# The Hall–Petch and inverse Hall–Petch relations and the hardness of nanocrystalline metals

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## Abstract

We review some of the factors that influence the hardness of polycrystalline materials with grain sizes less than 1 µm. The fundamental physical mechanisms that govern the hardness of nanocrystalline materials are discussed. The recently proposed dislocation curvature model for grain size-dependent strengthening and the 60-year-old Hall–Petch relationship are compared. For grains less than 30 nm in size, there is evidence for a transition from dislocation-based plasticity to grain boundary sliding, rotation, or diffusion as the main mechanism responsible for hardness. The evidence surrounding the inverse Hall–Petch phenomenon is found to be inconclusive due to processing artefacts, grain growth effects, and errors associated with the conversion of hardness to yield strength in nanocrystalline materials.

## Introduction

*H*) has been related to the compressive flow stress of a material by the following relation:

*d*, in polycrystalline materials according to the following relation:

*k*is a measure of the local stress needed to initiate plastic flow at a grain boundary and

*σ*

_{0}is the resistance to dislocation motion in the grain interior [13]. This relationship has been explained by a dislocation pile-up model for the stress concentration at the tip of a slip band [13, 14], but recently, the validity of this relationship has been debated [15].

In the 1980s, Gleiter et al. [17] pioneered research into polycrystalline materials whose grains are of nanometre size. It was thought then that these materials would exhibit superior hardness as well as superior wear resistance and fracture strength compared with their coarse-grained counterparts due to the large volume fraction of grain boundaries they contain as grain boundaries were known to govern the response of metals to deformation [18, 19]. Since that time, ultrafine-grained materials have been defined as having grain sizes in the range 100 nm < *d* < 500 nm, and nanocrystalline materials as having grain sizes less than 100 nm. There have been reports of nanocomposite coatings with Vickers microhardness (*Hv*) of up to ~ 40 GPa [20], which is of the same order of magnitude as diamond (*Hv**~* 70–90 GPa) [21]. This ‘super-hardness’ of nanocrystalline materials is of interest to the biomedical [22], military and electronics industries [23, 24].

^{10}s

^{−1}) whereas the strain rate for normal indentation experiments is quasistatic (typically 10

^{−3}s

^{−1}). Recent studies by Gurrutxaga-Lerma and colleagues have shown that the quasistatic theory of dislocations is not valid for shock plasticity [36, 37]. This is because a quasistatic analysis ‘ignores the finite time for elastic signals to travel in the medium’ so that the ‘stresses created by dislocations behind the shock front are felt instantaneously by [dislocation] sources ahead of the shock front’ [36]. The practical outcome of applying a quasistatic analysis is that ‘dislocation sources [are] activated ahead of the shock front’ (Fig. 3), which does not happen.

In “The role of dislocations in the deformation of nanocrystalline materials” section of this review, the evidence for and against key theories that have been developed to explain the deformation mechanisms operating in nanocrystalline materials are discussed along with recent reports claiming the absence of the Hall–Petch effect in grain size strengthening. In “The inverse Hall–Petch phenomenon” section, the inverse Hall–Petch phenomenon is discussed and the mechanisms postulated to explain grain size weakening are summarised. In “Synthesis of ‘super-hard’ nanocrystalline materials” section, the main methods used to synthesise nanocrystalline materials are summarised and the importance of grain boundary structure on the hardness of metals is discussed. “Summary and conclusions” section presents the overall conclusions reached.

## The role of dislocations in the deformation of nanocrystalline materials

### Extending the classic dislocation pile-up mechanism

### Expansion of a single dislocation loop against the grain boundary resistance

*l*is the loop diameter (taken to be equal to the grain diameter),

*τ*

_{0}is the multislip shear stress for deformation within grain volumes,

*b*is the Burgers vector,

*G*the shear modulus, and

*m*is the Taylor orientation factor. In constructing this equation, a term

*τ*

_{c}(the shear stress required to penetrate through the grain boundary) was added to the equation of expansion of a circular dislocation loop [15, 44]. This theory has been supported by several experimental studies [45, 46], as shown in Fig. 6.

