Interface effect on the formation of a dipole of screw misfit dislocations in an embedded nanowire with uniform shear eigenstrain field

https://doi.org/10.1016/j.euromechsol.2014.12.006Get rights and content

Highlights

  • Gurtin surface/interface elasticity as well as classical solutions.

  • Generation of a screw misfit dislocation dipole at a nanowire-matrix interface.

  • Remarkable interface effects on the critical radius.

  • Dependence of the critical radius on the shear eigenstrain inside the nanowire.

  • Dependence of the critical radius on the stiffness of the nanowire and matrix.

Abstract

The critical condition for the generation of a screw misfit dislocation dipole (MDD) at the interface between a nanowire (NW) with uniform shear misfit strain and its surrounding unbounded matrix within surface/interface elasticity theory is of particular interest. The analysis is carried out using the complex potential variable method. It is shown that the critical radius of the NW corresponding to the onset of the MDD generation decreases with the increase in the uniform shear eigenstrain inside the NW as well as when the stiffness of the NW increases with respect to the matrix. The critical radius strongly depends on the non-classical interface parameter. Comparison is made with classical solution, which can be obtained as the special case of the surface/interface elasticity theory.

Introduction

In recent years, extensive studies of embedded and core–shell nanowires (NWs), which are considered as prominent elements of devices of nano- and optoelectronics, photonics, plasmonics, sensors, etc. (Garnett et al., 2009, Garnett et al., 2011, Han et al., 2012, Hayden et al., 2005, Hochbaum and Yang, 2009, Kenry and Lim, 2013, Lauhon et al., 2002, Lauhon et al., 2004, Law et al., 2006, Lin et al., 2011, Lu et al., 2008, Qian et al., 2004, Qian et al., 2008, Xiang et al., 2006, Yan et al., 2011), have stimulated much attention to the stability of their structure and unique electronic and optic properties. In particular, the problem of misfit dislocations (MDs) appearing on interfaces between NWs and surrounding media, has been studied by many authors, both theoretically (Aifantis et al., 2007, Chu et al., 2011, Chu et al., 2013, Colin, 2010, Enzevaee et al., 2013, Fang et al., 2008a, Fang et al., 2009c, Gutkin et al., 2000, Gutkin et al., 2011, Haapamaki et al., 2012, Liang et al., 2005, Ovid'Ko and Sheinerman, 2004, Raychaudhuri and Yu, 2006, Salehzadeh et al., 2013b, Sheinerman and Gutkin, 2001a, Sheinerman and Gutkin, 2001b, Wang et al., 2010, Zhao et al., 2012, Zhao et al., 2013) and experimentally (Dayeh et al., 2013, Goldthorpe et al., 2008, Haapamaki et al., 2012, Kavanagh et al., 2011, Kavanagh et al., 2012, Lin et al., 2003, Perillat-Merceroz et al., 2012, Popovitz-Biro et al., 2011, Salehzadeh et al., 2013a, Salehzadeh et al., 2013b). The most of the MD models have dealt with either straight edge MDs (Enzevaee et al., 2013, Fang et al., 2009c, Gutkin et al., 2000, Liang et al., 2005, Raychaudhuri and Yu, 2006, Sheinerman and Gutkin, 2001a, Sheinerman and Gutkin, 2001b, Zhao et al., 2012, Zhao et al., 2013) or prismatic (Aifantis et al., 2007, Colin, 2010, Gutkin et al., 2011, Haapamaki et al., 2012, Liang et al., 2005, Ovid'Ko and Sheinerman, 2004, Raychaudhuri and Yu, 2006, Salehzadeh et al., 2013b) and glide (Chu et al., 2011, Chu et al., 2013) MD loops, although only two problems for screw MDs (Fang et al., 2008a, Wang et al., 2010) have been solved until now.

It is well known that the formation of a contact between two different materials is accompanied with the appearance of residual elastic strains and stresses caused by differences in the material structures and properties. The scale and distribution of these strains and stresses depend on many factors such as the contact characteristics, properties, shapes and sizes of the materials in contact. One mechanism of misfit eigenstrains (residual strains) relaxation is the nucleation of different type of defects at the interface, in particular, misfit dislocations. Some possible mechanisms of misfit screw dislocation dipole formation at the nanowire-matrix interface when the nanowire is subjected to the anti-plane eigenstrain field are demonstrated in Fig. 1.

