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Dual phase nano-particulate AlN composite — A kind of ceramics with high strength and ductility
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Ceramics International journal homepage: www.elsevier.com/locate/ceramint
Dual phase nano-particulate AlN composite — A kind of ceramics with high strength and ductility Yinbo Zhaoa,b, Xianghe Penga,∗, Bo Yanga, Cheng Huanga, Ning Hua, Cheng Yanb,∗∗ a
College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, 2 George Street, G.P.O. Box 2434, Brisbane, Australia
b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanocomposite AlN Shear band Phase transformation Interface Molecular dynamics simulations
Ceramics are widely used in many fields due to their excellent properties. However, the brittle fracture is a short board restricting their applications. To understand their deformation mechanism and explore a way to enhance both the strength and ductility, we investigated the mechanical behaviour of dual-phase AlNs composed of amorphous AlN matrix and crystalline nanoparticles under compression via molecular dynamics simulations. The stress concentration exists at the interface of nanocomposite AlN, where the particles and matrix are in the tensile and compressive states of stress, respectively. Strain hardening occurs when crystalline nanoparticle fraction fv ≥ 40.9%, attributed to the intersection between shear bands. The phase transformation from wurtzite structure (B4) to graphene-like structure (GL) is observed in the crystalline phase, as a result of high hydrostatic stress. After phase transformation, the particle might be cut into half during further compression along with the recovery of the GL structure to the wurtzite structure that could still bear load. The investigation of the effects of the volume fraction, surface-to-volume ratio, distribution pattern of the crystalline nanoparticles indicates that the dual-phase AlN nanocomposite with fv ≥ 40.9% and triangle distribution of particles would possess both higher strength and ductility.
1. Introduction Ceramics own many excellent physical and mechanical properties like low density, high melting point, high hardness, etc., however, its brittle nature strongly restricts their applications [1]. A lot of effort has been devoted to preparing ceramics of high toughness [2], among which the introduction of second ductile phases seems promising, such as fibers [3], graphene platelets [4], polymer [5], carbon nanotubes (CNT) [6], and metal infiltration [7], etc. However, these methods can toughen the ceramics by triggering crack defects instead of suppressing crack initiation with less ductility [8]. Similar issue also occurs in metallic glasses, which are a class of amorphous materials that possess high strengths and high elastic strain limits (∼2%) [9]. Dual-phase metallic glass was also developed to achieve better ductility [10]. Recently, Wu et al. [11] designed a dual-phase metallic glass which exhibits near-ideal strength as the localized shear bands are impeded by the crystalline phase. Different from the metallic glass, the amorphous ceramics can endure notable plastic deformation [12] but have lower strength
∗
compared with their crystalline counterparts [13,14]. Based on the concept of dual-phase metals, a dual-phase amorphous carbon ceramic was synthesized, which exhibits high strength and ductility under compression [15]. Although experimental work has shown the important role of second phases in the mechanical performance of the dual-phase ceramics, it is difficult to observe interactions and evolutions of the internal structures, such as shear band and inclusions, and to uncover the mechanism for the high strength and ductility. Molecular dynamics simulation (MD) is a suitable means that has been widely used to investigate the responses and the corresponding atomic structures (including the shear bands [14,16], dislocations [17,18], and phase transformation [19], etc.) as well as their evolutions during the deformations at nanoscale. Aluminium nitrides (AlNs) have attracted considerable attention due to their distinctive properties - high hardness [20], wide band gap [21], high piezoelectric coefficients [22], etc., which have been extensively applied in the fabrication of different kinds of devices in micro electromechanical system (MEMS) [23], high power electronics [24], and optoelectronic materials [25], etc. To achieve higher ductility and
Corresponding author. Corresponding author. E-mail addresses: [email protected] (X. Peng), [email protected] (C. Yan).
