Applied Surface Sciencehome.iitk.ac.in/~anandh/papers/ASS2016.pdf · techniques have been used by investigators for the computation of surface tension of slabs. Needs and Godfrey

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Applied Surface Science 371 (2016) 343–348

Contents lists available at ScienceDirect

Applied Surface Science

jou rn al h om ep age: www.elsev ier .com/ locate /apsusc

wo scale simulation of surface stress in solids and its effects

anesh Iyer a, Deb De b, Arun Kumar c, Raj Pala b, Anandh Subramaniam a,∗

Department of Materials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, IndiaDepartment of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, IndiaDepartment of Physics, Central University of Haryana, Mahendragarh 123029, India

r t i c l e i n f o

rticle history:eceived 12 December 2015eceived in revised form 22 February 2016ccepted 24 February 2016vailable online 27 February 2016

a b s t r a c t

Surface stress in solids can have profound effects in semi-infinite and nanoscale materials. The currentwork pertains to the simulation of surface stress, using the concept proposed by Shuttleworth [Proceedingsof Physical Society 63 (1949) 444–457]. A two-scale approach is used for the simulation of surface stress.Density functional theory is used to compute the lattice parameter of a free-standing layer (or two layers

eywords:pitaxial overlayerurface stress/tensionensity functional theoryinite element method

in the case of (1 1 0) surface) of atoms, which is further used as an input into a finite element model.Aluminium is used as a model material for the computation of surface tension of (1 0 0), (1 1 1) and (1 1 0)planes and the results of the simulations are validated by comparison with results from literature. Theutility of the model developed is highlighted by demonstrating the effect of surface tension on the: (i)stress variation in a thin slab & (ii) lattice parameter of nanoscale free-standing crystals.

© 2016 Elsevier B.V. All rights reserved.

. Introduction

Surface stress in solids can have profound effects, in the regionslose to the surface in bulk materials and in nanoscale materials [1].urface tension (�) is the average of two orthogonal surface stressomponents (� = (�x + �y)/2) and its origin can be conceived in theurface free energy arising from unsaturated bonds (higher energyf the surface atoms with respect to the bulk atoms) [2]. In thease where these orthogonal components are equal, the surface isn a state of equi-biaxial stress and the terms surface stress andurface tension are equivalent. It is to be noted that surface stresss the appropriate parameter to describe solid surfaces and thaturface tension is a quantity better suited to liquids [3]. The impor-ance of the concepts of surface energy and surface tension can beeen from the recent work of Hui and Jagota [4], where attempt isade to distinguish these quantities in non-equilibrated systems.

urface stress in general has both tensile and shear components,owever for surfaces with rotational symmetry higher than a 2-

old (3, 4, 6-fold), the shear component is zero [5]. This implies thator (1 1 1) and (1 0 0) surfaces surface tension and surface stress arequivalent.

∗ Corresponding author. Fax: +91 512 259 7505.E-mail address: [email protected] (A. Subramaniam).

ttp://dx.doi.org/10.1016/j.apsusc.2016.02.201169-4332/© 2016 Elsevier B.V. All rights reserved.

Knowing the value of the surface free energy (�) and its variationwith strain (εxy), surface tension (�xy) can be calculated using therelation by Herring [6]:

�xy = �ıxy + ∂�∂εxy

(1)

where ıxy is the Kroneker delta. Prior to this work, The scalar formof this equation was given by Shuttleworth [5]: � = F + A

(dF/dA

),

where F is the Helmholtz free energy and A is the surface area.The effects of surface tension can be best appreciated in free

standing nanocrystals, where surface tension can lead to a reduc-tion in the lattice parameter, as compared to that of the bulk crystal[7–10]. Researchers have used experimental methods as well astheoretical approaches to study the variation of lattice parameterwith the size of the nanocrystal. The lattice parameter reductionwith size has been studied for both spherical [7–9] and facetedcrystals (octahedral, tetrahedral and cubic [9]). Huang et al. [9] havepointed out that with an increasing shape factor the decrease in lat-tice parameter is more pronounced. Qi and Wang have derived thefollowing relation to account for the shape of the particles [8]:

�a

a= 1

1 + 2Kr(2)

where G is the shear modulus and K = G√˛/� (� is the shape fac-

tor). Medasani and Vasiliev [10] have computed surface energy,surface stress and lattice contraction in Al nanoparticles using ab-initio density functional theory (DFT) and empirical computationaltechniques. Woltersdorf et al. [11] using the moiré fringe technique

