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Mechanical properties of nanocrystalline materials
M.A. Meyers
, 
, A. Mishra and D.J. Benson
Department of Mechanical and Aerospace Engineering, Materials
Science and Engineering Program, Mail Code 0411, University of California, San
Diego La Jolla, CA 92093, United States
Received 1 November 2004;
revised 1 May 2005; accepted 1 August 2005. Available online 17
November 2005.
The mechanical properties of nanocrystalline materials are reviewed, with emphasis on their constitutive response and on the fundamental physical mechanisms. In a brief introduction, the most important synthesis methods are presented. A number of aspects of mechanical behavior are discussed, including the deviation from the Hall–Petch slope and possible negative slope, the effect of porosity, the difference between tensile and compressive strength, the limited ductility, the tendency for shear localization, the fatigue and creep responses. The strain-rate sensitivity of FCC metals is increased due to the decrease in activation volume in the nanocrystalline regime; for BCC metals this trend is not observed, since the activation volume is already low in the conventional polycrystalline regime. In fatigue, it seems that the S–N curves show improvement due to the increase in strength, whereas the da/dN curve shows increased growth velocity (possibly due to the smoother fracture requiring less energy to propagate). The creep results are conflicting: while some results indicate a decreased creep resistance consistent with the small grain size, other experimental results show that the creep resistance is not negatively affected. Several mechanisms that quantitatively predict the strength of nanocrystalline metals in terms of basic defects (dislocations, stacking faults, etc.) are discussed: break-up of dislocation pile-ups, core-and-mantle, grain-boundary sliding, grain-boundary dislocation emission and annihilation, grain coalescence, and gradient approach. Although this classification is broad, it incorporates the major mechanisms proposed to this date. The increased tendency for twinning, a direct consequence of the increased separation between partial dislocations, is discussed. The fracture of nanocrystalline metals consists of a mixture of ductile dimples and shear regions; the dimple size, while much smaller than that of conventional polycrystalline metals, is several times larger than the grain size. The shear regions are a direct consequence of the increased tendency of the nanocrystalline metals to undergo shear localization.
The major computational approaches to the modeling of the mechanical
processes in nanocrystalline metals are reviewed with emphasis on molecular
dynamics simulations, which are revealing the emission of partial dislocations
at grain boundaries and their annihilation after crossing them.
The landmark paper by Gleiter [1] redirected a significant portion of the global research efforts in materials science. The importance of this paper can be gauged by its 1300+ citations and the thousands of papers that appeared on this topic since its publication. Actually, this paper was preceded by an earlier, lesser known Gleiter paper, from 1983 [2]. In this paper, Gleiter points out the outstanding possibilities of what he called then “microcrystalline materials”. The name “nanocrystalline” has since taken over. The mechanical behavior of nanocrystalline materials has been the theme of approximately 500 publications. A significant number of review articles have been published. Table 1 shows the most important review articles as well as their foci.
Principal review articles on nanostructured materials [only first author named]
Author Year Title Gleiter [1] 1989 Nanocrystalline materials Birringer [6] 1989 Nanocrystalline materials Gleiter [349] 1992 Materials with ultrafine microstructures: retrospectives and perspectives
Suryanarayana [3] 1995 Nanocrystalline materials: a critical review Lu [39] 1996 Nanocrystalline metals crystallized from amorphous solids: nanocrystallization, structure, and properties Weertman [361] 1999 Structure and mech. behavior of bulk nanocrystalline materials Suryanarayana [350] 2000 Nanocrystalline materials—current research and future directions Valiev [56] 2000 Bulk nanostructured materials from severe plastic deformation Gleiter [22] 2000 Nanostructured materials: basic concepts and microstructure
Furukawa [66] 2001 Processing of metals by equal-channel angular pressing Mohamed [351] 2001 Creep and superplasticity in nanocrystalline materials: current understanding and future prospects Kumar [352] 2003 Mechanical behavior of nanocrystalline metals and alloys Veprek [353] 2005 Different approaches to superhard coatings and nanocomposites Wolf [354] 2005 Deformation of nanocrystalline materials by molecular-dynamics simulation: relationship to experiments? Weertman [363] 2005 Structure and mechanical behavior of bulk nanocrystalline materials Weertman [374] 2002 Mechanical behavior of nanocrystalline metals
Nanocrystalline materials have been the subject of widespread research over the past couple of decades with significant advancement in their understanding especially in the last few years [3]. As the name suggests, they are single or multi-phase polycrystals with nano scale (1 × 10−9–250 × 10−9 m) grain size. At the upper limit of this regime, the term “ultrafine grain size” is often used (grain sizes of 250–1000 nm). Nanocrystalline materials are structurally characterized by a large volume fraction of grain boundaries, which may significantly alter their physical, mechanical, and chemical properties in comparison with conventional coarse-grained polycrystalline materials [4], [5] and [6], which have grain sizes usually in the range 10–300 μm. Fig. 1 shows a schematic depiction of a nanocrystalline material. The grain-boundary atoms are white and are not clearly associated with crystalline symmetry.
Fig. 1. Two-dimensional model of a nanostructured material. The atoms in the centers of the crystals are indicated in black. The ones in the boundary core regions are represented as open circles [22].
As the grain size is decreased, an increasing fraction of atoms can be ascribed to the grain boundaries. This is shown in Fig. 2, where the change of the volume fraction of intercrystal regions and triple-junctions is plotted as a function of grain size. We can consider two types of atoms in the nanocrystalline structure: crystal atoms with neighbor configuration corresponding to the lattice and boundary atoms with a variety of interatomic spacing. As the nanocrystalline material contains a high density of interfaces, a substantial fraction of atoms lie in the interfaces. Assuming the grains have the shape of spheres or cubes, the volume fraction of interfaces in the nanocrystalline material may be estimated as 3Δ/d (where Δ is the average interface thickness and d is the average grain diameter). Thus, the volume fraction of interfaces can be as much as 50% for 5 nm grains, 30% for 10 nm grains, and about 3% for 100 nm grains.
Fig. 2. The effect of grain size on calculated volume fractions of intercrystal regions and triple junctions, assuming a grain-boundary thickness of 1 nm [124].
Nanocrystalline materials may exhibit increased strength/hardness [7], [8] and [9], improved toughness, reduced elastic modulus and ductility, enhanced diffusivity [10], higher specific heat, enhanced thermal expansion coefficient (CTE), and superior soft magnetic properties in comparison with conventional polycrystalline materials. This has been the incentive for widespread research in this area, and lately, with the availability of advanced tools for processing and characterization, there has been an escalation of work in this field.
Nanostructured materials provide us not only with an excellent opportunity to study the nature of solid interfaces and to extend our understanding of the structure–property relationship in solid materials down to the nanometer regime, but also present an attractive potential for technological applications with their novel properties [11]. Keeping this incentive in mind, the purpose of this paper is to provide an overview of the basic understanding of the mechanical properties of these materials.
A number of techniques have surfaced over the years for producing nanostructured materials, but most of them are limited to synthesis in small quantities. There has been a constant quest to scale up the process to bulk processing, and lately, a few advances seem to hold technological promise. This has made research in this area exciting to a higher level. The most important methods are presented in Section 4.
The synthesis and use of nanostructures are not new phenomena. In 1906, Wilm [12] observed age hardening in an Al–Cu–Mg–Mn alloy. Merica et al. [13] proposed in 1919 that the age hardening was caused by the precipitation of submicrometer-sized particles, which were later confirmed by X-ray and transmission electron microscopy (TEM). The precipitates are known as GP zones, GPII zones (θ″) and metastable (θ′) precipitates, and are typically 10 nm in thickness and 100 nm in diameter. In particular, the GP zones (named after Guinier and Preston, who suggested their existence through diffuse X-ray scattering) have thicknesses on the order of 1 nm. The accidental introduction of these precipitates into aluminum in the early 1900s revolutionized the aluminum industry, since it had a dramatic effect on its strength which enabled its widespread use in the burgeoning aircraft industry. Many important defects and phenomena in the mechanical behavior of materials take place at the nanoscale; thus, the realization that nanoscale is of utter importance has been a cornerstone of materials science for the past half century.
The quest for ultrafine grain sizes started in the 1960s by Embury and
Fischer [14]
and Armstrong et al. [15].
The driving force behind this effort was the possibility of synthesizing
materials with strengths approaching the theoretical value (G/10) by
reducing the grain size, a reasonable assumption from the Hall–Petch
relationship. A great deal of effort was also connected with superplasticity,
since it is known that the smaller the grain size, the higher the strain rate at
which this phenomenon is observed. Langford and Cohen [16]
and Rack and Cohen [17]
carried out detailed characterization of Fe–C and Fe–Ti wires cold drawn to true
strains of up to 7. They observed a dramatic reduction in the scale of the microstructure, with grains/subgrains/cells with sizes as low as
300 nm. This reduction led to significant increases in the flow stress,
shown in Fig.
3(a). The flow stress was increased to 1 GPa. The early effort by
Schladitz et al. [18]
to produce polycrystalline iron whiskers is also noteworthy. These whiskers, a
section of which is shown in Fig.
3(b), had grain sizes between 5 and 20 nm. One could say that this is
the first nanocrystalline metal. Jesser et al. [19]
calculated the strength using the H–P equation
(σ0 = 70 MPa;
k = 17 MPa m−1/2) and arrived at a
predicted value of 5.5 GPa for d = 10 nm.
Unfortunately, these whiskers, produced by CVD, have diameters not exceeding
20 μm.
Fig. 3. (a) Strength of wire drawn and recovered Fe–0.003C as a function of transverse linear-intercept cell size [17]; (b) Schladitz whisker, which can be considered the first nanocrystalline metal. The whisker is comprised of “onion-skin layers” with approximately 100 nm; these layers are composed of grains with diameters in the 5–20 nm range (from [19]).
Nanostructured materials as a major field in modern materials science did not start, however, until 1981 when Gleiter synthesized nanostructured metals using inert gas condensation (IGC) and in situ consolidation [20]. This involved generating a new class of materials with up to 50% or more of the atoms situated in the grain boundaries. Since the landmark paper of Gleiter, there has been increasing interest in the synthesis, processing, characterization, properties, and potential applications of nanostructured materials. Accordingly, a number of techniques have been developed to produce nanoscale particles as well as bulk nanostructured materials. They are briefly described in Section 4, since the synthesis method has a direct and important bearing on the resultant mechanical properties.
Siegel [21] classified nanostructured materials into four categories according to their dimensionality: 0D—nanoclusters; 1D—multilayers; 2D—nanograined layers; 3D—equiaxed bulk solids. For the major part of this review, we will focus our attention on 3D equiaxed bulk solids. We will not include nanocrystalline coatings. For information on this, the reader is referred to Verpek [336]. However, nanowires, that are one-dimensional nanostructures, have important electronic properties.
Classification can also be made based on the grain size: ultrafine grain sized materials, where the grain sizes are above approximately 500 nm (usually in the sub-micrometer range) and nanograined materials, where the grain sizes are below 500 nm and usually in the vicinity of 100–200 nm. Based on the starting material from which nanomaterials are made, they can be further classified as nanomaterials crystallized from amorphous solid or nanomaterials made from other methods where the starting material is usually crystalline.
Gleiter [22] further classified the nanostructured materials according to composition, morphology, and distribution of the nanocrystalline component as shown in Fig. 4. He used three shapes: rods, layers, and equiaxed grains. His classification includes many possible permutations of materials and is quite broad. According to the shape of the crystallites, three categories of nanomaterials may be distinguished: layer-shaped crystallites, rod-shaped crystallites (with layer thickness or rod diameters in the order of a few nanometers), and nanostructures composed of equiaxed nanometer-sized crystallites. Depending on the chemical composition of the crystallites, the three categories of nanomaterials may be grouped into four families. In the simplest case, all crystallites and interfacial regions have the same chemical composition. Examples of this family are semicrystalline polymers or nanomaterials made up of equiaxed nanometer-sized crystals, e.g., of Cu. Nanomaterials belonging to the second family consist of crystallites with different chemical compositions. Quantum well structures are the most well known examples of this type. If the compositional variation occurs primarily between the crystallites and the interfacial regions, the third family of nanomaterial is obtained. In this case, one type of atom segregates preferentially to the interfacial regions so that the structural modulation is coupled to the local chemical modulation. Nanomaterials consisting of nanometer-sized W crystals with Ga atoms segregated to the grain boundaries are an example of this type. An interesting new example of such materials was recently produced by co-milling Al2O3 and Ga. The fourth family of nanomaterials is formed by nanometer-sized crystallites dispersed in a matrix of different chemical composition.
Fig. 4. Classification scheme for nanostructured materials according to their chemical composition and their dimensionality (shape) of the crystallites (structural elements) forming the nanostructure. The boundary regions of the first and second family are indicated in black to emphasize the different atomic arrangements in the crystallites and in the boundaries [22].
Nanocrystalline materials can be synthesized either by consolidating small
clusters or breaking down the polycrystalline bulk material into crystalline
units with dimensions of nanometers. These approaches have been classified into
bottom-up and top-down. In the bottom-up approach we have
to arrange the nanostructure atom-by-atom, layer-by-layer. In the
top-down approach we start with the bulk material and break down the microstructure into a nanostructure. The principal synthesis methods are:
Inert gas condensation
Mechanical alloying
Electrodeposition
Crystallization from amorphous material
Severe plastic deformation
Cryomilling
Plasma synthesis
Chemical vapor deposition
Pulse electron deposition
Sputtering
Physical vapor deposition
Spark erosion [344]
The inert gas condensation technique, conceived by Gleiter [1], consists of evaporating a metal (by resistive heating, radio-frequency, heating, sputtering, electron beam heating, laser/plasma heating, or ion sputtering) inside a chamber that is evacuated to a very high vacuum of about 10−7 Torr and then backfilled with a low-pressure inert gas like helium (Fig. 5(a)). The evaporated atoms collide with the gas atoms inside the chamber, lose their kinetic energy, and condense in the form of small particles. Convection currents, generated by the heating of the inert gas by the evaporation source and by the cooling of the liquid nitrogen-filled collection device (cold finger) carry the condensed fine powders to the collector device. The deposit is scraped off into a compaction device. Compaction is carried out in two stages: (a) low pressure compacted pellet; (b) high pressure vacuum compaction. The scraping and compaction processes are carried out under ultrahigh vacuum conditions to maintain the cleanliness of the particle surfaces and to minimize the amount of trapped gases. The inert gas condensation method produces equiaxed (3D) crystallites. The crystal size of the powder is typically a few nanometers and the size distribution is narrow. The crystal size is dependent upon the inert gas pressure, the evaporation rate, and the gas composition. Extremely fine particles can be produced by decreasing either the gas pressure in the chamber or the evaporation rate and by using light rather than heavy inert gases (such as Xe).
Fig. 5. (a) Schematic drawing of the inert gas condensation technique for production of nanoscale powder [365]; (b) bright field TEM micrograph of TiO2 nanoparticles prepared by inert gas condensation [366].
A great deal of the early work on mechanical properties of nanocrystalline materials used the inert gas condensation technique. One shortcoming is the possibility of contamination of powders and porosity due to insufficient consolidation. There is also the possibility of imperfect bonding between particles, since most of the early work used cold consolidation. Nevertheless, the results obtained using specimens prepared by this method led the foundation of our understanding. The important contributions of Weertman, Siegel, and coworkers [23], [24], [25], [26] and [27] have used materials produced by this method. They were the first systematic studies on the mechanical properties of nanocrystalline metals (Cu and Pd) and were initiated in 1989. Fig. 5(b) shows the bright field image TEM micrograph of TiO2 nanoparticles prepared by this technique.
Nanocrystalline alloys can also be synthesized by evaporating the different metals from more than one evaporation source. Rotation of the cold finger helps in achieving a better mixing of the vapor. Oxides, nitrides, carbides, etc. of the metals can be synthesized by filling the chamber with oxygen or nitrogen gases or by maintaining a carbonaceous atmosphere. Additionally, at small enough particle sizes, metastable phases are also produced. Thus, this method allows the synthesis of a variety of nanocrystalline materials. The peak densities of the as-compacted metal samples have been measured with values of about 98.5% of bulk density. However, it has been established that porosity has a profound effect on the mechanical strength, especially in tension.
Mechanical alloying [28], [29], [30] and [31] produces nanostructured materials by the structural disintegration of coarse-grained structure as a result of severe plastic deformation. Mechanical alloying consists of repeated deformation (welding, fracturing and rewelding) of powder particles in a dry high-energy ball mill until the desired composition is achieved. In this process, mixtures of elemental or pre-alloyed powders are subjected to grinding under a protective atmosphere in equipment capable of high-energy compressive impact forces such as attrition mills, shaker mills and ball mills. Fig. 6(a) shows the set-up for ball milling process. It has been shown that nanometer-sized grains can be obtained in almost any material after sufficient milling time. The grain size decreases with milling time down to a minimum value that appears to scale inversely with melting temperature. It was suggested by Fecht et al. [29] that localized plastic deformation creates shear bands that show evidence of rotational dynamic recrystallization similar to the ones obtained in high-strain rate deformation (that are discussed in Section 7.5). Fig. 6(b) shows a dark-field TEM of an Al–Mg alloy processed by ball milling at 77 K and annealing at 150 °C. The grain size distribution varying from 20 to 200 nm is clearly shown. Cryomilling is a variation of ball-milling that has been extensively used by Lavernia and coworkers [32], [33], [34] and [35].
Fig. 6. (a) Mechanical milling as a means of synthesis of nanostructured material. (b) Dark field image of nanocrystalline Al–Mg alloy synthesized by cryogenic ball milling and annealed at 150 °C for 1 h [367].
The electrodeposition technique has significant advantages over other methods for synthesizing nanocrystalline materials: (1) potential of synthesizing large variety of nanograin materials—pure metals, alloys and composite systems with grain sizes as small as 20 nm, (2) low investment, (3) high production rates, (4) few size and shape limitations, and (5) high probability of transferring this technology to existing electroplating and electroforming industries.