Higher order continuum theories are well known to enhance the solutions in the vicinity of defects. A variant of gradient elasticity to deal with the strain and stress singularities at the dislocation line and crack tips has been given by Ru and Aifantis (1993). Altan and Aifantis (1997) showed that the theoretical treatment of Ru and Aifantis (1993) can be recovered from the more general nonlocal theory of Eringen and Edelen (1972). There are numerous papers devoted to the elimination of the stress (strain) singularity on the dislocation and disclination lines, and at the crack tips; Gutkin, 2006, Aifantis, 2011, Lazar and Maugin, 2005, and Askes and Aifantis (2011). Shodja et al. (2008) and Davoudi et al., 2009, Davoudi et al., 2010) have studied the interaction of dislocations and surfaces/interfaces. Aifantis and Askes (2007) have considered interfaces as surfaces of discontinuity for the strain gradient within strain gradient theory.

Gurtin surface/interface elasticity theory is known to capture the surface/interface effects properly when dealing with nano-size mediua. Recently, Rezazadeh Kalehbasti et al. (2014) have employed this theory to study wedge disclinations in the shell of a core–shell nanowire. Shodja et al. (2013) considered wedge disclination dipole in an embedded nanowire within surface/interface elasticity.

The case of screw MDs is still of interest due to the following reasons. First, the previous solutions (Fang et al., 2008a, Wang et al., 2010) left some open questions to be resolved: Fang et al. (2008a) used the approach of classical theory of elasticity to describe the generation of an MD dipole (MDD) and did not take into account the dislocation core energy contribution to the energy balance; both of these factors can lead to large errors in considering tiny NWs with only a few nanometers in diameter. Under such circumstances, on one hand, the MDs in the dipole are so close to each other that their core energy is comparable with their elastic strain energy, on the other, for ultra-thin NW the interface-to-bulk ratio is quite large and consequently the interface properties play a significant role. Wang et al. (2010) also used classical theory of elasticity and neglected the effect of the misfit strain. Moreover, they considered the stability of a pre-existing dislocation dipole in an elastically inhomogeneous core–shell NW. In the absence of misfit strain, they provided the critical shell thickness at which the dislocations are pushed out of the shell, but did not look into MDD formation at/near the core–shell interface.

Second, the framework of the surface/interface elasticity, which was first introduced by Gibbs (1906) and developed gradually over years (Cammarata, 1994, Goldstein et al., 2010, Gurtin and Murdoch, 1975, Gurtin and Murdoch, 1978, Gurtin et al., 1998, Shuttleworth, 1950), occurred very productive for revealing new characteristic features in surface/interface phenomena (Sharma and Ganti, 2004, Sharma et al., 2003, Tian and Rajapakse, 2007). Within this theory, the strain dependent surface energy Γ(εαβS) is related to the surface/interface stress tensor ταβS as:ταβS=τSδαβ+ΓεαβS,where εαβS is the 2 × 2 surface strain tensor, τS is the residual surface tension, δαβ is the Kronecker delta, and Greek indices take values 1 and 2. Based on this approach, some boundary-value problems for dislocations have been previously solved (Ahmadzadeh-Bakhshayesh et al., 2012, Enzevaee et al., 2013, Fang and Liu, 2006a, Fang and Liu, 2006b, Fang et al., 2008b, Fang et al., 2009a, Fang et al., 2009b, Feng et al., 2011, Gutkin et al., 2013a, Gutkin et al., 2013b, Liu and Fang, 2007, Luo and Xiao, 2009, Moeini-Ardakani et al., 2011, Ou and Pang, 2011, Shodja et al., 2011, Shodja et al., 2012, Zhao et al., 2012, Zhao et al., 2013), including three recent problems for MDs (Enzevaee et al., 2013, Zhao et al., 2012, Zhao et al., 2013). In particular, Zhao et al. (2012) considered individual edge MDs at the interface between an embedded nanotube and an infinite matrix (see our criticism of this work in Ref. Enzevaee et al., 2013) and in a core–shell NW (Zhao et al., 2013). Enzevaee et al. (2013) addressed both the cases of edge MD and MDD in a core–shell NW. The critical conditions for the MD generation were studied and discussed in detail in the context of non-classical surface/interface elasticity, capturing the surface/interface effects for fine cores of radius smaller than ∼20 interatomic distances. It was shown that the positive and negative surface/interface Lamé constants mostly make the generation of MDs easier and harder, respectively. Moreover, the positive (negative) residual surface/interface tensions mostly make the generation of MDs harder (easier). It was also concluded that the formation of individual MDs is energetically preferential in finer two-phase NWs, while the formation of MDDs is more expectable in the coarser ones.