∗∗
https://doi.org/10.1016/j.ceramint.2019.06.239 Received 10 May 2019; Received in revised form 14 June 2019; Accepted 22 June 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Please cite this article as: Yinbo Zhao, et al., Ceramics International, https://doi.org/10.1016/j.ceramint.2019.06.239
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where it can be seen that there exist several peaks in the curve of nanoparticles, which indicates that the nanoparticle is of crystalline structure. In contrast, the curve corresponding to the matrix does not show the characteristic of long range order, indicating that the matrix is amorphous. Using the melt-quench-duplicate approach, an a-AlN sample is also prepared for the comparative MD simulations [14]. It is crucial to select an appropriate interatomic potential in MD simulations. Xiang et al. [33] compared different kinds of interatomic potentials for AlN and found that Vashishta potential [34] is more suitable to describe the mechanical properties of AlN. This potential has been used to explore the deformation mechanism of crystalline AlN during indentation [35] and shock loading [36], and the mechanical responses of a-AlN under uniaxial loading [14,28]. In this work, the mechanical responses of nc-AlN samples subjected to uniaxial compressions are investigated. The compression is applied in z-direction at a strain rate of 108 s−1, PBCs are maintained in the yand z-directions while free surface condition is imposed in the x-direction. A microcanonical (NVE) ensemble is adopted with the temperature kept at 10 K using Langevin thermostat. Atomic stresses is calculated through the virial stress [37]. The local strain shear invariant
toughness, Heard et al. [26] introduced high hydrostatic pressure condition and Guo et al. [27] decreased the sample size to promote the plastic flow in AlN. We found that amorphous AlN (a-AlN) possesses high ductility, which could be ascribed to the self-repairing capability [14] and less sensitivity to fissures [28] compared with wurtzite structured AlN (w-AlN). However, a-AlN has low strength due to its amorphous structure, which is different from crystalline materials, in which there are grain boundaries, twin boundaries and dislocations, which may induce strengthening. In this work, we introduce crystalline AlN nanoparticles into amorphous AlN matrix for the purpose to obtain dual-phase AlN composites with both high strength and ductility. Although much effort has been made in the research of the interaction between crystalline nanoparticle and shear band in metallic glass composites to uncover the mechanisms [16,29,30], the mechanism of high strength and ductility in the dual-phase AlN ceramic remain unclear. How do shear bands develop and how do they react with crystalline nanoparticle in dual-phase ceramic composite? How do the crystalline particles in the amorphous matrix deform during loading? What kind of roles do interfaces play in the enhancement of the mechanical performance of dual-phase composites? To solve the problems mentioned above, the mechanical response of the dual phase AlN nanocomposites under compression is simulated using MD simulations. The effects of volume fraction, surface-to-volume ratio, and distribution patterns of the crystalline nanoparticles are also taken into consideration for better performance of the dual-phase nanocomposites (DPNC). The content of this article is arranged as follows: In Section 2, the simulation details are briefly introduced. In Section 3, the simulated results are exhibited and the corresponding discussion is made. The work in this article is summarized and some conclusions are drawn and given in Section 4.
1
ηMises [38] is computed with ηMises = 2 (ηij − ηm δij )(ηij − ηm δij ) , where ηij and ηm represent local Lagrangian strain and its hydrostatic component at an atom, respectively. For clarity, the atoms with ηMises > 0.2 are considered as those in a shear transformation zone (STZ). Identify diamond structure (IDS) method [39] is used to distinguish different atomic structures in AlN. The coordination number (CN) is utilized to identify the phase structure in crystalline AlN. All the post-processing analyses and graphical visualization are performed by using the software Ovito [40]. 3. Results and discussion
