Wedge disclination dipole in an embedded nanowire within the surface/interface elasticity

1 Introduction

In recent years, fabrication and application of nanowires in electronics, optoelectronics, optics, sensors, etc., has increased extensively (see, e.g., the reviews [1–11] and references therein). For practical use, nanowires are often covered by nanolayers of other materials (core-shell nanowires) and/or embedded in bulk parts of devices or matrices. As a result, in the process of fabrication and utilization, they are subjected to elastic stresses and strains of different origin, which can relax through generation of various defects. For example, it was theoretically shown that misfit strains in the core-shell and embedded nanowires can relax by straight misfit edge [12–15] and screw [16] dislocations, misfit wedge disclinations and dislocation walls [17], prismatic [14, 15, 18–21] and glide [22] dislocation loops, and penny-shape cracks [21]. Experimental evidence of misfit dislocations was demonstrated in GaP-GaN [23], Ge-Si [24], AlN-GaN [25], and InAs-GaAs [26–28] core-shell nanowires.

The role of wedge disclinations in structural transformations, misfit strain relaxation, and crack nucleation in various micro- and nanowire structures was theoretically described by many authors (see, e.g., refs. [17, 29–51]). Some of them used atomic simulations [30, 37, 38, 42–45], whereas the others dealt with the usual continuum approach within the classical theory of elasticity [17, 29, 31–36, 39–41, 46–51]. The experimental images of disclinated micro- and nanowires can be found, for example, in refs. [36, 39, 48, 50].

The continuum description of disclination behavior in micro- and nanowires is commonly based on the well-known solution of the boundary-value problem in the classical theory of elasticity for a wedge disclination in an infinite cylinder with a free surface (see ref. [32] for details). There are also classical elasticity solutions for disclination dipoles interacting with cylindrical [52] and angular [53] inhomogeneities in infinite matrices. However, when characteristic sizes of a problem are in the order of the intrinsic length scale of the material, the classical theory of elasticity ceases to hold. For example, the defect core size may be in the same order of magnitude as the nanowire radius. In contrast, the defect structures are generally nucleated near free surfaces and interfaces that demonstrate remarkable diversities with the bulk material. With decreasing the characteristic sizes to the scale of some nanometers, the ratio of the number of atoms on the surface/interface to the number of atoms in the bulk drastically increases so that the surface/interface energy cannot be neglected in comparison with the bulk energy. The nanoscopic description of phenomena, which are strongly influenced by the surface/interface effects, may be addressed, in particular, by using the so-called surface/interface elasticity approach.

On the basis of the general concept of surface/interface stress in solids by Gibbs [54] and developed for solving the elastic problems by Gurtin and Murdoch [55, 56] and Gurtin et al. [57], this approach has widely been used to study the elastic fields of nanosized inclusions and inhomogeneities [58–60], the elastic behavior of dislocations inside [61, 62] and near [63–66] embedded circular [61–64] and elliptical [65, 66] nanoinhomogeneities, inside [67–69] and near [70, 71] embedded core-shell nanowires, at the nanotube/matrix interface [72], and in free-standing nanotubes [73, 74] and core-shell nanowires [75, 76], and the elastic behavior of wedge disclination dipoles near an embedded circular nanoinhomogeneity [77] and in the shell of a core-shell nanowire [78].

It is seen from this list that the surface/interface effects on the disclination solutions are studied much less as compared with dislocations. Nevertheless, the first works in this field show new interesting results. In particular, Lou and Liu [77] revealed that when disclinations are placed near the interface, some non-classical phenomena such as extra repelling or attracting image forces, their size dependence, and new equilibrium positions of the disclination dipole can be detected. Our recent study [78] has demonstrated that surface/interface effects strongly depend on the type of the surface/interface as well as on the bulk nanowire parameters. For example, for negative values of the surface/interface modulus and relatively small values of the ratio of the shell and core shear moduli, the surface/interface effect manifests itself through non-classical stress oscillations along the surface/interface. However, if the surface/interface modulus is positive, the surface/interface effects are rather weak. On the basis of these results, it seems reasonable to consider some more disclination problems within the surface/interface elasticity approach, keeping also in mind their potential application in modeling structural transformations and plastic deformation in various heterogeneous nanostructures.

In the present work, we apply the surface/interface elasticity approach to the case of a wedge disclination dipole in a nanowire embedded in an infinite matrix. The governing equations of the interface elasticity are solved by means of a complex variable method. In contrast with the solution obtained within the classical theory of elasticity (the classical solution), our results demonstrate that the stress field of the disclinated nanowire is strongly dependent on the interface effect. The non-classical digressions from the classical solution become more pronounced by moving the dipole in the vicinity of the interface. The wedge disclination can be repelled or attracted by the interface.