Fig. 7(a) shows schematically the pulse electrodeposition sequence. As the current spikes, the metal cations are deposited in crystalline and amorphous patches. Fig. 7(b) shows the TEM micrograph of pulse electrodeposited Ni sample. Commercially synthesized (Integran) 5 mm thick plates are available in a range of compositions.
Fig. 7. (a) Pulsed electrodeposition set-up for synthesizing nanocrystalline materials. (b) Pulsed electrodeposited Ni. (Courtesy of M. Goeken, Univ of Erlangen, Germany.)
Over the past few years, Erb et al. [36] have studied the synthesis, structure and properties of nanocrystalline nickel synthesized by pulse electrodeposition. They demonstrated that grain refinement of electroplated nickel into the nanometer range results in unique and, in many cases, improved properties as compared to conventional polycrystalline nickel. Electrodeposition of multilayered (1D) metals can be achieved using either two separate electrolytes or much more conveniently using one electrolyte by appropriate control of agitation and the electrical conditions. Also, 3D nanostructure crystallites can be prepared using this method by utilizing the interface of one ion with the deposition of the other. It has been shown that electrodeposition yields grain sizes in the nanometer range when the electrodeposition variables are chosen such that nucleation of new grains is favored rather than growth of existing grains. This was achieved by using high deposition rates, formation of appropriate complexes in bath, addition of suitable surface-active elements to reduce surface diffusion of ad-atoms, etc. This technique can yield porosity-free finished products that do not require subsequent consolidation processing. Furthermore, this process requires low capital investment and provides high production rates with few shape and size limitations. Recent results by Shen et al. [37] and Lu et al. [38] indicated that a highly twinned structure can be produced under the right electrodeposition condition. This high annealing twin density is responsible for the enhancement of ductility which will be discussed later.
The basic principle for the crystallization method from the amorphous state [39] is to control the crystallization kinetics by optimizing the heat treatment conditions so that the amorphous phase crystallizes completely into a polycrystalline material with ultrafine crystallites. The metallic glasses can be prepared by means of the existing routes, such as melt-spinning, splat-quenching, mechanical alloying, vapor deposition, or electrodeposition [40]. Crystallization of amorphous solids has been successfully applied in producing nanometer-sized polycrystalline materials in various alloy systems, e.g., in Fe-, Ni-, and Co-based alloys [41], [42], [43] and [44], as well as some elements. The complete crystallization of amorphous solids is a promising method for the synthesis of nanocrystalline materials because it possesses some unique advantages, the most important being porosity-free product and the ease of synthesizing nanocrystalline, intermetallics, supersaturated metallic solid solutions, and composites.
The amorphous solids are in thermodynamic metastable states and they transfer into more stable states under appropriate circumstances. The driving force for the crystallization is the difference in the Gibbs free energy between the amorphous and crystalline states. Usually, amorphous solids may crystallize into polycrystalline phases when they are subjected to heat treatment [45], irradiation [46], or even mechanical attrition. Of these techniques, conventional thermal annealing is most commonly utilized in investigations of amorphous solids.
TEM images and the selected area diffraction patterns of Ni–25at%W alloys annealed at 723 K and 873 K for 24 h in vacuum show that extremely small sized grains can be crystallized from amorphous materials as shown in Fig. 8. However, nanocrystalline structures are unstable at high temperatures because of the large excess free energy and significant grain growth has been observed. On the other hand, stabilization of the nanocrystalline grain structure was observed in many materials after continuous annealing.
Fig. 8. TEM images and selected area diffraction patterns in the Ni–25.0at%W alloy annealed from amorphous state at (a) 723 K and for (b) 873 K for 24 h in vacuum. In (a), grain sizes between 5 and 8 nm is observed. (b) shows the random orientation of the grains [368].
Grain growth is described by the equation:
Severe plastic deformation breaks down the microstructure into finer and finer grains. As early as 1960, Langford
and Cohen [16]
and Rack and Cohen [17]
demonstrated that the microstructure in Fe–0.003%C subjected to high strains by wire drawing
exhibited sub-grain sizes in the 200–500 nm range. The use of severe
plastic deformation (SPD) for the processing of bulk ultrafine-grained materials
is now widespread [47],
[48],
[49],
[50],
[51],
[52],
[53],
[54],
[55],
[56],
[57],
[58],
[59],
[60],
[61]
and [62].
Again, this is not a new technology, since piano wire, known for over a century,
owes its strength to an ultrafine grain size. Although any means of introducing
large plastic strains in metals may lead to the reduction of the grain size, two
principal methods for subjecting a material to severe plastic deformation have
gained acceptance: these are known as equal-channel angular pressing (ECAP) [47],
[63],
[64],
[65],
[66],
[67],
[68]
and [69]
and high-pressure torsion (HPT).
ECAP was first proposed in the Soviet Union in the 80s. As illustrated in Fig.
9(a), ECAP uses a die containing two channels, equal in cross-section,
intersecting at an angle Φ that is generally close to 90°. The test
sample is machined to fit within these channels. It is pushed down from the
upper die by a piston (as shown by arrow) and is forced around a sharp corner.
The strain imposed on the sample in ECAP is dependent upon both the channel
angle between the two channels, and the angle defining the outer arc of
curvature where the two channels intersect. It can be shown that an equivalent
strain close to 
1 is introduced when the channel angle is 90° for all values of the
angle defining the arc of curvature. Since the cross-sectional dimensions of the
sample remains unchanged on passage through the die, repetitive pressings may be
used to attain very high strains. Fig.
9(b) shows a copper specimen subjected to eight repetitive passes in ECAP by
rotating the specimen by 90° at each stage (route BC). The TEM
reveals a structure containing grains of approximately 200 nm. Although
grains as small as 50 nm can be reached in Al alloys, the more common size
is 200 nm. In a strict sense, one calls this “ultrafine” grain size.
Fig. 9. (a) A section through an ECAP die showing the two internal angles
and Ψ. Notice the front end shape of sheared part of the sample. (b) Bright field image of Cu processed by 8 ECAP passes using route BC in a 90° die (transverse section sample).
An alternative procedure to introduce high plastic strains, illustrated in Fig.
10(a), is called high pressure torsion (HPT) [70]
and [71].
A small sample, in the form of a disk, is held under a high pressure and then
subjected to torsional straining. Processing by HPT has the advantage of
producing exceptionally small grain sizes, often in the nanometer range
(<100 nm), and the ability to process brittle materials such as
intermetallics and semiconductors. Nevertheless, HPT has the disadvantage that
the specimen dimensions are generally fairly small, with maximum disk diameters
of 20 mm and thickness of 1 mm. Fig.
10(b) shows as an illustration, the TEM image of a Ni specimen subjected to
HPT. The grain size shows a bimodal distribution with the smaller grains less
than 100 nm and the larger grains with approximately 500 nm size.
Fig. 10. (a) Schematic of high pressure torsion set-up. (b) TEM
In this section, we review the principal mechanical properties of nanocrystalline metals: yield stress, ductility, strain hardening, strain-rate sensitivity and dynamic response, creep and fatigue. At the outset, it should be emphasized that porosity is of utmost importance and can mask and/or distort properties. The early “bottom-up” synthesis methods often resulted in porosity and incomplete bonding among the grains.
Processing flaws like porosity are known to be detrimental to the properties of nanocrystalline materials. Fig. 11 shows the Young’s modulus as a function of porosity for nanocrystalline Pd and Cu as shown by Weertman et al. [72]. This decrease in Young’s modulus with porosity is well known and is indeed expressed in many mechanics simulations. One of the equations is Wachtman and MacKenzie [73] and [74]:
| E=E0(1-f1p+f2p2) | (2) |
.gif){kind=small}
. The yield stress and tensile ductility
are simultaneously affected. Fig.
12 shows as an illustration, a plot of the yield stress as a function of
density for Cu and Pd. The decrease in strength is obvious. The existing pores
provide initiation sites for failure.
Fig. 11. Young’s modulus as a function of porosity for nanocrystalline Pd and Cu [72].
(20K)
Fig. 12. Compressive yield strength of Cu and Pd as a function of consolidation density. (Data plotted from Youngdahl et al. [27].)
Grain size is known to have a significant effect on the mechanical behavior of materials, in particular, on the yield stress. The dependence of yield stress on grain size in metals is well established in the conventional polycrystalline range (micrometer and larger sized grains). Yield stress, σy, for materials with grain size d, is found to follow the Hall–Petch relation:
| σy=σ0+kd-1/2 | (3) |
n 0.7.
The mechanical properties of FCC metals with nano-range grain sizes have been estimated from uniaxial tension/compression tests and micro- or nano-indentation. Often micro-size tensile samples are used to avoid the influence of imperfections [72], e.g., voids that might adversely influence the mechanical response of the material.
The compressive yield stresses of nanocrystalline Cu and Pd samples
synthesized by IGC are summarized in Table
2 [27],
and the plot is given in Fig.
12. Weertman and coworkers [72]
observed that nanocrystalline Cu and Pd samples were remarkably stronger than
their coarse-grained counterpart and this was a strong function of density.
Their strain to failure was also higher. Suryanarayana et al. [75]
reported compressive yield strength of 500 MPa from their strongest nano Cu sample. Table
2 gives the values of the Vickers hardness, Hv divided by
3, which approximates to the yield strength if the work-hardening is not large.
Unlike the case of tensile yield strength, the compressive values of
σy scale well with Hv/3. Weertman et al. [76],
observed a large increase in hardness for the nanocrystalline Cu and Pd samples
made by IGC as compared to the annealed coarse-grained samples. It was difficult
to separate the magnitude of the strengthening effect of the small grain size
from the weakening effect due to the bulk sample defects which are inherent to
the IGC synthesis method (mainly pores).
Compressive yield strength of nanocrystalline Cu and Pd synthesized by inert gas condensation method (from [27])
Sample # Compaction temperature (°C) Density (% theor.) Grain size (nm) σy (GPa) Hardness/3 (GPa) Pd1 335 98.5 54 1.15 1.0 Pd2 183 97.9 38 1.10–1.13 1.1 Pd3 RT 95.3 24 0.75 0.75 Cu1 106 92.5 19 0.65 0.77 Cu2 106 98.4 20 0.85 0.87
In the conventional grain size regime, usually a reduction in grain size leads to an increase in ductility. Thus one should expect a ductility increase as the grain size is reduced to nanoscale. However, the ductility is small for most grain sizes <25 nm for metals that in the conventional grain size have tensile ductilities of 40–60% elongation [77]. Koch [78] identified three major sources of limited ductility in nanocrystalline materials, namely: (1) artifacts from processing (e.g., pores); (2) tensile instability; (3) crack nucleation or shear instability. It is difficult to process nanostructured materials free from the artifacts that mask the inherent mechanical properties. As a result, molecular dynamics simulation has been considered to be a valuable tool in aiding our understanding of their deformation mechanism [79], [80], [81], [82], [83] and [84]. This is treated in greater detail in Section 9. The results of the atomistic simulations have allowed several investigators to suggest different plastic deformation mechanisms as a function of grain size [85] and [86]. There seems to be agreement in the existence of three regimes: (a) grain size d > 1 μm regime in which unit dislocations and work hardening control plasticity; (b) smallest grain size d < 10 nm regime, where limited intragranular dislocation activity occurs and grain-boundary shear is believed to be the mechanism of deformation. The intermediate grain size regime (10 nm–1 μm) is less well understood. This will be discussed in detail in Section 7. These mechanisms are thought to affect ductility significantly.
Fig. 13(a) shows data on normalized yield strength (strength/strength of conventional polycrystalline) versus percentage elongation in tension for metals with grain sizes in the nanocrystalline range. There is a clear decrease in ductility as strength is increased. By comparison, ultrafine grained materials (100–500 nm), Fig. 13(b), exhibit increased yield strength along with good ductility in comparison to nanograined materials.
Fig. 13. (a) Compilation of yield stress versus % elongation showing the reduced ductility of nanocrystalline metals [96]. (b) Compilation of yield strength versus % elongation of various ultrafine grained metals [96].
Zhang et al. [87],
[88]
and [89]
varied the microstructure of nanostructured/ultrafine grain size of Zn by changing
the milling times. A very dramatic modulated cyclic variation of hardness was
observed as a function of milling time at liquid nitrogen temperature. The
sample cryomilled for 4 h exhibited an optimum combination of strength and
ductility. The grain size distribution in this sample contained 30% volume
fraction of grains larger than 50 nm along with the smaller nano-scale
grains. This optimum microstructure, which exhibits more strain hardening than samples milled
for either shorter or longer time, combined the strengthening from the reduced
grain size along with the strain hardening provided by dislocation activity in
the larger grains. This strain hardening, in turn, provided ductility. Thus, a
bimodal grain size distribution is a possible means to increase ductility.
Nonequilibrium grain boundaries [90] have also been proposed as a mechanism to enhance ductility. It has been argued that such boundaries provide a large number of excess dislocations for slip [91] and can even enable grains to slide or rotate at room temperature, leading to a significant increase in the strain hardening exponent. These boundaries will be discussed further in Section 7.2. Another way of increasing ductility is to decrease the strain rate in order for the specimen to sustain more plastic strain prior to necking [92].
Fig. 14(a) [93] shows the expected ductility of metals as a function of a normalized strength (strength in the conventional grain size domain). As expected, as the strength increases, the ductility decreases. This defines the grey region. However, there are five data points above this boundary. They all apply to copper. Three factors contribute to and in fact determine ductility: the work hardening, the strain rate sensitivity and thermal softening. The increased ductility that is exhibited in some cases comes, basically, from the inhibition of shear localization. The strain rate sensitivity, m, can be expressed as [94]:
.gif){kind=small}
is indicative of a smaller activation
volume, as pointed out by Lu et al. [94].
This, in turn, is connected to a change in the nanostructure (presence of
twins). Thus, ductility, strain rate sensitivity, and deformation mechanisms are
connected and it is possible, through the manipulation of the nanostructure to
increase ductility. Zhu and Liao [93]
were able to increase ductility of their nanocrystalline metals by increasing
significantly the density of growth (annealing) twins. This will be discussed in
Section 7.7.2.
Fig. 14. (a) Increased ductility in the nanocrystalline regime as indicated by experimental points in right-hand side of diagram [93]; (b) reduction of ductility as grain size is reduced for ball milled Zn tested at a constant strain rate of 10−4–10−3 s−1 at room temperature [87].
Fig. 14(b) shows the mechanical response of nanocrystalline zinc samples with different grain sizes. There is a significant drop in ductility as the grain size goes down from 238 nm to 23 nm. Zhang et al. [120] suggested that the reduction of elongation with the reduction of grain size could be an inherent property of nanocrystalline materials given that there is no porosity and bonding was complete during synthesis. Earlier results have shown that the mechanical properties of nanocrystalline materials can be misinterpreted because of the lack of attention to the details of the internal structure [62]. As mentioned earlier, contaminates and porosity are found to be extremely detrimental to ductility.
Table 3 gives a partial list of publications on the phenomenon of inverse Hall–Petch. For ease of reading, the H–P plots in this section are expressed in (nanometers)−1/2 and for rapid conversion into linear dimensions, we provide the conversion chart of Table 4. It should be noted that the entire conventional (microcrystalline) range (d > 1 μm) corresponds to d−1/2 < 0.031 nm−1/2.
Partial list of papers on inverse Hall–Petch relationship [only first author named]
Author Year Title Chokshi [97] 1989 On the validity of the Hall–Petch relationship in nanocrystalline materials Fougere [355] 1992 Grain-size dependent hardening and softening of nanocrystalline Cu and Pd Lu [356] 1993 An explanation to the abnormal Hall–Petch relation in nanocrystalline materials Malygin 1995 Breakdown of the Hall–Petch law in micro- and nanocrystalline materials Konstantinidis [252] 1998 On the “anomalous” hardness of nanocrystalline materials Song [357] 1999 A coherent polycrystal model for the inverse Hall–Petch relation in nanocrystalline materials Schiotz 1999 Softening of nanocrystalline metals at very small grain sizes Chattopadhyay [358] 2000 On the inverse Hall–Petch relationship in nanocrystalline materials Conrad [250] 2000 On the grain size softening in nanocrystalline materials Takeuchi [359] 2001 The mechanism of the inverse Hall–Petch relation of nanocrystals Wolf 2003 Deformation mechanism and inverse Hall–Petch behavior in nanocrystalline materials
Table 4.Conversion chart as an aid for reading H–P plots
d−1/2 (nm−1/2) d (nm) 0.025 1600 0.031 1000 (microcrystalline limit) 0.05 400 0.1 100 0.2 25 0.32 10
The Hall–Petch relationship predicts that the yield stress increases with the
inverse of the square root of the grain size (Eq. (3)).
However, experimental results on materials reveal that the Hall–Petch
relationship recorded at large grain sizes cannot be extrapolated to grain sizes
of less than 1 μm. Fig.
15 shows the Hall–Petch plot for Cu taken from different sources. As can be
clearly seen, there is ambiguity in the trend of the plot as the grain size
falls down to a value below 25 nm (d−1/2 = 0.2). While some
results predict a plateau, others show a decrease. The Hall–Petch trend for
different nanocrystalline samples crystallized from amorphous solids is plotted
in Fig.
16(a). Again, the two trends are seen: the formation of a plateau and a
decrease in σy as d is decreased below 25 nm. One
simple rationalization for this behavior is provided by Fig.
16(b) for pure Ni and Ni–P alloy [95].
The curve was extended all the way to the amorphous limit, which corresponds to
a hardness of 6 GPa. It is evident that this is the correct approach: the
amorphous state is the lower limit of the nanocrystalline state. The plot shows
a slight decrease. The breakdown in the Hall–Petch trend has been attributed to
different deformation mechanisms that become dominant once the grain size is
reduced down below a critical value [96].
Fig. 15. Compiled yield stress versus grain size plot for Cu from various sources ranging from coarse to nanograin size. The plots show different trend as the grain size falls below a critical size.
(51K)
Fig. 16. (a) Hall–Petch plots for different nanocrystalline samples crystallized from amorphous solids [39]; (b) Hardness as a function of grain size for pure Ni and Ni–P alloy going all the way to amorphous limit [95].
