Mechanics of Materials and Structures

Transcription

1 Journal of Mechanics of Materials and Structures NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY Changwen Mi and Demitris A. Kouris Volume Nº 4 April 6 mathematical sciences publishers

2 JOURNAL OF MECHANICS OF MAERIALS AND SRUCURES Vol. No. 4 6 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY CHANGWEN MI AND DEMIRIS A. KOURIS We investigate the effect of surface/interface elasticity in the presence of nanoparticles embedded in a semi-infinite elastic medium. he work is motivated by the technological significance of self-organization of strained islands in multilayered systems. Islands adatom-clusters or quantum dots are modeled as inhomogeneities with properties that differ from the ones of the surrounding material. Within the framework of continuum elasticity theory the Papkovitch Neuber displacement methodology coupled with Gurtin s surface elasticity yields an analytical solution. he elastic field is expressed in terms of four sets of spherical and cylindrical harmonics. Surface elasticity introduces an additional length scale and results suggest that local stresses are significantly affected by the size of the nanoparticles.. Introduction Self-assembly presents a promising alternative in fabrication of quantum dots. he main requirements for applications are size uniformity and high spatial density of dots. Currently uniformity is on the order of % and no systematic approach has been proposed for its improvement. Covering of the first generation of islands or dots by a cap layer say of a substrate material and growth of subsequent generations of dots result in the ultimate development of a two-dimensional superlattice. he spatial uniformity of island structures was first addressed by ersoff et al. [996]. heir islands were modeled as spherical inclusions same properties as the surrounding material in an elastic half-space a problem that was originally solved by Mindlin and Cheng [95]. ersoff et al. [996] chose to effectively treat the islands as point sources centers of dilatation in a semi-infinite matrix. In most material systems used in practice however quantum dots and the surrounding material do not have identical material properties. he presence of inhomogeneities in elastic media has been a well studied subject in applied mechanics for several decades [Mura 987]. Until recently however almost all such studies have been concerned with either an infinite medium or a semi-infinite body without including surface effects either on the plane boundary Keywords: surface/interface effects inhomogeneity half-space. 763

3 764 CHANGWEN MI AND DEMIRIS A. KOURIS or at the matrix/inhomogeneity interface. Mindlin and Cheng [95] first studied the thermoelastic stress in a semi-infinite solid containing a spherical inclusion subjected to dilatational thermal expansion. suchida [97; 97] derived a threedimensional solution for displacements and stresses in a half-space having a spherical cavity for various loading conditions. he classical solution for the elastic state of a spherical inhomogeneity in a semi-infinite elastic body without incorporating surface effects was obtained by sutsui et al. [974]. hese problems were solved within the context of classical elasticity without accounting for size effects. It is well known however that under certain circumstances surface elasticity can be of importance. For material structures with one or more dimensions at the nanometer scale for example thin film and nano-sized inhomogeneities the surface area to volume ratio becomes significant. Consequently surface effects have a major influence on the local stress-strain field. he physical origin of the surface stress can be explained by the nature of the chemical bonding of those atoms close to the surface. he loss of the translational symmetry in the direction normal to surface results in a different bonding configuration in the vicinity of the surface when compared to the one in the bulk. he atoms at and close to the surface lose some of their neighbors or are bonded to some wrong neighbors. As a result the equilibrium interatomic distance for such atoms is different from the bulk atoms. his phenomenon results in an excess stress for atoms close to the surface. he theory of isotropic surface elasticity was established by Gurtin and Murdoch [975a; 975b; 978]. Cahn and Larche [98] studied the mechanical and chemical equilibrium of a small spherical precipitate in an infinite matrix of a different phase. Recently Sharma et al. [3] derived closed-form expressions for the displacements and stresses of a strained spherical inhomogeneity by using a variational approach. Sharma and Ganti [4] also studied the elastic state of spherical and cylindrical inhomogeneities embedded in an infinite elastic medium by combining the classical Green s function method with linear isotropic surface elasticity theory. Duan et al. [5b] developed explicit forms of the stress concentration tensor for spherical and circular inhomogeneities with surface effects. While surface/interface effects were incorporated all of these investigations have been limited to the study of an infinite elastic medium. In the present work we model a nanoparticle or island as a spherical inhomogeneity problem inside an elastic half-space. he material properties of the inhomogeneity are different from those of the surrounding material matrix. he system is subjected to biaxial tension applied at the remote boundary of the matrix parallel to the free surface. he ensuing three-dimensional problem is solved with the help of the Papkovitch Neuber displacement potential theory. Section illustrates the formulation of the problem of an embedded inhomogeneity in a half-space including

