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Self-organization processes during deformation of nickel single crystals
Journal Pre-proof Self-organization processes during deformation of nickel single crystals Alfyorova Ekaterina, Filippov Andrey PII:
S1044-5803(19)31712-7
DOI:
https://doi.org/10.1016/j.matchar.2019.110007
Reference:
MTL 110007
To appear in:
Materials Characterization
Received Date: 26 June 2019 Revised Date:
7 November 2019
Accepted Date: 8 November 2019
Please cite this article as: A. Ekaterina, F. Andrey, Self-organization processes during deformation of nickel single crystals, Materials Characterization (2019), doi: https://doi.org/10.1016/ j.matchar.2019.110007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.
Self-organization processes during deformation of nickel single crystals † 1 2
Alfyorova Ekaterina 1, Filippov Andrey 2
National Research Tomsk Polytechnic University, Tomsk, Russia
Institute of Strength Physics and Materials Sciences SB RAS, Tomsk, Russia e-mail: †[email protected], [email protected]
Abstract In this work, plastic deformation of nickel single crystals with highly symmetric orientations was examined. Revealed scale levels of plastic deformation self-organization and the corresponding deformation mechanisms. It was established that the shear traces and mesobands are formed due to dislocation glide, macro-bands are formed due to dislocation glide in and near-surface misorientation of local areas, and for folds the disorientation inside the crystal is additionally developed. Based on the morphology of folded structures, it is shown that there is a correlation between the wavelength of the folds, the distance between the misoriented bands and the dislocation substructure. The Hurst exponent and correlation lengths correlate with the stages of the stress-strain curve and, consequently, the type of dislocation substructure. Keywords: FCC single crystal; plastic deformation; microstructure; scaling; deformation relief self-organization.
1. Introduction Plastic deformation is currently defined as a self-consistent process at various structural levels. Synergetics believes that self-organization is the process of ordering in an open system, which is due to the coordinated interaction between the set of its constituent elements. The one of importance things is this process occurs without specific actions from outside. The result is the appearance of a unit of the next quality level. For a single crystal, structural levels are dislocations, shear zones, dislocation groups, and slip systems. The new unit of the next quality level is bands of shear traces, mezo bands, macro bands and folders for FCC single crystals [1-3]. In the case of plastic deformation, self-organization is aimed at preserving the integrity of the material for as long as possible. Internal processes occurring during loading manifest themselves on the crystal’s surface as a deformation relief of various scale. At the initial degrees of deformation, of special significance are dislocation mechanisms (translational motion of dislocations along close-packed planes), which lead to the formation of thin shear traces on the surface. As early as the beginning of the 20th century, Rosenhain, Ewing, Hirth and Lote [4] showed that lines on the surface of a deformed sample are stages on the surface. They arise on the surface as a result of the collective movement of many dislocations.
Traces of shear appear at the initial degree of deformation under different types of loading in both single and polycrystals. Consequently, they can be considered the basic structural element of the deformation relief (SEDR). With further loading, the shear traces are ordered into the next scale level system (pack slip bands, meso- and macro bands, corrugated structures of various types (folds), persistent slip band, etc.). The role of large-scale structural levels increases accordingly with increasing degree of deformation. These levels involve the higher scale meso-volumes in the translationalrotational movement, including the ensembles of grains in the case of polycrystals. When researchers consider the deformation relief as a source of information about internal processes in a crystal during plastic deformation, they traditionally focus on the height of the shear, the length of slip lines and the distance between them. Mader, Kronmyuller, Pfaff and Mitchell [5-6] were among the first scientists to measure the height of the shear. They showed that it does not depend on the degree of deformation and remains practically unchanged within the second stage of the stressstrain curve. The manifestation of shear deformation in relief is diverse [7-16]. There is a body of literature that describes processes occurring with single slip lines or with a group of slip lines, i.e. these researchers consider the movement of isolated dislocation groups [17-18]. If we consider a smaller scale level, then these will be works that explore the processes of self-organization directly in the dislocation substructure. In the works [19-32], based on statistical processing of the parameters of the dislocation substructure (cell size, the distance between the geometrically necessary boundaries (GNB) and the misorientation angle between the cells), their scaling was established. The works [33-34] describe the self-affine nature of the cellular dislocation structure in Cu and KCl single crystals at different scale levels. The works indicate that self-affinity is characteristic of highly symmetric single crystals, where a large number of slip systems operate simultaneously. Similar results are described in the paper [35]. Hughes reports that various materials demonstrate similarities in the microstructural reliefs arising from monotonic deformation. Hughes identified scaling laws in the dislocation structure, which indicate that similar dislocation mechanisms work for a wide range of materials and deformation conditions, and showed that a sufficient three-dimensional dislocation mobility to create three-dimensional structures is a necessary condition [35]. The results were obtained for polycrystals of Ni, Cu, Al and 304L steel, as well as on Cu and Al single crystals under various loading conditions (compression, rolling, forging, torsion and friction). Oudriss et al. discuss the scaling parameters of the dislocation structure in tension of a [001] nickel single crystal. The author proves the presence of scaling in a large range of plastic deformations. In addition, the article demonstrates a series of scaling laws regarding shear stress
for cell size, dislocation cell wall thickness, and dislocation density in dislocation walls and cells [31]. Godfrey, Hughes [36] provide physical parameters (distances between geometrically necessary boundaries, etc.) connecting deformation microstructures in a wide range of scales and indicate that there is universal scaling. This scaling proves the existence continuous process that create new bounders, and those that destroy them from the macro to the nanoscale. In addition, the authors note that microstructural evolution occurs under conditions of dislocation slip down to the nanolevel. High-resolution electron microscopy also confirms the dominant role of dislocations. This experiment showed the presence of a large number of gliding dislocations in the layers between geometrically necessary boundaries with an individual step up to 5 nm [35] The scaling by the distance between the boundaries of the dislocation structure does not depend on the initial grain size and the distribution of grain sizes. Similar results were obtained for single crystals (aluminum) and polycrystals (nickel, aluminum) in [36] and for grains inside a polycrystalline sample (aluminum) [30]. The analysis performed shows that similar physical mechanisms underlie the plastic deformation of a wide range of different materials and loading conditions. A common factor in each case is that plastic deformation leads to microstructural gliding due to the formation of a three-dimensional dislocation network structure. The main physical conditions for the formation of such microstructures is that the deformation must occur through dislocation slip with the work of at least three non-mutually planar Burgers vectors and sufficient three-dimensional mobility of these dislocations [19, 23]. When determining the parameters of the dislocation structure (cell size), it should be borne in mind that the results of metallographic studies are sensitive to the parameters used when conducting transmission electron microscopy (TEM). This provides a natural explanation for large systematic and not yet understood discrepancies in relation to the average cell size reported by different authors, as also described in [24]. Thus, the currently available experimental material indicates the similarity in the microstructural regularities and dislocation mechanisms of deformation that occur when different FCC metals and alloys are loaded. At the same time, of particular interest is the study of self-organized deformation reliefs on a specific surface area. However, researchers pay much less attention to the correlation between shear traces in the far distance. To describe the integral picture, it seems reasonable to use fractal methods and such attempts are made in many papers [19, 37-39]. However, many of these works are reduced to obtaining the value of the fractal dimension of a particular surface, but the question of the deformation process self-organization and the identification of
scale levels remains open. One of the options for fractal analysis, which allows to identify scale-invariant reliefs of plastic deformation is the correlation function "height-height". The method of calculating and analyzing this function is described in detail in the section “Material and methods”. In the works [40-42] the results obtained using the correlation function "height-height" are described. With this method, was established the existence of several scale levels with different types of loading. Lyapunova et al. established the presence of two zones with Hurst exponent 0.3–0.4 and 0.5–0.6, for an aluminum alloy after dynamic loading. These zones represent the destruction surface. The interval of a certain scale is 1–18 µm [40]. Oborin et al. found that the Hurst exponent in the fracture zone is 0.64–0.81, depending on the experimental conditions. A number of papers give the Hurst exponent greater than 0.5 when describing the deformation relief in the case of various methods of loading [41-42]. This indicates that, on certain scales, the system has a persistent (trend-resistant) character. It is important to note that the size of the zones where the measurements were made, and the resolution of the equipment used makes significant adjustments to the results obtained. In particular, a number of studies discussed here were carried out at sites of several millimeters or even tens of millimeters, which made it possible to establish scaling at the macro level. An analysis of the literature has shown that, on the one hand, there are works that study the processes of plastic deformation self-organization separately at the level of the dislocation substructure, separately at the level of the description of the morphology of the relief. On the other hand, a tool allows us to combine the results and trace the hierarchical relationship between the scale levels. In this connection, this case study seeks to examine the scale-invariant reliefs of the plastic deformation development at different scale levels in nickel single crystals with different crystallographic orientation based on data on the evolution of the deformation relief. 2. Material and methods The findings are obtained on nickel single crystals with the compression axes oriented in the angles of a standard stereographic triangle with various sets of sides (Fig.1).
