Softening of nanocrystalline metals at very small grain sizes

Jakob Schiøtz, Francesco D. Di Tolla and Karsten W. Jacobsen
Center for Atomic-scale Materials Physics and Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark

Present address: SISSA, Via Beirut 2-4, I-34014 Grignano (TS), Italy.

This paper was published in Nature vol. 391 p. 561 (5 Feb 1998)

Nanocrystalline solids, in which the grain size is in the nanometre range, often have technologically interesting properties such as increased hardness and ductility. Nanocrystalline metals can be produced in several ways, among the most common of which are high-pressure compaction of nanometre-sized clusters and high-energy ball-milling.[1,2,3,4] The result is a polycrystalline metal with the grains randomly orientated. The hardness and yield stress of the material typically increase with decreasing grain size, a phenomenon known as the Hall-Petch effect.[5,6] Here we present computer simulations of the deformation of nanocrystalline copper, which show a softening with grain size (a reverse Hall--Petch effect[7,3]) for the smallest sizes. Most of the plastic deformation is due to a large number of small 'sliding' events of atomic planes at the grain boundaries, with only a minor part being caused by dislocation activity in the grains; the softening that we see at small grain sizes is therefore due to the larger fraction of atoms at grain boundaries. This softening will ultimately impose a limit on how strong nanocrystalline metals may become.

To simulate the behavior of nanocrystalline metals with the computer, we construct nanocrystalline ``samples'' with strucures similar to what is observed experimentally: essentially equiaxed dislocation-free grains separated by narrow straight grain boundaries[1]. Each sample contains 8 to 64 grains in a 10.6nm cube of material, resulting in grain sizes from 3.3 to 6.6 nm. The grains are produced by a Voronoi construction:[8] a set of grain centers are chosen at random, and the part of space closer to a given center than to any other center is filled with atoms in an fcc (face centered cubic) lattice with a randomly selected crystallographic orientation. A typical ``sample'' is shown in Fig. 1a. To mimic that the system is deep within the bulk of a larger sample, the system is replicated infinitely in all three spatial directions (periodic boundary conditions). The forces between the atoms are calculated with the Effective Medium Theory,[9,10] which suitably describes many-atom interactions in metals. The metal chosen for these simulations is copper, very similar results were obtained with palladium. Before deforming the system we ``anneal'' it by running a 50ps molecular dynamics simulation at 300K, allowing unfavorable configurations in the grain boundaries to relax. Doubling the length of the annealing does not have any significant effect, nor does an increase of the temperature to 600K.

  

Figure 1: A nanocrystalline copper sample (a) before and (b) after 10% deformation. The system contains 16 grains and approximately 100000 atoms, giving an average grain diameter of 5.2 nm. Atoms in the grain boundaries are colored blue, atoms at stacking faults are colored red. We clearly see stacking faults left behind by partial dislocations that have run through the grains during the deformation processes. Such stacking faults would be removed if a second partial dislocation followed the path of the first, but this is not observed in the present simulations. In the left side of the system a partial dislocation on its way through a grain is seen (green arrow).

The main part of the simulation is a slow uniaxial deformation while minimizing the energy with respect to all atomic coordinates. The deformation is applied by expanding the simulation cell in one direction, while the size is allowed to relax in the two perpendicular directions.

The initial and final configurations of such a simulation with a total strain of 10% are shown in Fig. 1. We see how the grain boundaries have become thicker, indicating that significant activity has taken place there. In the grains a few stacking faults have appeared. They are the signature of dislocation activity within the grains.

To facilitate the analysis of the simulations, we identify which atoms are located at grain boundaries and which are inside the grains, by determining the local crystalline order.[11,12] Atoms in local fcc order are considered ``inside'' the grains, atoms in local hcp (hexagonal close packed) order are classified as stacking faults. All other atoms are considered belonging to the grain boundaries. Unlike conventional materials, where the volume occupied by the grain boundaries is very small, a significant fraction (30-50%) of the atoms are in the grain boundaries, in agreement with theoretical estimates.[2]

While the deformation takes place, we calculate the average stress in the sample as a function of the amount of deformation. For each grain size the deformation of seven different initial configurations were simulated. Fig. 2a shows the obtained average deformation curves. We see a linear elastic region with a Young's Modulus around 90-105GPa (increasing with increasing grain size), compared to 124 GPa in macrocrystalline Cu[13]. This is caused by the large fraction of atoms in the grain boundaries having a lower Young's Modulus[14,15]. A similar reduction is seen in simulations where the nanocrystalline metal is grown from a molten phase[16]. The elastic region is followed by plastic yielding at around 1GPa, and finally the plastic deformation saturates at a maximal flow stress around 3GPa.

  

Figure 2: a: The average stress versus strain for each grain size. Each curve is the average over seven simulations. The curves show the response of the material to mechanical deformation. In the linear part of the curve (low strains) the deformation is mainly elastic: if the tensile load is removed, the material will return to the original configuration. As the deformation is increased irreversible plastic deformation becomes important. For large deformations plastic processes relieve the stress, and the curves level off. We see a clear grain size dependence which is summarized to the right. b and c: The maximal flow stress and the yield stress as a function of grain size. The yield stress decreases with decreasing grain size, resulting in a reverse Hall-Petch effect. The maximal flow stress is the stress at the flat part of the stress-strain curves. The yield stress is defined as the stress where the strain departs 0.2% from linearity.