Chokshi et al. [97] were the first to report the negative Hall–Petch effect by performing measurements on nanocrystalline Cu and Pd samples made by IGC. Both metals exhibited a negative slope, shown in Fig. 17(a). This landmark paper has received close to 300 citations. They attributed this negative trend to diffusional creep in nanocrystalline samples at room temperature analogous to grain-boundary sliding in conventionally-grained samples at high temperature. There have been reports of a similar trend in the Hall–Petch relationship from other sources [98], [99], [100], [101] and [104]. Fig. 17(b) shows, in contrast, results obtained by Weertman [102] which do not show this trend through hardness measurements, although they show it in tensile results. The decrease in σy at smaller grain sizes was attributed by Weertman [102] to the presence of flaws. In their synthesis technique, inert gas condensation method was used, followed by ambient temperature densification through uniaxial pressing. The data points are also later shown in Fig. 46(b), where they are discussed in connection to the core-and-mantle mechanism. Weertman [102] suggested that the negative slope obtained by Chokshi et al. [97] was due to the use of a single sample subjected to repeat anneals to change the grain size. Thus, it was a heat treatment artifact.
Fig. 17. (a) Inverse Hall Petch trend for Cu and Pd as shown by Chokshi et al. [97]. (b) Positive Hall–Petch slope with higher values for compressive (from hardness measurements) than for tensile strengths [27].
Chokshi et al. [97] argued that the negative slope for nanocrystalline copper arose from the occurrence of rapid diffusion creep at room temperature. Coble creep was considered as the deformation mechanism,
.gif){kind=small}
is the strain rate, Ω is the atomic
volume, δ is the grain-boundary width, Dgb is the
grain-boundary diffusion, σ is the stress, k is Boltzmann’s
constant and T is the absolute temperature. Chokshi et al. [97]
assumed:10−3 s−1) corresponds to grain sizes around
20 nm. These points are marked in plots. It has to be noted that the role
of plastic deformation is ignored in the Chokshi et al. [97]
analysis. However, plastic deformation is required for the grains to slide past
each other. This plastic accommodation has been treated by Fu et al. [103]
and this will be discussed later (Section 7.2).
Fig. 18. Log–log plot of strain rate versus grain size for stresses of (a) 100 MPa and (b) 1000 MPa for Coble creep as used by Chokshi et al. [97].
The Hall–Petch trends for a range of grain sizes from the micro to the
nanocrystalline are plotted in Fig.
19 for four different metals: Cu, Fe, Ni and Ti. Data points have been
collected from different sources for grain sizes ranging from micrometer to
nanometer range. Note that the data points in the conventional polycrystalline
range for most of these plots overlap while they are more spread out in the
nanocrystalline range. The Hall–Petch curve for the nanocrystalline range
clearly shows a deviation from the regular trend in the microcrystalline range;
there is a significant decrease in the slope for small grain sizes. However,
there is no clear evidence on the nature of the curves at grain sizes below 10–15 nm. Though researchers have debated the existence of the
negative Hall Petch effect, there is insufficient information to validate the
existence of this effect. The most probable behavior is that the yield strength
plateaus below a critical grain size. The real trend is still to be determined
along with the knowledge of whether it varies for different materials.
Fig. 19. Plots showing the trend of yield stress with grain size for different metals as compared to the conventional Hall–Petch response: (a) copper, (b) iron, (c) nickel and (d) titanium.
Nanocrystalline and ultrafine grained materials cannot generally sustain
uniform tensile elongation. Several reports show virtually no strain hardening
after an initial stage of rapid strain hardening over a small plastic strain
regime (1–3%) which is different from the response of coarse grained
polycrystalline metals [77],
[78],
[79],
[105]
and [106].
The density of dislocations in a nanocrystalline sample saturates due to
dynamic recovery or due to the annihilation of dislocations into the grain
boundaries. This is discussed in Section 7
and leads to a low strain hardening rate. It is only during large additional
strain that work hardening is observed. Dynamic recovery is known to occur
during severe plastic deformation [56],
[107]
and [108].
Due to the rise in the temperature, recovery converts the deformed microstructure into ultrafine grains having both low-angle and high-angle
grain boundaries. Low strain hardening behavior has been observed for samples
processed by both equal angular channel pressing and powder consolidation [107].
Fig.
20 shows the stress–strain curves in compression and tension for UFG copper
produced by ECAP (8 passes). The work hardening (in compression) is virtually
absent. This leads to necking at the yield stress (in tension), and the net
results is a low tensile ductility. Such a trend has also been observed for ECAP
Cu [108].
Fig. 20. Compressive and tensile stress–strain curves (tensile:engineering) for copper subjected to ECAP.
The stress–strain response of a nanocrystalline metal, e.g., copper, under tension shows a rapid peak and subsequent softening due largely to necking. The absence of strain hardening (dσ/dε = 0) causes localized deformation leading to low ductility. Flat compression curves have also been observed for other nanocrystalline metals including Fe (BCC) [109] and Ti (HCP) [107]. Necking is observed in most cases with the severe case of instability, and shear bands form in the consolidated Fe [109] and [110].
Room temperature dynamic recovery is common in nanocrystalline samples. Competition between the generation of dislocations during plastic deformation and the annihilation during recovery determines the steady state dislocation density. Though still debatable, the dislocations during deformation are thought to be generated at the grain boundaries which also act like sinks. It is, however, possible that dislocations are still the carriers of plastic deformation. As can be expected, the role of dislocations in the deformation process is difficult to determine since they arise and disappear in the large volume fraction of grain boundaries available. Other deformation mechanisms are not expected to be dominant until the grain size is in the range of 10–15 nm [80] and [111]. These mechanisms are discussed in detail in Section 7.
There have been reports of both increased and decreased strain rate
sensitivity with decreasing grain size in metals. Iron, which is normally rate
sensitive, with a strain rate exponent m (defined as .gif){kind=small}
or .gif){kind=small}
) hardening exponent on the order of 0.04,
goes down in the value to 0.004 when the grain size is 80 nm. Malow et al.
[112]
prepared nanocrystalline Fe using ball milling and consolidation, and found a
low m 0.006 at d 20 nm.
An opposite effect was found by Gray et al. [113]
on ultrafine grained FCC metals produced by ECAP: Cu, Ni and Al–4Cu–0.5Zr. Their
mechanical response was found to depend on the applied strain rate, which ranged
from 0.001 to 4000 s−1 as shown in Fig.
21. The strain rate sensitivity, m, based on the above strain rate
range, was measured to be 0.015 for Cu, 0.006 for Ni and 0.005 for Al–4Cu–0.5
Zr. Gray et al. [113]
attributed the relatively high rate sensitivity, coupled with the nearly zero
post-yield work-hardening rates in the ultrafine grained Cu and Ni, to a high
pre-existing dislocation density.
Fig. 21. Stress–strain response of ultrafine grained: (a) Cu, (b) Ni [113].
Higher values of m have also been reported for nanocrystalline Ni [114],
nanocrystalline Au [115],
and electrodeposited Cu [116].
However, these observations were only made in creep tests where grain-boundary
deformation is dominant [114].
In normal quasistatic tensile tests, none of these nanocrystalline metals
exhibited a high m. These small values of m are insufficient to
keep material deformation from shear localizing, which is similar to the
response of amorphous alloys [106].
A Newtonian viscous metal, which does not localize, has m 1.
Results by Wei et al. [369] show in a clear fashion that the strain rate sensitivity is increased at grain sizes below a critical value. These results are shown in Fig. 22. This enhanced strain rate sensitivity has also been measured through nanoindentation hardness tests. Results by Göken and coworkers [118] are shown in Fig. 23. The strain-rate sensitivity observed in pulse electrodeposited Ni remains about the same down to d = 60 nm (Fig. 23(b)). However, the slope for the d = 20 nm specimens is higher. A similar effect is observed for aluminum produced by ECAP. The ultrafine grain sized Al has a slope m = 0.027, whereas the strain-rate sensitivity for conventional polycrystalline Al is less than one third of this (m = 0.007) [117]. The increased strain rate sensitivity is directly related, Fig. 23(a) to a change in the rate controlling mechanism for plastic deformation. It can be seen through Eq. (4) that mαV−1, i.e., m is inversely proportional to the activation energy. Conventional FCC metals have a large activation volume, V:
2 passes (γ 2): m = 0.06
16 passes (γ 16): m = 0.14
Fig. 22. Strain rate sensitivity plot for Cu as a function of grain size [113] and [369].
(49K)
Fig. 23. (a) Comparison of strain rate sensitivity (from hardness measurements) for conventional and ultrafine grain sized aluminum. (Courtesy M Göken, Univ Erlangen-Nürnberg, Germany.) (b) Strain rate sensitivity plot (from hardness measurement) for pulse electrodeposited Ni. (Courtesy M Göken, Univ Erlangen-Nürnberg, Germany.)
The effects of temperature, strain rate and grain size on the flow behavior of Zn (representative HCP metal) have been studied to reveal the deformation mechanisms in UFG and nanocrystalline HCP structured metals [120]. Tensile test results for 3 h ball-milled Zn samples tested at different temperatures, but the same strain rate (10−4 s−1), are shown in Fig. 24. The grain size after 3 h ball milling was 238 nm. The yield stress (σy), as well as the strain (ε) to failure decreased with an increase of test temperature. The strain hardening under quasistatic conditions in samples tested at different temperatures was low [120]; at 200 °C, it ceased to exist. By increasing the ball milling time, the grain size was progressively decreased. This, on its turn, leads to an increase in yield stress and a decrease in work hardening, as seen earlier in Fig. 14(b). Jump tests (strain rate changes by factor of 2) also were performed at 20, 40, and 60 °C on the ball milled Zn samples. The results are shown in Fig. 25. The calculated m values were 0.15 for tests at 20 and 40 °C and about 0.17 for test at 60 °C. As can be noticed, these values are significantly higher compared to the value of m for ultrafine Cu and Ni discussed earlier. However, one should bear in mind that Zn is HCP, while Cu, Ni and Al are FCC.
Fig. 24. Tensile stress–strain curve for ball milled (3 h) Zn tested at 20, 40 and 200 °C at a constant strain rate of 10−4 s−1 [87].
(25K)
Fig. 25. Strain-rate jump tests (compression) performed on ball milled (3 h) Zn at 20 and 60 °C [87].
Nanocrystalline iron has been the focus of widespread research primarily to understand the mechanism of shear band formation [110] and [121]. Fig. 26 shows the true stress–strain curves of the consolidated iron with various average grain sizes, obtained from quasistatic and dynamic compression (split Hopkinson bar) tests [121]. The plot suggests that strain hardening decreases with decreasing grain size which is surprising in the polycrystalline regime but expected in the nanocrystalline regime. As seen earlier, this is probably due to a change in the deformation mechanism at smaller grain sizes. On the other hand, the reported influence of strain rate on strain hardening is insignificant, which is also typical of BCC metals. The calculated strain rate sensitivity values m are: 0.009 for grain size of 80 nm, 0.012 for grain size of 138 nm, 0.023 for grain size of 268 nm, 0.045 for grain size of 980 nm and 0.08 for grain size of 20 μm. In BCC metals, the activation volume for plastic deformation is much lower than FCC metals and therefore the change in rate controlling mechanism is not expected to occur.
Fig. 26. Typical stress–strain curves obtained for consolidated iron under quasistatic and high-strain rate uniaxial compression for different grain sizes [121].
Jia et al. [121] used a physically-based constitutive model that describes the rate-dependent behavior of BCC iron over the entire range of strain rates from 10−4 to 104 s−1:
The model represents the application of the MTS constitutive equation by Kocks et al. [122]. The first three terms in Eq. (7) are the athermal components of the stress:
The last term is the thermal component.
The strain hardening function g(γ) is included in the athermal part because experimental evidence indicates that the strain hardening is not affected by strain rate or temperature. However, this is not typical of BCC metals, as shown by Zerilli and Armstrong [123].
Creep in coarse grained materials has been widely studied for approximately one century and accurate models exist that capture deformation features and explain mechanisms involved therein. Creep in nanocrystalline materials has been studied only in recent years owing to several complications involved. First of all is the limitation of synthesizing bulk nanomaterial free of defects (porosity and impurities) with uniform grain size distribution that could provide reliable data to explain the deformation process. Second is the significant increase in the volume fraction of grain boundaries and intercrystalline defects such as triple lines and quadruple junctions that renders the creep mechanism complicated and leads to associated challenges in developing a model that could explain the deformation process. Third, grain growth occurs at much lower temperature as compared to coarse-grained materials limiting the testing temperatures to a low fraction of the melting point.
Since the volume fraction of grain boundaries is high, diffusion creep is considered to be significant. The high temperature deformation of crystalline materials is given by the following (Bird–Dorn–Mukherjee) equation:
.gif){kind=small}
is the strain rate, A is a
dimensionless constant, G is the shear modulus, b is the magnitude
of the Burgers vector, k is Boltzmann’s constant, T is the
absolute temperature, p is the inverse grain size exponent, and n
is the stress exponent. Among the established diffusional creep mechanisms in
coarse grained materials are the Nabarro–Herring creep that involves vacancy
flow through the lattice and Coble creep that involves vacancy flow along the
grain boundaries. The associated equations are:Palumbo et al. [124]
considered a regular 14-sided tetrakaidecahedron as the grain shape to estimate
total intercrystalline component and showed that it increases from a value of
0.3% at a grain size of 1 μm, to a maximum value of 87.5% at a
2 nm grain size (see Fig.
2). In accessing the individual elements of the intercrystalline fraction,
it was noted that the triple junction volume fraction displays greater grain
size dependence than grain boundary. Wang et al. [125]
modified the standard diffusion creep equation to accommodate for diffusion
along triple lines and this leads to the following expression for triple-line
diffusion creep:
3, p = 0 and
D = DL in the Bird–Mukherjee–Dorn equation.
Among the first reports on creep of nanocrystalline materials are the ones by Wang et al. [127] on 28 nm grain size Ni–P alloy (Fig. 27(a)), Deng et al. [128] on the same alloy, and Nieman et al. [129] on nanocrystalline Pd. In Fig. 27(a), one can see that the slope of the plot, n = 1, supports either Coble or Nabarro–Herring creep (Eqs. (10) and (11)). Wang et al. [127] attributed the creep response to grain-boundary diffusion and in their following work [128] concluded that while grain-boundary diffusion is the operating mechanism in nanocrystalline creep; a combined mechanism involving dislocation creep and grain-boundary diffusion governs deformation in coarse grained materials.
Fig. 27. Stress versus strain rate plots for (a) nanocrystalline (d = 28 nm) Ni–P [127] and (b) nanocrystalline TiO2 [130].
Hahn et al. [130] performed tests on compressive creep response of TiO2. The value of n obtained from their results is shown from the slope in Fig. 27(b): n = 2. Nieman et al. [129] reported no significant room temperature creep for nanocrystalline Pd under loads much larger than the yield stress of a coarse-grained Pd sample. They concluded that grain-boundary diffusional creep is not an appreciable factor in directly determining room temperature mechanical behavior in nanocrystalline Pd. Sanders et al. [131] carried out creep tests over a range of temperatures (0.24–0.64Tm) and stresses on samples of nanocrystalline Cu, Pd, and Al–Zr made by inert gas condensation and compaction. The experimentally observed creep rates were two to four orders of magnitude smaller than the values predicted by the equation for Coble creep. The predicted creep rates as a function of temperature for different grain sizes are shown in Fig. 28. The figure shows calculated creep curves assuming grain-boundary diffusion for different grain sizes (notice increase in strain rate by six orders of magnitude when grain size is decreased from 1 μm to 10 nm). Sanders et al. [131] concluded that prevalence of low-energy grain boundaries together with inhibition of dislocation activity caused by small grain sizes is responsible for low strain rates and higher than expected creep resistance.
Fig. 28. Calculated creep curves assuming grain-boundary diffusion for different grain sizes; notice increase in strain rate by six orders of magnitude when grain size is decreased from 1 μm to 10 nm [131].
Ogino [132]
compared the lower (by several orders of magnitude) strain rates experimentally
observed to the theoretically calculated values from the theory of Coble creep.
He attributed the effect to an increase in grain-boundary area due to
deformation of grains at an initial stage of creep. Wang et al. [133]
studied the effect of grain size on the steady state creep rate of
nanocrystalline pure nickel that was synthesized by electrodeposition. They
argued that at high stress levels, grain-boundary sliding becomes the major
deformation mechanism at room temperature. The contribution from diffusion creep
mechanisms through intercrystalline regions can be significant for smaller grain
sizes. It was suggested that dynamic creep should be taken into account when
analyzing the stress–strain curves of nanocrystalline materials. Cai et al. [134]
conducted tensile creep of nanocrystalline pure Cu with an average grain size of
30 nm prepared by electrodeposition in the temperature range 20–50 °C
(0.22–0.24Tm). The steady state creep rate was found to be
proportional to an effective stress
σe = σ − σ0,
where σ is the applied stress, and σ0 is the threshold
stress. Using σe, the creep rates were found to be of the same
order of magnitude as those calculated from the equation for Coble creep (Eq. (11)).
The existence of a threshold stress implied that the grain boundaries do not act
as perfect sources and sinks of atoms (or vacancies). The mechanism of creep was
identified as ‘interface controlled diffusional creep’. Li et al. [135]
refuted the result of Cai et al. [134]
by attributing the high value of minimum creep rate (that led to the proposal of
Coble creep) to premature fracture. Grabovetskaya et al. [136],
using nanostructured copper, nickel, and Cu–Al2O3
composite as examples, studied characteristic features of creep of
nanostructured materials produced by severe plastic deformation at temperatures
T < 0.3Tm. They concluded that the
steady-state creep rate of nanostructured copper and nickel is well described by
a power law where the exponent n is equal to 5.5 for copper and 8 for nickel. They postulated the development of meso- and macrobands
of localized deformation during deformation of fine grained materials. They also
estimated the apparent creep activation energies of nanostructured copper and
nickel in the temperature interval of 0.2–0.3Tm to be 2.5
times lower than the coarse-grained counterpart. Using direct experimental
methods, they showed that this difference is due to the significant
contributions to the overall deformation of grain-boundary sliding controlled by
grain-boundary diffusion.
