pss/RRN-01-021

Fractal Dislocation Patterning in Plastically Deformed NaCl Polycrystals

M. Zaiser (a,b), S. Marras (a), E. Nadgorny (a), H.P. Strunk (c), and E.C. Aifantis (a)

(a) Michigan Technological University, Houghton, MI 49931, USA
(b) Max-Planck-Institut für Metallforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany
(c) Universität Erlangen, Inst. für Werkstoffwissenschaften, Cauerstr. 6, D-91058 Erlangen, Germany

(Received May 3, 2001; accepted May 11, 2001)
Subject classification: 61.72.Lk; 62.20.Fe; S9.11

Introduction

In severely deformed metals, strain hardening goes along with the formation of cellular dislocation patterns. When many slip systems are active, these patterns exhibit multiscale behaviour: they are characterized by power-law distributions of cell sizes [1] and the cell arrangement has features of a self-similar `hole fractal'. Fractal cell patterning has been observed in fcc metals deformed in tension or compression along the [100] and [111] axes [1, 2] as well as in fcc polycrystals [3, 4]. For a general discussion of different types of dislocation patterns see [5].

In the present research note we report results of a fractal dimension analysis performed on dislocation cell patterns developing during severe deformation of polycrystalline NaCl. (For sample preparation and work-hardening behavior see [6].) The samples were deformed in compression at room temperature at a strain rate of 10^–5s^–1. Cellular patterns were observed inside the grains after deformation to a flow stress of 35.2 MPa (Fig. 1). The statistical characterization of these patterns is the subject of the present note.

Figure 1. Dislocation pattern in a grain of a NaCl polycrystal deformed at room temperature to a flow stress of 35.2 MPa (the adjacent grains are out of contrast)

Image Analysis and Results

TEM images of dislocation cell patterns such as the example shown in Fig. 1 were binarized and the resulting black-and-white patterns were analyzed using the box counting and correlation integral (`mass dimension') methods. For the box counting method, the cell pattern is covered with a grid of meshlength and the number of `boxes' which intersect the cell walls is plotted double-logarithmically vs. . For the correlation integral method, one determines the integral M(r) of the pattern autocorrelation function over a circle of radius r. This is proportional to the average number of black pixels contained in a circle of radius r around a black pixel chosen at random. For a fractal pattern of dimension D, the correlation integral increases as while the number of boxes decreases as . A detailed discussion of different methods of fractal dimension analysis applied to dislocation patterns is found in [2], and the relations between the fractal dimension D, which characterizes the distribution of length scales in the dislocation network, and more conventional characteristics of cell structures such as mean cell size and cell wall volume fraction are discussed in [2, 7].

Figure 2. Mass dimension analysis of TEM micrographs of cell structures in NaCl. The lower graph has been obtained from the micrograph shown in Fig. 1; for better visibility, the graphs have been shifted vertically

Results of the mass dimension analysis of two different micrographs taken from the same sample are shown in Fig. 2. To enhance variations in slope, M(r) has been divided by the area .

When , the slope of the double-logarithmic plot of (M(r)/A(r)) vs. r is m = D – 2. Accordingly, the analyzed micrographs yield mass dimensions and . The linear scaling regimes in the log–log plots extend in both cases over about one and a half orders of magnitude. The dimensions obtained by box counting from the same two micrographs are and , respectively. Hence both methods yield fractal dimensions around 1.8 which is close to the values that have been observed in high-symmetry orientated Cu single crystals [1, 2].

Discussion and Conclusions

Fractal cell patterns have been observed previously in high-symmetry orientated fcc single crystals and in fcc polycrystals. In these materials deformation is characterized by the simultaneous activity of a large number of slip systems. The deformation behaviour of NaCl polycrystals has many similarities with this situation [6]: While alkali halide single crystals usually exhibit macroscopic single glide on {110} glide planes, NaCl crystals deformed in [111] directions and NaCl polycrystals exhibit extensive glide on auxiliary slip planes (e.g. {111} or {100}) [6, 8]. The dislocation network is characterized by the presence of Burgers vectors of all orientations in approximately equal amounts, and the flow stress is governed by a forest mechanism [6]. All these findings are very similar to the situation in [100] or [111] orientated fcc single crystals or in fcc polycrystals. The observation of fractal cell patterns in NaCl polycrystals demonstrates that such patterns are not confined to fcc metals and at the same time corroborates the conjecture [9] that a large number of active slip systems is a necessary prerequisite for this type of dislocation patterning.

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