The main deformation mode is illustrated in Fig. 3b where the relative motion of the atoms is shown. We clearly see that most of the deformation occurs in the grain boundaries in the form of a large number of small sliding events, where only a few atoms (or sometimes a few tens of atoms) move with respect to each other. Occasionally a partial dislocation is nucleated at a grain boundary and moves through a grain. Such events are responsible for a minor part of the total deformation, but in the absence of diffusion they are required to allow for deformations of the grains, as they slide past each other. No dislocation motion is seen in Fig. 3, as none occured at that time of the simulation. As the grain size is reduced a larger fraction of the atoms belongs to the grain boundaries, and grain boundary sliding becomes easier. This leads to a softening of the material as the grain size is reduced (Fig. 2).

  

Figure 3: Snapshot of grain structure, displacements and stresses at 8% deformation. (a) The position of the grain boundaries (blue) and stacking faults (red) at this point in the simulation. (b) The relative motion of the atoms in the z direction (up in the plane of the paper) during the preceding 0.4% deformation. The green atoms move up, red atoms move down. We see many small, independent slip events in the grain boundaries, this is the main deformation mode. (c) The stress field (the $\sigma_{zz}$ component) in the grains. Shades of red indicate tensile stress, shades of blue compressive stress (dark colors corresponds to high stresses). The stress in the grain boundaries is seen to vary considerably on the atomic scale, and the average stress is approximately 10-20% lower than in the grains.

The observed deformation mode is in some ways similar to the way grain boundaries carry most of the deformation in superplastic deformation.[17,18] The main difference is that in superplasticity the grain boundary sliding is thermally activated, whereas here it occurs at zero temperature driven by the high stress. This is consistent with recent simulations of flow speed in nanocrystalline metals.[25]

In conventional metals an increase in hardness and yield strength with decreasing grain size is observed. This is called the Hall-Petch effect, and is generally considered to be caused by the grain boundaries impeding the generation and/or motion of dislocations as the grains get smaller; this behavior extends far into the nanocrystalline regime.[3,19] The grain sizes in the present simulations correspond to the smallest grain sizes that can be obtained experimentally. In that regime the Hall-Petch effect is often seen to cease or even to reverse, but the results depend strongly on the sample history and on the method used to vary the grain size. Many mechanisms have been proposed for this reverse Hall-Petch effect: increased porosity at small grain sizes,[3] suppression of dislocation pileups,[20] dislocation motion through multiple grains,[21] sliding in the grain boundaries[22], and enhanced diffusional creep in the grain boundaries.[7] Direct measurements of the creep rates seem to rule out the latter mechanism[19,20], but otherwise no consensus has been reached. The present simulations indicate that a reverse Hall-Petch effect is possible even in the absence of porosity, and that it may be caused by sliding in the grain boundaries even in the absence of thermally activated processes. We cannot, however, see a cross-over from this ``reverse'' behavior to the normal Hall-Petch regime at larger grain sizes in our simulations, because they become too computationally expensive at these larger sizes. For the same reason, we cannot provide a direct comparison with the behavior of the bulk metal.

A direct quantitative comparison between the simulations and the experimental results should be done with caution. The main difference between the simulated strain-stress curves and experimental curves is the level of the yield stress, which is approximately twice what is observed in experiments on low-porisity samples with grain sizes (400 MPa)[23]. This value is however obtained for a grain size of around 40nm, extrapolation to 7nm grains gives a yield strength around 800MPa,[23] assuming that the Hall-Petch behavior persists to these grain sizes. Experimentally produced nanocrystalline samples typically contain voids and surface defects reducing the strength of the material. Surface defects alone have been shown to be able to reduce the strength of nanocrystalline palladium by at least a factor of five.[19,24]

Another difference is the absence of thermally activated processes in the simulations. These processes give rise to a strain rate dependence of the mechanical properties, leading to higher yield stresses at higher strain rates where there is less time available for the activated processes to occur. Thermally activated processes are not included in the simulations because the energy minimization procedure quickly removes all thermal energy. No time scale or strain rate can be directly defined in the simulations, but in a sense the procedure corresponds to a slow strain at very low temperatures: thermally activated processes are excluded but the energy created by the work is carried away fast. The above-mentioned creep measurements[19,20] indicate that diffusion does not play a major role during deformation.

During the later part of the simulated deformation larger average stresses build up within the grains than in the grain boundaries (10-20%), and larger stresses build up in the larger grains (see Fig. 3c). This results in a larger variation in the maximal flow stress than in the yield stress (Fig. 2b): when the grain size is increased the maximal flow stress increases both because the stresses in the grains increase, and because the number of atoms within the grains becomes a larger fraction of the total number of atoms.

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ACKNOWLEDGMENTS. The authors wish to thank Jens K. Nørskov, Torben Leffers, Ole B. Pedersen, Anders E. Carlsson and James P. Sethna for many fruitful discussions. The Center for Atomic-scale Materials Physics is sponsored by the Danish National Research Foundation.

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This paper was published in Nature vol. 391 p. 561 (5 Feb 1998)