There have also been attempts to study creep in nanostrutured materials closer to the real situation. Estrin et al. [137] studied diffusion controlled creep in nanocrystalline materials under conditions of concurrent grain growth. The Nabarro–Herring and Coble mechanisms were modified to account for the effect of attendant vacancy generation on creep.
Several models for creep in nanocrystalline materials have been proposed
based on molecular-dynamics simulations. Yamakov et al. [138]
simulated fully three-dimensional, nanocrystalline face-centered cubic metal microstructures to study grain-boundary (GB) diffusion creep. The
simulations were performed at elevated temperatures where the distinct effect of
GB diffusion is clearly identifiable. In order to prevent concurrent grain
growth and thus to enable steady-state diffusion creep, the microstructure was tailored to have uniform grain shape (nm size) and to
contain only high-energy grain boundaries. Results indicated that under
relatively high tensile stresses these microstructures exhibited steady-state diffusion creep that is
homogeneous, with a strain rate that agrees with that given by the Coble creep
equation. The grain size scaling of the creep was found to decrease from
d−3 (typical of Coble creep) to d−2 (typical
of Nabarro–Herring creep) when the grain diameter was of the order of the
grain-boundary thickness. Direct observation of grain-boundary sliding as an
accommodation mechanism for the Coble creep, known as Lifshitz sliding, was also
reported. Haslam et al. [139]
in their MD simulation accounted for the effect of concurrent grain growth on
grain-boundary diffusion creep (like the study by Estrin et al. [137])
and grain-boundary sliding during high-temperature deformation of a
nanocrystalline Pd model microstructure. Prior to the onset of significant grain growth, the
deformation was shown to proceed via the mechanism of Coble creep accompanied by
grain-boundary sliding. While grain growth is generally known to decrease the
creep rate due to the increase of the average grain size, the results obtained
in this study revealed an enhanced creep rate at the onset of the grain growth,
when rapid boundary migration occurs. The enhanced creep rate was shown to arise
from topological changes during the initial growth phases, which enhanced both
the stress-induced grain-boundary diffusion fluxes and grain-boundary sliding.
Dislocations generated as a result of grain-rotation-induced grain coalescence
and grain-boundary decomposition in the vicinity of certain triple junctions
were also shown to contribute to the deformation.
In recent work, experimentally measured grain size compensated diffusion creep rates were shown to be identical in cubic, tetragonal and monoclinic zirconia by Chokshi [140], suggesting a similarity in the absolute magnitudes of their grain-boundary diffusion coefficients. Grain growth in tetragonal zirconia was shown to be substantially slower due to significant grain-boundary segregation.
The recent creep test results by Yin et al. [141] showed that both minimum creep rate and creep strain significantly decrease with increasing sulfur or by doping nanostructured nickel with boron. The stress exponent, n in the expression of Coble-type creep increased to around five at 373 K and 473 K from two at room temperature. A model for grain-boundary sliding, in which grain-boundary dislocations and back stress are introduced, was proposed to explain the large stress exponent. The calculated back stress indicated that the interstitials in grain boundaries effectively retard the sliding of grain-boundary dislocations.
There have not been many reports on the fatigue properties of nanocrystalline materials. Among the earliest study is Whitney et al. [142] on tension–tension cycling of nanocrystalline copper prepared by inert gas condensation, with a maximum stress that ranged from 50% to 80% of the yield stress. The minimum stress was 10 MPa. After several hundred thousand cycles, a moderate increase in grain size was observed (approximately 30%). The samples were shown to elongate slightly in the course of a prolonged fatigue test. The amount of strain is similar to the room temperature creep strain observed previously in nanocrystalline copper under a constant stress comparable to the maximum cyclic stress. The cyclic deformation appeared to be elastic. However, an elastic modulus was measured that is a factor of 2 smaller than the modulus for ordinary copper.
Yan et al. [143] performed tensile fatigue tests of nanosized yttria stabilized zirconia with average grain size of 100 ± 20 nm. Samples of the same chemical composition with submicron grains were simultaneously tested for comparison. It was documented by AFM imaging that localized superplastic deformation of the 100 nm grains at and near the fatigue fracture surfaces was generated while in the submicron grain-sized samples, the grains retained their original equiaxed morphology. The micromechanism responsible for the above mentioned phenomenon was thought to be essentially governed by grain-boundary diffusion. It was also suggested that a minor role might be played by dislocation climb and multiplication.
Vinogradov et al. [144] investigated cyclic behavior of commercial purity ultrafine-grained titanium obtained by severe plastic deformation through equal channel angular pressing (ECAP). It was shown that fine grained Ti processed by ECAP revealed considerable increase in fatigue life and fatigue limit under constant load testing when compared with those in the coarse-grain state. The S–N (Wöehler) plot for ECAP Ti is shown in Fig. 29. Contrary to typical wavy-slip materials such as copper and Al-alloys, which show significant degradation in strain-controlled cyclic properties as suggested by Pelloux [145], it was shown that Ti did not demonstrate any reduction in its fatigue performance under constant plastic strain cyclic testing as is evidenced by nearly the same Coffin–Manson behavior in the fine-coarse grain size conditions. It was concluded that both grain-refinement and the work hardening due to the increase of the average dislocation density play important role in resultant properties of materials obtained by severe plastic deformation.
Fig. 29. S–N diagram of ECAP titanium (inset box shows SN diagram for conventional polycrystalline Ti with grain sizes d = 9, 32, and 100 μm [144]).
Patlan et al. [146] and [147] studied the fatigue response of 5056 Al–Mg alloy and reported that while a certain enhancement of fatigue performance was observed [148] in the high-stress regime if compared with the conventional O-temper material, this advantage disappeared during the plastic strain controlled tests in the low-cyclic regime. They pointed out that the susceptibility of the ECAP materials to strain localization can be a main factor which limits their tensile and fatigue ductility and determined their fatigue weakness in comparison with conventional materials. The suggestion was made that a short annealing at moderate temperatures may dramatically improve the ductility of the ECAP Al alloys mainly due to partial recovery of grain boundaries. They reported that annealing at 150 °C for 15 min resulted in considerable improvement of low-cyclic fatigue life at constant plastic strain amplitude.
Chung et al. [149]
studied the mechanical properties and fatigue behavior of solid solution treated
6061 Al alloy processed by ECAP. They reported a remarkable enhancement in
fatigue life, by a factor of about 10, compared to a T6 treated commercial 6061
Al alloy. This occurred in both low- and high-cycle regimes after a single ECAP
pass. Further deformation by ECAP, however, was reported to virtually eliminate
this improvement especially in the high cycle regime. A fine-grained microstructure with low-grain boundary misorientation angles, typical
after one ECAP pass, was proposed to yield the best result in the fatigue
performance in a 6061 Al alloy. It was pointed out that one needs to pay
attention to a single pass material rather than multi-passed material if
improvement in fatigue life is a primary concern for engineering applications.
Kim et al. [150] tested ultrafine grained low carbon (0.15 wt.% C) steel processed by ECAP for fatigue properties, including cyclic softening and crack growth rate. ECAP steel was shown to exhibit cyclic softening. After the first cycle, tension and compression peak stresses were shown to decrease gradually with number of cycles. The ECAPed steel was shown to exhibit slightly higher crack growth rates and a lower ΔKth with an increase in R ratio. This was attributed to a less tortuous crack path. Kim et al. [151] studied the effect of grain size (varying in the range of 1–47 μm) on the fatigue behavior of AISI 304 SS and showed that a beneficial effect on the fatigue behavior of the steel is obtained by decreasing grain size.
Fatigue strength of two ultrafine grained steels (grain sizes 0.8 and 1 μm) was investigated by Chapetti et al. [153] who showed that important improvement on fatigue strength was observed in ultrafine grained steels when compared with similar steels of coarser grains. The results were shown to obey the Hall–Petch relation observed between smooth fatigue limit and grain size d.
Chapetti et al. [157] showed for ultrafine grained steel (grain size 1 μm) that the threshold for fatigue crack propagation is relatively low which is in good agreement with previous evidence. However, it was shown that when compared at medium and high applied stress intensity range level and the same applied mean stress, the ultrafine grained steel showed a lower crack propagation rate than a SM 490 steel or a HT 80 steel. Lukas et al. [158] studied the fatigue notch sensitivity of ultrafine grained copper of purity 99.9% produced by ECAP as cylindrical specimens with circumferential notches of different radii and compared results with the notch sensitivity of conventional copper. It was demonstrated that the fatigue notch sensitivity of ultrafine grained copper produced by ECAP technique is higher than that of standard polycrystalline copper. The ECAP copper of 99.9% purity was shown not to be prone to grain coarsening during cycling. The grain structure within plastic zone around the cracks was shown to differ substantially from the outside the plastic zone: the grains were found markedly elongated, but their size was shown to be preserved.
Mughrabi et al. [154]
studied the cyclic deformation and fatigue behavior of ultrafine-grained metals
produced by ECAP to determine cyclic softening, microscopic shear banding and
fatigue lives. Low cycle (strain amplitude 10−3–10−2) and high cycle (strain amplitude 10−4) tensile microyielding, unloading and reloading tests
were carried out on high-purity UFG copper and on commercial purity UFG
aluminum. For copper, in the as-ECAP processed condition, the unloading and
reloading stress–strain curves for the ultrafine grained sample were reported to
be highly non-linear with an appreciable microplastic back–forward flow during
unloading/reloading, whereas the coarse grained material was shown to exhibit an
essentially linear elastic unloading/reloading stress–strain behavior. It was
concluded that the non-linear unloading/reloading behavior is characteristic of
ultrafine grained materials and that, in the ultrafine grained materials, much
larger internal back stresses are induced during straining than in coarse
grained material. The low cycle fatigue behavior was found to be worse than
coarse grained material while the high cycle fatigue behavior was reported to
show remarkable improvement over coarse grained sample. After an optimized
annealing treatment (which probably leads to a bimodal distribution of grain
size) ultrafine grained copper showed a markedly enhanced low cycle fatigue
life; in fact, better than coarse grained copper. Fig.
30 shows the fatigue lives of ultrafine grain copper, plotted as an
S–N Woehler plot (stress amplitude Δσ/2 versus
log Nf). Ding et al. [155]
viewed the ultrafine grained materials as a composite with soft grain interiors
and hard grain boundaries and considered the resistance of this microstructure to fatigue crack propagation. An important aspect of the
model is the incorporation of appropriate cyclic stress–strain
curve—“saturation” stress versus (Δεpl/2). The model provided
a good fit to experimental data.
Fig. 30. S–N (Woehler) plot of fatigue life of ultrafine grain copper, ECAP copper and coarse grained copper [154].
Kim et al. [156] examined the fatigue properties of fine-grained AZ31 magnesium alloy produced by ECAP and reported a reduction of fatigue life by more than a factor of 10 in both low- and high-cycle fatigue region compared to the unECAPed sample. The ECAPed sample was shown to demonstrate 13% reduction in fatigue endurance life at N = 107. The ultimate tensile and yield strengths of the ECAPed samples were demonstrated to be lower than those of the unECAPed samples and comparison of fatigue behavior between the two conditions indicated that the fatigue life has a correlation with the yield strength in AZ 31 alloy. Unlike earlier reports on fatigue of nanocrystalline materials, AZ alloy with fine grains was shown to exhibit higher crack threshold and lower crack growth rate. It was suggested that enhanced ductility due to grain refinement in the ECAPed AZ 31 alloy resulted in increasing crack growth resistance because of its better ability to accommodate plastic strains during cycling.
Hanlon et al. [152] investigated the fatigue response of electrodeposited nanocrystalline pure Ni and a cryomilled ultrafine-crystalline Al–Mg alloy. They showed that grain refinement can have a substantial effect on total life under stress-controlled fatigue and on fatigue crack growth. Fig. 31 shows the da/dN versus ΔK curve for the ultrafine grained Al–Mg alloy tested at R = 0.1−0.5 over the entire range of fatigue crack growth rates. This is compared with the crack growth data for commercial 5083 aluminum alloy at R = 0.33. Fully dense nanocrystalline and ultrafine grained Ni produced by electrodeposition exhibited much higher resistance to stress-controlled fatigue compared to conventional micro-crystalline Ni. However, fatigue growth rate results showed that grain refinement in the nanocrystalline range had a deleterious effect on the resistance to subcritical fatigue fracture. These results were corroborated by results on cryomilled Al–7.5Mg alloy.
Fig. 31. Variation of fatigue crack growth rate, da/dN, as a function of stress intensity factor range, ΔK, for cryomilled Al–7.5Mg at R = 0.1–0.5 at a fatigue frequency of 10 Hz at room temperature [152].
As a conclusion, it can be stated that there is fair agreement among the different results. The fatigue life, as measured by S–N (Wöehler) type plots, is enhanced for nanocrystalline metals by virtue of their higher yield stress. On the other hand, ΔKth decreased and da/dN is increased for nanocrystalline metals. This latter effect is attributed to the smoother fracture path in nanocrystals.
There is a wealth of information available on nanostructured ceramics and composites primarily focusing on different synthesis techniques for specific composition and structure driven by the fact that the nanostructured ceramics could demonstrate improved properties as compared to their micro/coarse grained counterparts. Table 5 lists publications on this topic classified according to the composition. The most often researched compositions are ZrO2, TiO2, BaTiO3, and SiC. Superhard coatings made of nanocomposites have also been widely researched. The article by Veprek [337] reviews the subject. In this section we will attempt to capture the advances that have taken place in the field of mechanical properties of nanoceramics and nanocomposites over the last few years built on the understanding developed over years of research in this field.
Partial list of papers on synthesis techniques of common nanocrystalline ceramics
Material First author Year Synthesis technique Grain size (nm) ZrO2 Thangadurai [169] 2004 Chemical co-precipitation 10–13 Zhang [166] 2004 Agglomeration 12 Ramaswamy [174] 2004 Amorphous citrate route 8 Feng [180] 2004 Reduction of ZrCl2 by Na 17 Luscalea [187] 2004 pH controlled nitrate–glycene gel- 5–8 Frolova [188] 2003 Combustion – Robert [203] 2001 Co-precipitation 10–20 Juarez [211] 2000 Pechini process 10 Popp [219] 1998 Nitrate–citrate combustion route 10–100 Lamas [220] 1998 Evaporation of solid mater with CO2 laser 12–13 Lamas [221] 1998 Nitrate–citrate gel route 10–20 Duran [225] 1996 Pyrolysis-nitrate sol. with citric 9 Qui [226] 1995 Acid/glycine 200 Demyanets [227] 1995 Co-precipitation 15 n-Butanol azeotropic distillation Chemical precipitation TiO2 Li [161] 2005 Sol–gel <60 Duan [159] 2004 High pressure sintering <100 Dhage [175] 2004 Gel conversion 10 Lee [192] 2003 Spark plasma sintering >1000 Liao [214] 1999 High P and low T sintering 38 Bykov [228] 1995 Sintering 15 BaTiO3 Buscaglia [162] 2004 Spark sintering 40–60 Luan [183] 2004 Spark plasma technique 13 Kim [186] 2004 Two-step sintering 1000 Ragulya [218] 1998 Non-isothermal and rate controlled decomposition of organic precursors 20–25 Mn3O4 Anilkumar [164] 2005 Gel to crystal conversion 50 Chang [165] 2004 Chemical bath deposition – Zhang [171] 2004 Solvothermal technique – Finocchio [167] 2004 Co-precipitation – SiC Cheng [193] 2005 CVD and liquid precursor infiltration to prepare – Zhang [224] 1996 Nanosize tubules followed by dissolution of alumina in HF 50 CeO2 Zhang [166] 2004 Agglomeration 18 Markmann [198] 2002 Yttrium doping and colloidal processing – MgO Ehre [168] 2004 Hot-pressing 10 PbTiO3 Forrester [172] 2004 Mechanical alloying/sintering 20 Lemos [177] 2004 Pechini method 100 YAG Pradhan [185] 2004 Co-precipitation 5–7 Lu [202] 2002 From aq. sol. of Al, Yt and NyCl2 Perovskite Xue [195] 2002 Mechanically activating oxides – BaSnO3 Lu [163] 2005 Hydrothermal reaction and crystallization 27.6 Hydroxyapa Han [182] 2004 Citric acid sol–gel combustion – TiB2 Gu [196] 2003 Benzene-thermal reaction of Na with B powder and TiCl4 15–40 TiC Xinkun 2001 Mechanical alloying 7 Meyers [362] 1995 Spark erosion 5–50
There has been extensive effort to fabricate nanostructured ceramics. Methods such as hot-pressing, hot isostatic pressing, spark plasma sintering etc. have been employed. Conventional pressureless sintering is the most common low cost approach to sinter ceramics but it is difficult to realize densification without grain growth. This has stimulated researchers to develop new and fancy synthesis methods that can partly/fully deal with the limitations of powder processing synthesis techniques.
Duan et al. [159]
demonstrated the preparation of Al2O3/TiO2
nanocomposite by high-pressure sintering where addition of nano-MgO powder
improved the densification and retarded grain growth. Fig.
32 is an illustration of a ceramic nanocomposite prepared by high-pressure
sintering. The SEM micrograph is taken from a fracture surface. The grains are
on the order of 100 nm. The nanocomposite contains grains of alumina,
titania, and magnesia. Although the pressure (1 GPa) and temperature (850 °C/30 min) are high, no significant grain coarsening
occurred because of the different compositions.
Fig. 32. SEM micrograph of fracture surface in Al2O3–TiO2–MgO nanocomposite consolidated by high pressure (P
Kear et al. [160] developed two production methods, one method called transformation assisted consolidation (TAC) that utilized hot pressing to consolidate a flame-synthesized metastable nanopowder which took advantage of a pressure-induced phase transformation to suppress grain growth during consolidation. The second method utilized plasma spraying of an aggregate powder feed to generate a splat-quenched metastable phase with extended solid solubility.