*n*= 1 (where

*n*is the number of dislocation loops), the pile-up model predicts a transition to a higher stress than Hall–Petch [46, 47]. The discontinuity in the prediction stems from the transition from the Hall–Petch equation (which assumes

*n*is large) to Eq. (3), when

*n*is small [46]. Lu et al. [48] tested nano-twinned copper, taking the twin thickness as the effective grain size. Their data can be seen in Fig. 7 to initially follow the Hall–Petch relation but with a lower gradient due to the properties of coherent twin boundaries. A reversal of the Hall–Petch relation can be seen in their data at smaller grain sizes, which they ascribed to grain boundary weakening. Armstrong suggested, however, that this may be an artefact of the preparation of the nano-twinned material [44]. This matter is discussed further in “The inverse Hall–Petch phenomenon” section. In conclusion, the data presented in Fig. 7 are not in agreement with the single dislocation loop model.

### Work-hardening models

*ρ*is the average dislocation density,

*α*is a property of the material, and

*σ*

_{0}is defined in Eq. (2).

*d*

^{−0.5}relation between the applied force and the individual grain size (Fig. 11). Since the strain is proportional to the dislocation density [54], the increase in flow stress is proportional to \( \sqrt \rho . \) The authors argued that the increase in hardness was due to an increase in the dislocation density rather than a decrease in the pile-up length. This observation supports the work-hardening model. Evidence of the activation of dislocation sources in adjacent grains was gathered from the analysis of grain size-dependent ‘pop-ins’ (discontinuities in the force–displacement curve) [55].

*L*is related to the number of dislocations

*n*by

*D*is the grain size, \( \tau \) is the external stress, and

*A*is a constant. If the stress for activation of sources in adjacent grains is constant (as assumed by Hall–Petch theory), the pop-in load must be higher for smaller grains. This is supported by the experimental data shown in Fig. 12.

The authors conclude that in ultrafine-grained nickel, hardness scales with dislocation density (rather than pile-up length) where the pile-up length is within the grain size (supporting the Orowan model). However, they also found clear evidence for dislocation source activation in adjacent grains, and hence, a higher external load is needed to nucleate dislocation sources in adjacent grains for smaller grain sizes.

Meyers et al. [18] argued against this idea, pointing out that when the spacing of ledges is in the nanometre range, there would be a grain size below which this deformation mechanism is no longer operational. We would postulate that for grains a few nanometres in diameter, the grain boundaries may no longer be sharp and therefore point defects could be seen rather than the ledges that can be seen in Fig. 9.

Cordero et al. [14] also claimed in their study that there was no direct evidence that links the density of grain boundary ledges (which are affected by grain size) to the density of the dislocations produced.

### The core and mantle model

*k*

_{MA}is a fitted parameter. Equation (6) is in agreement with the Hall–Petch dependence for micron-sized grains, but it predicts a reduction in slope for smaller grain sizes. Figure 14 appears to show an agreement between experimental data and the Meyers–Ashworth model. However, Li et al. [15] recently concluded after analysis of a larger body of data that there is little correlation between experimental data and elastic anisotropy models such as those discussed above.

*ρ*

_{SS}refers to ‘statistically stored’ dislocations (as would build up in a uniformly strained single crystal) and

*ρ*

_{GN}refers to GNDs. The approximation shown in Eq. (7) is valid in the limit of small strains where

*ρ*

_{GN}≫

*ρ*

_{SS}. The model predicts the following relationship between the Hall–Petch coefficient

*k*and the plastic strain

*ε*namely \( k \propto \sqrt \varepsilon \) in the limit of small plastic strains [14].

*k*was measured as a function of plastic strain. Cordero et al. note that a parabolic strain dependence was not found experimentally. Rather in the majority of cases, although

*k*increases with strain, it did so according to a number of other relations. They attributed the lack of a clear parabolic dependence to their invalid assumption of small strain in the historic studies they examined. Cordero et al. ascribed the small number of cases where

*k*did not increase either to sample processing effects or to the effects of twinning (as opposed to glide) as a deformation mechanism. Despite this, Cordero et al. [14] argued that Ashby’s model is the most consistent overall with their examination of the literature on the strain dependence of the Hall–Petch coefficient and with experimental observations of dislocation substructure. However, against this Li et al. [15] argued that the Ashby model is not consistent with the experimental data they obtained.

### The size effect

*k*

*~*1 and a variable \( \varepsilon_{0} \) best described the data and where

*ε*=

*σ*/

*Y*is the stress normalised by the elastic modulus. They also postulated that a random error in grain size determination explains the apparent agreement with Eq. (2).

