Hall–Petch relationship in nanocrystalline Ni and Be–B alloys

Intermetallics 13 (2005) 377–385 www.elsevier.com/locate/intermet

Hall–Petch relationship in nanocrystalline Ni and Be–B alloys T.G. Nieha,*, J.G. Wangb a

Department of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996, USA b Materials Characterization Lab, The Pennsylvania State University, University Park, PA 16802, USA Available online 21 January 2005

Abstract An overview of the plastic deformation of crystalline solids with different grain sizes are presented. Special emphases are on materials with a grain size less than w20 nm. This is the region where the classical Hall–Petch (H–P) relationship breakdowns (or the inverse H–P relationship) are often reported. In the present paper, two alloy systems, pure nickel and Be–B binary alloys, are studied and the results discussed. The nanocrystalline nickel and Be–B alloys were produced by electrodeposition and sputter deposition, respectively. In the case of n-Ni, we used both nanohardness and nanoscratch experiments to demonstrate successfully a H–P breakdown at a grain size of about 14 nm. In the case of Be–B alloys, we illustrated that an apparent H–P breakdown is, in fact, an artifact. The apparent inverse HP relation was actually caused by the presence of relatively soft amorphous Be–B phases when the grain size of Be was significantly refined by B alloying. q 2004 Elsevier Ltd. All rights reserved. Keywords: B. Glasses, metallic; B. Phase diagrams; C. Coatings, intermetallic and otherwise

1. Introduction The strength of polycrystalline materials is expected to increase with decreasing grain size, based on the classical Hall–Petch (H–P) relationship: sZs0Ckhdn, where d is the grain size, s the 0.2% yield strength (or hardness), s0 the lattice friction stress to move individual dislocations (or the hardness of a single crystal specimen, d/N), n the grain size exponent (normallyK1/2), and kh a constant, called H–P intensity parameter [1]. However, experimental results sometimes showed the opposite, namely, an inverse H–P relationship. Several models have been proposed to explain the observed inverse H–P relationship; these include the absence of dislocation pile-up [2], the occurrence of diffusional creep [3], rapid dislocation annihilation at grain boundaries [4], and softening caused by the presence of a significant amount of grain triple junctions [5]. Recent atomic-scale simulations [6,7] indicated that most of the plastic deformation is mainly due to grain boundary sliding, with only a minor part being caused by dislocation activity within the grains. The planar slide causes softening. Whereas each of the above models has its own merit, * Corresponding author. Tel.: C1-925-4239802; fax: C1-925-4238034. E-mail address: [email protected] (T.G. Nieh). 0966-9795/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2004.07.029

none has a solid proof to support the theory and can satisfactorily explain the existing experiments. Traditionally, the tensile ductility of nc-materials is expected to increase, as a result of a more uniform structure and reduced stress concentration. However, again, the experimental results indicated limited or essentially no ductility in tension for grain sizes less than about 100 nm [8,9]. The predicted increased ductility for metals, intermetallics, or ceramics by reducing their grain size to the nanoscale has not been realized. Many of these disappointing experimental results have been attributed to artifacts present in consolidated particulate samples prepared by so-called two-step processes. Two-step processes usually involve making powders and, then, compacting and densifying them using enhanced pressure and temperature. Similar results have also been often observed in nc-materials made by one-step processes such as severe plastic deformation (e.g. equal channel angular extrusion) [10], nanocrystallization [11], and electrodeposition [12,13]. In this case, the low tensile ductility was argued to be attributable to the existence of structural defects (e.g. nano-sized pores), a non-uniform grain size distribution, grain boundary impurities, or a specific but yet-to-understand deformation mechanism [14]. As recently summarized by Professor Weertman [15], one of the pioneers

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Fig. 1. Yield strength as a function of grain sizes. According to Hall–Petch relationship, properties of are classified into four regions.