2. Computational methods 3.1. Effect of crystal nanoparticle volume fraction Nanocomposite AlN (nc-AlN) can be prepared by two ways. For the first one, amorphous AlN (a-AlN) is prepared firstly using a meltquench-duplicate method [14], then, nc-AlN is constructed by replacing some areas in the a-AlN with crystalline AlN nanoparticles. This method can hardly guarantee the charge balance at the interface between amorphous matrix and crystalline nanoparticle. In the second method, a crystalline wurtzite AlN (w-AlN) is prepared at first, then it is heated to molten state, followed by quenching it at an extremely large cooling rate. The material obtained with this method could have crystalline cores and the amorphous surroundings (i.e., particulate nc-AlN), due to the non-uniform temperature field during the quench. To obtain a nc-AlN sample for MD simulations, a cuboidal cell of wAlN is built firstly, as shown in Fig. 1(a). It has the size of 295.45 Å (x) × 59.25 Å (y) × 298.7 Å (z), containing 501,600 atoms and with x, y and z along [1¯21¯0], [101¯0] and [0001], respectively. Then, the w-AlN cell except the central region (with radius of 9 nm) is heated to 3500 K and homogenized for 500 ps (to reach a uniform molten state) in isothermal-isobaric (NPT) ensemble by a Nose-Hoover thermostat, followed by cooling it down to 10 K in 17.45 ps [14]. During this time interval, the atoms in the central area (nanoparticle) are kept to be crystalline structure. Finally, the cell is relaxed at 10 K for 500 ps to make it stable and eliminate residual stress. During the heat treatment, periodic boundary conditions (PBCs) are applied in all the three directions. The shape of the nanoparticle is set to be circular (Fig. 1(a)) which is similar to the experimental results [31] where spherical crystalline AlN nanoparticles are randomly embedded in the amorphous matrix as shown in Fig. 1(b). The radius of the nanoparticle can be changed by controlling the number of the atoms in the area frozen in crystalline state during the melting and cooling processes. All the MD simulations are performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) [32]. The time step of 1.0 fs is used throughout our MD simulations. The nc-AlN model is validated with the radial distribution functions (RDFs), as shown in Fig. 1(c),
3.1.1. Stress concentration in nc-AlNs Four different nc-AlN samples are adopted for MD simulations, of which the radii of the nanoparticle of are r = 30 Å, 50 Å, 70 Å and 90 Å, corresponding to particle volume fractions of fv = 6%, 14.4%, 26.4% and 40.9% (after relaxation), respectively, as shown in the inset of Fig. 2. It can be found that after relaxation stress concentration exists in these nc-AlN samples, as shown in the inset in Fig. 3. The hydrostatic stress is adopted for analysis, which can be expressed as:
σh =
1 (σx + σy + σz ) 3
(1)
where σx , σz , σz are the normal components of the stress at an atom in the Cartesian coordinate system. The distributions of σh in the samples along z-axis are shown in Fig. 3, where the values of σh are the mean of that over a small region (10(x) × 59(y) × 6(z) Å3). It can be seen that the crystalline nanoparticle is in the tensile stress state, while the amorphous matrix is in compressive stress state, and at the interface σh varies from compressive to tensile states, as a result of strain incompatibility of these two phases. Such effect is stronger at the interface than at the centre, so the tensile hydrostatic stress at the interface is relatively larger compared with that at the centre of the crystalline particle. It is noteworthy that the stress concentration tends to decrease with the increase of fv, as shown in Fig. 3. 3.1.2. Effect of fv on elastic response The compressive stress-strain (σ-ε) curves of these nc-AlNs are shown in Fig. 2, where the σ-ε curve of a-AlN is also provided for comparison. In the initial stage, the slope of the curves, i.e., the Young's modulus (E) increases with the increase of fv. The effective Young's modulus of each nc-AlN is calculated as shown in Fig. 4(a). The effective Young's modulus of a dual-phase composite, Enc, can be estimated with Mori-Tanaka approach [41]: 2
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Fig. 1. (a) Flow chart of nc-AlN procedure; (b) TEM micrograph of crystalline AlN particles embedded in an amorphous matrix (reproduced from Ref. [31]); (c) RDFs corresponding to nanoparticle and matrix in nc-AlN, respectively.
B1 = fv D1 + D2 + (1 − fv )(S2211 + S3311) B2 = fv + D3 + (1 − fv )(S2222 + S3322) B3 = fv + D3 + (1 − fv )(S2233 + S3333) B4 = fv + D3 + (1 − fv )(D1 S2211 + S3311) B5 = fv D1 + D2 + (1 − fv )(D1 S2222 + S3322) B6 = fv + D3 + (1 − fv )(D1 S2233 + S3333) B7 = fv + D3 + (1 − fv )(D1 S3311 + S2211) B8 = fv + D3 + (1 − fv )(D1 S3322 + S2222) B9 = fv D1 + D2 + (1 − fv )(D1 S3333 + S2233)
(2c)
D1 = 1 + 2(μp − μm )/(λp − λm) D2 = (λm − 2μm )/(λp − λm) D3 = λm /(λp − λm)
with λm , μm and λp , μp the Lamé constants of the matrix and nanoparticle, respectively. The components of the Eshelby Tensor Sijkl are shown in the supplementary material. The variation of Enc against fv determined by Equation (2a) is shown in Fig. 4(a), where it can be seen that Enc develops almost linearly with the increase of fv and the value is smaller than the result by MD simulations. To verify this result, we also calculate the Enc using the Voigt approximation, Enc = Ep fv + Em (1 − fv ) (Mixture rule 1), and the Reuss approximation,
Fig. 2. Compressive stress-strain curves for a-AlN and nc-AlNs with various radii of crystalline nanoparticles.