4 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 765 surface/interface effects. Following [Gurtin et al. 998] the displacements on the surface/interface boundary are still considered to be continuous. he tractions both on the plane boundary and at the matrix/inhomogeneity interface however are discontinuous as a result of the well known Laplace Young effects [Adamson 98 Chapter II]. he problem is solved in Section 3 incorporating the surface/interface stress boundary conditions. Section 4 illustrates the impact of surface/interface effects as the location size and material properties of the inhomogeneity change. Finally a series of conclusions are presented in Section 5. he elaborate and quite tedious mathematical formulation presented in Sections and 3 is the result of a conscious decision to provide a complete self-contained document since the detailed methodology cannot be found elsewhere.. Displacement formulation Let us consider a spherical inhomogeneity in a semi-infinite elastic solid as shown in Figure. he center of the inhomogeneity coincides with the center of the coordinate system. Cylindrical r θ z and spherical R θ ϕ coordinates will be used alternatively throughout the manuscript. Following [Gurtin and Murdoch 978] the plane boundary z = c and the matrix/inhomogeneity interface R = a are modeled as thin films of vanishing depth; both surfaces adhere to the bulk without slipping. he model is based on the assumption of a linearly elastic and isotropic material system. he governing equations for the bulk are given by ν σ i j j = σ i j = G ν ε kkδ i j + ε i j ε i j = u i j + u ji coupled with the equations for surfaces and interfaces derived in [Gurtin and Murdoch 975a; 975b] ε S αβ = P i jε jk P kl + P i j ε jk P kl σ S αβ = τ δ αβ + µ S τ ε S αβ + λs + τ ε S κκ δ αβ + τ u S α Ș β [σ i j ]n j = σ S αβ Ș β. By convention the Roman subscripts denote the quantities belonging to the bulk and assume values from to 3 while the Greek refer to the surface/interface and assume values from to. Here u ε and σ denote displacement strain and stress fields in bulk; the material properties of the bulk are represented by the shear modulus G and Poisson s ratio ν; λ S µ S and τ refer to the surface Lamé constants and the residual surface stress when the bulk is unstrained respectively; [σ i j ] = σ i j out σ i j in and n is the outward unit normal to the surface; ε S and σ S denote the surface strain and surface stress fields on the surface. Since

5 766 CHANGWEN MI AND DEMIRIS A. KOURIS G ν a ŌO Ḡ ν r θ P r θ z x y G ν Ō c a Ḡ ν O x y z ϕ R P R θ z Figure. Spherical inhomogeneity in a half-space. the surface is defined as a two-dimensional continuous space the components of surface strain and surface stress with direction normal to the surface vanish. P i j = δ i j n i n j is the projection tensor which allows tensor transformation between the three-dimensional bulk space and the two-dimensional surface space. In terms of the projection tensor the surface strain is defined as the average value of the bulk strains from both sides projected onto the surface. he notion of surface gradient surface divergence of a vector and surface divergence of a superficial tensor can

6 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 767 be clarified by the following identities see [Gurtin et al. 998]: u S α Ș β = u ik P kj a α α S αβ Ș β = σ S βα α β Ș α = tr σ S βα α β Ș γ for every constant vector a where tr is the trace. For convenience the surface divergence of the surface stress tensor will be denoted by γ. Both in the embedded particle and surrounding matrix the displacement field satisfies the following Navier s equation without body forces: ν u j ji + u i j j =. In the case of axial symmetry without torsion its solution can be expressed as superposition of two displacement fields due to potentials φ and φ 3. In cylindrical coordinates these displacements and the corresponding strains and stresses are given by Gu r = φ r + z φ 3 r Gu z = φ z + z φ 3 z 3 4νφ 3 Gε rr = r Gε θθ = r φ r φ z z φ 3 r r z φ 3 z φ r + z φ 3 r r Gε zz = φ z ν φ 3 z + z φ 3 z Gε rz = φ r z ν φ 3 r + z φ 3 r z 3 σ rr = r σ θθ = r φ r φ z z φ 3 r r z φ 3 z ν φ 3 z φ r + z φ 3 r r ν φ 3 z σ zz = φ z ν φ 3 z + z φ 3 z σ rz = φ r z ν φ 3 r + z φ 3 r z u θ = ε rθ = ε θz = σ rθ = σ θz =

7 768 CHANGWEN MI AND DEMIRIS A. KOURIS where φ = φ 3 = and = r + r r +. Converting these equations z to spherical coordinates via z = R cos ϕ and r = R sin ϕ we get for the spherical components of displacements strains and stresses Gu R = φ R + µ R φ 3 R 3 4νφ 3 Gu ϕ = µ φ R µ µ φ 3 µ + 3 4νφ 3 Gε R R = φ R + µ R φ 3 R ν φ 3 R Gε θθ = R φ R µ R φ µ φ3 + µ R µ R φ 3 µ Gε ϕϕ = φ R φ R R + µ φ R µ φ 3 Rµ R µ φ 3 R Gε Rϕ = µ φ R µ R + ν + 3 4νµ φ R µ + ν φ 3 R µ φ 3 R µ + νµ R σ R R = φ R + φ 3 Rµ R νµ φ 3 R ν µ φ 3 R µ σ θθ = φ R R µ φ + νµ φ 3 µ R µ R ν + ν φ 3 R µ σ ϕϕ = φ R φ R R + µ φ R µ φ 3 Rµ R + νµ φ 3 R σ Rϕ = µ R φ µ R R φ 3 µ φ 3 µ + 3 νµ ν φ 3 R µ φ R µ + ν φ 3 R µ φ 3 R µ + νµ R φ 3 µ u θ = Gε Rθ = Gε θϕ = σ Rθ = σ θϕ = 4

