Half-loop dislocations

Another manifestation of the strain in the films is the presence of the half-loop dislocations extending out from the open tubes in the GaN films grown on the porous substrates, as discussed in the previous section. Regarding the origin of these half-loops, it is easy to see that open tubes (or voids) in a strained film will act as stress concentrators since the normal component of the stress is necessarily zero at the tube wall, the material near the wall will be displaced relative to its position in the absence of the void, and the tangential in-plane component of the stress is thereby increased. In other words, during growth, the shear stress field of the GaN film will be locally concentrated around these open tubes in the film. The open tubes provide a free surface where these half-loops can nucleate due to the increased stress. 

We have performed finite element analysis on a simple model to demonstrate the concentration of shear stress around the pore wall. Figure 5.12(a) shows the model used and Figure 5.12(b) is a plot of shear stress as a function of distance from the pore wall. It is evident that there is considerable stress concentration in the film around the substrate pores, which can lead to the formation of the half-loop dislocations. 

The change in the dislocation structure, material behavior, and density with variation of strain rates could explain the mechanisms of strain hardening, showing that round half-loop dislocations are replaced by straight and crossing dislocations and then by cell-structured dislocatitnis. 

In crystalline polymer systems the tough response, besides cavitation and crazing, is crystallographic in natme. Crystallographic slips ai-e the main plastic deformation mechanisms that require generation and motion of crystallographic dislocations. The concepts of generation of monolithic and half-loop dislocations plausibly explain the observed yield stress dependences on crystal thickness, temperatm-e and strain rate. 

It has also been suggested that flow might occur at lower stresses than those predicted above by movement of material within the individual layers (Chu and Barnett, 1995). This has been observed in pearlitic structures made up of alternating layers of ferrite and cementite, and observations in other multilayer systems suggest that that deformation might occur in this way (Gil-Sevillano, 1979). Two cases have been identified the first where only the movement of a pre-existing dislocation loop is required, the second where the activation of a dislocation source within the layer is needed. Gil-Sevillano (1979) showed that the extra stress, Atm, required to move a dislocation half-loop in a layer of width 1 is... 

Figure 5.8 Cross-sectional TEM images of GaN films on (a) nonporous SiC substrate, and (b)-(d) on porous SiC substrates. The label ps denotes a porous substrate and np denotes a nonporous substrate. The surface pore density of the substrate in (b) is 13 prrr2, and in (c) and (d) is 11.5 pm-2. The open tubes in the GaN films on porous substrates are marked by T (the tubes appear white in contrast near the top of the film where they are empty, and black near the interface where they are filled by Ga). The regions labeled D contain a relatively low number of threading dislocations originating at the interface (due to the lateral epitaxial growth over the substrate pores), and they contain dislocation half-loops gliding in from tubes. One half-loop is faintly seen to the right of a in (c). Reproduced from A. Sagar et al., J. Vac. Sci. Technol. B 21, 1812. Copyright (2003), with permission from the American Institute of Physics...

Some mechanisms have been proposed to explain the generation of type-I and type-II dislocations. The reported mechanisms for the generation of type-II dislocation are (1) the bending of the threading dislocation in the substrate parallel to the interface [36] and (2) the glide of dislocation from the surface, which forms a half-loop [29], The reported mechanisms for the generation of type-I dislocation are (1) the reaction of two type-II dislocations [37] and (2) the dislocation climb of the pure edge dislocation from the surface [29],... 

In Section 9.4 we present analyses of nucleation of both edge- and screw-dislocation half loops from edges and surfaces of lamellae in spherulites under stress in PE samples with a wide range of lamella thicknesses and compare the predictions with the plastic deformation rate. 

Fig. 9.17 Sketches depicting (a) a typical lamella in a great circle of a spherulite of HOPE and (b) the geometry of a typical lamella, showing the principal slip system (100) [001] and three modes A, B, and C, namely nucleation of a monolithic screw dislocation from a narrow edge a screw-dislocation half loop, again from a narrow edge and an edge-dislocation half loop from a wide surface of a lamella, respectively, under an applied shear stress (from Argon et al. (2005) courtesy of Elsevier).

Mode B nucleation of a screw-dislocation half loop from the narrow edge of a lamella... 

Plastic shear on the (100) [001] system in a lamella can also be initiated in a more general case by the nucleation of a screw-dislocation half loop from the narrow XA face of the lamella of Fig. 9.17. This process has been considered rigorously using the variational boundary-integral method developed by Xu and Ortiz (1993). The problem of interest here has also been solved by Xu and Zhang (2003), giving a stress dependence of the activation free energy of this mode shown in Fig. 9.19 as the upper curve. It is of the form... 

This mode of dislocation-half-loop nucleation has been considered by Xu (2002). For this, the resulting dependence of the activation free energy AG on the normalized shear stress p is shown in Fig. 9.19 as the lower curve. It is of the form... 

The activation volumes, normalized by b, predicted by the models using eqs. (9.30) are compared in Fig. 9.23 with some of those measured by Kazmierczak et al. (2005) in strain-rate-jump experiments. In the model calculation for the nucleation of the monolithic-screw-dislocation emission A = 20 nm X/b = 78.5) was used. The data for 13 up-jump strain-rate-change experiments are also plotted in Fig. 9.23. The details of these experiments are given by Kazmierczak et al. (2005). The measured Av /Z> for the jump experiments from s = 5.5 X 10 to 5.5 X 10 fall quite close to the models for nucleation of dislocation half loops. The data for jumps at considerably lower strain rates and smaller flow stresses fall a bit closer to the model of nucleation of monolithic screw dislocations but are in much less satisfactory agreement. In all cases it was assumed that the externally applied stresses directly apply locally, which is the assumption in the Sachs model. However, considering the generally confused morphology of... 

