Measurement of strain fields in an edge dislocation

ARTICLE IN PRESS Physica B 405 (2010) 171–174

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Measurement of strain fields in an edge dislocation Z.S. Dong, C.W. Zhao  College of Science, Inner Mongolia University of Technology, 010051 Hohhot, PR China

a r t i c l e in fo

abstract

Article history: Received 25 May 2009 Received in revised form 7 August 2009 Accepted 10 August 2009

We present a strain analysis of an edge dislocation core in gold. The strain fields around an edge dislocation were mapped with a geometric phase analysis of high-resolution transmission electron microscopy images. The strain measurement results were compared with the gradient elasticity dislocation model, the field theory of elastoplasticity dislocation model and the Peierls–Nabarro dislocation model. We then concluded that for the field theory of elastoplasticity, the non-dimensional constant e0 is more suitable when chosen as 0.25 than with 0.399. However, the Peierls–Nabarro model is the most appropriate theoretical model to describe the deformation fields of the dislocation core. & 2009 Elsevier B.V. All rights reserved.

Keywords: Dislocations Strain High-resolution electron microscopy Geometric phase analysis

1. Introduction The dislocation core is an important region in the theory of metallic plasticity because high stress in this region often provokes dislocations and material failure. Research on dislocation core is a classic topic of materials science. There are many dislocation models, such as the elastic theory model, discrete model, etc. The traditional description of an elastic field produced by dislocations is based on the classical theory of linear elasticity. However, the classical dislocation theory breaks down in the dislocation core region. The elastic fields calculated within the theory of elasticity contain singularities at the center of each dislocation, leading to the development of modified dislocation models that propose to eliminate these singularities. Examples of these theories include that of gradient elasticity as proposed by Gutkin and Aifantis [1,2] and the field theory of elastoplasticity as proposed by Lazar [3–5]. On the basis of the results from elasticity theory, a significant theoretical advance was proposed by Peierls [6] and later elucidated and extended by Nabarro [7,8]. In the Peierls–Nabarro theory, the width of a dislocation and the Peierls stress were meaningfully related to the crystal characteristics of the material. Even though there are arguments about the drawbacks, limitations and, in particular, the accuracy of the predictions, the fundamental concepts and the methods originating from the Peierls–Nabarro theory still play a unique role in the discussion of dislocation up to the present. The dislocation core properties of the Peierls–Nabarro model were also theoretically studied by Lu [9] and Schoeck [10].

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E-mail address: [email protected] (C.W. Zhao). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.08.051

Although a lot of research has already been performed in this field, there are still many problems that remain unsolved. For example, a direct measurement to verify the Peierls–Nabarro model has not yet been achieved because of a lack of experimental techniques with sufficient displacement sensitivity and spatial resolution. Recently, high-resolution transmission electron microscopy (HRTEM) has become a powerful tool for mapping the displacement field at the nanoscale level, in large part because of the development of improved quantitative image analysis methods [11]. One such technique is geometric phase analysis (GPA) [12], which has been applied to a wide variety of systems, including quantum dots [13], nanoparticles [14], Si heterostructure [15], and low-angle grain boundaries [16], etc. The GPA has also been applied to quantitative measurements of displacement fields of edge dislocation in aluminum [17]. Here, we present a study of an edge dislocation in gold. The strain fields of the dislocation core were mapped with experiment, gradient elasticity theory, elastoplasticity theory and the Peierls–Nabarro model were then compared.

2. Theory 2.1. Strain of edge dislocation given by gradient elasticity theory The gradient elasticity theory describes the strain of an edge dislocation as  b y 2yx2 y2  3x2 ð1  2nÞ 2 þ 4 þ 4cy exx ¼  4pð1  nÞ r r6 r    2    2y y r y2  3x2 r K2 pffiffiffi  pffiffiffi 2  n K1 pffiffiffi  2y 4 r r c r c c

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 b y 2yx2 y2  3x2 ð1  2nÞ 2  4  4cy 4pð1  nÞ r r6 r    2    2y x r y2  3x2 r pffiffiffi K  pffiffiffi 2  n K1 pffiffiffi þ 2y 2 r4 r c r c c

eyy ¼ 

ð1Þ

of lattice fringes. The phase of these local Fourier components, or geometric phase Pg(r), is directly related to the component of the displacement field, u(r), in the direction of the reciprocal lattice vector g Pg ðrÞ ¼ 2pg  uðrÞ

ð4Þ

where x and y are the right-angle coordinates, respectively, and centered on the dislocation core position. r2 ¼ x2+y2. b is the Burgers vector, and n the Poisson ratio. The gradient coefficient c pffiffiffi can be estimated as c  a=4 [18], in which a is the lattice constant. Kn is the modified Bessel function of the second kind and of order n.