Fig. 1. Stereographic triangle and corresponding deformation relief
In the research, we analysed nickel single crystals grown by the Bridgman method. The specimens had a shape of a tetragonal prism with a height-to-width ratio of two. The orientation was carried out on an IRIS 3 X-ray generator using Laue back-reflection photographs with a ±1° accuracy and further adjusted to a ±0.02° accuracy, using a DRON-3 X-ray diffractometer. The surface of the specimens was prepared by mechanical grinding and polishing and final electrolytic polishing in a saturated solution of chromic anhydride in ortho-phosphoric acid at 20V. Compression was carried out on an Instron ElektroPuls E10000 testing machine at a speed of 1.4·10-3 s-1 at room temperature. A Leica DM 2500P optical microscope and an Olympus LEXT OLS4100 confocal laser scanning microscope were used to analyze the deformation relief. The size of the scanned region in each case was 0.066 mm2 (0.256 x 0.256 mm2). The depth resolution was 0.06 µm. The size of the scanned region in this case sets a limit on the selforganization scale being established. Experiments were carried out in the strain interval of 1.5–32%. An EBSD attachment to the Tescan Vega II LMU microscope was used to determine the extent of misorientation of local areas. We are discussing the results obtained from the areas in the center of the sample faces . Objects for transmission electron microscopy (TEM) were made in the form of thin foils using the EM-09100IS sample preparation system (JEOL Ltd, Japan) by ion thinning. The foils were prepared according to the cross-section technique. The dislocation structure carried out using a TEM instrument JEM-2100 (JEOL Ltd, Japan). To reveal scale-invariant regularities of plastic straining of nickel single crystals under compression, we used fractal analysis of the deformation relief based on the height–height correlation
function H(r) (1) [43-45], where Z(r) is the function of the surface height over all pairs of points separated by a fixed distance r; angle brackets indicate averaging over all pairs of points. ,
,
(1)
The slope of the initial segment of the curve describing the height–height correlation function H(r) and plotted in logarithmic axes can be used for determining the Hurst exponent (H) and correlation length (L). The correlation length is determined by the projection of the linear segment onto the abscissa. Analysis of the behavior of the resultant height–height correlation functions H(r) has made it possible to single out segment L1 and L2 on the curve on which the experimental points can be approximated by straight lines with different slopes relative to the axes (Fig. 2). This makes it possible to determine the Hurst exponent and the upper boundaries of the correlation length for several segments. On segment L3, it is impossible to approximate experimental data with the help of a straight line because of their fluctuation behavior. This indicates the lack of stable correlation on this size scale (Fig. 2).
Fig. 2. Height–height correlation function in the region of mezo bands formation, [110] nickel single crystal ε = 32% (dashed curves are approximations of the segments of the solid curves)
Hurst exponent H estimates the degree of chaotization (self-organization) of the system. The Hurst exponent value less than 0.5 indicates the antipersistent, ergodic nature of the system, i.e. the system tends to return to the average value, and the degree of the system stability depends on how close the value of H is to zero. A value of H = 0.5 indicates no correlation. If the value is greater than 0.5, it means/denotes the presence of long-scale correlations [40]. Moreover, for the self-affine surface for the function H (r) the following holds true:
~
2
, ,
≪ξ ≫ξ
where α is the scaling coefficient, Hurst exponent (H), or roughness index, ξ is the lateral correlation length, it defines the upper limit of the region of self-affinity and characterizes surface irregularities in the lateral direction. Height fluctuations at r << ξ (small scale) are correlated, dependent, and at r >> ξ (large scale) - uncorrelated, random. ω - the width of the interface is a characteristic of the irregularity of the surface in the vertical direction, it is also called the correlation length in the vertical direction or the mean square surface roughness (ω = Sq) [43-44, 46-48]. Once again, in the framework of this work, the deformation relief was considered as an external manifestation of plastic deformation processes occurring inside the crystal. We use the terms self-organization of the deformation relief and self-organization of plastic deformation as very similar in meaning, meaning that the processes of plastic deformation self-organize during crystal loading and, therefore, appear on the surface in the form of a self-organized relief. 3. Results The results are obtained on nickel single crystals with various compression axes in the strain interval of 1.5–32%. In an earlier paper, we described the evolution of the deformation relief [3]. The deformation relief after the initial loading (stage I of the stress-strain curve) is slip lines formed by individual slip traces. During the transition to the subsequent loading stages, the deformation relief depends on the crystallographic orientation. The development of shear traces, their coarsening and self-organization in packs is characteristic for [001] single crystals. Formation of mezo bands is characteristic for [110] single crystals and the formation of macro bands and folded structures for [111] single crystals (Fig.1). Fig. 3 shows the stress-strain curves for the single crystals with various compression axes. We can clearly observe the orientational dependence of the curves. In the present study, we calculate the values of the Hurst exponent for all the cases specified in the “Materials and Methods” section.
Fig. 3. The stress-strain curves
3.1. Deformation self-organization at the micro level Fig. 4 shows the values of the Hurst exponent, H1. In all the cases, the Hurst exponent is significantly less than 0.5, which indicates the antipersistent, ergodic nature of the system, i.e. the system tends to return to the average profile height. The degree of stability of the system depends on how close the value of H is to zero. For all the cases, the correlation length L1 is within 1.5–2.5 µm (on average, 2.2 ± 0.2 µm) and does not depend on the degree of deformation or crystallographic orientation. The correlation length L1 shows that the scale level under consideration refers to the micro level and the system's ability to self-organize at this level is controlled by the dislocation subsystem. At the micro level, self-consistent motion of dislocations deforms the crystal throughout the loading time.
Fig. 4. Hurst exponent H1
3.2. Deformation self-organization at the mezo level Let us move to the next scale level and consider the deformation relief on the single crystal's surface, which reflects the processes occurring inside the crystal. Fig. 5 shows a graph of the change in the Hurst exponent H2 depending on the deformation degree. For most cases, the tendency to self-organization is strongly pronounced. The Hurst exponent H2 is less than 0.5. Only in some cases for the orientation of [111] (110) and [001] (100) H2 increases slightly, which indicates a decrease in correlation. At the same time, when passing the value ɛ = 23%, the value of the Hurst exponent H2 decreases again.