8 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 769 where µ = cos ϕ and the harmonic operator now takes the form = R + R R µ R µ + µ R µ. For the purpose of completeness we now derive general expressions for surface/interface quantities both on the plane boundary z = c and on the matrix/inhomogeneity interface R = a. In cylindrical coordinates the projection tensor for the plane boundary is given by P rr = P θθ = all other P i j =. 5 Substituting Equations 3 and 5 into the first equation of and the first equation of the surface strain tensor and the surface gradient of the surface displacement on the plane boundary are found to be εrr f s = u rr f s = φ 4G r r + φ z ε f s θθ = u f s θθ = φ 4G r r + z r ε f s rθ = u f s rθ = φ 3 r + z r z= c φ 3 r + z φ 3 z z= c where the superscript f denotes the plane boundary. By the use of these relations and the second equation of it is found that σ f s rr = τ f χ f σ f s θθ = τ f + σ f s rθ = χ f r φ r r + χ φ f z + χ z φ 3 f r r + χ f z φ 3 z z= c φ r χ φ f z + χ z φ 3 f r r χ f z φ 3 z z= c where τ f denotes the deformation-independent surface stress defined on the plane boundary. he length scale parameters χ f χ f and χ f are defined in Equation 8 in the Appendix. he surface divergence of the surface stress tensor can be obtained by transforming the second equation of into cylindrical coordinates. An explicit formula for evaluating this vector can be found in [Duan et al. 5a.]. As a result the surface divergence of the surface stress γ f on the plane boundary is given by γr f 3 φ = χ f r z + z 3 φ 3 r z z= c γ f θ 6 = γ f z =. 7 For the matrix/inhomogeneity interface R = a the projection tensor is P θθ = P ϕϕ = all other P i j =.

9 77 CHANGWEN MI AND DEMIRIS A. KOURIS he relevant interface quantities can be derived from this equation together with and 4: εθθ s = us θθ = φ 4G R R µ φ R µ + µ φ 3 R µ φ 3 R µ R=a εϕϕ s = us θθ = µ φ 4G R µ φ R φ R R φ 3 Rµ R µ φ 3 R ε s θϕ = us θϕ = + ν + 3 4νµ R σθθ s = τ φ + χ R + χ φ R R χ µ φ R µ χ Rµ φ 3 R + χ µ φ 3 R νχ + χ 3 4νχ µ R σϕϕ s = τ φ + χ R χ φ R R + χ µ φ R µ χ Rµ φ 3 R χ µ φ 3 R σ s θϕ = + νχ + 3 4νχ χ µ R φ 3 µ R=a φ 3 µ φ 3 µ R=a R=a γ R = τ R + χ 3 φ R R + χ 3µ φ 3 R + νχ 3 µ φ 3 R µ R=a 8 γ θ = γ ϕ = χ µ φ µ R R + χ φ R R χ R 3 φ + µ φ 3 χ µ R + χ φ 3 R R 4 νχ R φ 3 φ 3 4νχ R + 3 8νχ + χ R φ 3 R R=a where τ is the deformation-independent interface stress defined at the spherical interface. he interface length scale parameters χ χ χ and χ 3 are defined in Equation 9 of the Appendix. Given the expressions for the surface divergence of the surface stress 7 the boundary conditions at the free surface can be expressed in cylindrical coordinates

10 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 77 as At infinity σ rz z= c = γ f r σ zz z= c =. 9 σ rr r = σ θθ r = σ rz r = σ zz r = corresponding to the far-field biaxial tension. he boundary conditions at the matrix/inhomogeneity interface consist of the displacement continuity condition and the third equation in. hese can be rewritten as follows using Equations 8 for the γ s: u R R=a = ū R R=a u ϕ R=a = ū ϕ R=a σ R R R=a σ R R R=a = γ R σ Rϕ R=a σ Rϕ R=a = γ ϕ where the quantities denoted by an over bar refer to the inhomogeneity. In cylindrical and spherical coordinates the general solution to the harmonic equation can be expressed in terms of cylindrical or spherical harmonics respectively. Based on these basic solutions four sets of displacement potentials are selected to represent the solutions to our problem. he first represents the biaxial tension applied at infinity and is given by φ = ν + ν R P µ φ 3 = + ν R P µ. Here P n µ is the Legendre function of the first kind of order n with P µ = P µ = µ P µ = 3µ / and so forth. In terms of cylindrical coordinates can be rewritten as φ = ν + ν z r φ 3 = + ν z. he disturbance due to the presence of the inhomogeneity can be expressed by the following three sets of displacement potentials: φ = φ = n= ψ λj λre λz dλ A n P n µ R n+ φ 3 = for the matrix with z > c R > a and φ = n= Ā n R n P n µ φ 3 = n= φ 3 = P n µ B n ; 3 Rn+ λψ λj λre λz dλ 4 n= B n R n P n µ 5

