Phase Mixture Models for Metallic Nanomaterials 1588830640

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Encyclopedia of Nanoscience and Nanotechnology

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Phase Mixture Models for Metallic Nanomaterials Yuri Estrin Clausthal University of Technology, Clausthal-Zellerfeld, Germany

Hyoung Seop Kim Chungnam National University, Daejeon, Korea

Mark B. Bush The University of Western Australia, Crawley, Australia

CONTENTS 1. Introduction 2. Phase Mixture Model for Plasticity of Nanocrystalline Materials 3. Phase Mixture Model for Elastic Modulus 4. Young’s Modulus of a Nanocomposite 5. Conclusions Glossary References

1. INTRODUCTION The mechanical properties of bulk materials are known to change with grain refinement. Extreme changes are expected when the average grain size is of the order of 100 nm or less. At the time when nanostructured materials (often defined as polycrystals with the average grain size of less than 100 nm) are becoming a major focus of materials research, the attention of researchers is turning more and more to their mechanical performance. Mechanical properties of nanostructured materials were recently reviewed by D. G. Morris [1]. His report provided an analysis of experimental literature and a comprehensive overview of the relevant deformation mechanisms. Still, the mechanical behavior of nanocrystalline materials remains a controversial

ISBN: 1-58883-064-0/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

issue and the roles of the different deformation mechanisms, such as dislocation glide and grain boundary processes, are not fully understood. Most recent molecular dynamics simulations [2] suggest that mechanical twinning may also play an important part in the plasticity of nanocrystalline materials, thus adding to the complexity of the emerging picture of the operating deformation mechanisms. Grain boundary sliding is often considered a deformation mechanism in nanostructured materials that permits a high level of strain before failure and may be responsible for superplastic forming (see [1] and references therein). Yamakov et al. [3] observed grain boundary sliding in their molecular dynamic simulations. As this mechanism would require grain boundary diffusion as an accommodation mechanism, it is claimed that grain boundary diffusion and grain boundary sliding “are closely connected and, perhaps, represent one and the same deformation mechanism”—a view the present authors subscribe to. In this chapter, we give an account of the mechanical properties of ultra-fine grained materials from our current viewpoint. It is based on recent modeling that appears to provide a conclusive description of the phenomenology and mechanisms underlying the mechanical properties of nanostructured materials. Common to the proposed models is the concept of a “phase mixture” in which the grain boundaries are treated as a separate phase. The volume fraction of this “phase” may be quite appreciable in a nanostructured material. The issues covered include strength, ductility, and the elastic properties of single-phase materials as well as the elastic properties of nanocomposites.

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 8: Pages (489–498)

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Phase Mixture Models for Metallic Nanomaterials

Experimental data on the mechanical properties of nanocrystalline metallic materials are appearing in literature at an ever-increasing rate. However, the range of data is still somewhat limited, except in the case of copper, where an extensive experimental database already exists. The various published mechanical properties of nanocrystalline Cu are summarized in Table 1. The table highlights a fundamental difficulty associated with the interpretation of experimental measurements made on nanomaterials. Residual porosity is an omnipresent feature of specimens manufactured using common production techniques. Porosity influences the observed bulk mechanical properties and, in fact, may dominate the behavior. Care must therefore be taken when drawing conclusions based on the bulk behavior. However, as will be seen later, porosity may be included in the phase mixture model to permit the effects of pores to be isolated from the grain size effects.

