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Sample Dimensions Influence Strength and Crystal Plasticity

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Science  13 Aug 2004:
Vol. 305, Issue 5686, pp. 986-989
DOI: 10.1126/science.1098993

Abstract

When a crystal deforms plastically, phenomena such as dislocation storage, multiplication, motion, pinning, and nucleation occur over the submicron-to-nanometer scale. Here we report measurements of plastic yielding for single crystals of micrometer-sized dimensions for three different types of metals. We find that within the tests, the overall sample dimensions artificially limit the length scales available for plastic processes. The results show dramatic size effects at surprisingly large sample dimensions. These results emphasize that at the micrometer scale, one must define both the external geometry and internal structure to characterize the strength of a material.

A size-scale effect can be defined as a change in material properties—mechanical, electrical, optical, or magnetic—that is due to a change in either the dimensions of an internal feature or structure or in the overall physical dimensions of a sample. For metals, size-scale effects related to changes in internal length scales are readily observed and are often exploited for industrial use. For example, it is well known that the yield strengths of metallic alloys can be improved through refinement of the grain size (13), where the yield strength is proportional to the inverse square root of the average grain diameter, and this relation is generally valid for grains that range in size from millimeters to tens of nanometers. By comparison, changes in the mechanical response of materials due solely to the physical geometry of a sample have been largely overlooked. Large increases in yield strength (approaching the theoretical limit) were observed over 40 years ago in tension testing of single-crystal metallic whiskers having micrometer-scale diameters (46). However, whisker testing is restricted to materials that can be grown in that form. Conversely, no changes in strength and only mild decreases in work hardening were observed during the deformation of simple metals at submillimeter sample diameters (710), but those studies only started to explore the gap between millimeter and whisker dimensions.

There remains a fundamental challenge to systematically investigate external length-scale effects in the submillimeter-to-nanometer size regime. Such small dimensions are pervasive in modern devices and also encompass the size range in which dislocation-based plasticity mechanisms occur. External length-scale effects may be observed at multiple stages over this wide range of sizes, because the mechanisms associated with dislocation storage, multiplication, motion, pinning, and nucleation are generally active over different length scales. Without such an understanding, it is impossible to know the appropriate material properties to use in the design of small devices. At present, one can question whether features having micrometer-sized dimensions should be designed using the extraordinary strengths of defect-free “whiskers” or using behavior more akin to that of bulk metal crystals.

Recently, size-scale effects in materials mechanics received renewed attention under conditions where deformation gradients are imposed at the micrometer scale (1114). These studies explore the evolution of geometrically necessary dislocations (GNDs) (15, 16) that are required to accommodate the plastic strain gradients that may be induced by the test condition or by the internal structure of the material. For example, the permanent change in the profile of a surface during indentation testing, which is due to the deformation gradient imposed by the indentation tip, may be wholly accommodated by the generation and motion of GNDs. These studies find that gradient-induced increases in defect evolution result in concomitant local changes in the strength and hardening rates of materials. However, the studies do not consider other changes in the fundamental deformation mechanisms associated with limiting the physical dimensions of the deforming volume; that is, one might speculate that the deformation micromechanisms themselves are affected by the size of the deforming volume. We suggest that a more complete understanding of size effects for a given material can only be realized after testing for geometric effects and under test conditions that minimize imposed deformation gradients, thus limiting the GND density.

We have developed a test methodology (17) that allows the exploration of size-scale effects in virtually any bulk inorganic material, using a focused ion beam (FIB) microscope for sample preparation, together with mechanical testing that is a simple extension of nanoindentation technology. The FIB is used to machine cylindrical compression samples into the surface of a bulk crystal, leaving the samples attached to the bulk at one end. Samples were prepared in the size range from 0.5 to 40 μm in diameter and with an aspect ratio ranging from 2:1 to 4:1. Once prepared, the samples were tested using a conventional nanoindentation device outfitted with a flat-punch indentation tip. Nanoindentation systems are normally used for depth-sensing indentation experiments using a sharp tip, but here the same platform is used to perform conventional uniaxial compression tests at prescribed displacement rates ranging from 1 to 5 nm/s. This technique can be used to study external size effects in single crystals in the absence of grain boundaries, which are strong internal barriers to dislocation glide. Although there have been notable advances in mechanical test methods that operate on micrometer-sized samples (1820), these test techniques use samples that have been fabricated with wafer processing methods that are specific to the microelectronic industry. The microstructures of those samples are typically polycrystalline, having a submicrometer grain size, which can complicate the interpretation of observed external size effects (21).