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44 G. Iyer et al. / Applied Surfa

o measure the lattice parameter of small particles, have shown aecrease in lattice parameter with size of the particle. They havelso determined the surface tension using the equation proposedy Stoneham [7]. It is to be noted that this technique is good tobtain broad trendlines, but not as good as using high resolution

attice fringe imaging for the determination of lattice parameter.Multiple techniques have been used by investigators for the

omputation of surface tension of slabs. Needs and Godfrey [12]ad used first principles pseudo-potential calculations to deter-ine the surface stress of a nine layered (thin) slab of aluminium

(1 1 1) and (1 1 0) surfaces). They also performed calculations on1 1 1) surfaces and described surface stress as a driving force foreconstruction [13]. For a nine layer slab Feibelman et al. [14] hasaken forward the method proposed by Needs and Godfrey andave computed the surface stress of the nine layered (1 1 1) sur-

ace, considering a linear combination of atomic orbitals. Shiiharat al. [15] have also computed the surface stress for thin Al slabsof thickness varying from nine to sixteen layers) with (1 1 1) sur-ace. They have reported oscillatory behaviour of stress, from theurface to the centre of the slab and referred to as friedel typescillations. Shenoy [16] has used embedded atom potentials toompare the surface stresses of (1 1 1), (1 0 0) and (1 1 0) surfacesn relaxed state. Wan et al. [17] studied the effect of relaxation onurface tension and surface energies of finite slabs using modifiedmbedded atom method (MEAM). As expected, the surface stressn two mutually orthogonal directions on the (1 1 0) surface ([11̄0]nd [0 0 1]) are unequal for both the unrelaxed and relaxed cases16,17]. Finite element method has been widely used for the mod-lling nanostructures with inhomogeneties [18,19] and study ofechanical systems like beams and micro-cantilevers with surface

tress [20] using the Gurtin–Murdoch continuum approach [21]. Inhe method of Gurtin–Murdoch the surface is prescribed a separateonstitutive relation.

The present investigation is divided into two parts: (i) simu-ation of surface tension and (ii) application of the methodologyeveloped to study the effect of surface tension. For the simula-ion of surface tension the conceptual approach of Shuttleworths used [5]. The specific tasks undertaken in the current work are:i) compute the lattice parameter of a free-standing layer (or twoayers in the case of the (1 1 0) surface) of atoms using density func-ional theory, (ii) use the DFT results to formulate a simple andntuitive finite element model to compute surface stress/tension,iii) to study the effect of surface tension on the lattice parameterf nanoscale free-standing particles (and its variation across thearticle). The methodology is tested using Aluminium as a modelaterial and is further validated by comparison with available the-

retical, computational and experimental results from literature.luminium has been considered a model material as its surfaceoes not undergo reconstruction and sufficient data is available in

iterature for comparison and validation. The following assump-ions are made in the current work: (i) Surface reconstruction hasot been considered, (ii) change in surface tension with curvatureas been ignored and (iii) simple crystal shapes have been used.

. Computational methodology

Surface tension/stress of selected surfaces ((1 0 0), (1 1 1) &1 1 0) surfaces of Al) are computed using the concept proposedy Shuttleworth [5], wherein a free-standing layer of atoms of

material (e.g. a metal like aluminium), is ‘stretched’ to bringt into registry with a bulk crystal [22]. The free-standing layer

f atoms considered has a lower lattice parameter and becomeshe surface layer after the aforementioned procedure. A two steppproach is followed to simulate surface tension: (i) density func-ional theory (DFT) is used for determining the lattice parameter of

ence 371 (2016) 343–348

a free-standing layer of atoms (two layers in the case of the (110)surface) and (ii) finite element method (FEM) is utilized to bringthis surface in registry with the bulk crystal.