studying the mechanical behavior of nanocrystalline materials, ‘Nanocrystalline materials are not as strong as expected and their ductility are always unexpectedly low.’ The scientific question arises as to why nanocrystalline materials do not have significantly improved strength and ductility properties over conventional materials. The mechanical behavior of a polycrystalline pure solid varies with grain size and can be schematically summarized in Fig. 1. The figure is divided into four regions. Region I (dO1 mm), where materials have been widely studied, is characterized by a relatively strong work hardening (caused by dislocation interactions), relatively low strength, and high ductility. Plasticity is controlled primarily by dislocation motion within grains. Material strength in this region follows the classical H–P relationship, namely, yield strength increases with decreasing grain size. Tensile failure initiates at macroscopic necking and the fracture mode is intragranular. In Region II (1 mmOdO20 nm), the H–P relationship still prevails and the strength of a material continues to increase as a result of reducing grain size. However, both the strain hardening rate and the tensile ductility decrease. There is also a gradual transition of fracture mode from intragranular to intergranular. Another important observation was that shear deformation becomes localized [16]. As the grain size further reduces, one enters into Region III—a region where only limited reliable experimental data are available [17]. However, recent computer simulations indicate that materials in this region are characterized by an inverse H–P relationship, i.e. strength decreases as grain size decreases [6]. Materials exhibit negligible strain hardening in this region. Plasticity occurs primarily within

grain boundary region in which the sliding of atomic planes is the dominant mode. Region VI (marked by an arrow) corresponds to amorphous materials (also known as metallic glasses), which have been extensively explored in recent years [18, 19]. Experimental results showed that a metallic glass in compression [19] exhibits no strain hardening and behaves like a perfectly plastic material. In tension, on the other hand, the material is highly elastic and essentially brittle [18]. The fracture of metallic glasses occurs by highly localized shear banding. The mechanical characteristics in the four regions can be conveniently summarized in Table 1. It is apparent in both Fig. 1 and Table 1 that there is a consistent trend of mechanical behavior as the grain size gradually scales down from mm in a conventional material to amorphous. For example, strain hardening effect and tensile ductility decrease with decreasing grain size. Fracture mode changes from intragranular to intergranular and, eventually, localized shear, as the grain size is reduced. This is somewhat expected since the grain boundary and triple junction volume fractions increases sharply in the vicinity of dw20 nm, as indicated in Fig. 2 [17,20,21]. This microstructural transition causes a dramatic change in deformation mode, from a lattice-dislocation-base to grain-boundary-base. The observed brittleness of nanocrystalline materials has been attributed to restricted dislocation activity and sample preparation artifacts including flaws, contamination, and residual stress [8,9]. Recently, Lu et al. [9] reported a large tensile ductility (O20%) in nanocrystalline Cu prepared by electrodeposition. This result appears exciting but is slightly misleading because most of the grain boundaries in their materials were, in fact, low-angle. In other words, grains are not true grains but subgrains, in the conventional sense. The ‘nanocrystalline’ Cu was actually single crystal containing many subgrains. This explains the ‘unusual’ results of low strength and high ductility. Here is just one example. There are many papers in literature studying the strength and ductility of nanocrystalline materials. One must exercise extreme precaution on citing the literature data. The intrinsic ductility property of a nanostructured material is still unanswered. Because of the limit of scope, in the present paper, we will only discuss the strength issue, in particularly the H–P relationship in nanocrystalline materials, i.e. Region 3 and 4

Table 1 Mechanical characteristics in different grain size regions Reg.

Grain size

Strength

Ductility

Strain hardening

Fracture

Dislocation activity

Grain-boundary activity

I II

Strong Low

Negligible Moderate

Low

Negligible

Negligible

Dominant

0

w0

None

Transgranular, ductile fracture Transition from transgranular to intergranular Sliding of atomic planes in grain boundaries Fracture by localized shear band formation

High Moderate

IV

Low Decreases with grain size Increases with grain size High

High Moderate

III

O1 mm !1 mm, O20 nm !20 nm

None

Practically 100%

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cross-sections of the electrodeposited foils; the reported results are average values of more than ten indents. Abrasive nano-scratch experiments were performed on n-Ni using the Nanoindenter-XP over a travel distance of 500 mm, with a ramping normal load from 20 to 1.5! 105 mN. The diamond tip was oriented such that an edge of the Berkovich pyramid was pointing in the direction of travel. At least three identical scratches were performed on each material. To minimize the substrate effects, the penetration depth of the indent during both hardness and scratch tests was always less than one tenth of the film thickness. 2.2. Microstructure of n-Ni

Fig. 2. Bright-field TEM images of nanocrystalline Ni with an average grain size dZ14 nm. The grain size distribution is quite uniform.

in Fig. 1. We will present our study of two systems, pure Ni and binary Be–B, to illustrate the validity of H–P breakdown.