Taking into account the lack of suitable solutions for screw MDs and motivated by our results on edge MDs (Enzevaee et al., 2013), the current paper employs the surface/interface elasticity approach to the case of an NW embedded in an infinite matrix. It is worth noting that in application to devices, NWs are often embedded in matrices which typically have inhomogeneous architecture (Garnett et al., 2009, Xiang et al., 2006, Yan et al., 2011) and are much larger than NWs, and hence the matrices can be approximated as infinite solids with some appropriate effective elastic constants. Furthermore, we assume that due to the uniform shear misfit strain within the NW, a screw MDD can be generated at the NW-matrix interface. Using the complex variable method, we provide an exact analytical expression for the energy criterion of the MDD generation and discuss in detail the conditions of its fulfillment.

Section snippets

Statement of the problem and formulations

The geometrical parameters of the considered embedded circular nanowire in an infinite matrix as well as the screw MDD at the intersections of the nanowire-matrix interface and the x-axis are shown in Fig. 2. The nanowire is denoted as domain 1 and has radius R, shear modulus μ1, and the Poisson's ratio ν1. The surrounding matrix is referred to as region 2 and has elastic constants μ2 and ν2. Throughout the current developments, the field quantities with superscripts k = 1,2 are pertinent to

Results and discussion

For convenience, we introduce the normalized parameters μ¯=μ1/μ2, γ = (μSτS)/μ2 and R¯=R/b. The energy change is normalized as ΔW¯=ΔW/(μ2b2). In numerical calculations, we take r0 = b.

Fig. 3(a) shows the dependence ΔW¯(R¯) calculated within classical theory of elasticity, when μS = 0 and τS = 0, for different values of ε and μ¯. The normalized critical radius R¯c of the NW is determined by equation ΔW¯=0. As is seen, R¯c increases when ε decreases. Depending on μ¯, R¯c varies from about 7.5

Conclusions

In summary, the critical conditions for the generation of a screw MDD at the NW-matrix interface due to a uniform shear eigenstrain within the NW have been studied in the framework of both classical and surface/interface elasticity theories. It has been shown that the NW critical radius decreases with increasing of the eigenstrain (misfit strain) as well as the increasing of NW to matrix shear moduli ratio. Non-classical interface effect is that, positive (negative) values of the interface

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      Thus, the study of the relaxation mechanisms and the critical conditions of their activation is of primary importance for these nanostructures. In particular, the mechanisms and critical conditions for the onset of misfit stress relaxation through generation of misfit dislocations (MDs) on interfaces between nanowires (NWs) and surrounding media were studied by many authors, both experimentally (Dayeh et al., 2013; Glas, 2015; Goldthorpe et al., 2008; Haapamaki et al., 2012; Kavanagh et al., 2011, 2012; Lin et al., 2003, 2017; Perillat-Merceroz et al., 2012; Popovitz-Biro et al., 2011; Salehzadeh et al., 2013a, 2013b) and theoretically (Aifantis et al., 2007; Chu et al., 2011, 2013; Colin, 2010, 2015a, 2015b, 2016a, 2016b; Colin and Grilhé, 2002; Enzevaee et al., 2014; Fang et al., 2008, 2009; Glas, 2015; Gosling, 1996; Gosling and Freund, 1996; Gutkin and Smirnov, 2015, 2016; Gutkin et al., 2000, 2003, 2011; Haapamaki et al., 2012; Kolesnikova and Romanov, 2004; Krasnitckii et al., 2018; Liang et al., 2005; Ovid'ko and Sheinerman, 2004 ; Raychaudhuri and Yu, 2006; Salehzadeh et al., 2013b; Sheinerman and Gutkin, 2001a, 2001b; Shodja et al., 2015; Wang et al., 2010; Zhao et al., 2012, 2013, 2017). Most of the studies were done for the case when NWs and surrounding media play the roles of the core and the shell, respectively, in a joint core-shell NW.

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