2. PHASE MIXTURE MODEL FOR PLASTICITY OF NANOCRYSTALLINE MATERIALS It is generally believed that grain refinement leads to enhancement of strength of metallic materials. Indeed, the classical Hall–Petch relation predicts an inverse square root dependence of the yield strength on the average grain size. However, it is clear that there must be a limit to an increase of strength with decreasing grain size. Grain refinement will reduce the characteristic length for vacancy diffusion. Mass transport by diffusion across the grains and along grain boundaries therefore becomes more significant and may eventually prevail over dislocation glide-controlled plastic flow. In addition, the very mechanism of dislocation glide may become inoperative when the grain size drops below a certain critical level, perhaps around 10 nm [4], when dislocation sources in the grain interior or in grain boundaries cannot be activated. The rate of plastic flow will then be controlled by vacancy diffusion through the bulk (the Nabarro– Herring mechanism) or along grain boundaries (the Coble

mechanism). In both cases, the plastic strain rate is proportional to stress and inversely proportional to the average grain size, d, raised to the power n, where n = 2 for the Nabarro–Herring mechanism and n = 3 for the Coble mechanism [5]. Hence, for a fixed strain rate, the stress turns out to increase with the average grain size as d 2 or d 3 , respectively. There is evidence in literature [6–7] that a pronounced departure from, or even inversion of the Hall–Petch behavior, is observed for nanostructured materials. Unfortunately, the picture is clouded by the inevitable presence of porosity, as fully dense specimens of very finegrained materials are difficult to prepare. In such cases, mathematical modelling offers a unique opportunity to provide further insight into the behavior of such materials. Ductility usually decreases with increase of strength, but expectations have been expressed in literature [8] that for the grain size in the submicrometer range, enhancement of strength without loss, or even with an increase, of ductility can be achieved. Such expectations were raised [9], particularly with regard to materials that have undergone severe plastic deformation, for example, by equal-channel angular pressing. The challenge in modelling the effects of ultra-fine grained structures is not only to capture the relevant physical phenomena, including dislocation glide and diffusion mechanisms, but also to correctly represent their relative contributions to the mechanical behavior of the material. In modelling the behavior of actual test specimens, porosity or other embedded flaws may represent an important component of the model. A constitutive model has been proposed that provides an adequate description of the grain size dependence of strain hardening in a broad range of grain sizes, down to the nanometer scale [10–12]. A distinctive feature of the model is that grain boundaries are treated as a distinct separate phase [4, 13], which is assumed to deform by a diffusion mechanism [12]. The grain interiors are considered to represent another phase. This phase is assumed to deform by a combination of two mechanisms: dislocation glide and diffusion transport. A transition between the regimes where the different mechanisms prevail can therefore be incorporated.

Table 1. The mechanical properties of nanocrystalline copper. Property Young’s modulus E Ultimate tensile strength Tensile elongation

Micro-hardness

(Grain size, property value, density)

Ref.

(10 nm, 108 GPa, 97.6%), (16 nm, 113 GPa, 98.6%), (22 nm, 116 GPa, 98.9%), (100
m, 131 GPa, 100%) (16 nm, 340 MPa, 98.6%), (26 nm, 425 MPa, 99%), (49 nm, 460 MPa, 99.1%), (110 nm, 415 MPa, 99.4%), (22 nm, 480 MPa, 98.8%), (20
m, 290 MPa, 100%) (16 nm, 2.4%, 98.6%), (26 nm, 1.8%, 99%), (49 nm, 1.6%, 99.1%), (110 nm, >8%, 99.4%), (22 nm, 1.6%, 98.8%), (20
m, >8%, 100%) (15 nm, 1.52%), (20 nm, 0.79%), (25 nm, 6.3%), (50 nm, 2.2%), (61 nm, 2.2%) (209 nm, 10%), (206 nm, 17%) (29 nm, 0%), (32 nm, 2%), (31 nm, 7%), (41 nm, 5%), (58 nm, 5%), (175 nm, 5%) (26 nm, 0.7%), (39 nm, 0.3%), (49 nm, 1.2%), (110 nm, 10%), (20000 nm, 15%) (210 nm, 5%) (34 nm, 797 MPa), (26 nm, 802 MPa), (25 nm, 779 MPa), (24 nm, 767 MPa), (22 nm, 752 MPa), (22 nm, 773 MPa), (18 nm, 696 MPa), (18 nm, 748 MPa), (18 nm, 769 MPa), (19 nm, 801 MPa), (21 nm, 822 MPa), (21 nm, 884 MPa), (18 nm, 889 MPa), (16 nm, 877 MPa), (16 nm, 854 MPa), (16 nm, 810 MPa), (16 nm, 812 MPa), (12 nm, 849 MPa), (11 nm, 773 MPa), (10 nm, 793 MPa), (10 nm, 828 MPa), (7 nm, 806 MPa) (7.8 nm, 144 MPa), (8.4 nm, 173 MPa), (14 nm, 207 MPa), (15 nm, 223 MPa)