2 Model and solution

Consider an infinitely extended elastically isotropic nanowire of radius a embedded in an infinite matrix (Figure 1A). The shear modulus and Poisson’s ratio of the nanowire are μ₁ and v₁, and those of the matrix are μ₂ and v₂, respectively. Let the nanowire contain a two-axes dipole of wedge disclinations with Frank vectors +ω and -ω, and their position vectors be c_pw and c_nw, respectively. The nanowire/matrix interface, Ω, is characterized by Lamé constants μ_Ω and λ_Ω and the interface residual stress τ_Ω.

Figure 1

(A) A two-axes dipole of wedge disclinations with arbitrary orientation of its arm in a nanowire embedded in an infinite matrix. (B) The equivalent configuration of two disclination dipoles with orthogonal arms. (C) Finite walls of infinitesimal edge dislocations modeling the two disclination dipoles.

As in the classical theory of the plane-strain elasticity, the stress and displacement fields in the bulk are defined in terms of two complex potentials, Φ(z) and Ψ(z) [79]:

where k=3–4 ν and the complex variable z is related to the Cartesian coordinates (x, y) as z=x+iy.

If the interface adheres to the bulk regions without slipping, then in the absence of body forces, the equilibrium and constitutive equations can be summarized as follows [55, 57]:

for the bulk, and

for the interface. Here, κ_αβ represents the curvature tensor of the interface, n_i is the normal vector on the interface, ε_ij is the infinitesimal strain tensor, σ_ij is the corresponding stress tensor, δ_αβ is the Kronecker delta, and is the interface stress tensor. The symbol 〈X〉 denotes the jump in the value of the quantity X across the interface. It should be noted that the Greek indices take on values 1 and 2, whereas Latin subscripts adopt values 1, 2, and 3.

By combining Eqs. (1)–(3) and (6)–(8), two interface conditions in terms of the complex potentials are obtained:

in which

and

Hereinafter, Φ₁ and Ψ₁ are the complex potentials pertinent to the nanowire, whereas Φ₂ and Ψ₂ denote those of the matrix.

To model a wedge disclination dipole with an arbitrary orientation of the dipole arm, it is convenient to decompose the dipole into a “horizontal” and a “vertical” dipole (Figure 1B). The horizontal (vertical) dipole can be modeled as a finite wall of continuously distributed infinitesimal edge dislocations (Figure 1C), which are characterized by the Burgers vector magnitude δb_y (δb_x) and the interdislocation spacing δl. Any field quantity Θ^∇(x, y) of the arbitrary disclination dipole is then calculated as

where the densities of distribution of the dislocations in the walls vary linearly as ρ(x′)=ρ(y′)=1/δl, and Θ^⊥(x-x′, y-y′) is the elastic field of an edge dislocation with Burgers vector magnitude δb_x=δb_y placed at the point (x′, y′). The vector c_pw=(c_px, 0) shows the location of the positive wedge disclination along the x-axis, whereas the vector c_nw=(c_nx, c_ny) denotes the position of the negative wedge disclination.

The potential functions needed to describe the elastic fields of an edge dislocation with Burgers vector b=b_xe_x+b_ye_y at the point c=(c_x, c_y) inside the nanowire are defined as follows:

with γ=μ₁(b_y-ib_x)/[4π(1-ν₁)]. Expanding the non-analytical parts of and into Maclaurin series and employing the boundary conditions together, the unknown coefficients and are determined.

3 Results

As mentioned earlier, the wedge disclination dipole is modeled by a wall of edge dislocation distribution over the dipole arm. The integrations in Eq. (11) are carried out with respect to the edge dislocation distributions over the components of the dipole arm along x- and y-directions. Consequently, by similar integrations of the potential functions associated with a single edge dislocation, Φ^⊥(z) and Ψ^⊥(z), the potential functions for the corresponding wedge disclination dipole can be obtained. The numerical study in the present work will concentrate on the case of a wedge disclination dipole located on the x-axis inside an embedded nanowire, for which the potential functions read as follows:

where λ=μ₁ω/[4π(1-ν₁)] and the radial distance from the center of the nanowire to the positive and the negative wedge disclinations would be referred as c_pw and c_nw, respectively. The unknown coefficients of the complex potentials are calculated by a simple integration of the coefficients and with respect to the parameter c; for example,

In the context of the present study, the interface modulus is defined as K_Ω=2μ_Ω+λ_Ω-τ_Ω. In the classical theory, τ_Ω=μ_Ω=λ_Ω=0, and so the interface effect is discounted.