11 77 CHANGWEN MI AND DEMIRIS A. KOURIS for the inhomogeneity R < a. In these expressions J ν λr is the Bessel function of the first kind of order ν; A n B n Ā n and B n are unknown coefficients of the spherical harmonics; and ψ λ and ψ λ are unknown functions of the integral variable λ. With these choices of the displacement potentials the boundary condition is satisfied automatically. he unknown constants A n B n Ā n B n and the unknown functions ψ λ ψ λ are to be determined by enforcing the boundary conditions 9 and. 3. Solution of the problem he displacement potential 3 is expressed in spherical coordinates. In order to satisfy the boundary condition 9 it must be transformed into cylindrical coordinates. From [Sneddon [95] p.54] it is found that for z P n µ R n+ = λ n J λre λz dλ. Replacing z with z µ with µ and with the aid of the parity property of Legendre polynomials it is found that for z P n µ R n+ = n J λre λz λ n dλ. Using this the displacement potential 3 can be rewritten as where φ = ω λj λre λz dλ n= φ 3 = λω λj λre λz dλ 3 ω λ = A n n λ n / ω λ = B n n λ n /. 6 At this point the expressions for stress and the surface divergence of the surface stress on the plane boundary can be obtained readily by substituting potentials 3 and 4 into 3 and 7 see Equations 3 and 4. By satisfying the plane boundary conditions 9 we get the relations λ ψ + cλ + νψ e cλ ω + cλ + νω e cλ J λr dλ = n= χ λ 3 ψ cλψ e cλ + ω cλω e cλ J λr dλ λ ψ + cλ + νψ e cλ + ω + cλ + νω e cλ J λr dλ =.

12 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 773 his is satisfied identically if χ f λ+ψ χ f cλ +cλ +νψ = χ f λω e cλ +χ f cλ cλ +νω e cλ ψ + cλ+ νψ = ω e cλ cλ +νω e cλ. Solving this for ψ λ and ψ λ we obtain ψ λ = c+b λ 3 4νω + 4 ν ν cc+b λ ω e cλ / + b λ ψ λ = ω 3 4ν + c+b λω e cλ / + b λ 7 where b = νχ f. hese are the conditions necessary to satisfy Equation 9. he stress components on the plane boundary are reduced to 5 by substituting Equations 6 and 7 in 3. he next task is to satisfy the boundary conditions on the matrix/inhomogeneity interface. For this purpose we must transform the potential 4 into spherical coordinates. From [Morse and Feshbach 953 p. 38] it is found that for z e λz J λr = λr n P n µ. n= Replacing z with z µ with µ and with the aid of the parity property of Legendre polynomials it is deduced that for z e λz J λr = n λrn P n µ. n= With the use of this equation the displacement potential 4 can be expressed in spherical coordinates as where α n = φ = α n R n P n µ n= n λ n φ 3 = β n R n P n µ 4 n= ψ λ dλ β n = n λ n λψ λ dλ. Substituting 6 and 7 into these two equations expanding / + b λ in powers of λ for small b and using Euler s integral of the second kind λ n e cλ dλ = /c n+ n c >

13 774 CHANGWEN MI AND DEMIRIS A. KOURIS see [Gradshteyn and Ryzhik 994 p. 357] we can write where α n = fm n A m + gm n B m β n = m= qm n = m+n m! m+n m! = k= m + q n m+ A m + h n m B m 8 m= λ m+n e cλ + b λ dλ m + c m+n+ if b = m+n+k b k m + n + k! m!c m+n+k+ if b = f n m = 3 4νqn m + c + b m + q n m+ g n m = 4 ν νqn m m cc + b m + q n m+ h n m = 3 4νqn m c + b m + q n m+. he expressions for displacement stress and the interface divergence of the interface stress on the matrix/inhomogeneity interface can be derived by substituting displacement potentials into Equations 4 and 8. he elastic fields in the matrix are obtained by superposing potentials 3 and 4 while those in inhomogeneity are given by 5. hese expressions are listed in the Appendix. o satisfy the boundary condition we must have n= n= sa A n + s B B n + s B B n+ + s α α n + s β β n + s β β n+ Ɣ P sā Ā n + s B B n + s B B n+ n µ νa = 3 + ν P µ + a 3 P µ 9 ta A n + t B B n + t B B n+ + t α α n + t β β n + t β β n+ Ɣ P n tā Ā n + t B B n + t B B n+ µ = a 3 P µ i sa + i A A n + i sb + i B B n + i sb + i B B n+ + i sα + i α α n + i sβ + i β β n + i sβ + i β β n+ P n µ + Ɣ i sā i Ā Ā n + Ɣ i s B i B B n + Ɣ i s B i B B n+ n= τ = a + 4 νχ 3 χ3 P 3a + ν 3 µ + 3a + P 3 µ