Li et al. [161] used anatase titania nanopowder with mean particle sizes of 7, 14, 26 and 38 nm synthesized by sol–gel technique to sinter bulk TiO2 nanoceramics. Buscaglia et al. [162] synthesized dense nanocrystalline barium titanate (BaTiO3) ceramic by spark plasma sintering technique (at 800 °C) of nanopowders produced by a wet chemical process. Lu et al. [163] prepared a BaSnO3 powder with a crystallite size of 27.6 nm through a hydrothermal reaction of a peptised SnO2 · xH2O and Ba(OH)2 at 250 °C and following crystallization of the hydrothermal product at 330 °C. Anilkumar et al. [164] followed a simple gel to crystal conversion route for the preparation of nanocrystalline tetragonal Mn3O4 powders at 80–100 °C under refluxing conditions. Various other methods for preparing nanocrystalline Mn3O4 powders such as chemical bath deposition [165], solvothermal [166] and coprecipitation [167] have also been reported. Densification of pure nanocrystalline MgO powder with 10 nm particle size by hot-pressing was investigated by Ehre et al. [168] in the temperature range 700–800 °C, applied pressure range 100–200 MPa, and for duration of up to 240 min. The final grain size was shown to decrease with increasing applied pressure.
Thangadurai et al. [169]
prepared La2O3 doped nanocrystalline zirconia
(ZrO2) by chemical co-precipitation method for 3, 5, 8, 10, 15, 20
and 30 mol% concentrations of La2O3. Gaikwad et al.
[170]
applied a simple coprecipitation technique for the preparation of pure ultrafine
single-phase CaBi2Ta2O9 (CBT). Ammonium
hydroxide and ammonium oxalate were used to precipitate Ca2+,
Bi3+ and Ta5+ cations simultaneously. Zhang et al. [171]
weakly agglomerated nanocrystalline ceramic oxides (ZrO2,
CeO2, and Y2O3) powders in the presence of
anionic starburst dendrimes poly (amidoamine), PAMAM with surface carboxylate
groups. The PAMAM dendrimer was shown to be an inhibitor for crystal formation
and affected the crystal morphology and particle size during the preparation.
The pressure-less sintering behavior of PbTiO3 powders synthesized by
mechanical alloying TiO2 and PbO was investigated by Forrester et al.
[172]
using dilatometry and X-ray diffraction. As-synthesized, the powders were shown
to be nanocrystalline with a mean particle size of 20 nm. Pressureless
sintering in the range 500–1050 °C gave single phase ceramics with
densities of 85–90% and crystallite sizes in the range 80–400 nm. Silva et
al. [173]
synthesized nanocrystalline CoFe2O4 powders using metallic
nitrates dispersed in aqueous media precipitated by stoichiometric amount of
NH4OH. The obtained amorphous powder was washed, dried at 80 °C
and heat-treated at various temperatures for 2 h for crystallization of the
material. Ramaswamy et al. [174]
synthesized nanocrystalline zirconia powder with a fairly narrow particle size
distribution by the amorphous citrate route. The powder obtained was shown to
have a high surface area of 89 m2 g−1. Rietveld
refinement of the powder XRD profile of the zirconia sample suggested
stabilization of zirconia in the tetragonal phase with around 8% monoclinic
impurity. Smaller crystallite size was shown to play a role in the stabilization
of zirconia into the tetragonal phase. Dhage et al. [175]
followed a simple gel to crystal conversion route for the preparation of
TiO2 at 80–100 °C under refluxing conditions. Freshly prepared
titanium hydroxide gel was allowed to crystallize under refluxing and stirring
conditions for 6–12 h. Formation of nanocrystals of anatase was confirmed
by X-ray diffraction (XRD) study. Chen et al. [176]
prepared nanocrystalline titanium diboride (TiB2) by the reaction of
TiCl4 with NaBH4 in the temperature range of
500–700 °C in an autoclave. TEM images showed particle morphology, with
average grain size of 15 nm for the powder obtained at 600 °C. Lemos
et al. [177]
synthesized ytterbium modified lead titanate ceramics by the Pechini method. The
materials were calcinated under flowing oxygen at different temperatures, from
300 °C to 700 °C. Nanostructured PYbT was obtained by high-energy
milling and investigated using different techniques that revealed that the
obtained materials were formed by nanometric particles. Rajesh et al. [178]
prepared nanocrystalline lanthanum phosphate having average crystallite size
10–12 nm by a controlled precipitation reaction from lanthanum nitrate and
orthophosphoric acid. The precipitate obtained was calcined in the range
300–800 °C and further sintered as compacted discs at a temperature of
1250 °C. An average grain size of 1–2 μm was obtained on sintering at
1250 °C. Li et al. [179]
synthesized nanosized
Sr0.5Ba0.5Nb2O6 (SBN50) ceramic
powder by an aqueous organic gel route. Homogeneous Sr–Ba–Nb precursor gels on
calcinations at 800 °C for 2 h produced a pure tungsten bronze SBN
phase and the corresponding average particle size was reported to be in the
range of 30–50 nm. Feng et al. [180]
synthesized cubic ZrO2 nanocrystals through a novel
reduction–oxidation route, in which metal zirconium was reduced from zirconium
tetrachloride (ZrCl2) by sodium, and then oxidized in atmosphere.
Using X-ray powder diffraction and TEM analysis, it was shown that cubic
nanocrystalline ZrO2 with high crystallinity was obtained, which
consisted dominantly of spherical particles with an average size of 17 nm
or so. Zhang et al. [181]
prepared nanocrystalline Ce0.8Gd0.2O2−δ
powder from a homogeneous precipitation system using hexamethylenetetramine as a
precipitate. The powder calcined at 700–800 °C was shown to exhibit optimal
sinterability, and could be sintered to nearly full density at a temperature as
low as 1250 °C. Han et al. [182]
used the citric acid sol–gel combustion method for the synthesis of
nanocrystalline hydroxyapetite (HAP) powder from calcium nitrate, diammonium
hydrogen phosphate and citric acid. The HAP powder was sintered into microporous
ceramic in air at 1200 °C with 3 h soaking time. Luan et al. [183]
obtained fine-grained BaTiO3 ceramic at a low sintering temperature
of 900 °C within a short sintering period by means of a new sintering
technique called spark plasma sintering (SPS). The ceramic was shown to be
highly densified to more than 99% of the theoretical X-ray density with
homogeneous microstructures, signifying that the SPS process is effective for
densification. Piticescu et al. [184]
demonstrated that polar ceramics with high homogeneity, high sinterability and
nanocrystalline structure can be obtained by hydrothermal procedures. They
studied the influence of the hydrothermal reaction conditions on the properties
of a PZT powder synthesized from soluable salts of Pb(II), Ti(IV) and Zr(IV).
Chemical quantitative analysis and microstructural investigations were performed
and based on these results a mechanism of hydrothermal synthesis of PZT was
proposed. Technique for processing of transparent nanoceramics was presented by
Pradhan et al. [185]
who synthesized Nd:YAG nanoparticles by coprecipitation. The powders prepared by
the coprecipitation technique were shown to display significantly less
agglomeration of crystallites, indicating higher sinterability. Crystallite size
dependence on calcination temperature was shown to suggest optimum temperature
of 1100 °C, at which phase purity of YAG nanopowder with highest
sinterability could be obtained. It was demonstrated that the optimum
temperature for vacuum-sintering is about 1785 °C for uniaxially pressed
samples. This resulted in transparent ceramic with uniform particle size of
about 5–7 μm, and above this temperature, large grain growth occurred and
facilitated large pores formation in the intergrain region that made the ceramic
fragile at the grain boundary. Kim et al. [186]
employed two-step sintering method to consolidate nanosize BaTiO3
powders. This comprised initial heating at relatively higher temperature and
following low-temperature sintering for a long period of heating. Pressed green
bodies of BaTiO3 were first hearted to 1300 °C to achieve an
intermediate density, then cooled down and held at 1100 °C for 0–20 h
until they became fairly dense. Slight grain growth was shown to occur during
the second step (with the final grain size of 1 μm), whereas the density
was shown to decrease significantly. Luscalea et al. [187]
investigated the synthesis of nanocrystalline
ZrO2–15mol%CeO2 powders by a pH-controlled nitrate–glycene
gel-combustion process. It was claimed that the metastable tetragonal phase
could be fully retained due to the small crystallite size of the powders
(5–8 nm). Frolova [188]
used coprecipitation method to produce stable sols of the binary oxides
(ZrO2–GeO2). Binary oxides exhibited high dispersity and
homogeneity. A heat-treatment of ZrO2–GeO2 xerogels (dried
in air sols) was shown to result in transformation of amorphous materials into
homogeneous nanocrystalline glass–ceramic composition. Dhage et al. [189]
applied a simple co-precipitation technique for the precipitation of pure
ultrafine single phase SrBi2Ta2O9. Ammonium
hydroxide was used to precipitate Sr+, Bi3+ and
Ta5+ cations simultaneously. No pyrochlore phase was found while
heating powder at 800 °C and pure
SrBi2Ta2O9 phase confirmed by X-ray
diffraction. Synthesis of nanocrystalline titanium carbonitride materials at
different pyrolysis temperatures via polymer-to-ceramic transformation of
synthesized poly(titanylcarbodiimides) and their structural and nanochemical
characterization using high-resolution and analytical electron microscopy, in
combination with quantum mechanical calculations was reported by Lichtenberger
et al. [190].
Pyrolysis at 800 °C was shown to form a mixture of amorphous carbon and
titanium nitride as crystalline particles of about 4 nm in size while
pyrolysis at 1100 °C yielded titanium carbonitride as crystalline particles
of 20–30 nm in size. Nersisyan et al. [191]
investigated the combustion process of TiO2–Mg and
TiO2–Mg–C systems with NaCl as an inert diluent. The values of
combustion parameters and temperature distribution on a high-temperature wave
according to the amount of sodium chloride were obtained by thermocoupling
technique. The leading stages of combustion processes were reported and the
sizes of reactionary zones were estimated. It was shown that the introduction of
NaCl in an initial mixture promotes the formation of a nanocrystalline structure
of the final product.
Lee et al. [192]
investigated the effect of spark plasma sintering (SPS) on the densification of
TiO2 ceramics. A fully-dense TiO2 specimen with an average
grain size of 200 nm was reported by spark plasma sintering at 700 °C for
1 h. In contrast, it was reported, that a theoretical density specimen
could only be obtained using conventional sintering above 900 °C for
1 h with an average grain size of 1–2 μm. Cheng et al. [193]
used polymeric and oligomeric carbosilanes having Si atoms linked by methylene
(-CH2) groups to prepare nanosized tubules and bamboo-like SiC
structures by both CVD and liquid precursor infiltration and pyrolysis inside of
nanoporous alumina filter disks, followed by dissolution of the alumina template
in HF(aq). In the case of the CVD-derived SiC nanotubes, annealing
these structures up to 1600 °C in an Ar atmosphere was shown to yield a
nanocrystalline β-SiC or β-SiC/C composite in the shape of the original
nanotubes, while in the case of the liquid precursor-derived nanostructures,
conversion to a collection of single crystal SiC nanofibres and other small
particles was reported. Panda et al. [194]
reported synthesis of nanocrystalline powders of stoichiometric and
non-stiochiometric strontium bismuth tantalite starting from an aqueous
precursor solution. Complete dehydration of the precursor solution was shown to
generate a precursor which on direct calcinations at 700–750 °C for
2 h resulted in phase-pure
Sr1−xBi2+yTa2O9
powders, with particle size in the range of 23–33 nm.
Xue et al. [195] synthesized a large number of ceramic perovskites by mechanically activating constituent oxides at room temperature. The activation-synthesized ferroelectrics and piezoelectrics were shown to exhibit a nanocrystalline structure that is not shown by the materials prepared by any other processing route.
Synthesis of titanium diboride (TiB2) via a benzene-thermal reaction of metallic sodium with amorphous boron powder and titanium tetrachloride at 400 °C in an autoclave was reported by Gu et al. [196]. The processing temperature was reported to be much lower than that of traditional methods. Nordahl et al. [197] demonstrated that seeding nanocrystalline transition alumina powders is a viable option for producing high quality, alumina based ceramic where by using α-Al2O3 seed particles, the sintering temperature was reduced from 1600 °C for unseeded γ-Al2O3 to 1300–1400 °C in dry pressed powders. Markmann et al. [198] exhibited that the combination of yttrium doping and colloidal processing allowed for the synthesis of dense nanocrystalline cerium oxide by pressureless sintering. Cerium oxide was synthesized by the direct homogeneous precipitation method using hexamethylenetetramine (HMT). Kim et al. [199] prepared nanocrystalline indium tin oxide powder with different particle size using a coprecipitation process. They examined the sintering characteristics of the powder at different heating rates. Decrease of particle size in nanosized powder regime promoted the densification in normal rate sintering as temperature increased, while this was shown to retard severely the densification at high temperature in rapid rate sintering. Bab et al. [200] synthesized nanocrystalline HfN powders by mechanically assisted gas–solid reaction. The results showed the formation of an interstitial solid solution α-Hf, prior to its transformation to cubic HfN. Zhu et al. [201] investigated the synthesis of TiC powder by mechanical alloying and showed that the nanocrystalline TiC powder is fabricated in a very short time at room temperature and that the mechanism is self-propagating reactive synthesis conducted by mechanical alloying. Lu et al. [202] claimed to develop a new generation of solid-state laser and optical materials on the basis of highly transparent nanocrystalline yttrium aluminum garnet Y3Al5O12 (YAG) ceramic that was synthesized using aqueous solutions of aluminum, yttrium and neodymium chlorides.
Laberty-Robert et al. [203] used a Pechini process for preparation of yttria-stabilized zirconia gels and powders. The decomposition of this gel which was shown to be based on a thermally induced anionic redox reaction yielded nanostructured powders at 325 °C that is agglomerated with an average of 10–20 nm primary particles.
Li et al. [204] reported synthesis of nanocrystalline CeO2 powders of high sinterability by a mimic alkoxide method, which employed alcohols as solvent, cerium nitrate hexahydrate as cerium source and diethyl-amine (DEA) as precipitant.
An overview of the emerging cost-effective electrostatic spray-assisted vapour deposition (ESAVD) based method for the synthesis of nanocrystalline oxide and non-oxide ceramic films and powders was presented by Choy [205]. ESAVD process involves spraying atomized precursor droplets across an electric field and if the droplets undergo heterogeneous chemical reaction near the vicinity of the heated substrate, a stable solid film with excellent adhesion onto a substrate results. This produces highly pure materials with structural control at low processing temperatures. Different films that were studied are Y2O3:Eu film, TiO2 film, ZnS film, and CdS film. It was shown that the structure and properties of the synthesized materials can be optimized by varying processing parameters. The process, apparently, is superior to chemical vapor deposition (CVD) and physical vapor deposition (PVD) methods since the depository efficiency is higher. Also, ESAVD based methods can be performed in an open atmosphere without the use of sophisticated reactor and vacuum system.
Zawadzki et al. [206] prepared nanocrystalline and nanoporous ceramics composed of a network of alumina and zinc aluminate doped with Tb ions. It was claimed that the textural properties make these structures interesting as a host lattice for active systems such as rare earth ions. Kear et al. [160] developed two methods for the production of nanostructured bulk ceramics. The first method involved hot pressing to consolidate a flame-synthesized metastable powder that took advantage of a pressure-induced phase transformation to suppress grain growth during consolidation. This process called transformation assisted consolidation (TAC) was used to prepare test samples of single phase nanocrystalline ceramic. The second method employed plasma spraying of an aggregated powder feed to generate a splat-quenched metastable phase with extended solid solubility.
Jose et al. [207] developed a single step process for the synthesis of nanoparticles of YBa2ZrO5.5, a complex perovskite ceramic oxide which did not involve a calcination step. The powder obtained by self-sustained combustion of a precursor complex of the respective metal ions, was reported to be in the range of 5–50 nm.
Das [208] reported an innovative sucrose process for synthesizing a variety of nanocrystalline ceramic powders using metal ion–sucrose solution. The process involved the dehydration of metal ion–sucrose solution to a highly viscous liquid, which on decomposition produced the precursor material. Calcination of the precursors at low temperatures was reported to produce nanocrystalline ceramic powders. The examples discussed included PbZr0.6Ti0.4O3 (PZT) and spinel ferrites MFe2O4 (M = Co, Ni, Zn). Lu et al. [209] claimed to have developed an innovative colloid-gel process using inorganic salts as starting material for preparing ferroelectric strontium bismuth tantalate (SrBi2Ta2O9) powder. The technique involved preparation of a mixture of Ta(OH)5 colloid with the aqueous solutions of strontium and bismuth cations, followed by the addition of ethylene glycol and citric acid as gel-forming reagents. During the calcination process, the fluorite phase was formed as an intermediate that was subsequently converted into the SrBi2Ta2O9 phase. After calcination at 750 °C, pure ultrafine SrBi2Ta2O9 polycrystalline powder with a narrow particle size distribution was obtained (40 nm). Li et al. [210] reported synthesis of nanocrystalline α-alumina powders with a primary mean particle diameter of 10 nm from alumina nitrate and ammonia solution using a precipitation method. The combined presence of 5 wt.% α-alumina seed crystals, 100 nm in diameter, and 44% ammonium nitrate was reported to reduce the θ-Al2O3 → α-Al2O3 transformation temperature from 1200 °C to 900 °C.
Juárez et al. [211]
presented a nitrate–citrate combustion route to synthesize nanocrystalline
yttria-doped zirconia powders for tetragonal zirconia polycrystal (TZP) ceramic.
The route was based on the gelling of nitrate solutions by the addition of
citric acid and ammonium hydroxide, followed by an intense combustion process
due to an exothermic redox reaction between nitrate and citrate ions. Yue et al.