Enc =
Em 1 + fv [A5 − νm (A 4 + A6 )]/ A
(2a)
where E m and v m are the Young's modulus and the Possion's ratio of the matrix, fv the volume fraction of the particle, and
A 4 = D1 (B4 B9 − B6 B7) + B1 (B6 − B9) + B3 (B7 − B4 ) A5 = D1 (B3 B7 − B1 B9) + B4 (B9 − B3) + B6 (B1 − B7) A6 = D1 (B1 B6 − B3 B4 ) + B9 (B4 − B1) + B7 (B3 − B6) A = B1 (B5 B9 − B6 B8) + B2 (B6 B7 − B4 B9) + B3 (B4 B8 − B5 B7)
(2d)
Ep Em
(Mixture rule 2), respectively, as shown in Fig. 4(a). It can be seen that Enc by the Mori-Tanaka approach is located between the two mixing rules. As fv < 0.34, the Enc by MD simulation is even larger than that by Voigt approximation, which is regarded as the upper bound of Enc. As these theoretical approaches are obtained based on the concept of continuum mechanics, the effects of the interface are not taken into account, which might take effect at nanoscale and lead to the enhancement elastic properties.
Enc =
(2b)
in which 3
(1 − fv ) Em + fv Em
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Fig. 3. Hydrostatic stress σh distributions corresponding to (a) nc-AlN (r = 30 Å); (b) nc-AlN (r = 50 Å); (c) nc-AlN (r = 70 Å); (d) nc-AlN (r = 90 Å) after relaxation. Regions for the amorphous matrix, the interface, and the crystalline particle are highlighted in light red, green and blue, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
A criterion of the effective yield stress of dual-phase composite was proposed to identify the start of plastic deformation: it depends on the value of σym/cm and σyp/cp, where σym and σyp are the yield stress of the matrix and nanoparticle, respectively, and cm and cp the average stress concentration factors of the matrix and the nanoparticles, respectively [42]. If σym/cm < σyp/cp, the matrix would yield first compared with the particle [42]. The values of cp and cm are nearly the same, as the average stress concentration only depends on the chemical composition [43]. Therefore, the ductile a-AlN matrix in the nc-AlN would yield first, which agrees with our simulation results, as illustrated in Figs. 5(a1)-(e1). As was mentioned above, the yield strength of the ncAlNs increases with the increase of fv, i.e., the larger the fv, the sooner the nc-AlN yields, as shown in Fig. 2. To gain an insight into the difference between the distributions of PSTZs in nc-AlNs and that in a-AlN, the polar coordinate frame (ρ, θ) is introduced, in which the shear stress component σρθ can be expressed
3.1.3. Effect of fv on plastic deformation The yield strength of the nc-AlNs increases with the increase of fv, as shown in Fig. 4(b), which can be ascribed to that, as the bond energy of the crystalline AlN is larger than that of the amorphous AlN, the deformation of the former needs more energy than the latter under the same strain. The nc-AlNs with fv = 0, 6%, 14.4%, 26.4% and 40.9% begin to yield (Fig. 2) at ε = 0.134, 0.110, 0.086, 0.074 and 0.069, with the corresponding distributions of ηMises shown in Figs. 5(a1)-(e1), respectively. It can be found that the preliminary shear transformation zones (PSTZs) develop randomly in pure a-AlN, which is in stark contrast to the case of nc-AlNs, where most PSTZs form at the interface between the crystalline nanoparticle and amorphous matrix. This phenomenon becomes more obviously with the increase of fv. In pure aAlN, as there is no pre-existing defect that would result in stress concentration, the flow units would distribute randomly in a-AlN [14], STZs would be activated randomly.