2.1. Density functional theory

DFT calculations are performed using the Vienna ab initio sim-ulation package (VASP). Calculation were done within the DFTframework with exchange correlation energy approximated in theGGA and electron ion interaction is described by using projectoraugment wave method (PAW) with three valance electrons for Al[23]. The energy cut off for the plane wave basis set was kept fixedat 400 eV for Al. The k-points are obtained by the Monkhust–Packscheme [24]. A 4 × 4 × 4 k-point grid is used for bulk and 4 × 4 × 1grid of k-points is used for surface. Calculation on bulk Al is per-formed as a validation of the method. The (1 1 1), (1 0 0) and (1 1 0)planes are constructed using the calculated equilibrium latticeconstant of 4.05 Å. Surfaces are modeled using periodic slabs andvacuum thickness of 10 Å. A (1 × 1) surface unit cell is consideredfor the calculation. The minimization of electronic free energy wasobtained using an effective iterative matrix diagonalization rou-tine based on a sequential band by band residuum minimizationmethod (RMM) [25]. The optimization of different atomic config-urations is based upon a conjugate gradient minimization methodtill the force on all atoms is 1mRy/bohr [26]. To check the effectof the k-point grid size on the computed lattice parameter of thesurface, the grid size was refined to 8 × 8 × 1, for all the three kindsof surfaces ((1 1 1), (1 0 0) and (1 1 0)) under consideration.

2.2. Finite element methodology

Fig. 1c–e. Slabs (Fig. 1a) with (1 1 1), (1 0 0) and (1 1 0) surface ori-entation are modelled by choosing the appropriate surface latticeparameter and the surface thickness (discussed further in Section3). Thick slabs (representing bulk materials) are simulated using2D plane strain conditions (Fig. 1a), while thin finite slabs are sim-ulated using 3D models (Fig. 1b).

Three kinds of finite particles are considered to study the effectof surface tension on the lattice parameter of the particle: (i) octa-hedral with (1 1 1) facets (Fig. 1c), (ii) cubical with (1 0 0) facets(Fig. 1d), (iii) spherical particle (Fig. 1e). It is to be noted that the(1 1 0) surface has 2-fold symmetry and hence the misfit strain intwo orthogonal directions will not be equal. To simplify the inter-pretation of the results, the lateral surfaces in Fig. 1a, b and d are notmodeled with surface tension. Similarly, the top surface in Fig. 1f isnot modeled with surface tension.

In all the models considered surface tension is simulatedby imposing eigenstrains corresponding to the lattice mismatchbetween the surface layer (as computed using DFT) and the bulk.The strain is calculated as:

εm =(asurface − abulk

abulk

)(3)

where asuface is the interatomic spacing of a free-standing layer(s)(monolayer or bilayer) of atoms and abulk is the correspondingspacing in a bulk crystal. For the {1 1 1} and {1 0 0} planes thesecorrespond to <1 1 0> type of directions. These surfaces are in astate of hydrostatic stress (in 2D). For the (1 1 0) surface (with 2-fold symmetry) the orthogonal directions ([0 0 1] & [11̄0]) are notequivalent and hence different eigenstrains have to be imposedalong these directions. It is important to note that a single atomic

layer of the (1 1 0) surface consists of rows of atoms (along[11̄0]),which do not touch in the orthogonal direction (as shown in Fig. 2).Keeping this in view two layers are considered in the computationof surface tension of the (1 1 0) surface.

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G. Iyer et al. / Applied Surface Science 371 (2016) 343–348 345

Fig. 1. Schematic of a finite element models used to simulate surface tension. (a) 2D model of a slab with thickness h (=hb + hs). (b) 3D model of a slab of length ‘L’ (for thebottom face uz = 0). (c) 2D model of a spherical particle (1/4th of the domain with axi-symmetry), (d) 3D model of a cubical particle with (1 0 0) surface (1/8th of the domaini es areo sed inc

cotatts

bh

s shown; eigenstrains are imposed in top, right & front faces; the xy, yz and zx planctahedral particle with (1 1 1) surfaces (1/8th of the domain). Eigenstrains are impooordinates in (e) are aligned with respect to the surface.

The lattice parameter of a free-standing surface to be used inonjunction with a spherical particle is expected to be dependentn its orientation and hence it would be required to impose orienta-ion dependent eigenstrains. To simplify the simulation procedure

constant surface lattice parameter is assumed (corresponding tohe (1 0 0) surface). This procedure is expected to capture the broadrend-lines with respect variation of the lattice parameter with the

ize of the spherical particle.