[212]
reported preparation of a nitrate–citrate gel from metal nitrates and citric
acid by sol–gel process, in order to synthesize
Ni0.25Cu0.25Zn0.50Fe2O4
ferrite. The nitrate–citrate gel exhibited self-propagating combustion behavior
and after combustion, the gel directly transformed into single-phase, nanosized
NiCuZn ferrite particles with spinel crystal structure. The synthesized powder
was densified at temperature lower than 900 °C and the sintered body
possessed fine-grained microstructure, good frequency stability and high-quality factor compared
to the sample prepared by conventional ceramic route. Das et al. [213]
synthesized nanocrystalline (30 nm) lead zirconate–titanate (PZT) ceramics,
with Zr/Ti ratio 60:40 by a chemical method starting from a mixed metal
ion–tartarate (Zr4+, Ti4+)–EDTA (Pb2+) complex
solution. After complete evaporation of the mixed complex solution, a fluffy
dried mass, known as precursor material was obtained. Heat-treatments of the
precursor material at 400 °C for 2 h resulted in single phase PZT. The
process was shown to provide a technically simple route for the preparation of
nanocrystalline PZT powders at low preparation temperatures. Liao et al. [214]
produced bulk nanocrystalline TiO2 by high pressure/low temperature
sintering. Nanophase TiO2 powder with metastable anatase structure
and particle size 38 nm was used as the starting material. During sintering, the
anatase phase transformed to either rutile or srilankite phase, depending on the
pressure/temperature combination. Grain growth was limited by the low sintering
temperature and multiple nucleation events in the parent phase. The transformed
rutile phase was reported to decrease with sintering pressure, due to increasing
nucleation rate and decreasing growth rate with high pressure. It was
demonstrated that in contrast to previous researchers who found exaggerated
grain growth during sintering of nanocrystalline ceramics, it is possible to
produce a dense sintered compact with grain size even smaller than that of the
starting powder by proper selection of sintering parameters. Shantha et al. [215]
investigated the influence of mechanical activation on the formation of
Bi2VO5.5, bismuth vandate (BiV) phase, by ball milling a
stiochiometric mixture of bismuth oxide and vanadium pentoxide. Jiang et al. [216]
synthesized nanocrystalline zinc ferrite powders with a partially inverted
spinel structure by high-energy ball milling from a mixture of
α-Fe2O3 and ZnO crystalline powders in equimolar ratio.
The resulting ZnFe2O4 particles were reported to be in the
superparamagnetic state at ambient temperature.
Haber et al. [217]
showed that solution-phase processing affords nanoparticles and nanocomposites
of a wide range of materials, including metals, alloys, intermetallics, and
ceramics, with substantial control of particle size, particle morphology, microstructure, and composite microstructure. It was claimed that the particle size of nanocrystalline
Cu produced by the reduction of CuCl could be varied between 10 and 30 nm
by changes in reducing agent and solvent, while that of nano-Al could be varied
between 45 and 180 nm. The morphology of nano-AlN produced by nitridation
of nano-Al could be varied from almost entirely equiaxed to almost entirely
whisker-like. Nanocrystalline nickel aluminides were synthesized by reacting
NiCl2 and LiAlH4 in organic solvents, followed by solid
state heating. Reactions performed in aromatic solvents were shown to produce a
porous phase with highly sintered microstructure, while reactions performed in ethereal solvents yielded
nanosintered nickel aluminide particles in an amorphous alumina matrix.
Nanocrystalline composites of AlN and Al were prepared by suspending the AlN
particles in organic solvent, and performing solution-phase chemistry to produce
nano-Al. Ragulya [218]
reported synthesis of nanocrystalline powders of barium titanate by
non-isothermal and rate-controlled decomposition of organic precursors. The
nanograined barium titanate powder sintered under rate-controlled conditions
produced dense and fine-grained ceramic and it was pointed out that the thermal
activated processes should be optimal through the entire temperature–time
processing schedule. Popp et al. [219]
synthesized nanocrystalline ceramic powders of ZrO2,
Al2O3, Si3N4, AlN by evaporation of
solid materials in the focus of a CO2 laser followed by
recondensation in a carrier gas stream. The production rates were reported to be
in the range of 40–100 g h−1 (oxides) or more than
200 g h−1 (nitrides). The particle size distribution was
shown to be influenced by different parameters of the formation process like
laser power, area of the laser focus, pulse length in the case of pw mode,
streaming velocity and type of the carrier gas. The powder consisted of nearly
spherical particles with diameters in the range of 10–100 nm. Powders of
tetragonal zirconia polycrystals (TZP) ceramic were synthesized by Lamas et al.
[220]
using a nitrate–citrate gel combustion process. These powders were compacted by
uniaxial pressing and fired at 1400–1500 °C for 3–5 min, obtaining
fine-grained, dense ceramic. Lamas et al. [221]
utilized wet chemical methods to obtain yttria or calcia-doped zirconia powders
for synthesizing partially stabilized zirconia ceramics. This method was based
on pyrolysis of nitrate solutions with the addition of citric acid or glycine as
a fuel. Pinckney et al. [222]
produced nanocrystalline glass–ceramic with high elastic modulus and moderate
strength and toughness from a wide area in the
SiO2–Al2O3–MgO–ZnO–TiO2 system. Pal
et al. [223]
grew barium hexaferrite phase of composition BaO-6Fe2O3 of
dimensions in the range of 8.2–17.6 nm within a glass medium by subjecting
the latter to a suitable heat treatment. Large bulk and fully dense SiC dense
SiC based nanoceramics with average grain size of 50 nm and 20–30 wt.%
nanometer sized α-Sialon,
AlxSi3−xO6 and α-SiO2
interfacial phases were prepared by crystallization of interfacial glass, using
LMAS glass-coated SiC powder as starting material by Zhang et al. [224].
Durán et al. [225]
prepared powders of yttria-doped tetragonal zirconia (3 mol%) with a narrow
pore size distribution and ultrafine particle size (9 nm) by the mixed organic and inorganic precursors
coprecipitation method. The compaction behavior of almost agglomerate-free
calcined powders was studied, and their sintering behavior using both isothermal
and non-isothermal techniques were evaluated. Reports of fully dense nanoscale
ceramics with an average grain size below 95 nm were presented after
sintering at 1200 °C for 20 min. Qui et al. [226]
by using heterogeneous n-butanol azeotropic distillation in the treatment
of an aqueous coprecipitation gel, produced weakly agglomerated and sinterable
nano-ZrO2 powder with high density (99.5%) and fine-grain (200 nm). Kats-Demyanets et al. [227]
reported preparation of amorphous to nanocrystalline
ZrO2–CeO2 alloy powders by chemical precipitation. It was
shown that sintering at 1300 °C for 2.5 h was followed by grain growth
between 1000 °C and 1600 °C and for 2–20 h. Sintering of
nanocrystalline powder material—Al2O3–ZrO2
(grain size 15 nm) and TiO2 was studied by Bykov et al. [228].
Fracture toughness studies on nanocrystalline tetragonal zirconia with low yttria content carried out by Bravo-Leon et al. [229] showed that they could reach toughness between 16 and 17 MPa m1/2. It was shown that maximum toughness occurs just below the critical grain size of 90 and 110 nm, respectively, i.e. just before spontaneous transformation to the monoclinic phase occurs. Moving away from the critical grain size, towards smaller grains, was shown to decrease toughness monotonically up to a factor of 5. The critical parameter for maximizing toughness in nanocrystalline zirconia ceramics was pointed out to be proximity to the phase transformation boundary.
One of the attractive features of densified nanocrystalline ceramics is their ability to be subjected to superplastic forming at strain rates considerably higher than their conventional polycrystalline counterparts. This is a direct consequence of the added interfacial surfaces available for intergranular sliding. Mayo [241] synthesized ZrO2 powders by a chemical precipitation and agglomerated them by sinter forging. When they deformed these consolidated nanocrystalline ceramics under compression (d = 80 nm) they found that the strain rate obtained was 34 times higher than conventional zirconia (d = 0.3 μm). Fig. 33 shows the stress versus strain rate curves for the two grain sizes. The fact that the two lines are parallel indicates that the same deformation mechanism is operating in them. This increased strain rate is important in superplastic forming. Superplastic forming processes that would require impractically long times can be applied, if the grain size is sufficiently reduced.
Fig. 33. Superplastic deformation of nanocrystalline and submicron-grained ZrO2–3mol%Y2O3 [241].
Superplasticity in fine-grained partially stabilized zirconia was
demonstrated by Wakai et al. [230]
at lower temperatures, particularly when the grain size is well below
100 nm. Jimenez-Melendo et al. [231]
presented a list of all studies on the creep and superplastic deformation on
these ceramics at temperatures above 0.5Tm. Nanocrystalline
oxide powder synthesized by Mayo et al. [232]
as shown to exhibit superplastic strain rates 100 times faster than in comparable undoped systems, due to the
dopant’s role in lowering of the activation energy for diffusion along grain
boundaries. Enhanced specific grain-boundary conductivity study in
nanocrystalline Y2O3-stabilized zirconia carried out by
Mondal et al. [233]
showed that the specific grain-boundary conductivity of the nanocrystalline
samples is 1–2 orders of magnitude higher than that of the microcrystalline
samples since the conductivities of the bulk and the grain-boundary regions were
the same irrespective of grain size. This was also attributed to the low
Si-content and its grain size-dependent segregation in the nanocrystalline
samples. Ductility study of nanocrystalline zirconia based ceramic at low
temperature by Betz et al. [234]
showed that a maximum elongation of about 60% can be achieved with a
ZrO2 + 10 wt.%
Y2O3 + 12 wt.% A2O3
composite ceramic at 1250 °C. The results were related to a deformation
model based on mesoscopic grain-boundary sliding. Betz et al. [235]
observed total strains exceeding 0.2 and strain rates as high as
10−5 s−1 without failure in nanocrystalline
5 mol% yttria stabilized zirconia ceramics with relative densities of about
80% and grain sizes between 30 and 60 nm. Porosity was not found to be
critical for the ability to deform without cracking while the grain size
exponent in the constitutive equation for superplastic flow yielded a value of
1.3. Winnubst et al. [236]
showed that the deformation strain rate of nanocrystalline Y-TZP increased by a
factor of 4 if the grain size decreased from 200 to 100 nm. It was
demonstrated that these materials showed real superplastic deformation (strain
rate > 10−4 s−1) at relatively low
temperature (1100–1200 °C). Grain boundary analysis indicated partial
removal of an ultra-thin (1 nm), yttrium-rich grain-boundary layer after
deformation. He et al. [237]
reported synthesis of ultrafine grained TZP by a gel precipitation and
sinter-forging. The resulting structure was shown to have improved
grain-boundary structure with reduced flaws. Sánchez-Bajo et al. [238]
carried out microstructural analysis of yttria-partially stabilized zirconia
(Y-PSZ) as a high performance ceramic using X-ray materials. Creep study on
yttria-partially stabilized zirconia with a grain size of 40 nm in
compression at temperatures ranges between 1100 °C and 1200 °C and
strain rates between 5 × 10−7 s−1 and
10−4 s−1 as demonstrated by Gutiérrez-Mora et al. [239]
showed that these ceramics probably cannot be deformed superplastically. The
creep parameters reported were n = 1.4 and
Q = 660 kJ/mol. Mayo [240]
argued that the unusual trends in room temperature strain rate sensitivity of
nanocrystalline materials may reflect thermally activated dislocation glide past
synthesis-generated defects, rather than a true change in deformation mechanism
at ultrafine grain sizes.
The expansion of the understanding of deformation of conventional polycrystalline materials to materials with grain sizes in the range of nanometer is, at present, an evolving process. Though several mechanisms have been proposed, there still exists widespread disagreement in the research community. Most striking of all is the observation of “Inverse Hall–Petch” phenomenon which is still in question ever since Chokshi et al. [97] reported it in 1989.
This section reviews the fundamental physical mechanisms responsible for the specific behaviors. From the previous sections, it can be concluded that nanocrystalline materials exhibit a high strength, albeit not as high as the extrapolated Hall–Petch prediction from the conventional regime of grain sizes (100–300 μm). Concomitantly, the nanocrystalline materials tend to exhibit much reduced ductility, due in part to a low work hardening rate leading to early strain localization and failure and also due to a reduced ability of the materials to accommodate the progression of cracks by extensive plastic deformation.
The concept of pile-ups has been at the root of the traditional explanation for the H–P effect. As the grain size is decreased, the number of dislocations piled up against a grain boundary decreases, at a fixed stress level, since this number is a function of the applied stress and of the distance to the source. Conversely, an increased stress level is needed to generate the same number of dislocations at the pile-up. At a critical grain size, we can no longer use the concept of a pile-up to explain the plastic flow. Fig. 34(a) shows pile-ups for a grain size in the micrometer regime. The sources are assumed to be in the center of the grain, leading to positive and negative dislocation pile-ups generated by the activation of a Franck–Read source. As the grain size is reduced to the nanocrystalline regime, the number of dislocations at the pile-up is eventually reduced to one. Thus, the multiplying effect on the stress field is lost. This is shown in Fig. 34(b). This mechanism was first proposed by Pande et al. [242] and further developed by Armstrong and coworkers [243], [244] and [245].
Fig. 34. Breakup of dislocation pile ups: (a) microcrystalline regime and (b) nanocrystalline regime.
As seen in Section 5, Koch et al. [246] performed microhardness tests on Fe samples produced by ball milling and observed that the hardness increased as the grain size decreased, as typically observed in coarse grained samples. Nieman et al. [101] observed that the hardness for nanocrystalline Pd followed a Hall–Petch dependence with a smaller slope than for coarse-grained Pd. Iyer et al. [105] observed a similar reduction in slope for Cu in the nanocrystalline range. These results are in contrast to the study of Chokshi et al., which show not only a decrease in slope, but a negative one.
Wadsworth and Nieh [247] predicted the grain size in various materials at which the Hall–Petch relationship would break down by assuming that there is a point at which each individual grain in a polycrystalline sample will no longer be able to support more than one dislocation. The critical equilibrium spacing between dislocations, lc, from their calculation is
Another interesting pile-up breakup model proposed by Pande et al. [242] considers the stress at the tip of any number of dislocations in the pile-up and equates it to the barrier stress required to free the dislocations.
The stress at the tip is given as
| σtip=[(n+m-1)/m]σ | (16) |
| σ*=[(n+m-1)/m]σ | (17) |
The relationship between n, the number of dislocations and L, the pile-up length is given by
Fig. 35. Exact and approximate comparison of the Pande et al. model with Hall–Petch [242].
The phenomenon of superplasticity has led to detailed study of grain-boundary sliding as the dominant deformation mechanism [248]. Fig. 36 shows a schematic in which one layer of grains slides with respect to the other, producing a shear strain in the process. Plastic deformation has taken place by virtue of the top layer of grains translating to the right with respect to the bottom layer of grains. This requires grain-boundary sliding and is the principal mechanism in superplasticity. For nanocrystalline materials, this has been proposed to be the dominant deformation mechanism at grain sizes <50 nm. Hahn et al. [249] proposed two hardness relationships,
.gif){kind=small}
, where δ ≈ 3b is
the grain-boundary width, A = b2,
ν = νD(≈1013 s−1)
the Debye frequency. The free energy is expressed as:
ΔG = ΔF − Vτe, where
ΔF is the Helmholtz free energy and
V = b3 the activation volume. ΔF was
approximated by the energy for atom–vacancy interchange in the lattice or
grain-boundary diffusion. Inserting activation the above values into Eq. (22)
and considering both forward and backward jumps resulted in 0.5 and sinh x ≈ 1/2 exp x
for x 2. Eq. (23)
thus predicts τe proportional to d for small values of
.gif){kind=small}
and varying as ln(d) for large
values.
Fig. 36. Grain boundary sliding model: (a) initial position of grains and (b) position after top layer has slid to right.
Conrad [85] analyzed data on the effect of grain size d in the range of mm to nm on the flow stress of Cu. Three regimes were identified:
Regime I (d > 10−6 m): Dislocation pile-up is considered to be the dominant deformation mechanism. The rate controlling mechanism is given by
Regime II (d ≈ 10−8–10−6 m): Grain boundary shear promoted by the pileup of dislocations is considered to be the deformation mechanism. The governing equation is given by
Regime III (d < 10−8 m): Grain boundary shear is considered to dominate deformation in this regime. The corresponding equation is given by
1013 s−1), ΔF* is the
Helmholtz free energy. These three regimes are shown in Fig.
37. The continuous lines represent the application of Eqs. (24),
(25)
and (26)
above, while the experimental points shown represent the compilation of several
investigations. For Regime I, the lines are given for two values of strain,
γ: 0.01 and 0.2; the slope is smaller than −1/2. Since the dependence is
expressed in a log–log form, the magnitude of the slope in Regime II is higher.
However, for Regime III a negative slope is shown. This represents the regime
dominated by grain-boundary shear.
Fig. 37. Variation of flow stress with grain size for copper (300 K) on a log–log plot; notice three regimes I, II, III governed by different plastic deformation mechanisms [370].
The phase mixture model, or rule of mixture [251] and [252] is one common approach to explaining the deformation mechanism of nanocrystalline materials. Kim et al. [251] considered a nanostructured material as consisting of crystallites, grain-boundary surfaces, triple line junctions, and quadratic nodes. The volume fraction of each was expressed in terms of the grain size and the grain-boundary width. The composite hardness was expressed as
Van Swygenhoven et al. [255] and [258] carried out molecular dynamics simulations that revealed sliding along boundaries. They show that grain-boundary sliding is the primary deformation mechanism in nanocrystalline materials and that such a sliding mechanism results in a build up of stress across neighboring grains. This stress, in turn, is relieved by grain boundary and triple junction migration. The degree to which this occurs depends on grain size since for small grain sizes, less planar interfaces provides reduced steric hindrance to concurrent grain sliding. Such grain-boundary sliding activity is mainly facilitated by atomic shuffling and stress-assisted free volume migration [254].
Van Swygenhoven et al. [309] performed MD simulations on two model FCC metals, Ni and Cu, having different stacking fault energies to address the question of grain-boundary structure and of the mechanism of plastic deformation as a function of grain size (5–12 nm) [256]. Analysis on the atomic level showed that all the deformation is accommodated in the grain boundaries while at greater grain sizes, intragrain deformation was detected by the presence of stacking faults inside the grains.