2. Hall–Petch breakdown in nanocrystalline Ni 2.1. Materials and experiments Foils of nanocrystalline nickel (n-Ni) were fabricated by direct current (DC) electrodeposition, using standard procedures given in more detail in Ref. [22]. A nanocrystalline structure is induced through the inclusion of a small amount (5 g/l) of saccharine in the plating bath. Usually, the grain size, morphology, and texture of the plated foils can be tailored to some degree through the choice of bath pH and applied current density [22]. The as-deposited foils, about 50–100 mm in thickness, were characterized by X-ray diffraction, and their grain size determined by applying the Scherrer formula for peak broadening to the (200) (220), or (111) reflections after correction for instrumental line broadening using a silicon standard. Microstructures of the Ni deposit were examined using transmission electron microscopy (Philips CM300 and a JEM-4000EX). Indentation testing was performed in an MTS/Nanoinstruments (Oak Ridge, TN) Nanoindenter-XP, as well as a TriboIndenter instrumented nanoindenter (from Hysitron, Minneapolis, MN), in all cases with a diamond Berkovich indenter. Hardness was determined both by the Oliver– Pharr method and the continuous stiffness method [23], with the instantaneous contact area determined using the calibrated area function of the Berkovich tip. Indentations were performed both on the planar surface and polished

The planar microstructure of the Ni deposit is shown in Fig. 2. For this particular deposit the average grain size is about 14 nm and size distribution is relatively narrow (12–17 nm). Also, the majority of grain boundaries is highangle; this is illustrated by a high-resolution electron micrograph shown in Fig. 3. While grains are surrounded by high-angle boundaries, a close examination at boundary areas also indicates the absence of amorphous phase. However, there exist large residual stresses in the deposit as evidenced by the appearance of many bend contours. The presence of impurity can greatly affect the strength of a material. It can cause grain boundary segregation, solid solution strengthening, precipitation hardening, or dispersion hardening, depending upon the nature of the impurity and the chemical interaction between the impurity and the matrix element. It is therefore important to synthesize a material of high purity in order to study solely the grain size effect, or the effect of grain boundary on deformation. We employed atom probe field emission

Fig. 3. High resolution view of Fig. 2 showing the most of grain boundaries are high angle. Also, there is no amorphous phase.

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Fig. 4. A typical field ion image of electrodeposited nanocrystalline Ni. Fine grains are revealed. There exist some oxygen and carbon impurities but no grain boundary segregation.

microscopy (APFIM) and high-resolution electron microscopy with angular dark field (ADF) imaging capability, i.e. Z-contrast. A typical field ion image is shown in Fig. 4. The small grain size is confirmed in the figure. Some atom probe analyses have been performed and the results show some carbon, nitrogen and oxygen in the material. However, they are not related to the grain boundary, suggesting there is no strong impurity segregation and boundaries are relatively clean. Also, no fine (!100 nm) particles or precipitates were detected in the field ion images. In the case of HRTEM investigation, Fig. 5 is an ADF image (Z contrast) of the microstructure in which the bright area represents high Z (mainly pure nickel) region and the dark area represents low Z region (impurities, e.g. oxygen or carbon). The electron energy loss spectrum (EELS) from

Fig. 6. The electron energy loss spectrum (EELS) from Spot 1 (dark region in Fig. 5) is shown in Fig. 6 in which the O–K edge and Ni–L2,3 edge energy loss peaks can be observed.