[21] [21] [21] [26] [28] [27] [18] [29] [21]

[51]

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Phase Mixture Models for Metallic Nanomaterials

Combined with a rule of mixtures, the constitutive equations for the mechanisms considered provide a simple description of the deformation behavior [10–12]. This model is illustrated by application to ultra-fine grained Cu—the material for which a most extensive experimental database is available. The grain boundary phase is considered to deform by a vacancy diffusion mechanism [12]. The corresponding plastic strain rate is given by ˙GB = A

GB DGB kT d 2

(1)

Here GB is the stress acting in the grain boundary (GB) phase,  the atomic volume, T the absolute temperature, k the Boltzmann constant, DGB the self-diffusion coefficient for grain boundary diffusion, and A is a numerical constant. This formula represents an interesting “hybrid” of the Nabarro–Herring and the Coble equations in that the grain size dependence is that of Nabarro–Herring, while the (grain boundary) diffusivity is that of the Coble mechanism. A similar relation was recently obtained through molecular dynamics simulations [3]. Plastic flow of the crystalline grain interiors is considered to be carried by two mechanisms that operate in parallel: the dislocation glide mechanism (contributing the plastic strain rate ˙disl ) and the diffusion-controlled mechanism. In principle, both bulk and grain boundary diffusion need to be considered. However, for most cases of interest in connection with ultra-fine grained materials, the bulk diffusion contribution can be neglected [10]. Accordingly, the plastic strain rate for the grain interior (GI) phase can be written as ˙GI = ˙disl GI + ˙Coble

(2)

where ˙Coble = 14

GI w DGB · · 2 kT d d

(3)

Here GI is the stress acting in the grain interior and w is the grain boundary width. We note that the latter contribution to the strain rate is similar to the expression for the plastic strain rate of the grain boundary phase, Eq. (1)—apart from the exponent in the grain size dependence. The dislocation glide contribution to the plastic strain rate of the grain interior phase is described in terms of a model [14] based on the dislocation density evolution. The total dislocation density is considered to be an internal variable. The evolution of dislocation density is affected by the grain size through its influence on the dislocation mean free path. The relation between the plastic strain rate ˙disl and stress is given by 
GI mi −mi /2 Z  d − dc (4) ˙disl = ˙o o where Z denotes the dislocation density normalized with respect to its initial value, and the parameters ˙o , o , and mi can be considered constant for a given temperature. The values of these parameters for Cu are known [15]. The Heviside step-function  d − dc expresses the fact that for d below some critical value, dc , the dislocation mechanism becomes

inoperative and its contribution to plastic strain rate, ˙disl , drops to zero. Following Wang et al. [4], the value of dc for Cu was taken to be 8 nm. The evolution equation for the dislocation density is
 −1/n  √ ˙ dZ = ˙disl C + C1 Z − C2 disl Z dt ˙o

(5)

Here t denotes the time, C C1  C2 , and ˙o are constants, while n is constant if the temperature is fixed. Grain size effects, which are of main concern here, enter through the term C, which is inversely proportional to d [14]. It is further assumed that the strain in both phases is the same (and equal to the macroscopic strain). Henceforth, the subscripts “GI” and “GB” in the respective strain rates will be dropped. The stress is determined using a rule of mixtures,  = fGB + 1 − f GI