A discussion of grain-boundary sliding was introduced by Meyers et al. [257]. An important consideration in grain-boundary sliding not discussed by Kim et al. [251] or Conrad et al. [250] is the compatibility of deformation. These models are based on the Coble creep equation, which expresses the strain rate as proportional to a d−3 term. As d decreases, the contribution to plastic flow coming from grain-boundary sliding increases. However, the simple Coble equation ignores the necessity of plastic flow to accompany sliding in order to accommodate the grains. It is clear that sliding alone cannot accomplish this deformation in the simple schematic of Fig. 36. Ashby and Verall [261] addressed this problem successfully for superplastic deformation and proposed a neighbor exchange mechanism capable of effecting plastic strain. This gliding/plastic deformation/neighbor exchange is illustrated in Fig. 38.
Fig. 38. Ashby–Verall mechanism for accommodating plastic flow through grain-boundary sliding coupled with plastic deformation and neighbor exchange: (a) original position of four neighboring grains; (b) convergence of vertices and (c) exchange of positions-neighbors become next neighbors and vice versa.
Thus, a plastic accommodation term has to be added to grain-boundary sliding. Fu et al. [103] have shown that the incorporation of the plastic accommodation term into the diffusion results in significant changes in the stress–strain curve. Grain-boundary sliding can be represented by a viscous and a plastic accommodation term,
.gif){kind=small}
,Fig. 39. (a) Schematic representation of grain-boundary sliding in polycrystal; (b) grain-boundary sliding path; (c) idealized sinusoidal path with diffusional migration of boundary according to Raj and Ashby [260]; (d) decrease in wave amplitude h with increasing plastic strain [103].
The parameters λ and h are the wavelength and amplitude,
respectively, defining the sinusoidal boundary in Fig.
39(c); Ω is the atomic volume; δ is the thickness of the
boundary; DV and DB are the volume and
boundary diffusion coefficients, respectively. At temperatures significantly
below Tm/2, DB 
DV and
.gif){kind=small}
on h, in Eq. (32),
is a clear suggestion that deformation will localize as h decreases. As
plastic deformation proceeds, it is easily visualized how the jaggedness of the
interface, measured by h, decreases, as shown in the transition from Fig.
39(c) to (d). In nanocrystalline materials the flow localization, in the
form of shear bands, is indeed observed.
Fig.
40 shows a plot of .gif){kind=small}
, calculated from the diffusional Raj and
Ashby equation (Eq. (33))
as a function of grain size d. The stress levels were varied and are
obtained from the compressive flow stresses for the various grain sizes
calculated from the computational procedure. They vary from 100 to 900 MPa,
for grain sizes of 100 μm to 26 nm, respectively. These stresses are
converted into shear stresses assuming a von Mises criterion (i.e., dividing
them by .gif){kind=small}
). The values of the other parameters are:
δ(3b) = 0.75 nm [249];
Ω = 0.0087 nm3(= 4/3πr3),
r being the atomic radius for copper; T = 298 K.
The grain-boundary diffusion coefficient room temperature was taken as
DB = 10−21 m2/s at
T/Tm = 0.225. This value is lower than the
extrapolation from Shewmon [259]
(10−22 m2/s) and higher than the one used by Kim et
al. [251]
(2.6 × 10−20 m2/s). The five calculated
strain rates are marked on plot and the data are well represented by a straight
line. Extrapolation to smaller grain sizes (shown by a dashed line) reveals
striking results: for D 10 nm, the strain rate due to diffusional sliding along
grain boundaries by a Raj–Ashby [259]
mechanism can reach significant values .gif){kind=small}
. It should be pointed out that plastic
flow, not considered by Raj and Ashby, can also assist in the accommodation
process at these temperatures. Thus, it is concluded that grain-boundary sliding
can contribute to the plastic deformation at grain sizes d 10 nm, corroborating recent calculations [250],
[251]
and [258].
Fig. 40. Calculated strain rates from diffusionally-driven grain-boundary sliding as a function of grain size (copper deformed at 300 K); extrapolated results indicated by dashed line [103].
In 1963, Li [259] proposed that grain-boundary ledges (which he had earlier called grain-boundary jogs [260]) acted as sources of dislocations at the onset of plastic deformation and that these dislocations created, by a double slip mechanism, a work hardened layer around the grain boundaries. A significant aspect of the Li [259], [260] and [262] proposal is that he was able to calculate the spacing between the ledges from the HP slopes. The values obtained reveal that the ledge spacing, l, is of a nanometer scale:
Cu: l = 50b
Al: l = 100b
Mo: l = (2.5 → 33)b
Fe, steel: l = (0.5 → 5)b
Zn: l = (2.5 → 10)b
Zr, Ti: l = 5b
This is insufficient to promote plastic deformation in the nanocrystalline regime. Thus, one can envisage a grain size below which ledges can no longer operate.
This concept was later extended by Ashby [263]
and [264]
who proposed that geometrically necessary and statistically stored dislocations
were involved in polycrystalline deformation. Subsequently Hirth [267]
analyzed the incompatibility stresses at the grain boundaries. Thompson [268]
and Margolin [269]
proposed similar models. Although these models have taken several forms, the
basic idea, illustrated in Fig.
41, is that the deformation within a grain is composed of two parts: (1) the
core, or grain interior, which is subjected to a more homogeneous state of
stress; and (2) the mantle, or grain-boundary region, in which several factors
contribute to increased resistance to plastic flow and work hardening:
grain-boundary sources, change in orientation in the plane of maximum shear,
elastic and plastic incompatibility. This leads to enhanced cross-slip, as shown
in Fig.
42 which depicts a grain in the conventional polycrystalline regime. In the
grain interior (core), slip system S1 is activated, corresponding to
easy glide and a low work hardening rate. In the grain-boundary mantle region,
cross-slip into system .gif){kind=small}
leads to increased work hardening.
Fig. 41. Core and mantle model, showing relative fractions of grain boundary and grain interior regions in the (a) microcrystalline and (b) nanocrystalline regimes.
(9K)
Fig. 42. Schematic depiction of a deformed grain showing the more intense cross slip and work hardening along the grain-boundary regions.
Fig. 41 shows how the relative fractions of these two regions vary with grain size. As the grain size is decreased into the nanocrystalline region, the mantle dominates the plastic flow process. Fig. 41(a) represents, in a schematic fashion, a grain with the scale of micrometers, whereas Fig. 41(b) represents a nanosized grain, in which the mantle predominates.
A model is presented here that was proposed by Meyers and Ashworth [266] and extended by Fu et al. [265] to the nanocrystalline regime. The compatibility requirement at the grain boundaries due to the differences in orientation applied to anisotropic crystals creates additional stresses, τI. Meyers and Ashworth [266] found for nickel that
| τI=1.37σAP | (35) |
As a result, while the grain interiors can be considered to harden by the classic easy glide/linear hardening/parabolic hardening sequence, the grain boundaries show a much faster rise in the dislocation density and the hardening rate. This is exemplified by the result shown by Hirth and Lothe [267] for a bicrystal. Although this is a compatible bicrystal, the activation of a second slip system around the boundary is seen. This conceptual frame is extended to a polycrystal in Fig. 43. The grain-boundary regions are shown with pronounced slip activity on two slip systems; this leads to a much higher hardening rate than the grain interiors. Recent results on copper polycrystals show a much greater tendency for slip on two or three systems in the regions close to the boundaries. Note that grain-boundary corners are regions especially prone to concentrated plastic deformation. Results by Gray [270] confirm the higher hardening in the regions adjacent to the grain boundaries and triple points. This serves a basis for the model developed by Meyers and Ashworth [266]. This predominance of dislocations along grain boundaries at early stages of plastic flow is eloquently illustrated by the transmission electron micrographs of Murr and Hecker [271]. This build up of plastic deformation has also been recently measured by Sun et al. [272]. The use of electron back-scattering diffraction in orientation imaging microscopy of an aluminum bicrystal deformed to a strain of 0.1 revealed a significantly higher dislocation density in the vicinity of the grain boundary.
Fig. 43. Core and mantle models for a polycrystalline aggregate: (a) Meyers–Ashworth model [266]; (b) Kim et al. model [251].
Fig. 44 shows the stages of deformation as the applied stress, σAP, is increased. As the applied stress increases, a work hardened layer along the grain boundaries is formed. Once this work hardened grain-boundary layer is formed, the stress within the polycrystalline aggregate homogenizes. Stages a–c in Fig. 44 represents the dominance of the elastic compatibility strains and the formation of a grain boundary work-hardened layer. Stages d–f represent the response of a composite material, consisting of dislocation-free grain interiors, with a flow stress σfG, and grain-boundary layers, with a flow stress σfgb. The flow stress of the grain aggregate is obtained in approximate fashion from
| σy=AGσfG+Agbσfgb | (36) |
Fig. 44. Sequence of stages in polycrystalline deformation, starting with (a,b) localized plastic flow in the boundary regions(microyielding), forming a grain-boundary work-hardened layer (c,d) that effectively reinforces the macrostructure, and leading to (e,f) macroyielding in which the bulk of the grains undergo plastic deformation [265].
Fig. 45 is an idealized representation of the aggregate. Grains are assumed to be spherical, with a diameter d; the grain-boundary layers are assumed to have a thickness t (in each grain, t < d/2). The diametral areal fractions are expressed by
.gif){kind=small}
, respectively, so that:Fig. 45. (a) Polycrystalline aggregate viewed as composite material composed of bulk and grain-boundary material, with flow stresses sgb and sfgb, respectively. (b) Idealized spherical grain of diameter D with grain-boundary layer of thickness t; sections S1, S2, S3, S4, and S5 reveal different proportions between the areas of the bulk and the grain-boundary material [265].
The variation of the thickness of the work hardened grain-boundary layer has to be considered. First, as the grain size is decreases, the stress field fluctuations vary with d, leading to a dependency t = k1d; secondly, as the dislocation spacing is unchanged, and the dislocation interactions will dictate a constancy in t, thus, a relationship t = k2d0. The geometric mean would be (k1k2d)1/2. This produces the expected Hall–Petch dependence, as shown below. Hence
| t=(k1k2d)1/2=kMAd1/2 | (42) |
| kHP=8kMA(σkgb-σfG) | (44) |
| dc=(4kMA)2 | (45) |
Fig. 46. σy versus D−1/2 relationship for (a) iron and (b) copper [265].
Kim et al. [273] used a phase mixture model analogous to Meyers–Ashworth [266] and developed an analytical model including the work hardened mantle regions and grain-boundary sliding. Fig. 43(b) shows the model that they have used, in which the grains are cubic. Kim et al. [251] considered a phase mixture model (discussed in Section 7.2) in which a polycrystalline material is regarded as a mixture of crystalline phase and a grain-boundary phase to describe the plastic deformation behavior of fine-grained materials. The mechanical properties of the crystallite phase was modeled as being viscoplastic, which involves dislocation activity and diffusion creep. Fig. 47(a) shows the calculated stress–strain curves for Cu of various grain sizes and at different strain rates. The flow stress increases with decreasing grain size. Fig. 47(b) illustrates the effect of d on the strength of the crystallite phase. For all strain rates, the stress first increased and then decreased as the grain size is reduced. The total strain rate of the crystallite was calculated by considering contributions from the dislocation, the boundary diffusion and the lattice diffusion mechanisms. The dominant deformation mechanisms for the crystallite phase as a function of grain size are depicted in Fig. 48(a).
Fig. 47. (a) Stress–strain curve for the crystalline phase of Cu for different crystallite sizes and strain rates. Solid and dashed curves are for the strain rates of 10−3 s−1 and 10−5 s−1, respectively; (b) Grain size dependence of the 0.2% offset stresses for a Cu crystallite (Hall–Petch curves) for the strain rates of 10−3, 10−4, and 10−5 s−1 [251].
(77K)
Fig. 48. (a) Grain size dependence of all three contributions to the total imposed strain (dislocation mechanism, grain-boundary diffusion and lattice diffusion mechanism) in Cu for the total imposed strain rates of 10−3 s−1. (b) Effect of grain size on the stress dependence of the creep rate for the grain-boundary phase of Cu (σam denotes the strength of amorphous Cu) [251].
The deformation mechanism of the grain-boundary phase was modeled as a diffusional flow of matter through the grain boundary. A plot of the strain rate of the grain-boundary phase versus the applied stress is presented in Fig. 48(b). The strain rate increases with the stress until the stress reaches the strength of amorphous material, σam. The slope is independent of grain size. The predicted results are shown in Fig. 49 and have both positive and negative Hall–Petch regions in comparison with experimental results from the literature. The strain rate has a significant effect on the positive/negative HP transition: the transition, which is in the nanometer size, is shifted to lower strain rates for larger grain sizes. These results are similar to the ones by Fu et al. [104] (Fig. 40).
Fig. 49. Average grain size dependence of σ0.2 for fine grained Cu at strain rates of 10−2, 10−3, 10−4, and 10−5 s−1 calculated using a log-normal distribution of the grain volume [251].
Recent observations by Ma and coworkers [107], [108], [109] and [276] point out the interesting possibility, that nanosized grains rotate during plastic deformation and can coalesce along directions of shear, creating larger paths for dislocation movement. Fig. 50 shows this in schematic fashion. The orientations of the slip systems with highest Schmid factors are represented by a short line in each grain (Fig. 50(a)). As plastic deformation takes place, two neighboring grains might rotate in a fashion that brings their orientation closer together (Fig. 50(b)). This leads to the elimination of the barrier presented by the boundary between them, providing a path for more extended dislocation motion (Fig. 50(c)). This mechanism can actually lead to softening and localization, and is consistent with the limited ductility often exhibited by nanocrystalline metals.
Fig. 50. Rotation of neighboring nanograins during plastic deformation and creation of elongated grains by annihilation of grain boundary.
One way by which grains can rotate is by disclination motion. Murayama et al. [274] first observed disclinations in a mechanically milled Fe sample and suggested that the generation of partial disclination defects provides an alternative mechanism to grain-boundary sliding, which possibly allows rotation of nanosized crystals during mechanical milling. A disclination is a line defect characterized by a rotation of the crystalline lattice around its line [275]. Motion of a disclination dipole, which is a combination of two disclinations, causes plastic flow accompanied by crystal lattice rotation behind the disclinations. Fig. 51 shows the rotation of a grain when a nanocrystalline metal is subjected to tension. This is accomplished by disclinations as postulated by Ovid’ko [275]. Two opposite arrows indicate the clockwise direction of rotation of a grain. The insert in Fig. 51 shows a magnified view of a grain with non-parallel lines representing disclinations.
Fig. 51. Rotation deformation of nanocrystalline material under loading; grain undergoing rotation is shown with two opposite arrows and is represented in magnified fashion inn upper right corner [275].
HRTEM observation by Murayama et al. [274] of milled Fe sample showed two wedge-shaped regions that together form a partial disclination dipole. The TEM image demonstrates that it is possible to directly observe the individual dislocations that constitute partial disclination dipoles in metals at the atomic level, even in mechanically milled powders that have undergone severe plastic deformation.
It was proposed that the set of terminating planes that constitute the
individual partial disclinations, such as the one circled and labeled I in Fig.
52(c), can also be considered terminating tilt grain boundaries [267]
and [276].
Terminating tilt grain boundaries contained missing dislocations, and these were
replaced by rotational elastic deformation in the crystal [277].
It was further suggested that such a configuration can be interpreted as a wedge
of material added to or removed from an ideal crystal, as illustrated in Fig.
53. This is evident from the wedge shape of the bent white lines in Fig.
52(b) and (c). The crystal rotation produced by the partial disclinations in
Fig.
52 is also evident. For example, the {1 1 0} planes located
between the two partial disclinations (labeled II in Fig.
52 (c)) are rotated 9° relative to the {1 1 0} planes located outside the dipole.
This observation was claimed to provide direct confirmation that crystals can
rotate during severe plastic deformation by the action of partial disclinations.
Fig. 52. (a) HRTEM image of mechanically milled, nanocrystalline Fe powder. (b) White lines shown superimposed periodically on the three sets of {1 1 0} planes in (a) highlight the distortion of the nearly horizontal set of {1 1 0} planes. (c) Nearly horizontal black and white lines in (b) magnified from the HRTEM image so that they are more clearly visible. Scale bar in (a), 1.0 nm [274].
(45K)
Fig. 53. Illustration of the elastic distortion associated with a partial wedge disclination (or terminating tilt grain boundary). (a) A set of planes in a perfect crystal, such as the perfect {1 1 0} planes seen in Fig. 52. A wedge-shaped piece of material is removed from the crystal in (b), and the new surfaces are allowed to close in (c) to fill the wedge. The resulting crystal in (c) contains a terminating tilt grain boundary, i.e., a terminating array of edge dislocations, and considerable elastic distortion, which increases the elastic strain energy of the remaining solid [274].
Murayama et al. [274] claimed that the large stress field associated with partial wedge disclinations makes it difficult for other deformation defects to move through the metal, thus imparting it higher strength. It was further argued that the reorientation associated with the generation and interaction of partial wedge disclinations assist in the grain break-down mechanism. Thus, disclinations contribute both to deformation and strengthening. It was suggested as an alternative deformation mechanism to grain-boundary sliding.
The deformation mode of nanocrystalline materials is known to change as the grain size decreases into the ultrafine regime. For all smaller grain sizes (d < 300 nm) shear band development is often observed to occur immediately after the onset of plastic deformation. This has been correlated to changes in strain hardening behavior at those grain sizes (discussed in Section 5.4), since the ability to work harden by the increase in dislocation density is lost. Shear bands have been observed by Wei et al. [371] in both low and high strain-rate tests. It was observed that under dynamic loading, conventional polycrystalline iron did not exhibit localized deformation. Fig. 54 compares the low-rate compressive deformation response of iron with two different grain sizes in the ultrafine grain size regime. The deformation characteristics in 54(b) (d = 268 nm) are significantly non-uniform unlike 54(a) (d = 980 nm) where deformation was found to be uniform. Shear band populations similar to Fig. 54(b) were observed in all specimens with grain sizes d < 300 nm. For 268 nm-Fe, the response under high-rate leading to a similar strain was found to be similar as under low-rate loading (Fig. 54(c)).
Fig. 54. Change in deformation mode of ultrafine grained consolidated iron with grain size for uniaxial compression: (a) uniform low-rate deformation with d = 980 nm, ε = 13.7%; (b) non-uniform low-rate deformation with d = 268 nm and (c) non-uniform high-rate deformation with d = 268 nm; Loading axis is vertical [371].