Spot 1 (dark region in Fig. 5) is shown in Fig. 6 in which the O–K edge and Ni–L2,3 edge energy loss peaks can be observed. There is no indication of oxygen segregation at grain boundaries. In fact, the relatively uniform image contrast in Fig. 5 indicates the distribution of oxygen impurity is also uniform throughout grains. At the present time, we have not quantified the oxygen concentration. By comparison, the EELS spectrum taken from Spot 2 (bright area in Fig. 5) only shows Ni–L2,3 edge energy loss, as shown in Fig. 7, suggesting relatively pure nano-grains. In summary, from both microstructural and chemical analyses, we feel comfortable to use the electrodeposited nanocrystalline nickel for the study of grain boundary effect. 2.3. Hardness near Hall–Petch breakdown in n-Ni The results of the instrumented indentation experiments are summarized as a function of grain size in the H–P plot

Fig. 5. Angular dark field image (Z-contrast) from n-Ni deposit, showing the presence of some low-Z impurities. Preferential segregation of oxygen or carbon impurities is absent.

Fig. 7. The EELS spectrum taken from Spot 2 (bright area in Fig. 5) only shows Ni–L2,3 edge energy loss, suggesting relatively pure nano-grains.

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Fig. 8. Hall–Petch plot of hardness (H) vs. reciprocal square-root grain size (dK1/2) for n-Ni and n-Ni–W alloys, compared with data from literature (from Ref. [22]).

of Fig. 8. For comparison, this plot also contains hardness data on nominally pure nickel from the work of Erb et al. [17], who used pulsed-current electrodeposition to produce materials, as well as Ebrahimi et al. [24], who prepared specimens by DC electrodeposition. To a grain size as fine as dZ14 nm and hardness as high as w6.4 GPa, the present data for n-Ni are in reasonable agreement with the classical H–P scaling behavior, and complement the data of Ebrahimi et al. [24]. However, between the smallest grain sizes of dZ14 and 12 nm, our data show a significant decrease in hardness, consistent with H–P breakdown anticipated at these grain sizes [2]. The peak hardness of w6.4 GPa observed for n-Ni at dZ14 nm is in agreement with the peak hardness measured by Erb et al. [17] near the same grain size (see Fig. 8), as is the observation of a H–P breakdown. However, our data reflect a rather abrupt breakdown point near dz14 nm, while the data of Erb et al. [17,25] show a much broader transition region, spanning grain sizes from 11–25 nm. A possible explanation for this discrepancy lies in the method of sample fabrication. Although both investigations employed electrodeposition techniques to prepare fully dense n-Ni specimens, the present study used direct current plating whereas Erb et al. [17,25] used pulsed current plating, in which the current is applied as a rectangular waveform. These methods produce grain structures with different morphologies; our DC electrodeposited foils have an elongated grain structure in the plating direction, as described in Ref. [22]. Additionally, pulsed-current plating gives a rather broad grain size distribution that would tend to broaden the observed H–P breakdown in Fig. 8.

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Fig. 9. Hall–Petch-style plot of abraded volume (V) vs. reciprocal squareroot grain size (dK1/2) for n-Ni after identical 500 mm scratches with a Berkovich diamond tip (from Ref. [22]).

2.4. Nano-abrasion near Hall–Petch breakdown in n-Ni Fig. 9 shows a summary of the major results from the nano-scratch experiments on n-Ni, the total volume of displaced material, V, as determined from atomic force microscope images of the scratches, plotted against dK1/2 in H–P style. The trend shown in this figure is similar to that observed in hardness (Fig. 8), with a generally linear relationship between V and dK1/2 from 15 mm down to w14 nm grain sizes. The H–P hardness breakdown observed below w14 nm in Fig. 8 is manifested here as a deviation from this linear relationship for the specimen with dZ12 nm. For a rigid asperity scratching a substrate, the amount of material dV displaced by abrasion is proportional to the distance of translation dx, and scaled by the projected area of the asperity in the direction of travel. For our experiments, the projected contact area is determined from the ideal geometry of a Berkovich tip, and the applied load increases as a linear function of position, with a proportionality constant kZ300 N/m. Under these conditions one can derive [22]: dV k Z 0:146 x dx H

(1)

This inverse relationship between hardness and abraded volume can be quantitatively validated by normalizing the wear data in Fig. 9 with the experimental hardness values (from Fig. 8). The solid points in Fig. 3 show the same data, now multiplied by the hardness according to Eq. (1). Whereas the volume of removed material varied by over a factor of five (open points in Fig. 8), normalization with