Fig. 5. Hurst exponent H2 However, the value of the correlation length L2 in this case is different depending on the crystallographic orientation (deformation relief) its change is shown in Fig. 6. In general, the magnitude of the correlation length L2 corresponds to the mesoscale and geometric size of the shear traces mesobands [3].
Fig. 6. Correlation length L2
Thus, the correlated shear in octahedral planes ensures self-organization at this scale. Depending on the crystallographic orientation, on the surface of a single crystal various elements of the deformation relief can form. Such as, individual shear traces and band of shear traces ([001]-single crystals), mesobands ([110]-single crystals), macrobands and folded (corrugated) structures ([111]-single crystals) [3, 49-50]. Next, we consider the values of the mean root-mean-square roughness surface. Fig. 7 shows the root-mean-square roughness and indicates that the interface width does not change with an increase in the degree of deformation for all studied orientations, except [001] (100) and [111] (110). For these cases, there is both a significant increase and a decrease in the width of the interface, depending on the degree of deformation.
Fig. 7. Interface width / root-mean-square roughness (rms) (ω/Sq)
It should be noted that for [001] and [111] -single crystals on the side faces (100) and (110), respectively, the Hurst exponent at the micro- and meso level considerably exceed the values established for other crystallographic orientations. At the same time, the Hurst exponent H1 and H2 for the indicated orientations differs most with a strain of 15-20%, which exactly corresponds to the transition between the second and third stages. At the micro level, this process is accompanied by the beginning of the formation and development of a disoriented band substructure from a cellular one. At the meso level, we can observe the development of shear traces and their combination into bundles. Interestingly, when examining [111] single crystals for the (112) face perpendicular to the (110) face described above, such a significant jump in the Hurst exponent is not observed. This face is occupied by folded (corrugated) structures that exhibit a slightly different behavior. Below, attention will be given to folded structures of various types.
3.3. The consistency of plastic deformation self-organization levels Let us return to the folded (corrugated) structures. They represent the most interesting and morphologically diverse type of deformation relief [2, 49-50]. The results of the calculation of the Hurst exponent described above were carried out for the folds corresponding to type 1 in Fig. 8, a-b. However, folded structures of a different morphology can be distinguished on the (112) face of [111] single crystals (type 2 in Fig. 8, c – d). Fig. 8 demonstrates a comparison of the morphology of folded structures. In the case of type 1, folded structures work together with macro bands (indicated by an arrow in Fig. 8, a) and are arranged perpendicular to them. The formation of a "hilly-shaped" surface is characteristic of type 2 folds [51]. When considering the folds of type 2, when the degree of deformation reached 18%, three correlation lengths can be identified on the height-height correlation function (L1 = 1.2 µm, L2 = 7 µm, L3 = 10 µm) and three the Hurst exponents (H1 = 0.11, H2 = 0.22, H3 = 0.27).
a)
c)
b)
d) Fig. 8. Relief morphology: 3D surface (a, c), surface profile along a secant 1-2 (b, d);
type 1 (a-b), type 2(c-d) Consider the folded structure type 2 in more detail. Fig. 9 shows the 3D image of the folded structure (Fig. 9, a), the relief of misorientation (Fig. 8, b), the distribution of the misorientation angles relative to the fixed point 1 (Fig. 9, c) and the dislocation structure (Fig. 9,
d). Simultaneous consideration of the above illustrations shows that between the wavelength of the folds, the distance between the misoriented bands and the dislocation substructure is a correlation. From the data obtained it follows that for the formation of one fold, it is necessary to have about 20 bands bounded by geometrically necessary boundaries (GNB). This amount allows you to get the angle of misorientation of the order of 10-11°. At the surface of the crystal, the angles of misorientation are more than 15 °, but the greatest number of boundaries are of the misorientation of about 8–11 °. When moving deeper into the sample, the fraction of boundaries with a misorientation value of more than 2 ° gradually decreases. The distance between the reoriented bands at the crystal surface is 45 ± 7 µm; at a depth of 700 µm from the surface, the value increases to 70–115 µm. This indicates that the deformation processes in the surface layer are more intense.
a)
c)
b)
d)
Fig. 9. Relief morphology: 3D surface (a), misorientation of areas (EBSD) (b), misorientation about point 1 from fig. b (c), dislocation substructure (TEM) (d)
This is due to the inclusion of an additional method of organizing plastic deformation at the level of the dislocation subsystem — the formation of misoriented microbands and misorientations of a larger scale. As shown by our studies obtained by the method of diffraction of reflected electrons, the most pronounced disorientation in [111] nickel single crystals reaches a depth of 350–380 µm from the sample surface on the (112) face [51-52]. Note that in [111] single crystals, octahedral planes do not have free exit to the crystal surface, they are limited to punches of the testing machine. Consequently, reverse stresses arise, which are absent in the formation of shear traces, shear traces bands and mesobands. Such difficult conditions for shear, contribute to the fact that in order to preserve the integrity of the crystal, it is necessary to connect an additional method of deformation (reorientation of local volumes) from earlier degrees of deformation. We also performed EBSD analysis for other single crystals. We did not observe misorientations in the formation of shear traces ([001] single crystals) and mesobands ([110] single crystals) in the considered degrees of deformation (Fig. 10, b – g). The EBSD analysis data showed the presence of small near-surface misorientations in addition to the octahedral slip in the case of the formation of macrobands ([111] single crystals, (110) face) (Fig. 10, h – j).
b)
c)
a)
d)
e)
h)
f)
i)
g)
j)
Fig. 10. EBSD-analyzes: scheme and stereographic triangle (a), meso bands (face (110), compression axis [110]), ε = 29% (b-d), shear traces (face (110) compression axis [001]), ε = 10% (e-g), macro band (face (110) compression axis [111]), ε = 16% (h-j)
4. Discussion 4.1. Micro level As mentioned in section 3.1, the magnitude of the correlation length L1 indicates that we are observing the microscale. In this case, the correlation length L1 does not depend on the degree of deformation or crystallographic orientation. Consider the behavior of the Hurst exponent H1 along the stress-strain curve. At the first stage of the stress-strain curve, plastic deformation processes are associated with the development of a shear in a single crystal volume and unrestricted material rotations [53]. Here, the main sources of dislocations are surface stress concentrators at the microscale level, and relaxation occurs with reverse shears. In this case (Fig. 4), very low values of the Hurst exponent can be observed. At the same time, for some crystals, Fig. 4 indicates a rather significant increase in the value of the Hurst exponent compared to other values. This refers to [001] single crystals with side faces (100) and to [111] single crystals with side faces (110). When ɛ = 17-18%, the value of H1 first increases tenfold, and then decreases again. This is apparently connected with the change of stages of deformation. In this case, with continuing loading of the crystal, the deformation increasingly reflects its self-organization features
in the way dislocations are
distributed [18, 20]. First, the excess dislocation density is redistributed and then accumulated. At the second stage, the cellular dislocation structure begins to form; and at the third stage, the cellular dislocation substructure continues to develop, and transition to the disoriented cellular dislocation structure occurs. At the second and third stage, cells in FCC materials with high and average values of the packing defect energy play a significant role in plastic deformation and hardening. Dislocation cells create barriers to the movement of dislocations. This happens at the third stage when these cells become disoriented, sub-boundaries are destroyed and new sub-boundaries appear which determines the development of the microband substructure. In this case, traces with a greater shear often appear. Fig. 9, d shows the dislocation substructure at the surface of the face occupied by the folds. Fig. 11, b shows the dislocation substructure at the surface of the face occupied by shear bands packs and meso bands (foils are cut perpendicular to the relief elements, arrows indicate the places of intrusion in the area of the formation of relief elements). The results indicate that, at
the surface, the dislocation substructure for folded structures (Fig. 9, e), shear tracks packs (Fig. 11, a) and meso bands (Fig. 11, b) is different. This difference is reflected in the values of the Hurst exponent H1.