To probe the models for nucleation-controlled plastic flow we compare the predicted temperature dependence of the tensile plastic resistance with the tensile-yield-stress experimental results of Brooks and Mukhtar (2000). For comparison the polyethylene PE3 of average molecular weight = 131000 with a crystallinity of only 0.673 and lamella thickness of 34.3 nm is chosen. For the predictions of the temperature dependence, eqs. (9.26)-(9.28) of Section 9.4.3 are used, where we take in the denominator of eq. (9.25) the factor (1 + K), since the experiments were performed in tension. Noting that the lamella thickness of this polymer type is 1 = 34.3 nm, which is thicker than that for mode A of monolithic-screw-dislocation nucleation, we consider only modes B and C involving nucleation of screw-dislocation half loops and edge-dislocation half loops, and, together with the results of Fig. 9.21, we state the expected tensile yield stress Oy to be... 

In the case of antiplane shear deformation, the behavior of a semicircular dislocation loop at a free surface with Burgers vector parallel to the surface under the action of a mismatch shear stress Tm, as depicted in Figure 6.37, is identical to the behavior of a circular loop in an unbounded solid due to the same shear stress. The traction on the boundary vanishes due to symmetry in the latter case. According to Hirth and Lothe (1982), the energy of formation of such a surface half loop of radius R is... 

For very small values of R/h, the behavior of the function is dominated by W R), and W R) increases from fF(0) = 0 with increasing R/h. On the other hand, for large values of R/b, the behavior is dominated by and W R) decreases to ever more negative values. The function assumes a maximum value at some intermediate value of R this maximum value represents the activation energy that must be supplied through random thermal fluctuations in order to nucleate a dislocation in the form of a surface half loop. The value of R at which this maximum occurs, say i2max> satisfies the equation iF (i2max) = 0 which implies that... 

Within the present scheme, the work needed to create a dislocation half-loop is proportional to a Such work does not depend on a for the edge component, while for the screw component it does depend on a. Competition between the positive term, which has the argument raised to the lower power, and the negative term, which has the argument raised to the higher power, determines the critical size of the hole and the critical load on the indenter. Both of these values are in good agreement with the experimental values. 

The traditional microindentation of the surface of ionic and covalent crystals allows one to study the effect of adsorption on the movement of the screw components of the dislocation half-loops formed, but only outside the contact zone. The capabilities are broadened with the use of the micro-sclerometric and ultramicrosclerometric (scratching) methods developed by Savenko and coworkers [46,68,70]. A step-by-step increase in the load applied to the indenter allows one to observe a transition from the reversible elastic contact to the appearance of the very first damage, that is, nearsurface dislocations, and further to the development of plastic deformations, and then to microcrack nucleation (Figure 7.42). The adsorption taking place from the active medium can both facilitate damageability and retard it. 

Figure 1.11 depicts three modes of dislocation nucleation-controlled processes of chain slip on the (100) [001] system. The process identified as A is the previously considered one of nucleation of a fully formed screw dislocation monolithically from the narrow face. Clearly, as the lamella thickness I increases, the ever increasing activation energy of this mode will no longer be kinetically possible. Thus, two other modes were considered a screw dislocation half loop nucleation, still from the narrow face, and an alternative process of edge dislocation half loop nucleation fix>m the wide face, depicted as processes B and C, respectively, in Figure 1.11. 

A series of considerations including a correct value of /x modulus and its dependence on temperature, the kinetic crystallographic shear strain rate expression encountered in crystal plasticity, and a Coulomb law where the shear resistance r is dependent on the normal stress glide plane, lead to detailed expressions for yield stresses for all three cases of monolithic, half loop screw and half loop edge dislocations engaged in plastic deformation. The comparison of compressive yield stress of all three mechanisms with experimental data of Kazmierczak et al [149] is illustrated in Figure 1.12. 

The data points for thin lamellae are close to the prediction of the model of monolithic screw dislocations. The plot suggests a sharp departure from the mode of screw dislocation line nucleation to the half loop modes at roughly aromid 16 nm lamella thickness. The experiments show a more gradual transition at a lamella thickness of roughly 28 mn. For lamellae thicker than 28 nm the data points fall in between the models for nucleation of edge half loops and screw half loops. 

Fig. 10. Deformation microstructures containing perfect dislocations (the confining pressure is 5 GPa). (a) Deformation temperature T = 293 °C (101) foil plane, weak-beam dark field (4.1g, g = 202). The dislocations nucleated at crack edges are of 1/2[1 0 i](l 11) type. These half-loops are elongated along the [3 21] direction (after Rabier and Demenet [62]). (b) In the bulk, the same dislocations tend to be aligned along several Peierls valleys < 112 > /30°, < 12 3 > /4T, and screw orientation (after Rabier et al. [62]). (c) Deformation temperature T = 150 °C. Same Peierls valleys as at room temperature some strong pinning points are indicated by arrows (after Rabier et al. [61]).

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