Summary of proposed mechanisms responsible for the grain size weakening effect

Mechanism name | Mechanism origin | Governing equation | References |
---|---|---|---|

Grain boundary sliding | Independent atomic shear events at the grain boundary. Thermally activated shear. Does not account for compatibility of deformation | \( \tau - \tau_{0} = \left( {\frac{kT}{V}{ \ln }\frac{{\delta v_{d} }}{{\dot{\gamma }}}} \right) + \frac{\Delta F}{V} + \frac{kT}{V}{ \ln }d \) where \( \Delta F \) is the Helmholtz free energy, \( \delta \) is the grain boundary width, v | |

Grain boundary shear dominates over dislocation plasticity as volume fraction of grain boundary increases. Predicts a ‘strongest size.’ Assumes dislocations are emitted from triple grain boundary junctions to satisfy compatibility. | \( \dot{\gamma } = \left( {\frac{3\delta }{d}} \right)\dot{\gamma }_{\text{gb}} + \left( {1 - \frac{3\delta }{d}} \right)\dot{\gamma }_{\text{D}} \) where \( \frac{3\delta }{d} \) is the volume fraction of the grain boundary region | ||

Grain boundary sliding described in terms of a viscous and a plastic accommodation term \( \tau_{\text{p}} \). Grain boundary sliding accounts for a third of the behaviour (see Fig. 22) | \( \tau_{0} = \left( {\eta_{i} + \eta_{\text{D}} } \right)\dot{\gamma } + \tau_{\text{p}} \) where | ||

Accommodation between adjacent grains through diffusional creep | \( \dot{\gamma } = \frac{{64\delta \varOmega D_{\text{B}} }}{kT}\left( {\frac{1}{{d^{3} }}} \right)\tau_{0} \) where | ||

Grain boundary diffusion | Competition between lattice dislocation slip and Coble creep mechanisms (see Fig. 23) | \( \tau = \tau_{0} + kd^{ - 0.5} + k_{1} + \frac{A}{d} + Bd^{3} \) \( k\left( {d^{*} } \right)^{ - 0.5} = \frac{A}{{d^{*} }} + B\left( {d^{*} } \right)^{3} \) where | |

Grain rotation | Grain rotation and translation through motion of dislocation quadrupoles and dislocation dipoles (see Fig. 24a, b) | \( \tau = \frac{Gb}{{2\pi \left( {1 - \nu } \right)d}}{ \ln }\left( {\frac{0.4\alpha d}{b}} \right) \) where | |

Amorphous limit | Transition to glasslike deformation behaviour. Rate and pressure sensitivity of nano-grained materials are characteristic of amorphous solids. For the smallest grain size, unstable localised plasticity occurs (shear banding) | [78] |

*k*and

*σ*

_{0}, their values were uniformly distributed in log

*σ*

*−*log

*d*space (Fig. 17), in agreement with Benford’s Law [64, 65]. Benford’s law is applicable to large data sets where the data points come from many different distributions that span several orders of magnitude [66, 67]. The law states that the probability that the first digit of a number is

*p*is given by

*k*in Fig. 16 and the higher

*k*values reported for fcc metals as compared to bcc metals [69].

*O*

_{n}= 2

^{n}

*P*

_{0}, where

*O*

_{n}are the odds that Eq. (8) is true and

*n*are the number of datasets that fall above the 1/

*d*line (Fig. 19). Therefore, even with a low prior probability

*P*

_{0}that such a well-established equation is incorrect, the odds were found to be overwhelmingly in favour of Eq. (8). Hence, Li et al. [15] argued that the grain size strengthening of metals is driven by constraints on the dislocation curvature and therefore that the pile-up, grain boundary ledges, and core and mantle models make a much weaker contribution to grain size strengthening than the dislocation curvature.

## The inverse Hall–Petch phenomenon

Although Chokshi et al. attributed the softening effect to the onset of Coble creep [71], researchers have since suggested several other theories to explain it including flaws in the synthesis of the nanocrystallites [26], the presence of disordered grain boundaries [31], or a transition to and from dislocation-based deformation to grain boundary sliding or rotation [72, 73, 74].