a)
b)
Fig. 11. Dislocation substructure [001] – single crystal, ε = 39% (stem) (a), [110] – single crystal, ε = 29% (b)
An example of an extensive overview of the formation and development of a dislocation structure in pure FCC metals is presented in [54]. Given that the processes of self-organization on the correlation length L1 are provided through cooperative processes at the level of a dislocation subsystem, of particular interest is how similar or different these cooperative processes are in different materials, and in what ways the dislocation subsystem is self-organised. The works [19-32] describe the statistical parameters of the dislocation structures for single crystals and polycrystals of different materials deformed under different conditions. It was found that such parameters (normalized to the most probable values) of the dislocation substructure as cell size, i.e. the distance between geometrically necessary boundaries (GNB), and the misorientation angle between cells obey the same distribution law. This is regardless of the type of material and deformation conditions. Therefore, we can talk about the scaling of these parameters. Pantleon et al. provided evidence of the correlations for FCC single crystals (Cu, Al) between misorientations within the boundaries of neighboring dislocation cells [22]. In addition, they compared the experimental results (EBSD and X-ray diffraction) with the model, and showed that misorientations in the neighboring dislocation cells also feature anticorrelation. The researchers proposed a model according to which the misorientation most frequently manifests itself through opposite signs in the adjacent cell boundaries. This fact leads to the alignment of the misorientation angle across several cell boundaries, which is confirmed experimentally. In
our paper, the Hurst exponent H1 is less than 0.5, which also indicates that any tendency tends to be replaced by the opposite one (anticorrelation). In other words, the accumulation of the disorientation of one sign, for example, tends to be replaced by the misorientation of the opposite sign; or the growth of the height of protrusions on the surface tends to decrease). Pantleon et al. explain the anticorrelation between the misorientations of the angles of the neighboring boundaries of dislocation cells by the limited free path of dislocations [22]. Pairs of dislocations with Burgers vectors of opposite sign are separated by a distance of 2λ, determined by the mean free path λ of an individual dislocation. If both dislocations were completely independent, any correlation would disappear. In general, the growing role of cooperative processes in the dislocation substructure ensures that, over the L1 correlation length, the Hurst exponent H1 increases slightly with increasing loading degree (Fig. 4). The crystal ability to self-organize at the micro level is very high in the whole considered range of deformation. All these processes of self-organization at the level of the dislocation substructure are reflected on the crystal surface. Therefore, the question of the relationship between the morphology of the surface caused by the deformation and the microstructure of the defect remains interesting. Above, we wrote about the self-similar structure of the dislocation subsystem for a wide range of materials and deformation conditions; many of these works concluded that dislocation reliefs in deformed metals can be characterized in terms of the universal cell size and distribution of the statistical characteristics of the cells. This leads us to the idea that there is no one-to-one correspondence between the defective microstructure and plastic flow reliefs, and that different mechanisms can regulate the spatial organization of slip and the formation of the relief caused by deformation. Similar results were discussed in the paper [37]. 4.2. Mezo level The magnitude of the correlation length of L2 indicates that the processes under consideration are characteristic of the mesoscale level. Comparing the Hurst exponent H2 and the correlation length L2 with the stress-strain curve, we can see the connection of these parameters with the stages. During the change of stages of the stress-strain curve, we observe an increase in the Hurst exponent H2 and the correlation length L2. The work [40] also established the relationship between the Hurst exponent and the correlation length with the stages of plastic deformation. If we combine the graphs H1 and H2 (Fig. 12), we can note that the values of H1 and H2 are very close at the initial degrees of deformation, at the initial stages.
Fig. 12. Hurst exponent Н1 and Н2 The leading role belongs to the translational motion of dislocations in close-packed planes. In this case, barriers that prevent the free sliding of dislocations (dislocation walls, clusters, etc.) have not yet formed, so this mechanism alone is sufficient to ensure self-consistent deformation at the micro- and meso level. The difference in the Hurst exponents H1 and H2 are becoming more significant with increasing load. In addition, this difference depends on the crystallographic orientation of the compression axis and the side faces. This dependency is discussed below. In all likelihood, we observe how the correlated shear of the material mesovolumes and the rotational component is involved in the formation of the deformation relief. The self-organization of the dislocation structure, the misorientation of local regions and the geometric parameters of the folded (corrugated) structures was discussed in the Section 3.3. Thus, using a number of parameters, one can trace the hierarchical interconnection of the self-organization processes at various scale levels. The micro level is a dislocation substructure, which can be characterized by the Hurst exponent H1, the correlation length L1, the misorientation angles, the distance between the geometrically necessary boundaries (GNB), the scaling of these distances. The mesoscale – shear planes bundles, Hurst exponent H2, L2 correlation length, macro level – structural elements of the deformation relief, the entire sample face, can be described by the Hurst exponent and correlation length obtained from regions of appropriate size and interface width (not shown in this paper). In addition, the authors have obtained data using other methods that indicate the presence of self-organization of plastic deformation processes at the macro level [3]. A number of papers, for instance [39-40] also show that the self-organization on macro scale differs from nature at the micro and meso levels described in this article. 4. Conclusion
Thus, for nickel single crystals with highly symmetric orientations of the compression axis was established: 1. The self-organization of the plastic deformation process occurs at different scale levels starting from the dislocation substructure scale through the macro strain relief one. The relationship between the scale levels has been revealed. 2. Using statistical processing of the deformation relief parameters based on the Hurst exponent, it was revealed that at the micro level the mutually consistent deformation is carried out due to the dislocation structure, at the meso-level - due to a correlated shear in parallel slip planes. 3. Traces of shear and meso bands, in the strain range under study, are formed due to dislocation glide; near-surface misorientation of local regions is additionally used to form macrobands, and also misorientation inside the crystal for corrugation. 4. It is established the relationship between changes in the type of dislocation substructure, the Hurst exponent, the correlation length, and the stages of the stress-strain curve. The obtained results supplement the already accumulated knowledge of metallic materials plastic deformation and from the point of view of the authors, it would be interesting to continue the research of plastic deformation processes self-organization based through the analysis of the deformation relief on a polycrystalline aggregate. The results described in this paper, obtained on single crystals of different crystallographic orientations, can serve as a fundamental basis for understanding the processes occurring in individual polycrystal grains. Acknowledgement The results of EBSD have been obtained with financial support from RFBR, according to the research project No. 16-32-60007 mol_а_dk. The preparation of samples, the results of laser scanning microscope have been obtained with financial support from Fundamental Research Program of State Academies for 2013-2020 (project No III.23.2.4 ). The research is supported by Tomsk Polytechnic University within the framework of Tomsk Polytechnic University Competitiveness Enhancement Program. The authors are grateful to Professor D. Lychagin for the samples provided for the investigation.
Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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1. The dislocation structure realize self-organization deformation at the microlevel 2. Correlated shear in parallel planes realize self-organization deformation at the mezolevel 3. Scale invariants connect with dislocation substructure, deformation relief, curve 4. The purpose of the deformation self-organization is to preserve the crystal integrity under load 5. Deformation mechanisms for different types of deformation relief was established
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S1044-5803(19)31712-7
DOI:
https://doi.org/10.1016/j.matchar.2019.110007
Reference:
MTL 110007
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Materials Characterization
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Please cite this article as: A. Ekaterina, F. Andrey, Self-organization processes during deformation of nickel single crystals, Materials Characterization (2019), doi: https://doi.org/10.1016/ j.matchar.2019.110007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.
Self-organization processes during deformation of nickel single crystals † 1 2
Alfyorova Ekaterina 1, Filippov Andrey 2
National Research Tomsk Polytechnic University, Tomsk, Russia
Institute of Strength Physics and Materials Sciences SB RAS, Tomsk, Russia e-mail: †[email protected], [email protected]
Abstract In this work, plastic deformation of nickel single crystals with highly symmetric orientations was examined. Revealed scale levels of plastic deformation self-organization and the corresponding deformation mechanisms. It was established that the shear traces and mesobands are formed due to dislocation glide, macro-bands are formed due to dislocation glide in and near-surface misorientation of local areas, and for folds the disorientation inside the crystal is additionally developed. Based on the morphology of folded structures, it is shown that there is a correlation between the wavelength of the folds, the distance between the misoriented bands and the dislocation substructure. The Hurst exponent and correlation lengths correlate with the stages of the stress-strain curve and, consequently, the type of dislocation substructure. Keywords: FCC single crystal; plastic deformation; microstructure; scaling; deformation relief self-organization.
1. Introduction Plastic deformation is currently defined as a self-consistent process at various structural levels. Synergetics believes that self-organization is the process of ordering in an open system, which is due to the coordinated interaction between the set of its constituent elements. The one of importance things is this process occurs without specific actions from outside. The result is the appearance of a unit of the next quality level. For a single crystal, structural levels are dislocations, shear zones, dislocation groups, and slip systems. The new unit of the next quality level is bands of shear traces, mezo bands, macro bands and folders for FCC single crystals [1-3]. In the case of plastic deformation, self-organization is aimed at preserving the integrity of the material for as long as possible. Internal processes occurring during loading manifest themselves on the crystal’s surface as a deformation relief of various scale. At the initial degrees of deformation, of special significance are dislocation mechanisms (translational motion of dislocations along close-packed planes), which lead to the formation of thin shear traces on the surface. As early as the beginning of the 20th century, Rosenhain, Ewing, Hirth and Lote [4] showed that lines on the surface of a deformed sample are stages on the surface. They arise on the surface as a result of the collective movement of many dislocations.
Traces of shear appear at the initial degree of deformation under different types of loading in both single and polycrystals. Consequently, they can be considered the basic structural element of the deformation relief (SEDR). With further loading, the shear traces are ordered into the next scale level system (pack slip bands, meso- and macro bands, corrugated structures of various types (folds), persistent slip band, etc.). The role of large-scale structural levels increases accordingly with increasing degree of deformation. These levels involve the higher scale meso-volumes in the translationalrotational movement, including the ensembles of grains in the case of polycrystals. When researchers consider the deformation relief as a source of information about internal processes in a crystal during plastic deformation, they traditionally focus on the height of the shear, the length of slip lines and the distance between them. Mader, Kronmyuller, Pfaff and Mitchell [5-6] were among the first scientists to measure the height of the shear. They showed that it does not depend on the degree of deformation and remains practically unchanged within the second stage of the stressstrain curve. The manifestation of shear deformation in relief is diverse [7-16]. There is a body of literature that describes processes occurring with single slip lines or with a group of slip lines, i.e. these researchers consider the movement of isolated dislocation groups [17-18]. If we consider a smaller scale level, then these will be works that explore the processes of self-organization directly in the dislocation substructure. In the works [19-32], based on statistical processing of the parameters of the dislocation substructure (cell size, the distance between the geometrically necessary boundaries (GNB) and the misorientation angle between the cells), their scaling was established. The works [33-34] describe the self-affine nature of the cellular dislocation structure in Cu and KCl single crystals at different scale levels. The works indicate that self-affinity is characteristic of highly symmetric single crystals, where a large number of slip systems operate simultaneously. Similar results are described in the paper [35]. Hughes reports that various materials demonstrate similarities in the microstructural reliefs arising from monotonic deformation. Hughes identified scaling laws in the dislocation structure, which indicate that similar dislocation mechanisms work for a wide range of materials and deformation conditions, and showed that a sufficient three-dimensional dislocation mobility to create three-dimensional structures is a necessary condition [35]. The results were obtained for polycrystals of Ni, Cu, Al and 304L steel, as well as on Cu and Al single crystals under various loading conditions (compression, rolling, forging, torsion and friction). Oudriss et al. discuss the scaling parameters of the dislocation structure in tension of a [001] nickel single crystal. The author proves the presence of scaling in a large range of plastic deformations. In addition, the article demonstrates a series of scaling laws regarding shear stress
for cell size, dislocation cell wall thickness, and dislocation density in dislocation walls and cells [31]. Godfrey, Hughes [36] provide physical parameters (distances between geometrically necessary boundaries, etc.) connecting deformation microstructures in a wide range of scales and indicate that there is universal scaling. This scaling proves the existence continuous process that create new bounders, and those that destroy them from the macro to the nanoscale. In addition, the authors note that microstructural evolution occurs under conditions of dislocation slip down to the nanolevel. High-resolution electron microscopy also confirms the dominant role of dislocations. This experiment showed the presence of a large number of gliding dislocations in the layers between geometrically necessary boundaries with an individual step up to 5 nm [35] The scaling by the distance between the boundaries of the dislocation structure does not depend on the initial grain size and the distribution of grain sizes. Similar results were obtained for single crystals (aluminum) and polycrystals (nickel, aluminum) in [36] and for grains inside a polycrystalline sample (aluminum) [30]. The analysis performed shows that similar physical mechanisms underlie the plastic deformation of a wide range of different materials and loading conditions. A common factor in each case is that plastic deformation leads to microstructural gliding due to the formation of a three-dimensional dislocation network structure. The main physical conditions for the formation of such microstructures is that the deformation must occur through dislocation slip with the work of at least three non-mutually planar Burgers vectors and sufficient three-dimensional mobility of these dislocations [19, 23]. When determining the parameters of the dislocation structure (cell size), it should be borne in mind that the results of metallographic studies are sensitive to the parameters used when conducting transmission electron microscopy (TEM). This provides a natural explanation for large systematic and not yet understood discrepancies in relation to the average cell size reported by different authors, as also described in [24]. Thus, the currently available experimental material indicates the similarity in the microstructural regularities and dislocation mechanisms of deformation that occur when different FCC metals and alloys are loaded. At the same time, of particular interest is the study of self-organized deformation reliefs on a specific surface area. However, researchers pay much less attention to the correlation between shear traces in the far distance. To describe the integral picture, it seems reasonable to use fractal methods and such attempts are made in many papers [19, 37-39]. However, many of these works are reduced to obtaining the value of the fractal dimension of a particular surface, but the question of the deformation process self-organization and the identification of
scale levels remains open. One of the options for fractal analysis, which allows to identify scale-invariant reliefs of plastic deformation is the correlation function "height-height". The method of calculating and analyzing this function is described in detail in the section “Material and methods”. In the works [40-42] the results obtained using the correlation function "height-height" are described. With this method, was established the existence of several scale levels with different types of loading. Lyapunova et al. established the presence of two zones with Hurst exponent 0.3–0.4 and 0.5–0.6, for an aluminum alloy after dynamic loading. These zones represent the destruction surface. The interval of a certain scale is 1–18 µm [40]. Oborin et al. found that the Hurst exponent in the fracture zone is 0.64–0.81, depending on the experimental conditions. A number of papers give the Hurst exponent greater than 0.5 when describing the deformation relief in the case of various methods of loading [41-42]. This indicates that, on certain scales, the system has a persistent (trend-resistant) character. It is important to note that the size of the zones where the measurements were made, and the resolution of the equipment used makes significant adjustments to the results obtained. In particular, a number of studies discussed here were carried out at sites of several millimeters or even tens of millimeters, which made it possible to establish scaling at the macro level. An analysis of the literature has shown that, on the one hand, there are works that study the processes of plastic deformation self-organization separately at the level of the dislocation substructure, separately at the level of the description of the morphology of the relief. On the other hand, a tool allows us to combine the results and trace the hierarchical relationship between the scale levels. In this connection, this case study seeks to examine the scale-invariant reliefs of the plastic deformation development at different scale levels in nickel single crystals with different crystallographic orientation based on data on the evolution of the deformation relief. 2. Material and methods The findings are obtained on nickel single crystals with the compression axes oriented in the angles of a standard stereographic triangle with various sets of sides (Fig.1).