3.1 Stress distribution

Figure 2 illustrates the variation of the stress component σ_rr along the interface of an embedded nanowire with a=2 nm, ν₁=ν₂=0.3, and μ₂/μ₁=0.5 and 2, due to the presence of a wedge disclination dipole located on the x-axis with c_pw=0.5 nm and c_nw=1.75 nm. The plots are given for different values of the interface modulus and residual stress. Hereinafter, the superscripts (1) and (2) over the stress components imply that the corresponding stress acts in the nanowire and the matrix, respectively. In Figure 2, and are in particular the radial stress values on the inner and outer sides of the interface, respectively. As expected, the classical solutions for and are identical here, whereas the interface-elasticity (non-classical) solution is discontinuous across the interface. The effects of the negative and positive interface modulus (2μ_Ω+λ_Ω=±0.3 μ₁ nm with τ_Ω=0) are studied in Figures 2A and B. The most obvious feature of the non-classical stresses is the oscillations in the stress field along the interface for K_Ω<0. The oscillation amplitude is higher in the matrix, especially when the matrix is softer than the nanowire (see Figure 2B). At the same time, the positive interface modulus only slightly shifts the radial stress () without any oscillations.

Figure 2

Distribution of the radial stress component along the nanowire/matrix interface for a=2 nm, ν₁=ν₂=0.3, μ₂/μ₁=2 (A, C) and 0.5 (B, D), and different values of K_Ω at τ_Ω=0 (A, B) and different values of τ_Ω at μ_Ω=λ_Ω=0 (C, D). Stress values are given in units of μ₁ω/10.

Similar non-classical oscillations were first observed in the case of an edge dislocation inside the wall of a nanotube [74]. The authors inferred that the oscillations are due to the rippling effect of the surface of the nanotube. Recently, similar stress oscillations have been revealed in various cylindrical nanostructures containing edge dislocations [69, 76] and wedge disclinations [78]. Gutkin et al. [69] have shown that the appearance of stress oscillations along the inner interfaces is caused by the negative values of the interface moduli, whereas the positive values of the interface moduli cause no oscillations. The same has occurred true for core-shell nanowires containing edge dislocations [76] or wedge disclinations [78] in their shells. In discussion of this interface effect in ref. [78], we have noted that the physical origin of stress oscillations under negative values of the interface moduli is not well understood yet, and supposed that the negative values stimulate some additional degrees of freedom in the form of shape instability for interfaces of enhanced curvature. In terms of work [74], it was treated as surface rippling that could be attributed to interfaces as well.

Figures 2C and D illustrate the effects of the residual stress on the radial stress along the interface for τ_Ω/μ₁=-0.1, 0, and 0.1 nm and μ_Ω=λ_Ω=0. In general, positive values of τ_Ω result in oscillations of and along the interface. For the case of softer matrix (here at μ₂/μ₁=0.5, see Figure 2D), the stress oscillations become remarkable, and again, the oscillation amplitude is higher in the matrix. However, in the case of softer nanowire (here at μ₂/μ₁=2, see Figure 2C), the oscillations are negligibly small. In contrast, the negative residual stress does not cause any oscillations in σ_rr.

Thus, the stress oscillations appear when either the interface modulus K_Ω is negative (at τ_Ω=0) or the residual stress is positive (at μ_Ω=λ_Ω=0).

Figures 3A and B depict the combined effect of the interface modulus, K_Ω, and the residual interface stress, τ_Ω, on the distribution of along the nanowire/matrix interface; the pairwise values of K_Ω and τ_Ω considered in these figures are (K_Ω, τ_Ω)=(±0.3μ₁, ±0.1μ₁) nm and (±0.1 μ₁, ±0.1 μ₁) nm. As it is observed, the negative modulus results in oscillatory interface stress distribution that is amplified for positive values of τ_Ω. Whereas, a positive interface modulus combined with a negative residual stress does not cause tangible digressions from the classical solutions. Comparison of the stress distribution in Figures 3A and B reveals that the effect of nanowire/matrix interface is more pronounced when the matrix is softer than the nanowire (μ₂/μ₁=0.5).

Figure 3

Distribution of the stress component along the nanowire/matrix interface for a=2 nm, ν₁=ν₂=0.3, μ₂/μ₁=2 (A) and 0.5 (B), and different values of K_Ω and τ_Ω. Stress values are given in units of μ₁ω/10.