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the hardness completely removes this variation to within experimental error. Thus, although the nanocrystalline nickel specimens used in this work exhibited a broad range of wear resistance, we find that these differences are quantitatively commensurate with the measured change in hardness, even in the limit of the finest grain sizes. Although the breakdown of H–P strengthening at these grain sizes may be related to a transition in the plastic deformation mechanism, the abrasive wear mechanism appears unchanged through this transition. It is important to emphasize, however, that we have observed an apparent breakdown of H–P strengthening in pure n-Ni only at a single grain size in this work.

described above. Also, to minimize the substrate effects, the penetration depth of the indent during hardness measurements was always less than one tenth of the film thickness. The 0.2–3 mm thick coatings are prepared for bright field imaging in plan view with a transmission electron microscope (TEM) by thinning the reverse side of the substrate through mechanical polishing followed by ion milling at glancing incidence. Specimen tilting at 308 is used to reveal any evidence of columnar growth. The bright field imaging and diffraction work is conducted using a JEOL 200CX electron microscope. The grain size (dg) of each coating is determined using the standard linear intercept method.

3. Nanocrystalline and amorphous Be–B system

3.2. Hardness of ultrafine-structured Be–B

3.1. Materials and experiments

To investigate the effect of B addition alone on the grain size refinement in Be-rich coatings, the pure Be and B targets are used to produce a set of Be1KxBx coatings for examination in cross-section (as seen in Fig. 10). The grain size is evidently reduced as the B concentration is increased as seen in the progression from images in Fig. 10a. through Fig. 10e. Beyond 11 at% B, the microstructure transforms from a nanocrystalline (seen in Fig. 10d.) to an amorphous structure (seen in Fig. 10e and f). This result is probably the most compositionally Be-rich, amorphous binary phase ever explored. There is no phase contrast indicated in the crosssection images (Fig. 10e and f, respectively) for the 16 at% B and 21 at% B coatings, other than the faint image artifact of film curvature through the very thin cross-section of the imaged area. Corresponding SADPs (shown in Fig. 11) are generated from each cross-sectioned Be1KxBx coating.

To investigate the mechanical properties of nanocrystalline Be alloys, Be–based coatings were synthesized by sputter deposition onto polished silicon wafers using planar magnetrons [26]. The sputter targets are fully dense powder compacts. The pure Be target (chemically assayed at 99.4 at% purity), and target alloys of Be0.90B0.10 and Be0.94Cu0.06 were produced by the hot-isostatic pressing of powder metal precursors in sealed, evacuated tantalum containers. The silicon substrates are located 6 cm beneath the center of the mini-source array. The nominal composition of each alloy coating is computed from calibrated deposition rates. Both the coating modulus and hardness are measured using a Nanoindenter XPTM. The measuring procedures were similar to that used for nanocrystalline Ni,

Fig. 10. TEM bright field images are shown as viewed in cross-section for sputter deposited coatings of (a) Be0.994 (b) Be0.97B0.03 (c) Be0.93B0.07 (d) Be0.89B0.11 (e) Be0.84B0.16, and (f) Be0.79B0.21. (from Ref. [26]).

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Fig. 11. SADPs taken from the cross-sectioned coatings of (a) Be0.994 (b) Be0.97B0.03 (c) Be0.93B0.07 (d) Be0.89B0.11 (e) Be0.84B0.16, and (f) Be0.79B0.21.

The interplanar d-spacings (listed in Table 2) for each crystalline deposit are consistent with the hcp phase of Be. Again, the incorporation of B appears to dilate the host lattice as seen in the increase of the (00.2) d-spacings. In general, the same is true for both the (10.0) and the (10.1) d-spacings with one exception. It appears that the (10.0) d-spacing decreases for the 11 at% B coating, i.e. just before the structural transition from a nanocrystalline to an amorphous phase. No intermetallic Be–B was detected in the coatings. Note that although the 16 at% B sample is amorphous, a faint inner ring with a lattice spacing of 0.227 nm is detectable in the SADP (Fig. 10e). This spacing actually corresponds to a native oxidation of the Be coating that occurs while being imaged in the microscope. For reference, a 0.2337 nm d-spacing corresponds to the hcp BeO phase (10.0). The average d-spacing that corresponds