It is assumed that the displacement field is constant along the propagation direction through the foil or that if there are small variations, they are averaged out. In the latter case, the displacement field is the projected displacement field averaged over the foil thickness. And by measuring two phase images, Pg1(r) and Pg2(r), the two-dimensional displacement field can be determined:

2.2. Strain of edge dislocation given by elastoplasticity theory

uðrÞ ¼ 

The elastoplasticity theory describes the strain of an edge dislocation as  b y 2x2 4 exx ¼  ð1  2nÞ þ 2 þ 2 4 ðy2  3x2 Þ 4pð1  nÞ r 2 r k r  2   y 2 2 2  n krK1 ðkrÞ  2 ðy2  3x2 ÞK2 ðkrÞ r r

Here a1 and a2 are the basis vectors for the lattice in real space corresponding to the reciprocal lattice defined by g1 and g2. Eq. (5) in matrix form is ! ! ! Pg1 a1x a2x ux 1 ¼ ð6Þ Pg2 uy 2p a1y a2y

 b y 2x2 4 ð1  2nÞ  2  2 4 ðy2  3x2 Þ 2 4pð1  nÞ r r k r  2   x 2 2 2  n krK1 ðkrÞ þ 2 ðy2  3x2 ÞK2 ðkrÞ r r

eyy ¼ 

ð2Þ

The factor k should be fitted by comparing predictions of the theory with experimental results. In general, the factor k can be used to determine the width of a dislocation and the amplitude of the force stress; k1 has the physical dimension of a length and therefore it defines an internal characteristic length (dislocation length scale) [5]. The internal characteristic length may be selected to be proportional to the lattice parameter a for a single crystal, i.e., k1 ¼ e0a, where e0 is a non-dimensional constant that can be determined by experiment [19]. For e0 ¼ 0 we recover classical elasticity theory and for e0 ¼ 0.25 we recover gradient elasticity theory. In [19,20] the choice e0 ¼ 0.399 and in [18] the choice e0 ¼ 0.25 are proposed.

1 ½Pg1 ðrÞa1 þ Pg2 ðrÞa2  2p

Plane strain can be written as: 8 @u > > exx ¼ x > > @x > > > < @uy eyy ¼ @y > >   > > @uy 1 @ux > > > exy ¼ þ : @x 2 @y

ð5Þ

ð7Þ

3.2. Electron microscopy The HRTEM sample is a gold crystal specimen that consists of an ultra-thin layer of gold grown epitaxially into a single crystal along the [0 0 1] direction. HRTEM experiment was performed on the JEM-2010 transmission electron microscope at 200 kV. Images were recorded on a Gatan 1k  1k slow scan CCD camera and processed using the software GPA Phase, developed by the Gatan DigitalMicrograph environment.

2.3. Strain of edge dislocation given by Peierls–Nabarro model According to the Peierls–Nabarro dislocation model, the strain of an edge dislocation along the x direction can be written as

exx ¼ 

b

ð1  nÞy

p 4ð1  vÞ2 x2 þ y2

ð3Þ

3. Experimental methods 3.1. Geometric phase analysis An HRTEM image formed at a zone axis of a crystal can be considered as a set of interference fringes corresponding to the atomic planes of the specimen. GPA analyses these interference fringes individually to extract the information concerning displacement. In particular, the technique measures the displacement of lattice fringes with respect to a perfect lattice (for example provided by a region of undistorted substrate). The method is based on the calculation the local Fourier components