The development of shear bands in quasistatic deformation was studied by Wei et al. [110] in quasistatic deformation (Fig. 55(a) and (b)); on increasing the strain from 3.7% to 7.8% in a nanocrystalline Fe sample, the number of shear bands increases from 3 to 8. The shear bands (Fig. 55(a) and (b)) were marked I, II, and III to compare the growth process of each individual band as the strain increased. Label A denotes a shear band tip. Fig. 55(c) shows a high-magnification view of a single shear band [121]. Fig. 55(d) shows an organized network of shear bands. It can be concluded that they play an important role in plastic deformation of ultrafine grained Fe. It was observed by Jia et al. [121] in ball milled fine-grained Fe samples that the basic mechanism of shear banding does not appear to change between the low-rate and high-rate loading and the failure mechanism is governed by grain size rather than strain rate.
Fig. 55. Evolution and development of shear bands in 268 nm-Fe: (a,b) observation of shear bands at the same location at different nominal strain levels: (a) 3.7%; (b) 7.8%. (c) Shear strain developed in shear band immediately after onset of plastic deformation (0.3% plastic strain). (d) Shear band network formed at strain of 7.8% [371].
TEM observations of the shear bands from outside and within the shear bands in 110 nm-Fe deformed in compression at low rates are shown in Fig. 56. A typical picture obtained from within the shear band showed elongated grains containing high dislocation densities. Fig. 56(b) obtained from a region outside the shear band, showed equiaxed grains. The distortion angle is approximately 7°, corresponding to a shear strain of 0.12.
Fig. 56. TEM micrograph taken within (a) and outside (b) a shear band in 138 nm-Fe [121].
These observations correspond to the mechanism of grain-boundary rotation
envisaged in Fig.
50. The equiaxed grain structure is replaced by elongated grains through
plastic deformation inside the shear band. It is interesting to mention at this
point that nanosized grains were also observed by Meyers et al. [279]
in adiabatic bands. Fig.
57 shows shear bands in AISI 304L SS where the original grain size of the as
received material was 30 μm. The grains within the band were equiaxed and smaller by two
orders of magnitude (100–200 nm) than the grains outside the band (original
microstructure). Similar ultrafine grained structures were observed in Ti
[280]
and [281],
Cu [282]
and [286],
Al–Li [283]
and [284],
and brass [285].
Fig. 57. (a) Ultrafine grain structure (d
Shear bands formed by the deformation of hat-shaped samples and the thick
walled cylinder technique [287]
and [288]
exhibiting equiaxed grains with diameters of 0.05–0.2 μm with relatively
low dislocation density, led Meyers and Pak [289]
to suggest that the structure was due to dynamic recrystallization. Dynamic
recrystallization in shock-conditioned copper was first observed by Andrade et
al. [282];
in tantalum it was first observed by Meyers et al. [290]
and Nesterenko et al. [291]
and confirmed by Nemat-Nasser et al. [292].
In Al–Li alloys, Chen and Vecchio [293]
and Xu et al. [294]
observed dynamic recrystallization. The microstructure of these different crystal structures (HCP, FCC, BCC) was
remarkably similar.
Meyers et al. [278], [280] and [283] suggested a rotational recrystallization mechanism in titanium, and this was supported by Meyers et al. [295], Andrade et al. [282] for copper, and Nesterenko et al. [291] for tantalum. Rotational recrystallization occurs as a series of steps in which the homogeneous distribution of dislocations rearranges itself into elongated dislocation cells, then as deformation progresses, these cells become elongated subgrains and finally these break up into approximately equiaxed micrograins. It is proposed that the relaxation of the broken-down elongated sub-grains into an equiaxed micro-crystalline structure can occur by minor rotations of the grain boundaries lying along the original elongated boundaries. The mechanism by which the ultrafine grained size is produced within an adiabatic shear band is thought to be similar to the one in UFD processes such as ECAP.
In conventional constitutive models, the stress is considered to be a function of strain, strain rate, and temperature. In gradient models, the stress is also assumed to be a function of the gradient of a variable. This gradient has taken several forms, depending on the researcher. In any case, the gradient introduces a spatial scale into the constitutive description. This length scale has been successfully used to describe the effect of grain size, all the way from the nanocrystalline to the monocrystalline domains, into the calculation of the strength.
An attractive approach to the analytical prediction of the grain-size dependence is through strain-gradient plasticity. Strain-gradient plasticity is a recent development that incorporates a length scale in the analytical treatment of plasticity problems. It enables the prediction of the effect of indentation size on the hardness of metals and ceramics, the effect of hard particles in the work hardening of metals, grain size, and other effects. Originally proposed by Aifantis [296] in the context of shear localization, it has been extended to a variety of problems by Fleck et al. [297] and [298] and Gao et al. [302] and [303]. From the dislocation perspective, the evolution of statistically stored and geometrically necessary dislocations, first proposed by Ashby [8], provides the underlying physics. The physical processes envisaged for the grain-size dependence of yield and flow stresses involve a length scale. Fleck et al. [297] and [298] apply the geometrically necessary (ρg) and statistically stored (ρs) dislocation densities to determine the athermal component of the flow stress, through the conventional Taylor expression:
| τμ=αGb(ρg+ρs)1/2 | (46) |
This theoretical framework does not incorporate the formation of a grain boundary work-hardened layer. Furthermore, it does not predict a grain-size dependence of yield stress, since both the curvature and plastic strain is zero prior to plastic deformation. As work hardening builds up, the effect manifests itself, resulting in an increased work hardening for decreasing grain size. It is the objective of this paper to provide some additional insights. The strain gradient theory does not take into consideration the difference in dislocation density observed between the grain interiors and the grain-boundary regions. A framework incorporating the strain gradient concepts and more attuned to the physics of the problem would be to track the dislocation densities in both grain interior and boundary. This has been done by Fu et al. [103], who introduced the gradient of the angle of maximum shear stress with respect to the external traction direction. This is described in Section 9.1.
Two types of twins are considered here: mechanical and growth (or recrystallization, or annealing) twins. Although they are crystallographically related, the phenomenology is completely different. Both play a role in nanocrystalline materials.
Mechanical twinning and slip can be considered as competing processes. Indeed, Meyers et al. [348] developed a constitutive treatment for twinning based on this. In FCC metals and alloys, the twinning stress is directly related to the stacking fault energy. As early as 1960, Venables [299] expressed the twinning shear stress, τT as a function of source size, l, and stacking-fault energy, γ:
Venables’ [34] equation predicts an increase in twinning stress with increasing stacking-fault energy and decreasing source size. Narita and Takamura [300] also found a similar relationship after analyzing several FCC metals and alloys:
Fig. 58 shows the relationship between a normalized twinning stress and a normalized stacking-fault energy. This fit was obtained by Voehringer [301]; the line was extended and the presumed twinning stresses for Ni and Al added. It is seen that the twinning stress for Al is quite high. The decrease in grain size is expected to render deformation twinning more difficult. At least, this is what conventional materials science predicts in the micrometer regime. This has been analytically expressed by Meyers et al. [306] in terms of two H–P relationships, one for twinning and one for slip:
| σT=σ0T+kTd-1/2 | (50a) |
| σS=σ0S+kSd-1/2 | (50b) |
kS. Fig.
59 shows the slip and twinning domains in a temperature versus strain rate
plot for iron. It is clear that, as the grain size is decreased, the twinning
domain shrinks. Eventually, it completely disappears.
Fig. 58. Effect of stacking fault energy on twinning stress.
(19K)
Fig. 59. Calculated effect of grain size on the twinning domain in pure iron [306].
This trend is confirmed by, e.g., El-Danaf et al. [307] who showed that, in a Co–Ni–Cr–Mo alloy, mechanical twinning could not be induced for grain sizes of 7 and 1 μm, while larger grain sizes twinned. In FCC metals, extreme deformation regimes (either low temperatures or high strain rates) have to be applied to produce twinning. Andrade et al. [282] showed that for small grain sizes even high shock pressures could not generate twinning. Aluminum, which has a stacking-fault energy higher than Cu, could not twin either at high shock pressures or by deformation close to 0 K.
It is therefore surprising that molecular dynamics simulations by Yamakov et al. [308] predicted mechanical twinning in the deformation of a nanocrystalline aluminum-like metal. Fig. 60 shows a snapshot at a plastic strain of 0.12 for aluminum with a grain size of 45 nm. The grey lines indicate regions where the stacking sequence has been altered. Several processes of plastic deformation involving grain boundaries are seen:
1. Heterogeneous nucleation of twin lamellae from the grain boundaries (e.g., τ1 in Grain 3).
2. Homogeneous nucleation of twin lamellae from the grain interior (e.g., τ2 in Grain 1).
3. Growth of twin lamellae to form a new grain (Grain A).
Fig. 60. MD simulation of nanocrystalline aluminum-like solid (d = 45 nm) having been subjected to plastic strain of 0.12; grey lines enclose regions with altered stacking sequence [308].
These MD simulations by Yamakov et al. [308]
and [309]
were followed, in 2003, with the first observations of twins in aluminum by Chen
et al. [310].
Fig.
61 shows TEM micrographs at increasing magnifications of aluminum that was
subjected to plastic deformation Fig.
61(a) shows planar defects within the nanograins. In Fig.
61(b) one can see that these defects are twins and stacking faults. The
atomic resolution micrograph of Fig.
61(c), which comes from the boxed area in (b) is clear evidence of twinning.
These twins in nanocrystalline aluminum were confirmed in TEM by Liao et al. [311],
among others. This finding has triggered a considerable debate, since aluminum
has among FCC metals, one of the highest SFE: 160 mJ/m2. Nanotwins are also being found in other
nanostructured metals, such as stainless steels (Fig.
62) [312].
Fig. 61. TEM micrographs of nanocrystalline aluminum deformed by grinding: (a) overall view of nanograins with planar features within grains; (b) multiple deformation twins and stacking faults and (c) blowup of box in (b) showing atomic resolution and zig-zag due to twinning [92].
(79K)
Fig. 62. Nanocrystalline AISI 305 SS produced within shear localization region in high-strain rate deformation [372].
It is early to propose a mechanism for deformation twinning in nanocrystalline metals, but some preliminary comments are of order. Meyers et al. [306] calculated the critical nucleus size for twinning as a function of applied stress using the Eshelby strain ellipsoid formalism. This analysis predicts a critical nucleus radius, rC, equal to
.gif){kind=small}
is the local (and not global) twinning
shear stress. The critical radius is plotted against the normalized shear stress
in Fig.
63 for Cu, Al, Ni, and Ag. At a local stress level of 500 MPa,
characteristic of nanocrystalline metals, the critical radius is on the order of
50 nm for Cu and Al; for Ni, it is even higher. What this indicates is that
the critical nucleus approach becomes untenable at grain sizes below
100 nm. There are at least three possible reasons for this: (a) the
conventional nucleation mechanism breaks down in the nanoscale; (b) there are
local stress concentrations (triple points, etc.) that raise the stress
significantly above the 500 MPa level, propitiating smaller critical radii;
(c) the partial dislocation separation increases in the nanoscale domain, aiding
twinning. The last possibility is discussed further in Section 7.8.
Fig. 63. Critical shear stress required for the activation of a twin nucleus as a function of size; notice that nucleus size is in the nanocraystalline domain for stresses on the order of the flow stress of nanocrystalline metals.
The effect of growth (or annealing, or recrystallization) twins has also been
investigated, and results are encouraging. Effort into optimizing strength and
ductility [278]
and [304]
has shown that annealing twins can play a significant role. Wang et al. [304]
produced microstructures consisting of a combination of nanoscale/ultrafine grains
(80–200 nm) along with about 25% volume fraction of coarser grains
(1–3 μm). They rolled the Cu to 93% at liquid nitrogen temperature and then
annealed it at temperatures below 200 °C. The original heavily cold worked
Cu had a high dislocation density along with some grains less than 200 nm.
Annealing resulted in a mixture of coarse and fine grained microstructure. The authors suggest that the excellent combination of
strength and ductility is the result of (1) multi-axial stress states in the
confined grains, (2) growth twins in the larger grains, and (3) preferential
accommodation of strain in the larger grains. The grain size distribution allows
for significant strain hardening which prevents localized deformation and
premature fracture.
Lu et al. [305] showed that a good trade-off between mechanical strength and electrical conductivity in copper could be made by introducing a high density of nanoscale growth twins. They demonstrated a tensile strength about 10 times higher than that of conventional coarse-grained copper, while retaining an electrical conductivity comparable to that of pure copper. They argued that the increased strength comes from the effective blockage of dislocation motion by numerous coherent twin boundaries that possess an extremely low electrical resistivity unlike other types of grain boundaries. Fig. 64 shows the plot of mechanical response of Cu sample with nanotwins in comparison with that for a coarse-grained and nanocrystalline sample. The uniform elongation increases from 1% to 10% with the introduction of nanotwins.
Fig. 64. Effect of nanotwin density on the mechanical response of copper [305].
Prompted by molecular dynamics simulations carried out primarily by Van Swygenhoven and coworkers [111], [255], [258], [313], [314], [315], [316], [317], [318], [319] and [320] and by TEM observations showing a low dislocation density after appreciable plastic deformation, a combined grain-boundary source-sink model is evolving. When the grain size is reduced to the nanocrystalline regime, the mean free path of dislocations generated at grain-boundary sources is severely limited. Rather than cross slipping and generating work hardening, these dislocations can run freely until they meet the opposing grain boundary, which acts as a sink. Thus, the dislocation density remains low throughout the plastic deformation process, and work hardening is not significant. Fig. 65 shows this model in a schematic fashion. Dislocations, which were generated at one grain boundary, run unimpeded until they encounter the opposing grain boundary (on right-hand side). It was seen in Section 7.3 that grain-boundary ledges are responsible for generation of plastic flow in the conventional polycrystalline regime. However, as the grain size falls below 20 nm, the grain boundaries will become virtually free of ledges, and intrinsic and extrinsic grain-boundary dislocations have to be “pushed out” into the grains. Another significant difference is that the mean free path of dislocations is limited by the grain size, and therefore dislocation reactions, cross slip, and other mechanisms of dislocation multiplication are effectively prohibited.
Fig. 65. Grain boundary source-sink model.
Molecular dynamics simulations in the recent past have helped the materials community to gain a better understanding of the deformation mechanisms in nanocrystalline materials, especially relating the role of the grain boundary in deformation process [256], [313], [314], [315], [316], [317], [318], [319] and [320]. Recent simulations have also reported a transition with decreasing grain size from plasticity related to dislocation activity towards a plasticity that is primarily accommodated in the grain boundary. This is treated in Section 7.2.
Fig. 66 [319] shows the emission of dislocations from grain boundary in specimens subjected to indentation. Nanoindentation techniques coupled with atomic scale simulation have shown that dislocations can be emitted, absorbed, and even reflected from grain boundaries [319]. The sample used was constructed using Voronoi polyhedra, with 15 grains having a mean diameter of 12 nm. The samples were indented with an indenter that had a size smaller than the grain size. The load was increased to a critical value whereby a dislocation homogeneously nucleated with a corresponding reduction in the indenter load. As indentation progresses (Fig. 66), a dislocation is emitted from just beneath the indenter, and propagates to the nearest grain boundary. On further indentation, the leading partial of the dislocation is completely absorbed by the grain boundary. On continuing the indentation, the trailing partial is also absorbed. The process of dislocation absorption is accompanied by an increase of the number of grain-boundary dislocations, rearrangement of grain-boundary dislocations, and a change in the grain-boundary structure. As indentation is continued, a large number of the dislocations that are emitted under the indenter start interact with the surrounding grain boundaries, some of them are absorbed, and some of them are reflected back to the plastic zone. Moreover, grain boundaries also act as sources of dislocations that are emitted into the plastic zone. This suggests that both the local stress and the coherency of a grain boundary play a combined role in the interaction between the grain boundary and the incoming dislocation.
Fig. 66. A series of snapshots, showing the atomic structure of a sample involving a dislocation as a function of indenter depth [319].
The processes identified through MD simulations have been studied analytically, and can indeed be predicted by using basic concepts from dislocation theory. It is instructive to calculate the stresses required for emission of dislocations from grain boundaries in the absence of ledges. Fig. 67 shows a dislocation being emitted form a grain boundary. To a first approximation, this dislocation is assumed to form, at the moment of release from grain boundary, a semicircle with radius r = d/2. The shear stress required to accomplish this is
α 1. The stress predicted from Eq. (52)
is quite high. For instance, for copper with a grain size of 50 nm, and
assuming α = 0.5, b = 0.25 nm, one
obtains τ = 242 MPa. For d = 20 nm,
τ = 603 MPa. These values are somewhat higher than
nanocrystalline strengths, but on the same order of magnitude.
Fig. 67. Perfect dislocation emitted from grain boundary as a loop.
Another, more realistic scenario is one in which a dislocation is emitted at a grain boundary and travels to the opposite grain boundary, leaving behind two sides, as shown in Fig. 68. Calculations were proposed by Liao et al. [325] and Asaro et al. [326]. We will present here a modified version of their analysis. Fig. 68 shows a dislocation emitted from the grain boundary AD. We assume a cubic grain, shown in Fig. 68(a). This dislocation, which we assume to be of edge character, leaves behind two trailing screw dislocations AB and DC. Fig. 68(b) shows the tridimensional picture. An array of such dislocations was emitted from the left-hand boundary and traveled an average distance x into the grain. The left-hand side of the grain is deformed by a shear strain γ. We will equate the work of deformation, dW, to the increase in energy due to the formation of the segments AB and CD on all loops, dE. This requires neglecting the frictional stresses as well as the energy required to emit the dislocations from boundary.
| dW=τγV | (53) |
| V=xd2 | (54) |
| γ=nb/d | (55) |
| dW=nbxτd | (56) |
| dE=2nxαGb2 | (57) |
Fig. 68. (a) Dislocation traveling through nanograin and leaving behind two segments AB and CD; (b) dislocations traveling through nanograin in parallel planes and creating a shear strain γ.
Eq. (58) is identical to Eq. (52). If one considers the emission of partials instead of perfect dislocations, one makes the substitution
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