Fig. 1. Stereographic triangle and corresponding deformation relief
In the research, we analysed nickel single crystals grown by the Bridgman method. The specimens had a shape of a tetragonal prism with a height-to-width ratio of two. The orientation was carried out on an IRIS 3 X-ray generator using Laue back-reflection photographs with a ±1° accuracy and further adjusted to a ±0.02° accuracy, using a DRON-3 X-ray diffractometer. The surface of the specimens was prepared by mechanical grinding and polishing and final electrolytic polishing in a saturated solution of chromic anhydride in ortho-phosphoric acid at 20V. Compression was carried out on an Instron ElektroPuls E10000 testing machine at a speed of 1.4·10-3 s-1 at room temperature. A Leica DM 2500P optical microscope and an Olympus LEXT OLS4100 confocal laser scanning microscope were used to analyze the deformation relief. The size of the scanned region in each case was 0.066 mm2 (0.256 x 0.256 mm2). The depth resolution was 0.06 µm. The size of the scanned region in this case sets a limit on the selforganization scale being established. Experiments were carried out in the strain interval of 1.5–32%. An EBSD attachment to the Tescan Vega II LMU microscope was used to determine the extent of misorientation of local areas. We are discussing the results obtained from the areas in the center of the sample faces . Objects for transmission electron microscopy (TEM) were made in the form of thin foils using the EM-09100IS sample preparation system (JEOL Ltd, Japan) by ion thinning. The foils were prepared according to the cross-section technique. The dislocation structure carried out using a TEM instrument JEM-2100 (JEOL Ltd, Japan). To reveal scale-invariant regularities of plastic straining of nickel single crystals under compression, we used fractal analysis of the deformation relief based on the height–height correlation
function H(r) (1) [43-45], where Z(r) is the function of the surface height over all pairs of points separated by a fixed distance r; angle brackets indicate averaging over all pairs of points. ,
,
(1)
The slope of the initial segment of the curve describing the height–height correlation function H(r) and plotted in logarithmic axes can be used for determining the Hurst exponent (H) and correlation length (L). The correlation length is determined by the projection of the linear segment onto the abscissa. Analysis of the behavior of the resultant height–height correlation functions H(r) has made it possible to single out segment L1 and L2 on the curve on which the experimental points can be approximated by straight lines with different slopes relative to the axes (Fig. 2). This makes it possible to determine the Hurst exponent and the upper boundaries of the correlation length for several segments. On segment L3, it is impossible to approximate experimental data with the help of a straight line because of their fluctuation behavior. This indicates the lack of stable correlation on this size scale (Fig. 2).
Fig. 2. Height–height correlation function in the region of mezo bands formation, [110] nickel single crystal ε = 32% (dashed curves are approximations of the segments of the solid curves)
Hurst exponent H estimates the degree of chaotization (self-organization) of the system. The Hurst exponent value less than 0.5 indicates the antipersistent, ergodic nature of the system, i.e. the system tends to return to the average value, and the degree of the system stability depends on how close the value of H is to zero. A value of H = 0.5 indicates no correlation. If the value is greater than 0.5, it means/denotes the presence of long-scale correlations [40]. Moreover, for the self-affine surface for the function H (r) the following holds true:
~
2
, ,
≪ξ ≫ξ
where α is the scaling coefficient, Hurst exponent (H), or roughness index, ξ is the lateral correlation length, it defines the upper limit of the region of self-affinity and characterizes surface irregularities in the lateral direction. Height fluctuations at r << ξ (small scale) are correlated, dependent, and at r >> ξ (large scale) - uncorrelated, random. ω - the width of the interface is a characteristic of the irregularity of the surface in the vertical direction, it is also called the correlation length in the vertical direction or the mean square surface roughness (ω = Sq) [43-44, 46-48]. Once again, in the framework of this work, the deformation relief was considered as an external manifestation of plastic deformation processes occurring inside the crystal. We use the terms self-organization of the deformation relief and self-organization of plastic deformation as very similar in meaning, meaning that the processes of plastic deformation self-organize during crystal loading and, therefore, appear on the surface in the form of a self-organized relief. 3. Results The results are obtained on nickel single crystals with various compression axes in the strain interval of 1.5–32%. In an earlier paper, we described the evolution of the deformation relief [3]. The deformation relief after the initial loading (stage I of the stress-strain curve) is slip lines formed by individual slip traces. During the transition to the subsequent loading stages, the deformation relief depends on the crystallographic orientation. The development of shear traces, their coarsening and self-organization in packs is characteristic for [001] single crystals. Formation of mezo bands is characteristic for [110] single crystals and the formation of macro bands and folded structures for [111] single crystals (Fig.1). Fig. 3 shows the stress-strain curves for the single crystals with various compression axes. We can clearly observe the orientational dependence of the curves. In the present study, we calculate the values of the Hurst exponent for all the cases specified in the “Materials and Methods” section.
Fig. 3. The stress-strain curves
3.1. Deformation self-organization at the micro level Fig. 4 shows the values of the Hurst exponent, H1. In all the cases, the Hurst exponent is significantly less than 0.5, which indicates the antipersistent, ergodic nature of the system, i.e. the system tends to return to the average profile height. The degree of stability of the system depends on how close the value of H is to zero. For all the cases, the correlation length L1 is within 1.5–2.5 µm (on average, 2.2 ± 0.2 µm) and does not depend on the degree of deformation or crystallographic orientation. The correlation length L1 shows that the scale level under consideration refers to the micro level and the system's ability to self-organize at this level is controlled by the dislocation subsystem. At the micro level, self-consistent motion of dislocations deforms the crystal throughout the loading time.
Fig. 4. Hurst exponent H1
3.2. Deformation self-organization at the mezo level Let us move to the next scale level and consider the deformation relief on the single crystal's surface, which reflects the processes occurring inside the crystal. Fig. 5 shows a graph of the change in the Hurst exponent H2 depending on the deformation degree. For most cases, the tendency to self-organization is strongly pronounced. The Hurst exponent H2 is less than 0.5. Only in some cases for the orientation of [111] (110) and [001] (100) H2 increases slightly, which indicates a decrease in correlation. At the same time, when passing the value ɛ = 23%, the value of the Hurst exponent H2 decreases again.