4. Results and discussion Fig. 1a shows an HRTEM image of an edge dislocation in gold. Geometric phase images were calculated for the two sets of {2 0 0} lattice fringes (Fig. 1c and d) using Gaussian masks and are shown in Fig. 1e and f. Taking the x-axis parallel to [0 1 0] and the y-axis parallel to ½1 0 0, the Burgers vector b ¼ 1/2[0 1 0]. The strain fields can be calculated from geometric phase images and are shown in Fig. 2a. Here we have Poisson’s ratio n ¼ 0.412, and the lattice constant set as a ¼ 0.408 nm. There is a convergence region of the strain field around the edge dislocation core. In the region of the extra half-plane (y40), the strains are negative and compressive and the lattice is expanding on the other side. The largest strain values occur in the immediate core region. The strain is smaller in regions farther from the dislocation core. Also shown in Fig. 2b–d are three theoretical models of exx strain fields. The color scale indicates strain changes of 5% to +5%. The strain maps of gradient elastic theory and elastoplasticity theory are butterfly-shaped. The strains achieve their extreme values at a distance 0.1 nm from the dislocation core. The strain map of the Peierls–Nabarro model is figure-eight-shaped. The strain of the

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Fig. 1. Geometric phase analysis of an edge dislocation in a gold: (a) HRTEM image in [0 0 1] orientation; (b) Fourier transform of image (a); (c) ð2 0 0Þ lattice fringes obtained by filtering; (d) (0 2 0) lattice fringes obtained by filtering; (e) geometric phase image of ð2 0 0Þ lattice fringes and (f) geometric phase image of (0 2 0) lattice fringes. Colors range from p to p rad.

Fig. 2. Experimental and theoretical strain field: (a) experimental exx strain field; (b) gradient elastic theoretic exx strain field; (c) elastoplasticity theoretic exx strain field (e0 ¼ 0.399) and (d) The Peierls–Nabarro dislocation model exx strain field.

Peierls–Nabarro prediction is very similar to the experimentallydetermined strain. The strain of the gradient elastic theory and elastoplasticity theory near the dislocation core region is much smaller than was experimentally. Qualitatively, the Peierls– Nabarro dislocation model is in the best agreement with the data. We thus determined that though the gradient elastic theory and the elastoplasticity theory usefully eliminate singularities in the dislocation core region, they are not very appropriate for modeling the dislocation core. Therefore, it would seem that the constant associated with these values (e0 ¼ 0.399) is inappropriate when modeling dislocation core dynamics. e0 ¼ 0.25, as used in elastoplasticity theory, is much more

effective. Therefore, in the elastoplasticity theory, choosing e0 ¼ 0.25 is more appropriate. To analyze the degree of agreement between the experiment and theory in greater detail we selected four different circles of radius r from the dislocation core to measure the variation of strain and compared them with three theoretical models (the measurement location, see Fig. 2d). The results are shown in Fig. 3a–d. The standard deviations of the strain difference between the experimental and theoretical exx strain components for different circle of radius r are shown in Table 1. The difference between the experimental and the Peierls–Nabarro dislocation model’s exx strain component is the smallest. Therefore, the

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Fig. 3. Comparison of angular variations of strain field: (a) r ¼ b; (b) r ¼ 2b; (c) r ¼ 4b and (d) r ¼ 8b.

Table 1 Standard deviations on the differences between the experimental and theoretical exx strain components.

Experiment and gradient elastic theory Experiment and elastoplasticity theory Experiment and Peierls–Nabarro model

r¼b

r ¼ 2b

r ¼ 4b

r ¼ 8b

0.077 0.090 0.016

0.032 0.034 0.0087

0.018 0.017 0.0053

0.0094 0.0092 0.0029

agreement between the experimental result and the Peierls– Nabarro dislocation model is best.

5. Conclusions A direct quantitative measurement of the strain field around an edge dislocation core in gold was achieved with a combination of HRTEM and GPA. The strain measurement region is 6  6 nm. The gradient elastic theory dislocation model, the elastoplasticity theory dislocation model and the Peierls–Nabarro dislocation model are all appropriate in regions far from the dislocation core. According to our analysis of the comparison results of three theoretical models with the experimental results in the vicinity of the dislocation core, the Peierls–Nabarro model is the most appropriate theoretical model in describing the strain field of the dislocation core in gold.

Acknowledgment This work was supported by the National Natural Science Foundation of China no. 10862002.

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