

On surfaces there can be long-ranged forcesbetween adsorbates, for instance, those mediated

by Shockley surface states on the (1 1 1) faces ofnoble metals, with an oscillatory form modulatedby a 1=d2 envelope at asymptotic adsorbate sepa-rations d. For this interaction a non-perturbativeanalytical estimate has been given [24], specified byexperimentally accessible Shockley-state parame-ters and the finite Fermi-level phase shift, whichcharacterize the standing-wave patterns observedin STM images.This electron-mediated force is oscillatory in

space, with a period related to the surface elec-trons’ Fermi wavelength. There is alternately at-traction and repulsion, as one atom ‘‘rides’’ theelectron waves produced by the other, leading toattractive and repulsive rings surrounding eachatom. This kind of indirect interactions betweenadatoms was suggested decades ago, but there aresevere experimental problems to detect such weaklong-range forces. In a new extensive and carefulSTM study, however, the properties of the scat-tered electron waves are shown to be in agreementwith the non-perturbative analytical predictions[25].Atoms and molecules adsorbed on metals affect

each other indirectly over considerable distances,not only electronically but also, e.g., elastically.For intermediate distances systematic density-functional calculations have been performed [26,28], to establish the nature and strength of suchinteractions, and to explain for what adsorbatesystems they critically affect important materialsproperties. The calculations are performed withunusually large supercells (up to 336 atoms) onone of the worlds biggest computers, but the sep-arations are still too small to make contact withthe asymptotic result mentioned above. However,as shown in Fig. 8, the adatom–adatom interactionis typically oscillatory and repulsive (except for theshortest (dimer; not shown in figure) bond withadatoms in adjacent sites, where the binding en-ergy is one order of magnitude larger (0.52, 0.26,and 0.26 eV for Al/Al(1 1 1), Cu/Cu(1 1 1), and Al/Al(1 0 0), respectively) [26].With interaction-modified KMC simulations of

epitaxial growth, a number of recent experimentalreports on anomalously low diffusion prefactors arerationalized with ‘‘normal-sized’’ prefactors. Fur-ther, the indirect adsorbate interactions are shown

Fig. 7. The island morphology calculated by KMC simulations

[22] for homoepitaxial growth on Al(1 1 1) at 80 K (left) reveals

dendrites with the same orientation as triangles grown at 200 K

(right). In both figures the coverage is 0.10 ML and the image

size 680� 570 �AA2.


to strongly alter the binding and motion of thedeposited adsorbates. It is explained, where it isimportant to take such long-ranged interactionsinto account, and the KMC simulations demon-strate the strong effects they can have on surfacemorphology (Fig. 9). The results of the KMCsimulations for low-temperature island nucleation

in homoepitaxial growth on Al(1 1 1) with andwithout interactions between the adatoms lookreally different [27,28]. With the repulsive interac-tions accounted for, there is a tendency towardsself-organization, and the island density is signifi-cantly higher. As a function of the inverse temper-ature, the logarithm of the island density is shownto grow both with and without interactions (Fig.10), however with values more than two orders ofmagnitude higher with interactions [26].

4.5. Mesoscale effects of interactions

Anomalous prefactors deduced from STM datahave been determined by STM island-densityanalysis using the nucleation-curve method, whenArrhenius parameters for surface diffusion aredetermined by gathering island-density data and

Fig. 8. Top view of adsorption geometries used in the DFT

calculations [26]. One adatom is placed at its preferred binding

site (hcp for Al and fcc for Cu; mid-size circles), and another

atom is then placed at successive binding sites and saddle points

(tiny circles) along the h110i direction (half the length of the

Al supercell is shown). The binding energy is defined as

E ¼ E11 þ E2

1 � E0 � E2, where the subscript denotes the numberof adatoms in the cell and the superscript identifies the indi-

vidual atomic positions, and is shown as a function of adsor-

bate separation d in terms of lattice sites. Both frozen (middle

graph) and relaxed (bottom graph) cases are truncated at short

separations to enhance resolution.

Fig. 9. KMC-computed surface morphology (500� 425 �AA2) of Cu/Cu(1 1 1) [27,28] without (left) and with (right) at T ¼ 25 K,

F ¼ 0:01 ML/s, H ¼ 5.

Fig. 10. Island density as a function of inverse temperature

from STM experiments and from KMC simulations performed

with and without adatom interactions [28].


then analyzing these data within mean-field nu-cleation theory (MFNT). According to MFNT,under certain circumstances the island density nxrelates to the monomer diffusivity D through ascaling law, nx � ðF =ðDN 2

0 ÞÞ1=3.

Adatom interactions are completely discardedin MFNT, and actually in all previous models ofhomogeneous nucleation. On the other hand, theabove KMC simulations are performed with‘‘normal’’ prefactor values and give island-densityresults like in experiments.As a first attempt to account for adsorbate–

adsorbate interactions in mesoscopic theory, therate equation (7) has recently been generalized[29]. The term r2n1 in Eq. (7) is replaced byr2ðn1ebVk Þ, where Vk is the repulsive interactionpotential and k the number of atoms in an islandthat interact with the adatom under consideration.The diffusion constant D is just a constant equal toD0 expð�E=kT Þ, but the jump rate in different di-rections depends on Vk. Exactly how it is affectedby Vk depends on what assumptions are madeabout the variation of the barrier height with Vk.Based on a rather crude approximation, namelythat all barriers have the same energy (relative tosome fixed energy of the system), and the fact thatrk is rather insensitive to the actual choice, pre-liminary results for the dependence of the islanddensity on the coverage h ¼ Ft are obtained (Fig.11). Also in a rate-equation description, interac-tion effects increases the island density significantly[29].

5. Conclusions

The importance of bridging length scales formaterials is stressed and illustrated by three ex-amples: (i) NLCs, where a field-theoretical modelshows strong macroscopic consequences for asimple microscopic input that can be made moreversatile, (ii) strength of materials, where applica-bility and limitations of a mesoscopic (Peierls–Nabarro) model for dislocations are illustrated,and (iii) epitaxial growth, where microscopic find-ings on adsorbate interactions are shown to callfor improvements of mesoscopic descriptions. Itshould be obvious from these examples that, like afamous continent, this area has a great future in amultitude of interesting and challenging thingsthat remain to be done.

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Fig. 11. Island density, calculated from Eq. (7) with the interaction correction as described in Section 4.5, as a function of coverage.

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Bridging between micro- and macroscales of materials by mesoscopic models

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