14 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 775 l sa + l A A n + l sb + l B B n + l sb + l B B n+ + l sα + l α α n + l sβ + l β β n + l sβ + l β β n+ P + Ɣ l sā l Ā Ā n + Ɣ l s B l B B n + Ɣ n µ l s B l B B n+ n= 3χ χ = P µ 3a 3 where P n µ = d P nµ/ dµ Ɣ = Ḡ/G and the coefficients s t i l are defined in the Appendix page 788. In Equations 9 and the displacement containing α β represents the rigid body motion of the matrix. We set α = β = by viewing the matrix as a reference state. he displacement containing Ā B corresponds to the rigid body motion of the inhomogeneity. As a result one of the two coefficients can be set to zero so we set Ā = B =. Equating the coefficients of P n µ P n µ on both sides of Equations 9 and with the aid of Equation 8 we obtain a set of linear algebraic equations leading to the unknown coefficients {A n B n Ā n B n }. 4. Results and discussion Results were obtained in order to illustrate the impact of surface and interface effects on the elastic field in the vicinity of the inhomogeneity nanoparticle. he key parameters are the shear moduli ratio Ɣ and the inhomogeneity position and size denoted by c and a respectively. he shear modulus of the matrix is set to G = 6 GPa while Ɣ ranges over different values of the nanoparticle s stiffness. he limiting value Ɣ = represents a spherical void while Ɣ corresponds to a rigid particle. Poisson s ratios are kept equal ν = ν =.5 for the matrix and the inhomogeneity. he far-field biaxial tension is set at = MPa. Numerical computations indicate that the coefficients A n B n Ā n and B n decay monotonically with increasing n. hese coefficients converge more rapidly for small ratios a/c. As a result for all ratios a/c less than terms are necessary to obtain the elastic field with an accuracy of five significant figures. When surface and interface effects are ignored the results are compared with the classical solution of a semi-infinite elastic body having a spherical cavity or a spherical inhomogeneity [suchida and Nakahara 97; sutsui et al. 974] and were found to be in perfect agreement. he impacts of interface and free-surface elasticity were studied separately. First only the effects of interface elasticity were considered. he interface elastic constants were selected as λ s =6.857 N/m µ s =.8345 N/m and τ =.984 N/m. hese quantities can be deduced from the manipulation of the surface properties of an aluminum [] free surface in [Miller and Shenoy ].

15 776 CHANGWEN MI AND DEMIRIS A. KOURIS 4.. Interface elasticity only: stress distribution. Normalized stress components with and without interface elasticity are shown in Figures 4. Stresses at three specific points of the matrix/inhomogeneity interface vary with the ratio a/c radius of the particle normalized by the distance from the free surface. he particle position is fixed at c = 5 nm. It is clear that the stress changes significantly when the particle size is small for both soft Ɣ =.5 and hard Ɣ = particles. In some cases the interface effects even change the sign of the stress for small ratios of a/c. As expected the stress fields converge to the classical solutions when the inhomogeneity size increases. Numerical calculations are also performed to study the impact of interface elasticity on the stress distribution at R = a when a spherical particle of fixed size approaches the plane boundary. Results were similar to the ones illustrated in Figures 4. As long as the particle size is of nanometer size interface elasticity significantly changes the particle/matrix interface stresses even if the particle is pretty far from the plane boundary free surface. o examine the stress distribution along the interface calculations were carried out for various shear moduli ratios Figures 5 6. For the matrix softer particles Ɣ < are affected more than harder particles Ɣ >. As seen in Figure 6 top the maximum shear stress τ converges to the classical solution as Ɣ increases when the inhomogeneity is hard Ɣ > while it deviates greatly from the classical solution when the inhomogeneity is soft Ɣ <. he maximum principal stress σ is compressive while its classical counterpart is tensile everywhere. he sign of σ has been altered by the interface effects. Unlike σ σ is more sensitive to harder particles than to softer ones. Compared to σ τ is only slightly affected by interface elasticity. It remains tensile at every point around the interface. he maximum variation of τ from the classical solution occurs at ϕ = π for soft inhomogeneity Ɣ <. Figure 7 illustrates the variation of stresses σ rr and σ θθ with r/c at the plane boundary. he impact of interface elasticity on σ rr and σ θθ is analogous to that on σ and τ : the softer the particle the more significant the interface elasticity becomes. However for large values of r/c 3 σ rr and σ θθ converge to the classical solution. By limit analysis the solutions of the present problem with only the interface effects should approach those of a spherical inhomogeneity with interface effects in an infinite medium when a c. With the ratio a/c. the elastic field at the spherical interface is in agreement with the one obtained for the infinite medium with an accuracy of at least five significant figures. 4.. Free-surface elasticity only: stress distribution. Calculations were performed to determine the free-surface elasticity effects on the stress distribution at z = c

16 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY classical solution with interface effects c = 5 nm =.5 =.5 =.5 = = = Radial stress RR Ratio a/c Radial stress RR classical solution with interface effects c = 5 nm =.5 =.5 =.5 = = = Ratio a/c Figure. Variation of σ R R top and σ R R bottom versus a/c around the spherical interface at ϕ = π/ π.