Koch et al. argued [26] that incomplete densification during synthesis of nanocrystallites via inert gas condensation (as employed by Chokshi et al. [25]) and ball milling methods [75] can lead to residual porosity in nano-grained materials and consequently to poor bonding between particles, resulting in a decrease in the strength of these materials [26]. Armstrong argued that a reduction in Hall–Petch slope could be caused by the presence of disordered grain boundaries in nanocrystalline materials, which would allow plastic flow to be transmitted more easily between grains [31].

### Mechanisms governing grain size weakening

*L*

_{gb}(which is assumed to be soft and prone to deformation through rotation and sliding), and a plastic ‘GI–GB’ phase of thickness

*L*

_{g}adjacent to the grain boundary which accounts for the transition from the ductile GB to the rigid GI phase due to the limited diffusion into the grain interior of dislocations and disclination dipoles generated at the grain boundary.

Zhang and Aifantis’ [79] gradient plasticity model includes an interface energy term \( \gamma_{\text{gb}} \) which allows the interface itself to follow its own yield behaviour. \( \gamma_{\text{gb}} \) is positive for microscopic grains as grain boundaries inhibit plastic flow (the yield stress of the GB phase is greater than the yield stress of the GI phase), whereas for nanometre-sized grains, \( \gamma_{\text{gb}} \) is negative because the grain boundaries behave plastically and are softer than the grain interior (the yield stress of the GB phase is less than the yield stress of the GI phase).

*L*

_{g}=

*ad*, where

*d*is the grain size and

*a*is a constant with a value lying between 0 and 1) and \( \sigma_{0} \) is defined in Eq. (2). Equation (10) was used by Zhang and Aifantis to analyse the data published by a number of authors (see Fig. 26). As the figure shows, it provides a good fit for seven different nanocrystalline metals and alloys.

However, it was pointed out by Zhang and Aifantis that the processing methods were not the same for the various experimental data sets they compared in Fig. 26 (see “Synthesis of ‘super-hard’ nanocrystalline materials” section of our review for a discussion about synthesis methods). We also note that they used Tabor’s hardness-yield stress relation to compare experimental data which could potentially introduce errors for nano-grained materials (this matter is discussed further in the “Discussion of the evidence” section).

*d*

_{c}at which peak material strength occurs. This critical size can be directly computed if the GB energy \( \gamma_{\text{gb}} \), the fraction

*a*of the GB thickness that yields and the Hall–Petch coefficient

*k*are known for the material.

### Discussion of the evidence

Molecular dynamics (MD) simulations [27, 84, 85] have predicted a peak in hardness for copper with grain sizes in the range 10 nm < *d* < 15 nm. The simulations also support the existence of the inverse Hall–Petch slope and deformation via grain boundary slip. Although MD simulations allow researchers to directly model atoms and investigate grain boundary structure for grains less than 10 nm in size [86], the simulated strain rates are so high as to be inaccessible experimentally [87] (see the discussion of Fig. 3 in “Introduction” section). Also due to computational limitations, simulations cannot handle samples larger than a few hundreds of nanometres in size and therefore cannot be related simply to macroscopic experiments [84].

For all the reasons mentioned above, the experimental evidence currently available for the inverse Hall–Petch relationship is inconclusive. So in order to prove the existence of the inverse Hall–Petch effect, and the mechanisms behind it, many more experimental studies need to be performed in which (1) consistent material processing methods are used, (2) direct yield stress measurements are made (rather than assuming the Tabor hardness-yield stress relation), and (3) attention is given to grain size coarsening effects during testing. If these investigations are carried out, the uncontrollable factors in the experiments performed so far will be minimised and reliable data will be generated for materials with typical grain sizes in the nanometre range.