The mechanical behavior of bulk single crystals of pure Ni is well known, so this material was used as a model system for the test method. A single-slip orientation was selected to simplify the defect evolution and hardening conditions. The stress-strain curves for Ni microcompression samples having diameters in the 20- to 40-μm range are similar to those for bulk samples (Fig. 1A), because the yield strength and overall work-hardening rates are within 30% of the measured properties of millimeter-sized specimens. After testing, fine discrete slip bands can be observed along the gauge length of the samples, which are also found in the bulk specimen tests (Fig. 1B).

Fig. 1.

Mechanical behavior at room temperature for pure Ni microsamples having a 〈134〉 orientation. (A) Stress-strain curves for microsamples ranging in size from 40 to 5 μm in diameter, as well as the stress-strain curve for a bulk single crystal having approximate dimensions 2.6 × 2.6 × 7.4 mm. (B) A scanning electron micrograph (SEM) image of a 20-μm-diameter microsample tested to ∼4% strain. The circle milled into the top surface of the microsample is a fiducial mark used during sample machining. (C) A SEM image of 5-μm-diameter microsample after testing, where the sample achieved ∼19% strain during a rapid burst of deformation that occurred in less than 0.2 s.

For samples 5 μm in diameter, there are distinct changes in the stress-strain curves that are indicative of physical size limitations. These samples display large strain bursts: very rapid flow to values up to 19% strain upon yielding, in contrast to bulk samples that show a smooth transition from elastic to plastic flow and a steady rate of work hardening. The yield stress for each of the four 5-μm-diameter samples is higher than that for microsamples that are equal to or larger than 20 μm in diameter and varies over a range of 70 MPa. Differences may also be observed in the appearance of the microsamples after testing. There are fewer slip bands, but those that exist appear to be much more active, as indicated by large single-slip plane displacements (Fig. 1C). Strain bursts are also observed for samples 10 μm in diameter, although the extent of these is typically less than 1% strain. For samples 10 μm in diameter and larger, most of the plastic deformation consists of short periods of stable flow with low work-hardening rates, separated by increments of nearly elastic loading. There is a gradual progression between bulk and size-limited behavior as the sample size decreases from 40 to 5 μm in diameter. These attributes are distinct from the common behavior of both bulk materials and whiskers. Whiskers of pure metals typically display much higher yield stresses than bulk materials. In one study, the strength of Cu whiskers 16 μm in diameter and smaller exhibited yield stresses in the range from 0.3 to 6 GPa (6), whereas the yield stress for bulk Cu is on the order of 10 to 50 MPa (depending on purity levels and heat treatment conditions). In addition, after yielding, whiskers do not maintain this high flow stress; rather, the flow stress drops to the level observed in bulk Cu or the whisker simply fractures. The reasons for this are understood to be related to the fact that, unlike most common metals, the whiskers start out defect-free before loading.

One interpretation of these results is that decreasing sample diameter affects the mechanisms for defect multiplication and storage that are associated with plastic flow, before the dislocation-source–limited regime attributed to whiskers is achieved. The increases in flow stress and extremely low hardening rates fall outside the regimes known for bulk tests but do not enter the regime of high stresses known for metal whiskers. The increase in the spread and the rise of the values of the yield stress for smaller samples suggest aspects of self-organization and criticality events at the elastic-plastic transition. That is, the transition appears to be stochastic, showing a progression toward a single catastrophic event as the ability to multiply dislocations or the number of dislocation sources is truncated. This occurs either through increasing levels of deformation or through shrinking the total volume of the sample.

The same method was used to examine an intermetallic alloy, Ni3Al-Ta, which is widely known to exhibit fundamentally different flow mechanisms. One physical manifestation of this behavior is an anomalous increase in strength with increasing temperature. There is considerable evidence that at temperatures in the anomalous flow regime, the mobility of screw-character dislocations is greatly influenced by the lateral motion of large jogs and kinks along the length of the dislocations (2224), and it is likely that dislocation kinetics are strongly influenced by the characteristic active line length of dislocations known to be on the order of a few micrometers (23, 25). The characteristic scales for multiplication are unknown. In the present study, the sample sizes are equivalent to the length scales for the physical processes governing flow.