to the amorphous phase halos (of 16 at% B and 21 at% B in Fig. 3e and f) is about 0.165G0.0015 nm. The mechanical properties of the Be and Be alloy coatings were measured using nanoindentation. The coating hardness is determined as that value corresponding to an indenter displacement beyond 100 nm. The coating hardness values are listed in Table 2 for each coating along with that of the reference of Be bulk. The sputter deposited pure Be is harder than the bulk, specifically, they are 8.5 and 28 GPa, respectively. This is expected since the grain size is much finer. With B addition, a peak HF value of 20.2 GPa is seen in the deposit of 7 at% B with a 17 nm grain size. The results for the 11 at% B, 16 at% B, and 21 at% B samples indicate that the coatings soften with the onset of amorphous phase formation. Similar trends are measured for the elastic modulus of the coatings as for the hardness values.

Table 2 Properties of Be and binary Be–B alloys Sample

Be foil

Be0.994

Be0.90B0.10

Be0.97B0.03

Be0.93B0.07

Be0.89B0.11

Be0.84B0.16

Be0.79B0.21

dg (nm) d(10.0) (nm) d(00.2) (nm) d(10.1) (nm) HF (GPa) HF/Ho EF (GPa)

2000 0.197 0.179 0.173 8.5G0.2 1.00 220G5

33G4 0.1967 0.1785 0.1721 9.5G0.3 1.12 260G11

15G3 0.1990 0.1808 0.1738 16.3G0.2 1.56 250G10

23G3 0.1974 0.1793 0.1725 16.8G0.5 1.85 248G4

17G2 0.2020 0.1831 0.1768 20.2G0.5 2.05 245G10

15G2 0.1924 – 0.1740 11.0G0.3 1.03 215G8

0 – – – 12.1G0.4 1.04 217G15

0 – – – 13.0G0.3 1.03 217G14

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3.3. Inverse Hall–Petch relationship in the Be–B system? The present findings for the Be1KxBx coatings indicate that a homogeneous (hence isotropic), and strong amorphous coating is formed for B concentrations greater than 11 at%, i.e. xO0.11. The observation of an amorphous Be-rich phase in Be1KxBx coatings (xO0.11) is consistent with the established binary phase diagram that shows a eutectic at a composition of 11 at% B [27]. The stabilization of amorphous phases usually occurs at a eutectic composition. The amorphous coating can be considered as a homogeneous and isotropic phase. To analyze the hardening effect measured in the Be1KxBx coatings, the hardness is plotted (in Fig. 12) as a function of the B concentration. An increase in the hardness with increasing B concentration (and decreasing grain size in the crystalline deposits) is only seen in the low B region for grain sizes greater than 17 nm. The addition of B to Be is expected to enhance hardness because of a solid solution strengthening. Also, because B can refine the grain size of Be, B alloying produces a grain size strengthening effect (i.e. H–P relationship). Assuming that the solid solution and grain boundary effects operate independently in the Be–B alloys, then the hardness of the alloys can be expressed as Ht Z Ho C DHs C DHg

(2)

where Ht is the total hardness, H0 is the hardness of a single crystal Be specimen, DHs is the hardness resulting from solid solution strengthening, and DHg is the hardness due to grain size refinement. The grain boundary effect for hardness can be described using the well-known H–P relationship DHg Z Hg K Ho Z kdK0:5

(3)

Fig. 12. The variation of hardness (GPa) with boron concentration (at% B) for sputter-deposited coatings of Be1KxBx (xZ0, 0.03, 0.07, 0.10, 0.11, 0.16, 0.21, 1.00) coatings. The apparent inverse Hall–Petch is an artifact, actually caused by the presence of soft amorphous Be–B phases.