Fig. 5. Hurst exponent H2 However, the value of the correlation length L2 in this case is different depending on the crystallographic orientation (deformation relief) its change is shown in Fig. 6. In general, the magnitude of the correlation length L2 corresponds to the mesoscale and geometric size of the shear traces mesobands [3].
Fig. 6. Correlation length L2
Thus, the correlated shear in octahedral planes ensures self-organization at this scale. Depending on the crystallographic orientation, on the surface of a single crystal various elements of the deformation relief can form. Such as, individual shear traces and band of shear traces ([001]-single crystals), mesobands ([110]-single crystals), macrobands and folded (corrugated) structures ([111]-single crystals) [3, 49-50]. Next, we consider the values of the mean root-mean-square roughness surface. Fig. 7 shows the root-mean-square roughness and indicates that the interface width does not change with an increase in the degree of deformation for all studied orientations, except [001] (100) and [111] (110). For these cases, there is both a significant increase and a decrease in the width of the interface, depending on the degree of deformation.
Fig. 7. Interface width / root-mean-square roughness (rms) (ω/Sq)
It should be noted that for [001] and [111] -single crystals on the side faces (100) and (110), respectively, the Hurst exponent at the micro- and meso level considerably exceed the values established for other crystallographic orientations. At the same time, the Hurst exponent H1 and H2 for the indicated orientations differs most with a strain of 15-20%, which exactly corresponds to the transition between the second and third stages. At the micro level, this process is accompanied by the beginning of the formation and development of a disoriented band substructure from a cellular one. At the meso level, we can observe the development of shear traces and their combination into bundles. Interestingly, when examining [111] single crystals for the (112) face perpendicular to the (110) face described above, such a significant jump in the Hurst exponent is not observed. This face is occupied by folded (corrugated) structures that exhibit a slightly different behavior. Below, attention will be given to folded structures of various types.
3.3. The consistency of plastic deformation self-organization levels Let us return to the folded (corrugated) structures. They represent the most interesting and morphologically diverse type of deformation relief [2, 49-50]. The results of the calculation of the Hurst exponent described above were carried out for the folds corresponding to type 1 in Fig. 8, a-b. However, folded structures of a different morphology can be distinguished on the (112) face of [111] single crystals (type 2 in Fig. 8, c – d). Fig. 8 demonstrates a comparison of the morphology of folded structures. In the case of type 1, folded structures work together with macro bands (indicated by an arrow in Fig. 8, a) and are arranged perpendicular to them. The formation of a "hilly-shaped" surface is characteristic of type 2 folds [51]. When considering the folds of type 2, when the degree of deformation reached 18%, three correlation lengths can be identified on the height-height correlation function (L1 = 1.2 µm, L2 = 7 µm, L3 = 10 µm) and three the Hurst exponents (H1 = 0.11, H2 = 0.22, H3 = 0.27).
a)
c)
b)
d) Fig. 8. Relief morphology: 3D surface (a, c), surface profile along a secant 1-2 (b, d);
type 1 (a-b), type 2(c-d) Consider the folded structure type 2 in more detail. Fig. 9 shows the 3D image of the folded structure (Fig. 9, a), the relief of misorientation (Fig. 8, b), the distribution of the misorientation angles relative to the fixed point 1 (Fig. 9, c) and the dislocation structure (Fig. 9,
d). Simultaneous consideration of the above illustrations shows that between the wavelength of the folds, the distance between the misoriented bands and the dislocation substructure is a correlation. From the data obtained it follows that for the formation of one fold, it is necessary to have about 20 bands bounded by geometrically necessary boundaries (GNB). This amount allows you to get the angle of misorientation of the order of 10-11°. At the surface of the crystal, the angles of misorientation are more than 15 °, but the greatest number of boundaries are of the misorientation of about 8–11 °. When moving deeper into the sample, the fraction of boundaries with a misorientation value of more than 2 ° gradually decreases. The distance between the reoriented bands at the crystal surface is 45 ± 7 µm; at a depth of 700 µm from the surface, the value increases to 70–115 µm. This indicates that the deformation processes in the surface layer are more intense.
a)
c)
b)
d)
Fig. 9. Relief morphology: 3D surface (a), misorientation of areas (EBSD) (b), misorientation about point 1 from fig. b (c), dislocation substructure (TEM) (d)
This is due to the inclusion of an additional method of organizing plastic deformation at the level of the dislocation subsystem — the formation of misoriented microbands and misorientations of a larger scale. As shown by our studies obtained by the method of diffraction of reflected electrons, the most pronounced disorientation in [111] nickel single crystals reaches a depth of 350–380 µm from the sample surface on the (112) face [51-52]. Note that in [111] single crystals, octahedral planes do not have free exit to the crystal surface, they are limited to punches of the testing machine. Consequently, reverse stresses arise, which are absent in the formation of shear traces, shear traces bands and mesobands. Such difficult conditions for shear, contribute to the fact that in order to preserve the integrity of the crystal, it is necessary to connect an additional method of deformation (reorientation of local volumes) from earlier degrees of deformation. We also performed EBSD analysis for other single crystals. We did not observe misorientations in the formation of shear traces ([001] single crystals) and mesobands ([110] single crystals) in the considered degrees of deformation (Fig. 10, b – g). The EBSD analysis data showed the presence of small near-surface misorientations in addition to the octahedral slip in the case of the formation of macrobands ([111] single crystals, (110) face) (Fig. 10, h – j).
b)
c)
a)
d)
e)
h)
f)
i)
g)
j)
Fig. 10. EBSD-analyzes: scheme and stereographic triangle (a), meso bands (face (110), compression axis [110]), ε = 29% (b-d), shear traces (face (110) compression axis [001]), ε = 10% (e-g), macro band (face (110) compression axis [111]), ε = 16% (h-j)
4. Discussion 4.1. Micro level As mentioned in section 3.1, the magnitude of the correlation length L1 indicates that we are observing the microscale. In this case, the correlation length L1 does not depend on the degree of deformation or crystallographic orientation. Consider the behavior of the Hurst exponent H1 along the stress-strain curve. At the first stage of the stress-strain curve, plastic deformation processes are associated with the development of a shear in a single crystal volume and unrestricted material rotations [53]. Here, the main sources of dislocations are surface stress concentrators at the microscale level, and relaxation occurs with reverse shears. In this case (Fig. 4), very low values of the Hurst exponent can be observed. At the same time, for some crystals, Fig. 4 indicates a rather significant increase in the value of the Hurst exponent compared to other values. This refers to [001] single crystals with side faces (100) and to [111] single crystals with side faces (110). When ɛ = 17-18%, the value of H1 first increases tenfold, and then decreases again. This is apparently connected with the change of stages of deformation. In this case, with continuing loading of the crystal, the deformation increasingly reflects its self-organization features
in the way dislocations are
distributed [18, 20]. First, the excess dislocation density is redistributed and then accumulated. At the second stage, the cellular dislocation structure begins to form; and at the third stage, the cellular dislocation substructure continues to develop, and transition to the disoriented cellular dislocation structure occurs. At the second and third stage, cells in FCC materials with high and average values of the packing defect energy play a significant role in plastic deformation and hardening. Dislocation cells create barriers to the movement of dislocations. This happens at the third stage when these cells become disoriented, sub-boundaries are destroyed and new sub-boundaries appear which determines the development of the microband substructure. In this case, traces with a greater shear often appear. Fig. 9, d shows the dislocation substructure at the surface of the face occupied by the folds. Fig. 11, b shows the dislocation substructure at the surface of the face occupied by shear bands packs and meso bands (foils are cut perpendicular to the relief elements, arrows indicate the places of intrusion in the area of the formation of relief elements). The results indicate that, at
the surface, the dislocation substructure for folded structures (Fig. 9, e), shear tracks packs (Fig. 11, a) and meso bands (Fig. 11, b) is different. This difference is reflected in the values of the Hurst exponent H1.