(6)

where f is the volume fraction of the grain boundary phase given by 
w 3 1−f = 1− d

(7)

Equation (7) reflects the assumption of cube-shaped grains. A schematic illustrating the phase mixture model is shown in Figure 1. The set of Eqs. (1–7) furnish a full constitutive description for a “single-phase” material, which, in fact, is treated at the microscopic level as consisting of two subphases. It has been applied to the case of Cu [10–12] for which the parameters pertaining to the dislocation mechanism and the grain boundary diffusion mechanism are known from previous work [5, 15–16]. As an example, the stress versus strain curves for tensile deformation with constant strain rate are presented in Figure 2. The curves show a good agreement with reported experimental data [17–18]. The dependence of the offset yield stress, 02 , on the calculated grain size using this model is plotted in the “Hall– Petch” diagram of Figure 3. It is seen that a Hall–Petch-like behavior in the range of large grain sizes breaks down for d below about 50 nm, and an “inverse” Hall–Petch behavior

Figure 1. Schematic of two-phase model of grain deformation. The contribution of the Nabarro–Herring (N–H) creep mechanism in the grain interior becomes insignificant for very small grain size, d. Reprinted with permission from [10], H. S. Kim et al., Acta Mater. 48, 493 (2000). © 2000, Elsevier Science.

492

Figure 2. Calculated and experimental stress-strain curves for Cu at the strain rate of 10−3 s−1 . The solid lines are the model predictions, while the symbols refer to experimental data [17–18]. Reprinted with permission from [10], H. S. Kim et al., Acta Mater. 48, 493 (2000). © 2000, Elsevier Science.

˙ The numerical coris predicted for small values of d and . respondence between the calculated curves and the experimental results is rather poor, but the main trends predicted seem to be confirmed by experiments. It should be noted, however, that the experimental data were obtained for specimens manufactured by different methods [17–21], but in all cases, residual porosity and other imperfections may be responsible for the over-prediction of strength by the model considered. The model can also be used to study the grain size dependence of the tensile ductility. Koch and Malow [22], in their compilation of ductility data, demonstrated that ultra-fine grained single-phase materials exhibit little roomtemperature ductility in tension. For elemental metals that are ductile when coarse grained, the data show a clear trend for decreasing tensile ductility with decreasing grain size (Fig. 4). A similar behavior was recently observed in Al alloys [23]. The effect of grain size on ductility was

Figure 3. Average grain size dependence of 02 for fine grain Cu at strain rates of 10−2 , 10−3 , 10−4 , and 10−5 s−1 calculated using the phase mixture model with a log-normal distribution of the grain volume. Also shown are the experimental data points (see Fig. 2). Reprinted with permission from [12], H. S. Kim et al., Mater. Sci. Eng. 316A, 195 (2001). © 2001, Elsevier Science.

Phase Mixture Models for Metallic Nanomaterials

Figure 4. Strain to necking (calculated) and experimental tensile room temperature ductility as a function of the average grain size d. Symbols:
Ref. [26],  Ref. [27], △ Ref. [28], ▽ Ref. [18], ♦ Ref. [29]. Reprinted with permission from [24], H. S. Kim and Y. Estrin, Appl. Phys. Lett. 79, 4115 (2001). © 2001, American Institute of Physics.

considered [24] using Hart’s condition for the onset of necking [25]:  ≤  1 − 1/m

(8)