## Synthesis of ‘super-hard’ nanocrystalline materials

Summary of common synthesis methods for nanocrystalline materials

Mechanism | Description of process(es) | Sample characteristics |
---|---|---|

Inert gas condensation | Metal is evaporated, condensed into a fine powder, and compacted | Porosity. Poor bonding between particles [18] |

Mechanical alloying | Powder particles are repeatedly ground in a dry, high energy mill | Porosity [26] |

Electrodeposition | A current is pulsed to deposit metal cations in crystalline and amorphous regions | Low porosity. Improved ductility due to growth twins [18] |

Crystallisation from amorphous solids | Heat treatments crystallise metallic glasses into nano-polycrystalline solids | Residual amorphous regions can remain [26] |

Severe plastic deformation (SPD) | Two main methods: equal channel angular pressing (ECAP) and high-pressure torsion (HPT). The sample of subjected to large plastic strains to break down the microstructure | High proportion of HAGBs, NGBs. Impurity segregation resulting in ‘super-hardness’ [22] |

### A larger fraction of high-angled grain boundaries (HAGBs)

HAGBs are more effective in impeding dislocation slip as there is greater crystallographic misalignment across the grain boundary. The fraction of HAGBs can be increased from 55 to 80% by increasing the number of high-pressure torsion (HPT) turns from one half to ten [88].

### Segregation of impurity and alloying elements at grain boundaries

Precipitation of alloying elements in grain boundary regions suppresses the emission of dislocations from grain boundaries. Additionally, the precipitates cause drag on GNDs [22, 89].

### Non-equilibrium grain boundaries (NGBs)

SPD produces more dislocations than geometrically necessary to accommodate plastic deformation at grain boundaries, causing an increase in grain boundary energy [22, 89].

Hu et al. [88] recently showed that careful use of annealing can result in the doubling of hardness of nano-grained nickel and nickel-molybdenum without altering the grain size (Fig. 28a). They found that the indentation produced little coarsening for their annealed samples. They thus concluded that structural relaxation and segregation of the molybdenum in the alloy causes relaxation of local stress levels at grain boundaries, which then become more stable to straining. This could reduce the threat of grain coarsening to the refinement process.

They also argued that grain boundary mediated deformation (which can cause softening) is replaced by deformation by the generation of extended partial dislocations at grain boundaries. The emission of partials is suppressed due to impurity segregation, similar to the suggestions by Valiev mentioned above [89], enhancing the formation of extended stacking faults. The large stresses required for nucleation of dislocations from stable grain boundaries results in a high hardness and a (1/*d*) grain size dependence. Hu et al. [88] argued that differences in grain boundary structure can explain the controversy over hardening and softening behaviour reported with decreasing grain size in previous studies. Their results could lead to the synthesis of further ‘super’-hard materials.

## Summary and conclusions

This article has reviewed the hardness of nanocrystalline metals, focusing on the theories describing dislocation plasticity, grain size weakening, and super-hardness effects. The main conclusions reached are outlined below.

### Deformation mechanisms

100 nm < *d* < 1 µm: Core and mantle type models best describe the deformation behaviour.

30 nm < *d* < 100 nm: Dislocation ledge spacing becomes large compared to the grain size; therefore, there is a transition from a dislocation-based plasticity to grain boundary sliding as the main mechanism responsible for hardness. There is a dearth of reliable hardness measurements in this grain size range, and therefore, the main accommodation mechanism cannot be distinguished.

*d* < 30 nm: Transition from nanocrystalline to amorphous behaviour.

### The relationship between hardness and grain size

Although Li et al.’s [15] arguments for a (log *d*)/*d* relationship are compelling, they require further statistical analysis and corroboration in order to overturn the large body of evidence that supports the \( d^{ - 0.5} \) relationship, which has been added to recently by Armstrong [13] and Cordero et al. [14].

### The inverse Hall–Petch effect

A transition from dislocation-based plasticity to a grain boundary sliding mechanism could explain the reversal in Hall–Petch slope. This transition has been seen to occur at grain sizes from around 100 nm [18] down to 10 nm [27, 84, 85, 88]. An analytical expression to predict the theoretical critical grain size was devised by Zhang and Aifantis on the basis of the grain boundary plasticity theory [79]. However, the inverse Hall–Petch effect could also result from processing artefacts or stress-induced grain growth during testing. Based on the available experimental evidence, the existence of the inverse Hall–Petch effect cannot be confirmed.

### Processing methods

Inert gas condensation and mechanical alloying can result in grain size weakening due to incomplete densification resulting in porosity. Severe plastic deformation can result in HAGBs, segregation of alloying elements, and NGBs which produce ‘super-hardness.’ Short annealing treatments have recently been used to increase the hardness of nickel-molybdenum alloys by up to 120% [88] by reducing the local stress levels at grain boundaries.

## Notes

### Acknowledgements

The authors thank the referees for making useful suggestions for improving the paper.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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