a)
b)
Fig. 11. Dislocation substructure [001] – single crystal, ε = 39% (stem) (a), [110] – single crystal, ε = 29% (b)
An example of an extensive overview of the formation and development of a dislocation structure in pure FCC metals is presented in [54]. Given that the processes of self-organization on the correlation length L1 are provided through cooperative processes at the level of a dislocation subsystem, of particular interest is how similar or different these cooperative processes are in different materials, and in what ways the dislocation subsystem is self-organised. The works [19-32] describe the statistical parameters of the dislocation structures for single crystals and polycrystals of different materials deformed under different conditions. It was found that such parameters (normalized to the most probable values) of the dislocation substructure as cell size, i.e. the distance between geometrically necessary boundaries (GNB), and the misorientation angle between cells obey the same distribution law. This is regardless of the type of material and deformation conditions. Therefore, we can talk about the scaling of these parameters. Pantleon et al. provided evidence of the correlations for FCC single crystals (Cu, Al) between misorientations within the boundaries of neighboring dislocation cells [22]. In addition, they compared the experimental results (EBSD and X-ray diffraction) with the model, and showed that misorientations in the neighboring dislocation cells also feature anticorrelation. The researchers proposed a model according to which the misorientation most frequently manifests itself through opposite signs in the adjacent cell boundaries. This fact leads to the alignment of the misorientation angle across several cell boundaries, which is confirmed experimentally. In
our paper, the Hurst exponent H1 is less than 0.5, which also indicates that any tendency tends to be replaced by the opposite one (anticorrelation). In other words, the accumulation of the disorientation of one sign, for example, tends to be replaced by the misorientation of the opposite sign; or the growth of the height of protrusions on the surface tends to decrease). Pantleon et al. explain the anticorrelation between the misorientations of the angles of the neighboring boundaries of dislocation cells by the limited free path of dislocations [22]. Pairs of dislocations with Burgers vectors of opposite sign are separated by a distance of 2λ, determined by the mean free path λ of an individual dislocation. If both dislocations were completely independent, any correlation would disappear. In general, the growing role of cooperative processes in the dislocation substructure ensures that, over the L1 correlation length, the Hurst exponent H1 increases slightly with increasing loading degree (Fig. 4). The crystal ability to self-organize at the micro level is very high in the whole considered range of deformation. All these processes of self-organization at the level of the dislocation substructure are reflected on the crystal surface. Therefore, the question of the relationship between the morphology of the surface caused by the deformation and the microstructure of the defect remains interesting. Above, we wrote about the self-similar structure of the dislocation subsystem for a wide range of materials and deformation conditions; many of these works concluded that dislocation reliefs in deformed metals can be characterized in terms of the universal cell size and distribution of the statistical characteristics of the cells. This leads us to the idea that there is no one-to-one correspondence between the defective microstructure and plastic flow reliefs, and that different mechanisms can regulate the spatial organization of slip and the formation of the relief caused by deformation. Similar results were discussed in the paper [37]. 4.2. Mezo level The magnitude of the correlation length of L2 indicates that the processes under consideration are characteristic of the mesoscale level. Comparing the Hurst exponent H2 and the correlation length L2 with the stress-strain curve, we can see the connection of these parameters with the stages. During the change of stages of the stress-strain curve, we observe an increase in the Hurst exponent H2 and the correlation length L2. The work [40] also established the relationship between the Hurst exponent and the correlation length with the stages of plastic deformation. If we combine the graphs H1 and H2 (Fig. 12), we can note that the values of H1 and H2 are very close at the initial degrees of deformation, at the initial stages.
Fig. 12. Hurst exponent Н1 and Н2 The leading role belongs to the translational motion of dislocations in close-packed planes. In this case, barriers that prevent the free sliding of dislocations (dislocation walls, clusters, etc.) have not yet formed, so this mechanism alone is sufficient to ensure self-consistent deformation at the micro- and meso level. The difference in the Hurst exponents H1 and H2 are becoming more significant with increasing load. In addition, this difference depends on the crystallographic orientation of the compression axis and the side faces. This dependency is discussed below. In all likelihood, we observe how the correlated shear of the material mesovolumes and the rotational component is involved in the formation of the deformation relief. The self-organization of the dislocation structure, the misorientation of local regions and the geometric parameters of the folded (corrugated) structures was discussed in the Section 3.3. Thus, using a number of parameters, one can trace the hierarchical interconnection of the self-organization processes at various scale levels. The micro level is a dislocation substructure, which can be characterized by the Hurst exponent H1, the correlation length L1, the misorientation angles, the distance between the geometrically necessary boundaries (GNB), the scaling of these distances. The mesoscale – shear planes bundles, Hurst exponent H2, L2 correlation length, macro level – structural elements of the deformation relief, the entire sample face, can be described by the Hurst exponent and correlation length obtained from regions of appropriate size and interface width (not shown in this paper). In addition, the authors have obtained data using other methods that indicate the presence of self-organization of plastic deformation processes at the macro level [3]. A number of papers, for instance [39-40] also show that the self-organization on macro scale differs from nature at the micro and meso levels described in this article. 4. Conclusion
Thus, for nickel single crystals with highly symmetric orientations of the compression axis was established: 1. The self-organization of the plastic deformation process occurs at different scale levels starting from the dislocation substructure scale through the macro strain relief one. The relationship between the scale levels has been revealed. 2. Using statistical processing of the deformation relief parameters based on the Hurst exponent, it was revealed that at the micro level the mutually consistent deformation is carried out due to the dislocation structure, at the meso-level - due to a correlated shear in parallel slip planes. 3. Traces of shear and meso bands, in the strain range under study, are formed due to dislocation glide; near-surface misorientation of local regions is additionally used to form macrobands, and also misorientation inside the crystal for corrugation. 4. It is established the relationship between changes in the type of dislocation substructure, the Hurst exponent, the correlation length, and the stages of the stress-strain curve. The obtained results supplement the already accumulated knowledge of metallic materials plastic deformation and from the point of view of the authors, it would be interesting to continue the research of plastic deformation processes self-organization based through the analysis of the deformation relief on a polycrystalline aggregate. The results described in this paper, obtained on single crystals of different crystallographic orientations, can serve as a fundamental basis for understanding the processes occurring in individual polycrystal grains. Acknowledgement The results of EBSD have been obtained with financial support from RFBR, according to the research project No. 16-32-60007 mol_а_dk. The preparation of samples, the results of laser scanning microscope have been obtained with financial support from Fundamental Research Program of State Academies for 2013-2020 (project No III.23.2.4 ). The research is supported by Tomsk Polytechnic University within the framework of Tomsk Polytechnic University Competitiveness Enhancement Program. The authors are grateful to Professor D. Lychagin for the samples provided for the investigation.
Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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1. The dislocation structure realize self-organization deformation at the microlevel 2. Correlated shear in parallel planes realize self-organization deformation at the mezolevel 3. Scale invariants connect with dislocation substructure, deformation relief, curve 4. The purpose of the deformation self-organization is to preserve the crystal integrity under load 5. Deformation mechanisms for different types of deformation relief was established
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