Here the strain hardening coefficient and the effective strain rate sensitivity parameter are defined as  = / ˙ and ˙  , respectively. It can easily be seen 1/m =  log / log  from this criterion that for extremely fine-grained materials, when plasticity is carried chiefly by the diffusion mechanisms and m tends to unity, tensile deformation will always be stable, strain-to-necking tending to infinity, as long as strain hardening is nonzero. That is to say, the material will never fail by necking, its ductility being limited by other failure mechanisms. In a real case, however, even for a very small grain size, the fulfillment of the necking condition, indicated by the equality in Eq. (8), will depend on the interplay of m, , and . For d in the range, where diffusion mechanisms of plasticity prevail over the dislocation glide mechanism, 1/m tends to unity and the strain-hardening coefficient tends to zero. Indeed, diffusion-controlled mechanisms do not exhibit any strain hardening. It is apparent that the loss of strain hardening as grain size is decreased cannot be compensated for by the stabilizing effect of the growing strain rate sensitivity, as it is not close enough to unity to outstrip the destabilizing effect. Numerical simulations of a strain rate jump test can be used to determine the dependence of the instantaneous strain rate sensitivity 1/m on the grain size. These numerical experiments show that the strain dependence of the rate sensitivity parameter 1/m is fairly weak. Substituting the values of , , and 1/m (calculated for strain of 0.2%) into Hart’s condition, the grain size dependence of strainto-necking was determined. This dependence is shown as a solid line in Figure 4. The results presented in Figure 4 demonstrate that the experimentally observed reduction of tensile room temperature ductility with grain refinement is in close quantitative agreement with the model. The loss of ductility with grain refinement found for room temperature deformation of copper does not necessarily represent a generic trend with all

493

Phase Mixture Models for Metallic Nanomaterials

materials at all temperatures. In fact, a simple linear stability analysis shows that the characteristic strain required for an incipient neck to grow to a sizable defect, growth , is of the order of 1 − / · 1/m. That is, it scales with 1/m. As seen in Figure 5, which shows the grain size and temperature dependence of 1/m for small grain sizes (below 50 nm, for instance), growth increases with temperature quite significantly and rapidly becomes more important than the strain to the onset of necking. In other words, a tensile specimen can sustain a significant strain as the pace of neck development is slow. The effect becomes more pronounced with decrease of the grain size. It can be stated that while nanostructured materials do not possess good room temperature ductility, their ductility at elevated temperatures is much better than for their coarse-grained counterparts. Thus, a common expectation that the temperature and strain rate range for superplastic forming may be moved to lower temperatures and higher strain rates by grain refinement down to the submicrometer range is qualitatively supported by these calculations.

3. PHASE MIXTURE MODEL FOR ELASTIC MODULUS There are some conflicting results in the literature regarding the variation of elastic modulus E with grain size in nanostructured materials [20–21, 30–34]. Most of the experimental results indicate a reduction in E when the grain size is reduced to nanoscale [20–21, 30–31], although some reports suggest there is only little, if any, grain size effect [32–34]. Many authors [20–21, 31–32, 35] attribute the decrement of E to porosity and assume the intrinsic elastic modulus of the material itself is independent of the grain size. For example, Sanders et al. [21] relate the decrement of E for nanocrystalline Cu to the degree of porosity by analogy with the porosity dependence of E for coarse-grained Cu. By contrast, several investigators [30, 36] assumed an intrinsic grain size dependence of E. Weller et al. [30] attributed a reduction in the shear modulus G for nanocrystalline Pd to increased atomic interaction distance in the interface region. Similarly, Chen [36] explained a reduction

Figure 5. Grain size and temperature dependence of 1/m. Reprinted with permission from [24], H. S. Kim and Y. Estrin, Appl. Phys. Lett. 79, 4115 (2001). © 2001, American Institute of Physics.

in E for nanocrystalline Fe in terms of increased spacing between atoms in the grain boundary. Since the elastic modulus is a measure of the bonding between atoms, an overall decrease in E is expected in nanostructured materials as a result of the increased volume fraction of the less stiff grain boundary phase. Porosity can be taken into account by including pores as a third phase with bulk modulus K and shear modulus G set to zero. The phase mixture model is combined with Budiansky’s self-consistent method [37] for a random mixture of isotropic constituents. Since the bulk modulus and shear modulus of the pore are both zero, the bulk modulus K and the shear modulus G of the composite are related, in Budiansky’s self-consistent approach, to the moduli Ki and Gi of the ith constituents by the following equations: fGI and