3.2 Strain energy

The strain energy of a wedge disclination dipole in a solid can be calculated as the work done to generate it in its own stress field [32]. When a wedge disclination is placed in the nanowire, the strain energy per unit disclination length can be calculated by making the disclination cut from the surface of the imaginary disclination core of radius r_c to the point (x=h, y=0) on the remote outer surface of the solid. Then, the strain energy of the dipole reads

By breaking down the integration limits over the nanowire and matrix regions, the strain energy is given by

In doing numerical calculations, we assume that the core radius r_c is negligibly small, h≈1 μm, a=4 nm, ν₁=ν₂=0.3, μ₂/μ₁=0.5 and 2, and the dipole arm is 1 nm. Figure 3 shows the dependence of the strain energy W on the position c_t of the central point of the dipole with respect to the nanowire axis. For μ₂/μ₁=2 (0.5), the strain energy monotonically increases (decreases) with increasing c_t. The effect of different interface moduli (K_Ω/μ₁=-0.3, 0, and 0.3 nm) in the absence of the residual interface stress, τ_Ω=0 is demonstrated in Figure 4A. As is seen, the positive (negative) interface modulus leads to a larger (smaller) value of the strain energy as compared with the corresponding classical solution. The differences between the classical and non-classical solutions are very small when the dipole is near the nanowire axis and gradually grow with increasing c_t, as the dipole approaches the interface. This non-classical interface effect was also reported by Gutkin et al. [69] for the case of an edge dislocation inside the core of a core-shell nanowire embedded in an infinite matrix.

Figure 4

Dependence of the strain energy of a wedge disclination dipole on its position c_t for a=4 nm, μ₂/μ₁=0.5 and 2, ν₁=ν₂=0.3, and (A) different values of K_Ω at τ_Ω=0, and (B) different values of τ_Ω at μ_Ω=λ_Ω=0. The energy values are given in units of μ₁a²ω²/100.

Figure 4B shows the interface effect on the disclination dipole strain energy in the two cases when τ_Ω/μ₁=±0.1 nm and λ_Ω=μ_Ω=0. One can see that the positive (negative) residual interface stress decreases (increases) the strain energy. It is worth noting that the interface effect is more pronounced when the nanowire is stiffer than the matrix.

3.3 Image forces

The image force f, acting on the wedge disclination dipole, can be obtained as follows:

where W is given by Eq. (18).

Figure 5 demonstrates the dependence of the image force acting on the disclination dipole of arm 1 nm, on its position c_t when K_Ω/μ₁=±0.3 nm and τ_Ω=0 (Figure 5A) and when τ_Ω/μ₁=±0.1 nm and λ_Ω=μ_Ω=0 (Figure 5B). The system is characterized by the following set of parameters: a=4 nm, μ₂/μ₁=0.5 and 2, and ν₁=ν₂=0.3. As is seen, the dipole is attracted to (repulsed from) the interface when the matrix is softer (stiffer) than the nanowire. The interface with positive (negative) K_Ω (at τ_Ω=0, see Figure 5A) exerts an extra repulsive (attractive) force on the dipole, whereas the positive (negative) τ_Ω (at λ_Ω=μ_Ω=0, see Figure 5B) exerts an extra attractive (repulsive) force on it. These non-classical interface effects are more pronounced in the vicinity of the interface. A similar interface effect was revealed for an edge dislocation inside the core of a core-shell nanowire embedded in an infinite matrix [69].

Figure 5

Dependence of the image force acting on a wedge disclination dipole on its position c_t for a=4 nm, μ₂/μ₁=0.5 and 2, ν₁=ν₂=0.3, and (A) different values of K_Ω at τ_Ω=0, and (B) different values of τ_Ω at μ_Ω=λ_Ω=0. The image force values are given in units of μ₁aω²/100.

4 Conclusions

The stress field and the strain energy of a two-axes dipole of wedge disclinations within an embedded nanowire are obtained in the context of both the surface/interface elasticity and classical elasticity theories. Moreover, the classical and non-classical image forces acting on the dipole are computed and compared in these frameworks. In doing so, an exact analytical method incorporating two complex potentials has been used.

In contrast with the classical expectations, the non-classical radial stress component is discontinuous along the interface. The negative interface modulus or positive residual stress causes oscillations in the stress field. In contrast, a positive interface modulus or a negative residual stress does not show strong digressions from the classical solutions.

The strain energy of the disclination dipole noticeably depends on its position in the nanowire, on the ratio of the shear moduli of the matrix and nanowire, and on the interface properties. The strain energy increases with the stiffness of the matrix with respect to the nanowire. When the nanowire is softer than the matrix, the disclination dipole tends to occupy its stable equilibrium position at the nanowire axis. In the opposite case, the disclination dipole tends to move toward the interface. The positive (negative) interface modulus leads to greater (lower) values of the strain energy; meanwhile, a negative (positive) residual stress augments (decreases) the strain energy. These effects become more pronounced when the disclination dipole approaches the interface.

The interface with negative modulus (positive residual stress) adds an extra attractive force to the image force exerted by the bulk material, whereas the interface with positive modulus (negative residual stress) repels the disclination dipole.