This region where eigenstrains are imposed, along with theoundary conditions are shown in Fig. 1 (regions with thicknesss). The surface thickness (hs) is not known uniquely. The thickness

constrained from moving in z, x and y directions respectively). (e) 3D model of an Regions marked ‘A’ (shaded grey with thickness hs) as elucidated in the text. Local

of the surface layer (‘hs’ in the figures) is an important parameterin the simulations. It is unfortunately a non-trivial task to ascribe aprecise value to this parameter. In the current work we have com-puted surface stresses for two values of ‘hs’: (i) equal to diameterof an atom (hs = 2r), (ii) equal to the distance till the second planeof atoms (hs = r + d, where ‘d’ is the interplanar spacing). The lateraccounts for the effect of one subsurface layer of atoms. The length

of the slab (‘L = 2683 Å’ in Fig. 1) is chosen such that the ‘edge effects’do not affect the region close to the centre of the domain (the valueof surface tension is determined from this region). A thick slab (’infi-

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346 G. Iyer et al. / Applied Surface Science 371 (2016) 343–348

Fig. 2. Atomic configuration of the (1 1 0) surface. It is to be noted that a mono-l(

nt

ceot�

l(Fcmsesbauotcrq

3

(tmta[dastshaa

sIss

[15]. In all the models the surface layer is in tension. In the model

ayer will consist of atomic rows, which do not touch along the orthogonal direction[0 0 1]).

ite slab’) is simulated by setting hb = 4664.5 Å. In this ‘infinite’ slabhe change in surface tension with thickness is small (less than 1%).

Isotropic material properties are used in the simulations for easyomparison with available results in literature and can be readilyxtended to anisotropic material properties. These properties arebtained by Voigt averaging single crystal data for Al [27]: a0 (lat-ice parameter of bulk Al) = 4.05 Å, d(1 1 1) = 2.863 Å, G = 26.18 GPa,

= 0.348.2D simulations are performed using 4-noded bilinear quadri-

ateral plane strain (model in Fig. 1a) and axisymmetric elementsFig. 1c), while 8-noded linear brick (for cubical and thin slabs,ig. 1b,d) and 4-noded linear tetrahedra (for octahedral parti-les, Fig. 1e) elements are used in the 3D simulations. In the 2Dodel misfit strains imposed are uniaxial, while in the 3D the

trains imposed are biaxial (the orthogonal strains imposed arequal for the (1 0 0) and (1 1 1) surfaces, but unequal for the (1 1 0)urface). The 3D models (with biaxial strains) are expected toe closer to real surfaces; however, 2D models are computation-lly economical. The finite element models are implemented bysing ABAQUS\Standard (6.81version, 2008) software. Variationf surface lattice parameter with size/curvature is neglected inhe models. In ionic materials (with polar surfaces) and inert gasrystals surface tension may have a negative value [5]. The cur-ent model is not directly applicable to such crystals. In additionuantum size effects cannot be captured using the model.

. Results and discussions

The in-plane lattice parameter for free-standing layers ((1 1 1),1 1 0) & (1 0 0)) of Al computed using DFT is listed in Table 1. Theable includes lattice parameters computed for two k-point grid

esh sizes: 4 × 4 × 1 and 8 × 8 × 1. The (1 1 0) surface comprises ofwo atomic layers as a single layer has isolated rows of atoms (i.e.toms touch each other along the [11̄0] direction, but not along the0 0 1]). It is to be noted that for the (1 1 0) surface the orthogonalirections ([11̄0] & [0 0 1]) have different lattice spacings. To make

meaningful comparison of the surface tension of (1 1 1) and (1 0 0)urfaces with that of the (1 1 0) surface, the interatomic spacing forwo free-standing layers of these surfaces are also tabulated. It iseen from the values in table that, k-point grid mesh refinementas small effect only in the case of monolayer of (1 1 1) and (1 0 0)toms. In addition, there is small change in the lattice parameterlong the [11̄0] direction of the (1 1 0) surface (less than about 1%).

The stress state (plot of �xx stress contours) of Al (1 1 1) orientedlab is shown in Fig. 3 for a: (a) nine layer slab and (b) thick slab.

n both the cases eigenstrains have been imposed only on the topurface. It is seen that there is very little influence of surface ten-ion on the bulk of the thick slab. In the case of the nine layer slab

Fig. 3. State of stress of (plot of �xx contours) due to surface tension: (a) slab of ninelayers, (b) in a thick slab.

(Fig. 3a), the slab interior is under compressive stress of the sameorder as the tensile stress on the surface.