1 − a + a KKGI fGI 1−b+

b GGFI

+

+

fGB 1 − a + a KKGB fGB 1−b+

b GGGB

+

fPORE =1 1−a

(9)

+

fPORE =1 1−b

(10)

where
 1 1+# 3 1−#
 2 4 − 5# b= 15 1 − # a=

(11) (12)

and #, Poisson’s ratio of the phase mixture, is, in turn, given by #=

3K − 2G 6K + 2G

(13)

Here, the pore volume fraction, fPORE , was introduced, in addition to the volume fractions of the GIs and the GBs. In this formulation, the volume fractions of the grain boundaries proper, the triple lines, and the quaternary nodes were lumped together in the quantity fGB . An alternative approach was adopted in [38], where these regions were treated separately. The Newton–Raphson method was used to obtain the solution of the set of coupled Eqs. (9–13). The model was applied to three metallic systems (Cu, Fe, and Pd), as data for these materials in the nanocrystalline range are available in literature. The local elastic constants for the grain boundary region were calculated by atomistic simulation to be around 70% of the values for the crystalline phase [39]. Thus, the elastic modulus of the grain boundary was taken to be 70% of that for the corresponding coarse-grained polycrystal. Figure 6 shows the calculated elastic modulus as a function of grain size for Fe, Pd, and Cu for various levels of porosity [21, 31]. It can be seen that the elastic modulus decreases very slowly with decreasing grain size as long as d remains in the range of relatively coarse grain size (>20 nm), but this decrease becomes very rapid when the grain size is below about 20 nm. It was also found that for a given level of porosity, the shape of the E versus d curve follows the grain size dependence of the volume fraction of the crystallite phase fCR . The effect of porosity is clearly much

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Phase Mixture Models for Metallic Nanomaterials

more significant than that of the grain size. Since nanostructured materials usually contain some porosity (whose level depends on the processing route and conditions), the effect of the grain size on the elastic modulus will, in many cases, be insignificant. The calculated values of E for the Cu and Pd specimens correlate reasonably well with the experimental data. In the case of Fe, the theoretical results show a good agreement with experimental data, both for relatively coarse grained (d = 100 nm, 90 nm, and 44 nm) and ultrafine-grained (d = 4 ∼ 20 nm) material.

4. YOUNG’S MODULUS OF A NANOCOMPOSITE Using the same concept of a phase mixture, Young’s modulus of a nanocomposite—a particle reinforced metal matrix composite with an ultra-fine grain size—can be evaluated [40]. The composite is now treated as a mixture of a metallic matrix phase, a reinforcing ceramic particle phase, and a boundary phase. Porosity—an inevitable companion of nanostructured materials—can be treated as a separate phase, but it can also be considered to reside in the boundary phase [40]. Indeed, residual pores are usually found at the interfaces between reinforcement particles and the matrix, as well as at the boundaries between matrix grains themselves. For example, Kizuka et al. [41] reported the evidence of voids residing on grain boundaries rather than in the grain interiors: an observation valid for many compacted MMCs. Figure 7 shows a schematic representation of the phase mixture model identifying a unit cell [40]. The material is considered to be constructed by a periodic reproduction of this unit cell in three dimensions. The unit cell consists of a reinforcing particle and the matrix material separated from each other by the boundary phase. It should be noted that the matrix is not considered to be a contiguous phase, but is rather taken to consist of compacted matrix material “particulate.” The model has been applied to a composite containing ceramic inclusions with an average diameter of 200 nm, a grain size of 247 to 126 nm, and an interparticle spacing of 330 to 137 nm [40].