where Hg is the hardness caused by grain size effects, d is the grain size, and k is a material constant. Using the two hardness values for pure Be (8.5 GPa for the 2 mm grain size, and 9.5 GPa for the 33 nm grain size) in Table 2, one readily obtains kZ6.6 GPa-nm0.5 and H0Z8.35 GPa. Using these values, one would predict hardness values of 9.95 and 9.73 GPa for grain sizes of 17 and 23 nm, respectively. These values are much lower than the values measured from Be0.93B0.07 and Be0.97B0.03, which are 20.2 GPa and 16.8 GPa, respectively. Obviously, grain size effects cannot solely account for the overall strengthening. The solid solution effect can be estimated as follows. For a dilute substitutional Be–B solid solution the hardness increase can be expressed using the relationship [28]: DHs Z 3Gð3s Þ3=2 c0:5 ð700ÞK1

(4)

where c is the volume fraction of B, G is the shear modulus, and 3s is a parameter combining size-effect and modulus interaction energies. By inserting appropriate values for each variable, DHs is estimated to be only 0.81 GPa. Evidently, this value is much too small to account for the observed hardness increase. Since the atomic radius of B is smaller than that of Be, the substitution of Be with B is expected to reduce the lattice parameter. However, adding B actually dilates the lattice spacing of the Be matrix, suggesting defects (such as vacancies) might have been created. These defects can also contribute to hardness, similar to solid solution hardening. Because of both the solid solution and grain boundary strengthening effects, it seems that the hardness of Be1KxBx coatings should increase monotonically with B concentration. However, the data did not indicate so. The hardness value peaks at a B concentration of about 7 at%, and a further increase in B concentration, in fact, causes a decrease in hardness. This decrease in hardness with increasing B concentration nearly coincides with the onset of the presence of amorphous phase. As pointed out earlier, when the B concentration is near the eutectic composition (11 at%), the amorphous Be–B phase is stabilized. Microstructures show (in Figs. 10 and 11) that the Be1KxBx structure is entirely amorphous when xO11 at%. There are overwhelming evidences indicating that an amorphous structure is mechanically softer than its crystalline counterparts. For example, it has been shown that amorphous alumina is softer than its crystalline counterparts. [29]. Also, it has recently been reported that an amorphous alloy containing nanocrystals (produced by crystallization of a metallic glass) is harder than the amorphous structure alone [30,31]. It is therefore reasonable to conclude that the decrease in hardness at high B concentrations (greater than 11 at%) is caused by the presence of the amorphous phase, and should not be viewed as the inverse H–P effect. It is particularly noted that at 11 at% B (in Fig. 12d), although the grain size in the sample is smaller than that in the 7 at% B sample (in Fig. 10c),

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the structure is likely to be an amorphous–crystalline mixture. As a result, the sample is softer than the fully crystalline 7 at% B sample. It is also noted in Fig. 12 that, despite different compositions, the Be1KxBx alloys show only a slight difference in hardness value in the ‘amorphous region’. In this region, since the diffraction results indicate that both Be and B are in glassy states, the material apparently can be treated as a stoichiometric mixture of its non-interacting glassy phase elements. This is illustrated by the straight dashed line shown in Fig. 12.

4. Summary and remark The deformation of two crystalline alloys, pure nickel and Be–B binary alloys, with grain sizes near and less than 20 nm were studied and the results presented. This is the grain size region where the classical H–P relationship breakdowns, or the inverse H–P relationship, are often reported. In the case of Ni produced by electrodeposition, we demonstrated that the materials were highly pure, the majority of grain boundaries was high-angle, and there was no impurity segregation. Subsequent nanohardness and nanoscratch tests were carried out and showed that H–P relationship did break down when the grain size is less than about 14 nm. In the case of Be–B alloys produced by sputter deposition, we found that the alloying of B can significantly refine the grain size of Be. However, when the B content is over the eutectic composition an amorphous Be–B phase begin to appear. An apparent H–P breakdown phenomenon was also observed. We illustrated that it is an artifact. The apparent inverse HP relation was, in fact, caused by the presence of the relatively soft amorphous Be–B phase.

Acknowledgements This work is supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy under contract DE-AC05-00OR22725 with UT-Battelle, and the Lawrence Livermore National Laboratory, under US Department of Energy contract W-7405-Eng-48. The authors would like to thank Drs C.A. Schuh, H. Iwasaki, and Alan Jankowski for their technical contributions.

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