The surface tension/stress values computed using the modelas in Fig. 1a (for a thick/infinite and finite slabs) is listed inTable 2. Values computed using two thickness of the surface (hs = 2r& hs = (r + d)). Results from literature are also included in Table forcomparison. The table includes surface tension values computedfor the two sets of lattice parameters as listed in Table 1 (corre-sponding to two k-point grid mesh sizes) for hs = 2r ((1 1 1) & (1 0 0)surfaces) and hs = 2 × 2r (for (1 1 0) surface). It is seen that the sim-ulated values are in reasonable agreement with the values citedin literature for finite slabs. The values cited in literature amongstother computational methods, also compare well with one another.As expected, the surface tension computed using hs = (r + d) has ahigher value than with hs = 2r. The 3D model, wherein biaxial eigen-strains have been imposed, gives a higher value of surface tensionas compared to the 2D model. For the (1 1 0) surface, the stress alongthe two principal axes ([0 0 1] & [11̄0]) are listed in Table 2. Further,it is seen that using the lattice parameters after k-point grid meshrefinement leads to a change in the surface tension values of lessthan about 6%. Though this change is not insignificant, this has to beviewed in the overall context of the accuracy of the methodology.

In some of the studies, surface tension is computed for bothrelaxed (R) and unrelaxed (UR) surfaces. The relaxation beingreferred to is normal to the surface. The FEM model has no con-straints normal to the surface, which implies that the strains areuniaxial (for a 2D model) and in-plane biaxial (for a 3D model).The value of the surface tension computed depends on the dimen-sionality of the model, though 2D models seem to give reasonablevalues. It is to be noted that (1 1 0) surface cannot be computedusing a 2D model. The value of surface tension computed using theFEM model for an infinite slab is slightly enhanced than that forthe nine-layer slab. This is to be expected as in the case of the infi-nite slab there is more of the bulk constraining the strain due tomismatch. Additionally, this brings out the utility of the method,wherein thick slabs and large domains can be simulated with littleadditional computational effort.

Fig. 4 shows the variation in stress (plot of �xx) across the thick-ness of a nine layer slab (model as in Fig. 1b with hs = 2r). The figurealso includes the results from Feibalman [14] and Shiihara et al.

of Shiihara et al. [15] some of the sub-surface layers are in slighttension, which has been ascribed to asymmetric charge distribu-tion at the surface. In the current model all the ‘core’ layers are in

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G. Iyer et al. / Applied Surface Science 371 (2016) 343–348 347

Table 1In-plane interatomic spacing of (1 1 1), (1 0 0) & (1 1 0) free-standing layer(s) of Al computed using DFT (for two k-point grid mesh sizes: 4 × 4 × 1, 8 × 8 × 1). The correspondingspacings for a bulk crystal are also included for comparison. Direction on the surface refers to the direction along which strains are imposed in the FEM model. Number ofatomic layers mentioned refers to the number used in the computation of surface tension using FEM.

Surface plane (1 1 1) (1 0 0) (1 1 0)

Direction on the surface [11̄0] [011] [001] [11̄0]Interatomic spacing (bulk) (Å) 2.863 2.863 4.05 2.863No. of atomic layers 1 (2) 1 (2) 2 2Interatomic spacing (DFT) (Å)K-point grid mesh: 4 × 4 × 1 2.658 (2.764) 2.633 (2.778) 3.55 2.51Interatomic spacing (DFT) (Å)K-point grid mesh: 8 × 8 × 1 2.67 (2.764) 2.66 (2.778) 3.55 2.55

Table 2Surface tension values computed using the model as in Fig. 1a (for finite and infinite slabs). Simulated results corresponding to different surface thicknesses are listed in thetable. Data from literature is also included for comparison. For the (1 1 0) surface, the surface tension along two orthogonal directions ([0 0 1] & [11̄0]) are listed. The valuesin brackets correspond to k-point grid mesh sizes of 8 × 8 × 1.