Figure 6. Effect of grain size and porosity on the elastic modulus E for Fe, Cu, and Pd. (a) Fe, data points from [31], experimental porosity level (%): (1) = 55, (3) = 10, (5) = 125, (7) = 185, (8) = 24, (9) = 28, (10) = 355, (a) = 4, (b) = 10, (d) = 18, (g) = 25, (h) = 5, (i) = 10, (j) = 115, (l) = 15, (m) = 20, (n) = 25, (A) = 0, (C) = 6, (D) = 19, (F) = 22, (H) = 25, (J) = 30; (b) Cu, data points from [21], experimental porosity level (%):  = 0,  = 11,  = 14,  = 24; (c) Pd, data points from [21], experimental porosity level (%):  = 0,  = 15,  = 21,  = 49. Reprinted with permission from [38], H. S. Kim and M. B. Bush, Nanostructured Mater. 11, 361 (1999). © 1999, Elsevier Science.

Figure 7. Phase mixture model for a particle reinforced composite with ultra-fine microstructure. Reprinted with permission from [11], H. S. Kim et al., Mater. Sci. Eng. 276A, 175 (2000). © 2000, Elsevier Science.

495

Phase Mixture Models for Metallic Nanomaterials

The model of Figure 7 has the provision for distinguishing between the properties of the boundaries between matrix particles and matrix/inclusion interfaces; however, the same properties were assumed for both kinds of interface. A cylindrical geometry was assumed for the ceramic particles. The overall effective relative density, Reff , of the composite can be represented in terms of the volume V and density & of each phase as [11]: Reff =

Vp &p + Vm &m + Vb &b V p &p + V m &m + V b &m

(14)

Here the subscripts m, p, and b refer, respectively, to the matrix, particles, and boundaries. From Eq. (14), the density of the boundary phase &b can be obtained when the volume and density of the other two phases and the overall density are known. The unit cell dimensions used in the calculations are based on experimental data [42]. Three kinds of ball-milled, compacted, and sintered composites with reinforcement volume fractions of 0, 10, 20, and 30% (referred to as C-00, C-10, C-20, and C-30) were prepared. A fully dense bulk Al material (referred to as “Al standard”) was used as a reference material. The details (volume fraction, hardness, grain size, and density) of these materials are summarized in Table 2. Although these materials are not fine scaled enough to be referred to as genuinely nanocrystalline, the general trends observed can be translated to nanocomposites. The particle radius was taken to be 100 nm. The unit cell size (radius) was considered to be 247 nm, 215 nm, 171 nm, and 149 nm, which corresponds to the particle volume fraction of 0, 10, 20, and 30%, respectively. The notion of a “generalized” boundary phase in the current model includes various imperfections, such as voids and cracks in the interface region. Hence, its properties need not be identical with those of the grain boundary proper. The effective width of the grain boundary was selected by taking into account the following considerations [16]. Using Auger electron spectroscopy of the surface products on atomized stainless steel powder particles, as well as on the particle boundaries that existed prior to compaction [43], the average oxide thickness was estimated to be 6 nm. This is much larger than a typical grain boundary thickness in elemental Table 2. Microstructure and measured properties of the composite materials investigated. Al Standard Volume fraction of reinforcement, % Volume fraction of boundary phase, % Matrix grain size (diameter), nm Unit cell size (radius), nm Overall relative density, % Local relative density of boundary phase, %, as obtained using Eq. (2)

C-00

C-10

C-20

C-30

0

0

10

20

30

0

6

9

13

15

—

247

218

162

126

247 958 294

215 924 111

171 952 584

149 969 771

— 100 100

metals usually taken to be of the order of 1 nm. The inhomogeneous surface region, as indicated by the variations of the chemical composition, reached an even larger depth of up to 10 nm. This value may be taken as representative of the effective boundary thickness scale for the case considered. Fortunately, the sensitivity of the mechanical properties of interest here to the boundary thickness is fairly weak: both the magnitude of the strength and the effective Young’s modulus of the composite vary with the boundary width only slightly, the general trends remaining unchanged. The calculations that follow were done for a constant thickness of the boundary phase (assumed to be amorphous and porous) set at 1, 6, and 10 nm. The dependence of mechanical properties on porosity can be represented by various empirical equations [44]. Spriggs’ empirical relation [45] for Young’s modulus, E, was adopted for the grain boundary phase where porosity was assumed to be concentrated [40]: EGB = E0 e−s 1−R