17 778 CHANGWEN MI AND DEMIRIS A. KOURIS.4. Hoop stress classical solution with interface effects c = 5 nm =.5 =.5 =.5 = = = Hoop stress.5.5 classical solution with interface effects c = 5 nm =.5 =.5 =.5 = = = Ratio a/c Figure 3. Variation of σ θθ top and σ θθ bottom versus a/c around the spherical interface at ϕ = π/ π.

18 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY angential stress.5.5 classical solution with interface effects c = 5 nm =.5 =.5 =.5 = = = angential stress classical solution with interface effects c = 5 nm =.5 =.5 =.5 = = = Ratio a/c Figure 4. Variation of σ ϕϕ top and σ ϕϕ bottom versus a/c around the spherical interface at ϕ = π/ π.

19 78 CHANGWEN MI AND DEMIRIS A. KOURIS.5 Maximum principal stress.5.5 classical solution with interface effects c = nm a = 8 nm =. =.5 = = º 3º 6º 9º º 5º 8º.5 classical solution with interface effects c = nm a = 8 nm =. =.5 = = Maximum principal stress º 3º 6º 9º º 5º 8º Figure 5. Variation of maximum principal stress σ top and σ bottom with ϕ at the spherical interface.

20 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 78.5 Maximum shear stress.5.5 classical solution =. with interface effects =.5 = c = nm a = 8 nm = º 3º 6º 9º º 5º 8º...9 classical solution with interface effects c = nm a = 8 nm =. =.5 = = Maximum shear stress º 3º 6º 9º º 5º 8º Figure 6. Variation of maximum shear stress τ top and τ bottom with ϕ at the spherical interface.

21 78 CHANGWEN MI AND DEMIRIS A. KOURIS Radial stress r r / classical solution with interface effects c = nm a = 8 nm =. =.5 = = Normalized distance r/c Hoop stress / classical solution with interface effects c = nm a = 8 nm =. =.5 = = Normalized distance r/c Figure 7. Variation of σ rr top and σ θθ bottom with r/c at the plane boundary.

22 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY Shear stress r z / classical solution with plane surface effects c = nm a = 8 nm =. =.5 = = Normalized distance r/c Figure 8. Variation of σ rz with r/c at the plane boundary. and R = a. We consider an aluminum [] free surface with properties λ f s = N/m µ f s = N/m and τ f = N/m [Miller and Shenoy ]. he most important character of the plane surface effects is the nonvanishing shear stress σ rz at the plane boundary as shown analytically in 5. In the classical solution the conditions of zero traction at the plane boundary force σ rz to vanish. Figure 8 shows the distribution of σ rz with the normalized radial coordinate r/c for c = nm a = 8 nm and various shear moduli ratios. When the particle stiffness is not too different from the surrounding material Ɣ =.5 surface elasticity is not important. Otherwise Ɣ =. its influence is significant. Nevertheless the magnitude of σ rz is small %. he impact of the free surface effects on σ rr and σ θθ is found to be of the same order as that of σ rz. he influence on the stress fields at the matrix/inhomogeneity interface is even smaller. One can safely conclude that compared to the applied tension at infinity the free surface elasticity may be neglected. 5. Summary and conclusions he axisymmetric problem of a spherical particle near the surface of a semi-infinite elastic body was solved incorporating the effects of surface/interface elasticity.

23 784 CHANGWEN MI AND DEMIRIS A. KOURIS he solution was obtained within the framework of linear elasticity through a displacement potential formulation. he free surface and the interface were modeled as two-dimensional continuous spaces with vanishing thickness. Aside from the bulk material properties the surface and interface boundaries have their own material constants for example the residual surface stress and surface Lamé constants. Numerical calculations indicate that the stress distribution along the plane boundary and the matrix/particle interface is significantly affected by surface/interface elasticity. Following the analysis one can make a few important observations: he impact of interface elasticity is a function of the particle size: the smaller the particle the more important interface elasticity becomes. he elastic fields converge to the classical solution as the particle size increases typically for a nm. he influence of interface elasticity depends on the softness or rigidity of the particles. For the matrix the interface effects on the stress distribution are more pronounced when the particles are soft. However the opposite is true for the particles themselves. In contrast with the zero traction requirement along the free surface in the classical solution plane elasticity yields a nonvanishing shear stress as shown in 5 and Figure 8. he plane surface effects become more important when the particle approaches the plane boundary. However for distances where continuum theory is considered valid typically 5 nm the magnitude of the disturbance due to the plane surface elasticity is small when compared to the applied load. Due to the presence of the residual interface stress see [He and Li in press] superposition will not be valid when one considers interface elasticity. his is critical because it significantly complicates the problem when multiple loads say misfit strain and mechanical load are applied separately. he study presented here illustrates that models incorporating nanoparticles near surfaces cannot ignore surface and particularly interface elasticity effects. his is especially true when one models self-organized adatom clusters and islands since most of these structures are of nanoscale size. he important problem of strained islands due to misfit strains will be addressed in a separate communication.