Literature & currentwork

Technique Dimensions (thickness) Surface tension (�xx) (N/m)

(1 1 1) (1 0 0) (1 1 0)

Current work FEM (9 layer slab) 2D (hs = 2r) 1.14 (1.08) 1.29 (1.20) –3D (hs = 2r) 1.54 (1.45) 1.75 (1.62) –3D (hs = 2(2r)) 2.57 2.92 3.61 [0 0 1]

(3.54)3.17 [11̄0](3.07)

2D (hs = r + d) 1.47 1.66 –3D (hs = r + d) 2.03 2.30 –3D (hs = 2(r + d)) 3.23 3.65 4.53 [0 0 1]

3.98 [11̄0]Current work FEM (numerically infinite slab) 2D (hs = 2r) 1.42 (1.37) 1.61 (1.55) –

3D (hs = 2r) 1.91 (1.85) 2.17 (2.09) –3D (hs = 2(2r)) 3.81 4.33 5.03 [0 0 1]

(4.96)4.71 [11̄0](4.66)

2D (hs = r + d) 1.99 2.24 –3D (hs = r + d) 2.74 3.10 –3D (hs = 2(r + d)) 3.96 4.52 5.44 [0 0 1]

5.11 [11̄0]Needs et al. [12,13] FP (9 layer slab) 3D 2.32 (R)1.24 (UR) 1.84(R) 1.98(R)Feibelman et al. [14] FP (9 layer slab) 3D (R) 1.44 – –Shihara et al. [15] DFT (9 layer slab) 3D 0.80 – –Shenoy [16] EAM (slab model) 3D(R) 0.91 (EA) 1.23(V) 0.57 (EA) 1.31(V) 1.03 (EA)

1.29 (V)Wan et al. [17] MEAM (slab model) 3D 2.77 (R)2.28 (UR) 1.88 (R)1.76 (UR) 2.23 (R)

2.49 (UR)

Key: FP—first principles, FEM—finite element method, EAM—embedded atom method, M& adams potentials, V—voter potential.

F(n

ct

The measurements of Woltersdorf et al. [11] underestimates the

ig. 4. Variation of stress across a nine layer slab: comparison of the FEM modelhs = 2r in Fig. 1b) with literature (Feibalman [14] and Shiihara et al. [15]). Theumbers on the x-axis refer to the centroid position of the various layers.

ompression, akin to the model of Feibalman [14]. It is seen thathe current model is in good agreement with the models from lit-

EAM—modified embedded atom method, R—relaxed, UR—unrelaxed, EA—ercolessi

erature and is able to capture most of the interesting features (withrespect to stress state) of a thin slab. The significant difference iswith respect to the stress at the interface between surface and bulkregions. In the 3D FEM simulation this plane is in nearly zero stress,which is reasonable to expect as this plane separates the tensile andcompressive regions in the slab.

The variation of lattice parameter of free-standing crystals(spherical, octahedral & cubic) with size of the particle (radius forthe sphere, centre to vertex distance for the octahedron and cube)is shown in Fig. 5. The plots correspond to surface thickness of 2r inthe FEM model. Results from literature (experimental [11], compu-tational [10] and theoretical [8]) are also overlaid on the plot. Thebest comparison can be made for spherical particles as experimen-tal and multiple theoretical studies are available. A good match isseen across all methods for the spherical particle, except for theexperimentally determined values, which has a lower magnitudeat small sizes as compared to the other studies. From the plot it isseen that significant change in lattice parameter occurs only whenthe crystallite size is less than 20 nm.

value of the lattice parameter for smaller sized particles (<20 nm)as compared to the other techniques and being based on the moiréfringe technique, might suffer from some error. It is to be noted that

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348 G. Iyer et al. / Applied Surface Science 371 (2016) 343–348

Table 3Surface tension and lattice parameters at the surface (asf) & centre of the particle (amid) for octahedral, cubical & spherical particles. The values tabulated are for two sizes ofthe particles: 40 nm (largest size in the current simulations) and 3 nm (smallest size in the current simulations).

Particle shape (surface plane) Largest particle (d = 40 nm) Smallest particle (d = 3 nm)

Surface tension (N/m) asf (Å) amid (Å) Surface tension (N/m) asf (Å) amid (Å)

Octahedral (1 1 1) 1.814 4.046

Cubical (1 0 0) 1.408 4.042

Spherical 2.112 4.037

Fig. 5. Variation in lattice parameter of free-standing particles with size (radius fortam

t(eopolF�

twNtesf

4

mlt

[[[[[[[[

[[[[[[[24] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188–5192.[25] G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15–50.[26] E. Hernández, M.J. Gillan, Phys. Rev. B 51 (10) (1995) 157–160.[27] W.F. Gale, T.C. Tottemeier (Eds.), Smithells Metals Reference Handbook,

eighth ed., Butterworth-Heinemann publishers, Oxford, UK, 2004.

he sphere and centre to vertex distance for the octahedron and cube). FEM modelsre compared with theoretical, computational and experimental models. The readeray refer to the text for the References.