We observed a dramatic size effect on strength for a Ni3Al-1% Ta alloy deforming under nominally single-slip conditions (Fig. 2A). The flow stress increased from 250 MPa for a 20-μm-diameter sample to 2 GPa for a 0.5-μm-diameter sample. These flow stresses are much higher than those found for bulk crystals, which themselves exhibit a flow stress of only 81 MPa. Although these stresses exceed those for the bulk material, the influence of sample size occurs at dimensions that are large by comparison to whisker-type tests. After testing, slip traces are very fine and are homogeneously distributed along the gage section (Fig. 2B), except for the 0.5- and 1-μm-diameter samples, because they have completely sheared apart during large strain bursts. Closer inspection of the loading curves for all of the tests before the large strain bursts show small events of plastic activity that occur sporadically during the loading of the sample, separated by nearly elastic loading, again akin to self-organized processes. These aspects of work-hardening behavior are similar to what we have observed in the smaller Ni samples but have not been reported for bulk samples.

Fig. 2.

Mechanical behavior at room temperature for Ni3Al-Ta microsamples having a 〈123〉 orientation. (A) Representative stress-strain curves for microsamples ranging in size from 20 to 0.5 μm in diameter, as compared to the behavior of a bulk single crystal having approximate dimensions 2.5 × 2.5 × 7.5 mm. (B) A SEM image of 20-μm-diameter microsample after testing, where the sample achieved ∼10% strain during the rapid burst of deformation. (C) A SEM image of 1-μm-diameter microsample after testing, where the top of the sample has completely sheared off during the rapid strain burst. This behavior is observed for both the 1- and 0.5-μm-diameter samples.

Examination of the flow stress in Ni3Al-Ta as a function of sample diameter (Fig. 3) shows two regimes of size-dependent strengthening that scale with the inverse of the square root of the sample diameter—coincidently similar to grain-size hardening. However, although such strength scaling in metals usually arises from the presence of internal kinematic barriers to flow, these samples have no known internal barriers. One may speculate that this remarkable behavior is associated with changes in the self-exhaustion or annihilation of dislocations, specifically those of screw character. That said, it is surprising that significant length-scale effects are observed for such large sample sizes; note that the transition to bulk behavior is predicted from the scaling relation in Fig. 3 to occur for samples greater than 42 μm in diameter.

Fig. 3.

Dependence of the yield strength on the inverse of the square root of the sample diameter for Ni3Al-Ta. The linear fit to the data predicts a transition from bulk to size-limited behavior at ∼42 μm. σys, the stress for breakaway flow.

Finally, we examined a Ni superalloy single crystal that consisted of a Ni solid-solution matrix having a high volume fraction of Ni3Al-based precipitates that are ∼250 nm in diameter and are uniformly distributed. Both solid-solution alloying and the precipitates provide additional strengthening mechanisms and help to determine internal deformation length scales. A 10-μm-diameter microcompression sample, which had about 30 precipitates spanning the width of the sample, displayed a mechanical response that matched the behavior of a bulk tension test (Fig. 4). The agreement is not surprising, because the strong internal hardening mechanisms that control plastic deformation operate at the dimensional scale of the precipitates and are still effective at this sample size, thus preempting influences from limited sample dimensions.

Fig. 4.

Mechanical behavior at room temperature of a Ni superalloy microsample having a near-〈001〉 orientation. (A) A stress-strain curve for a 10-μm-diameter microsample tested in compression as compared to the behavior of a bulk single tested in tension. The microsample was machined from an undeformed region of the grip region of the bulk sample after testing. (B) A SEM image of the microsample after testing.

We have demonstrated a method to characterize aspects of length-scale effects on deformation and strength by shrinking the traditional uniaxial compression test to the micrometer scale. From these tests it is clear that when the external dimensions of the sample become smaller than a few tens of micrometers, the basic processes of plastic deformation are affected; thus, it may not be possible to define the strength of a given material in the absence of physical conditions that are completely specified. The results show that such influences occur at much larger dimensions than are classically understood for metal whisker-like behavior (6). Emerging strain-gradient–based continuum theories of deformation (that is, models that incorporate a physical length scale into the constitutive relations for the mechanical response of materials) must carefully account for these fundamental changes of deformation mechanisms that extend beyond the gradient-induced storage of defects.

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