24 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 785 Appendix Elastic fields at the plane boundary. he stresses along the plane boundary are given below in terms of potentials 3 and 4: σrr z= c = λω cλω λj λr r J λr e cλ dλ νλ ω J λre cλ dλ λψ cλψ λj λr r J λr e cλ dλ + νλ ψ J λre cλ dλ σθθ z= c = λ r ω cλω J λr + νλω J λr e cλ dλ λ r ψ cλψ J λr νλψ J λr 3 e cλ dλ σzz z= c = λ ω cλ + νω J λre cλ dλ + λ ψ + cλ + νψ J λre cλ dλ σrz z= c = λ ω cλ + νω J λre cλ dλ + λ ψ + cλ + νψ J λre cλ dλ. he surface divergence of the surface stress on the plane boundary is γ f r = χ f λ 3 ω cλω J λre cλ dλ + λ 3 ψ cλψ J λre cλ λ. 4 σrr With Equations 6 and 7 Equation 3 is reduced to σθθ z= c = + n A n + n= 4 Fn+ + 4 ν r n B n 4 ν Fn+ n= z= c = + n A n + n= + 4c Fn+ F n+ F n 4 ν ν r 4 νc r 4 ν F n+ r 4ν Fn+ n B n 4ν ν Fn+ n= + + 4νc Fn+ 4 ν ν r F n + 4c ν r 5 F n+ F n+

25 786 CHANGWEN MI AND DEMIRIS A. KOURIS σrz z= c = 4b n F n+3 A n n= σ zz z= c = + B n ν Fn+ + c Fn+3 6 where Fν n = J ν λre cλ λ n dλ. 7 + b λ With the aid of the aylor s series expansion of / + b λ for small parameter b and the relation J ν λre cλ λ n n ν! c dλ = r + c n+/ Pν n c > v + n > r + c from [Gradshteyn and Ryzhik 994 p ] Equation 7 can be approximated as Fν n m b m m + n + ν! c r + c m+n+/ Pν m+n r + c m= where Pn ν µ is the associated Legendre function of order n and degree ν. he length parameters in 6 are defined as χ f = λ f s + µ f s /4G χ f = λ f s + τ f /4G χ f = χ f χ f 8 where λ f s µ f s denote the surface Lamé moduli for the plane boundary z = c. Elastic fields at the spherical interface. he displacements and stresses in the matrix at R = a are given by Gu R ν R=a = 3 a + ν δ n δ n + s A A n + s B B n + s B B n+ n= + s α α n + s β β n + s β β n+ P n µ Guϕ R=a = µ 3 aδ n + t A A n + t B B n + t B B n+ σr R R=a = σrϕ n= 3 δ n δ n + i A A n + i B B n + i B B n+ n= + t α α n + t β β n + t β β n+ P n µ + i α α n + i β β n + i β β n+ P n µ R=a = µ 3 δ n + l A A n + l B B n + l B B n+ n= + l α α n + l β β n + l β β n+ P n µ

26 σθθ σϕϕ NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 787 R=a = + R=a = + A n a n+3 B n n ν n a n+ β n a n n ν n n= B n+ n+ ν + n+3ν n+3a n+3 α n a n β n+a n n+3ν + n+ ν µp n n+3 µ δ n A nn+ a n+3 n= n= B n n ν n a n+ B n+n+ n+ ν + n+3ν n+3a n+3 An + β n a n nn ν n + α n a n n + β n+a n n+ n+ ν n+3ν n+3 a n+3 + B n n 3 ν n a n+ + β n a n n 3 ν n P n µ B n+ n+3 ν n+3 ν n+3a n+3 + α n a n β n+a n n+3 ν n+3 ν µp n n+3 µ 3 δ n+δ n A nn+ a n+3 n= B n n n ν n a n+ B n+n+ n+n + ν + n+3 ν n+3a n+3 α n a n n β n a n n nn +ν n β n+a n n+ n+n + +ν + n+3 ν P n µ. n+3 he displacements and stresses in the inhomogeneity at R = a are given by Ḡū R R=a = sā Ā n + s B B n + s B B n+ P n µ n=