(15)

Here, E0 is zero-porosity Young’s modulus of the grain boundary material, R is the relative density (the ratio of the actual and the theoretical density), and s is a material constant that was found to range between 3 and 4.5 [45]. Young’s modulus and Poisson’s ratio of the particle material (Al2 O3 , in the particular case of an Al-Al2 O3 composite considered) were taken to be 380 GPa and 0.22 [42], respectively. Fracture of particles and debonding between a particle and the matrix were not included. The modulus of elasticity of the grain boundary phase in the absence of porosity cannot be calculated from the available experimental data. The boundary phase was considered to be amorphous in character. Indeed, glass-like behavior in nanocrystalline alloys has been reported in the literature [46]. This provides certain albeit indirect support for the above assumption. It is generally accepted that the modulus of elasticity of an amorphous material Eam amounts to 60∼75% of that of the corresponding equilibrium crystalline alloy [47]. This value is further reduced by the effect of porosity, in accordance with Eq. (15). Thus, the modulus of elasticity of the Al matrix, the Al2 O3 reinforcement, and the boundary phase used in the calculations were 70, 380, and 47.3 GPa, respectively. The classical models of Voigt and Reuss determine, respectively, the upper and the lower bounds of Young’s modulus of the composite in the form Eupper = *fi Ei 

Elower = *fi /Ei −1

(16)

where fi and Ei are the volume fraction and Young’s modulus of the component i. In a nontextured polycrystalline material, the effective modulus often lies between these two bounds, as determined by Hashin and Shtrikman [48]. Results very similar to those by Hashin and Shtrikman may be obtained by making use of Hill’s approach [49], in which an effective Young’s modulus is determined simply as the average of the upper and the lower bounds: Eeff = Eupper + Elow /2

(17)

496 The volume fractions of the reinforcing particle, matrix, and boundary phases as a function of the unit cell size are shown in Figure 8 for the boundary width of 10 nm and the particle radius of 100 nm. As the radius of the unit cell ru increases, the volume fraction of the matrix increases as well, while those of the particles and the boundary decrease. The trends seen in Figure 8 are consistent with the results presented in Table 2. It should also be noted that as the volume fraction of reinforcing particles increases, the grain size of the matrix decreases, and the volume fraction of the boundary phase increases. While the density of the grain boundary phase taken in isolation is hardly accessible to experimental determination, it can be deduced from the overall density of the composite and the volume fraction of reinforcement particles. Figure 9 represents the contour plot for the relative density of the boundary phase as a function of the volume fraction of the particles and the relative density. As the overall relative density increases at a constant volume fraction of reinforcement particles, the local relative density of the boundary phase increases as well. When the volume fraction of the reinforcement particles is low, the relative density of the boundary phase is also low for a constant overall relative density. It should be noted that the effect of the overall relative density on the relative density of the boundary phase is much more significant than the effect of the volume fraction of the reinforcement particles. That is, the relative density of the boundary phase is inversely proportional to its volume fraction, since the volume occupied by pores, given by the product of the local porosity and the boundary volume, is constant. As shown in Table 2, the overall densities of the C-00, C-10, C-20, and C-30 specimens are 95.8, 92.4, 95.2, and 96.9% and the local relative densities in the boundary phase are 29.4, 11.1, 58.4, and 77.1%, respectively. The substantial variation in the derived density of the boundary phase may come as a surprise. However, it must be remembered that the model assumes that all the porosity resides in the grain boundary phase and the relative density of the grain interior is 100%. When the volume fraction of the grain boundary phase is small (For instance,