he results of the current investigation, along with that in literaturecited in Fig. 5) are in stark contrast to the results cited by Huangt al. [9]—i.e. there is a significant variation in lattice parametersf nanoparticles of size beyond 5 nm. Surface tension and latticearameters at the surface (asf) & centre of the particle (amid) forctahedral, cubical & spherical particles (for two particle sizes) are

isted in Table 3. It is seen that the surface tension computed usingEM for the slabs is in the same order as that for the particles (i.e.surface(111) > �surface

(100) ).The main advantage of the method developed is that surface

ension effects can be included in other FEM simulations like thatith dislocations, cracks, coherent precipitates, epitaxial films, etc.eedless to point out, the current methodology has its set of limi-

ations imposed by the assumptions used. The methodology can bextended for the computation of surface tension for other metallictructures (BCC, HCP). Improvements on these fronts can be scopeor the future work.

. Summary and conclusions

A combination of density functional theory and finite elementethod (a two-scale approach) offers a simple technique to simu-

ate the surface tension effects in metallic solids. This is based onhe concept proposed by Shuttleworth. In the current work the sur-

4.048 1.188 4.17 4.0294.044 1.028 3.94 3.984.045 1.397 3.915 4.007

face tension of (1 1 1), (1 0 0) and (1 1 0) surfaces of Aluminium (asmodel system) is computed using the methodology developed andis validated using available theoretical, computational and experi-mental results from literature. The simulations are utilized to studythe effect of surface tension on: (i) stress variation in a thin slab& (ii) lattice parameter of nanoscale free-standing crystals. Theimportance of the technique developed lies in the fact that, it canbe used in conjunction with finite element simulations involvingprecipitates, epitaxial films and dislocations.

References

[1] M.F. Ashby, P.J. Ferreira, D.L. Schodek, Nanomaterials, Nanotechnologies andDesign, Elsevier, Amsterdam, 2009.

[2] P. Müller, A. Saúl, Surf. Sci. Rep. 54 (2004) 157–258.[3] J.W. Gibbs, Collected Works 1 (1876) 315–317.[4] C.Y. Hui, A. Jagota, Langmuir 29 (2013) 11310–11316.[5] R. Shuttleworth, Proc. Phys. Soc. 63 (1949) 444–457.[6] C. Herring, The Physics of Powder Metallurgy, McGraw-Hill, 1951, pp. 143.[7] A.M. Stoneham, J. Phys. C: Solid State Phys. 10 (1977) 1175–1179.[8] W.H. Qi, M.P. Wang, J. Nanopart. Res. 7 (2005) 51–57.[9] Z. Huang, P. Thomson, S. Di, J. Phys. Chem. Solids 68 (2007) 530–535.10] B. Medasani, I. Vasiliev, Surf. Sci. 603 (2009) 2042–2046.11] J. Woltersdorf, A.S. Nepjko, E. Pippel, Surf. Sci. 106 (1981) 64–69.12] R.J. Needs, M.J. Godfrey, Phys. Scr. T19 (1987) 391–398.13] R.J. Needs, M.J. Godfrey, M. Mansfield, Surf. Sci. 242 (1991) 215–221.14] P.J. Feibalman, Phys. Rev. B 50 (1994) 1908–1911.15] Y. Shiihara, M. Kohyama, S. Oshibashi, Phys. Rev. B 81 (2010) 075441 (1-11).16] V.B. Shenoy, Phys. Rev. B 71 (2005) 094104 (1-11).17] J. Wan, Y.L. Fan, D.W. Gong, S.G. Shen, X.Q. Fan, Modell. Simul. Mater. Sci. Eng.

7 (1999) 189–206.18] W. Wang, X. Zeng, J. Ding, World Acad. Sci. Eng. Technol. 48 (2010) 865–870.19] L. Tian, R.K.N.D. Rajapakse, Comput. Mater. Sci. 41 (2007) 44–53.20] C. Liu, R.K.N.D. Rajapakse, A.S. Phani, J. Appl. Mech. 78 (2011) 1–10.21] M.E. Gurtin, A.I. Murdoch, Arch. Ration. Mech. Anal. 57 (1975) 291–323.22] R. Cammarata, K. Sieradzki, Annu. Rev. Mater. Sci. 24 (1994) 215–234.23] P.E. Blöchl, Phys. Rev. B 50 (17) (1994) 953–979.