27 788 CHANGWEN MI AND DEMIRIS A. KOURIS Ḡūϕ σr R σrϕ σθθ σϕϕ R=a = µ tā Ā n + t B B n + t B B n+ P n µ n= R=a = i Ā Ā n + i B B n + i B B n+ P n µ n= R=a = µ R=a = + n= n= R=a = n= n= l Ā Ā n + l B B n + l B B n+ P n µ n= Ā n a n + B n a n n ν n + B n+ a n n+3 ν + n+ ν n+3 Ā n a n n + B n a n nn ν n + B n+ a n n+ n+ ν n+3 ν n+3 Ā n a n + B n a n n 3 ν n B n+ a n n+3 ν n+3 ν µp n µ P n µ µp n µ n+3 Ā n a n n + B n a n n nn + ν n + B n+ a n n+ n+n + + ν + n+3 ν P n µ. n+3 n= he interface divergence of the interface stress at R = a is given by γ R = a τ + 4 νχ 3 δ n χ 3 3a+ν 3a δ n +i SA A n +i SB B n +i SB B n+ +i Sα α n +i Sβ β n +i Sβ β n+ +Ɣ i SĀ Ā n +i S B B n +i S B B n+ P n µ γ ϕ = µ 3χ χ δ n +l SA A n +l SB B n +l SB B n+ +l Sα α n 3a n= +l Sβ β n +l Sβ β n+ +Ɣ l SĀ Ā n +l S B B n +l S B B n+ P n µ. In these equations the coefficients are defined as

28 NANOPARICLES UNDER HE INFLUENCE OF SURFACE/INERFACE ELASICIY 789 s A = n+ a n+ t A = a n+ s B = nn+3 4ν n a n t B = n 4+4ν n a n s B = n+n+5 4ν n+3a n+ t B = n+5 4ν n+3a n+ s α = na n δ n t α = a n δ n s β = nn 4+4νan δ n n s β = n+n +4νan+ n+3 t β = n 4+4νan δ n n t β = n+5 4νan+ n+3 sā = na n tā = a n s B = nn 4+4 νan δ n n s B = n+n +4 νan+ n+3 t B = n 4+4 νan δ n n t B = n+5 4 νan+ n+3 i A = n+n+ a n+3 i SA = n+n+χ 3 a n+4 i B = nn +3n ν n a n+ i S B = n nn+ νn χ3 n a n+ i B = n+n+n+5 4ν n+3a n+3 i SB = n+n+n+5 4νχ 3 n+3a n+4 i α = n na n i Sα = n nχ 3 a n 3 i β = n nn 4+4νan n i Sβ = n nn 4+4νχ 3a n 3 n i β = n+n n νa n i Sβ = n+3 n+ nn+ + νn+ χ 3 a n n+3 i Ā = n na n i SĀ = n nχ 3 a n 3 i B = n nn 4+4 νan n i S B = n nn 4+4 νχ 3a n 3 n i B = n+n n νa n i S n+3 B = n+ nn+ + νn+ χ 3 a n n+3

29 79 CHANGWEN MI AND DEMIRIS A. KOURIS l A = n+ a n+3 l S A = n+nχ +χ a n+4 n 3 n ν n + 4 ν χ l B = n +ν n a n+ l S B = + n + 4 νn χ n a n+ l B = n+n+5 4ν n+3a n+3 l SB = n+n+5 4νnχ +χ n+3a n+4 l α = n a n l Sα = n n+χ χ a n 3 l β = n n 4+4νan n l β = n +n +νa n l Sβ = n+3 l Sβ = n n 4+4ν n+χ χ a n 3 n n 3 +n 5 4ν+n5 8ν 3 χ n+ 4 νn+ χ a n n+3 l Ā = n a n l SĀ = n n+χ χ a n 3 l B = n n 4+4 νan n l B = n +n + νa n l S n+3 B = he length scale parameters are defined as l S B = n n 4+4 ν n+χ χ a n 3 n n 3 +n 5 4 ν+n5 8 ν 3 χ n+ 4 νn+ χ a n n+3 χ = 4G λs +µ S χ = 4G λs +τ χ = χ χ χ 3 = χ + χ where λ S µ S are the interface Lamé moduli for spherical interface R = a. References [Adamson 98] A. W. Adamson Physical chemistry of surfaces Wiley New York [Cahn and Larche 98] J. W. Cahn and F. Larche Surface stress and the chemical equilibrium of small crystals. II. solid particles embedded in a solid matrix Acta Metall. 3: [Duan et al. 5a] H. L. Duan J. Wang Z. P. Huang and B. L. Karihaloo Eshelby formalism for nano-inhomogeneities P. Roy. Soc. Lond. A Mat. 46: [Duan et al. 5b] H. L. Duan J. Wang Z. P. Huang and Z. Y. Luo Stress concentration tensors of inhomogeneities with interface effects Mech. Mater. 37: [Gradshteyn and Ryzhik 994] I. S. Gradshteyn and I. M. Ryzhik able of integrals series and products 5th ed. Academic Press San Diego 994.

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