Fig. 4. Volume fraction effect of nanoparticle on E and σs of nc-AlN. 4
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Fig. 5. Evolution of von-Mises strain in the a-AlN and nc-AlNs with different crystalline volume fraction at different strain levels.
comparing the σρθ curve in Fig. 6(a) and those in Figs. 6(b)-(d). In Fig. 6(a), σρθ in the matrix is comparable to that in the nanoparticle, which could render the PSTZs to develop not only at the interface, but also in the amorphous matrix, as illustrated in Fig. 5(b1). In Figs. 6(b)(d), σρθ in the matrix is distinctly smaller than that at the interface, which can account for why the PSTZs mainly develop at the interface in these samples, as illustrated in Figs. 5(c1)-(e1). PSTZs are activated at the interface in the direction of 45° and 135° (Figs. 5(c1)-(e1)), which can be ascribed to the high shear stress (maximum resolved shear stress) along these two directions, as shown in Fig. 6. After yield, the stress decreases sharply from Point e1 to Point e2 in the σ-ε curve of nc-AlNr=90 Å as manifested in Fig. 2, which can be attributed to the formation of four embryonic shear bands (SBs) as shown in Fig. 5(e2). Analogously, σ also drops from Point d1 to Point d2 in the σ-ε curve of nc-AlNr=70 Å (Fig. 2) due to the formation of two mature
as:
σρθ = −sinθcosθσx + sinθcosθσz + [(cosθ)2 − (sinθ)2] σxz
(3)
z−b
where θ = tan−1 x − a , with (x,z) indicating the coordinate of each atom, and a and b are the half edge lengths of box along x and z directions, respectively. The distributions of shear stress σρθ of the nc-AlNs at the yield point (Fig. 2) are shown in Fig. 6. It is rather remarkable that the distribution of σρθ in the nanoparticle are symmetrical about the lines of θ = 135° and θ = 45°, respectively, and zero stress zones exist between them. The radial distributions of σρθ along the while line (135°) in the samples containing particles of r = 30, 50, 70 and 90 Å are given respectively on the right column in Fig. 6, in which each σρθ curve takes the shape of a swallow, i.e., it increases rapidly in the nanoparticle from its centre to outer radius and then decreases at the interface. Difference can be found in the distributions of σρθ in the amorphous matrix by 5
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Fig. 6. Radial distributions of shear stress σρθ corresponding to (a) nc-AlN (r = 30 Å) at ε = 0.110; (b) nc-AlN (r = 50 Å) at ε = 0.086; (c) nc-AlN (r = 70 Å) at ε = 0.074; (d) nc-AlN (r = 90 Å) at ε = 0.069. Regions for the amorphous matrix, the interface, and the crystalline particle are highlighted in light red, green and blue, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
SBs, as illustrated in Fig. 5(d2). For nc-AlNr=50 Å, σ declines gradually from Point c1 to Point c2 along with the development of SB, as indicated in Fig. 5(c2). There is no significant drop of stress in the σ-ε curve of ncAlNr=30 Å and that of pure a-AlN. The embryonic SBs take shape at the interface in nc-AlNs, but not in a-AlN until ε = 0.20, as shown in Figs. 5(b2) and (a2). It is evident in the σ-ε curve shown in Fig. 2 that σ increases to Point e3, i.e., work hardening occurs to nc-AlNr=90 Å,
attributed to the intersection between SBs, which leads to an “ear-like” extrusion of atoms, as shown in Fig. 5(e3), where high level of strain appear at the four corners. The shear band offset occurs for ncAlNr=70 Å and nc-AlNr=50 Å at ε = 0.19, as shown in Figs. 5(d3) and (c3), respectively. At ε = 0.30, there is no formation of mature SB in ncAlNr=30 Å and a-AlN, as shown in Figs. 5(b3) and (a3). In the previous part, we found that there is a concentration of σρθ at the interface in nc6
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Fig. 7. Distributions of (a) CN at ε = 0.220, 0.234, and 0.300, respectively; (b) hydrostatic stress at ε = 0.234 in nc-AlNr=90 Å; (c) The schematic of the wurtzite-tographene like (GL) phase transformation. Arrows represent the relative motion of atoms. The colour red and blue denote the atoms in tensile and compressive stress state, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
AlN samples along both 45° and 135° directions, however, different number of SBs form eventually for nc-AlNs samples containing nanoparticles of r = 50 Å, 70 Å and 90 Å, respectively, as manifested in Figs. 5(c3)-(e3). As the space for the extension of SB (in 45° and 135° directions) decreases with the increase of fv, free surface may suppress the progress of SB, which may trigger the STZs in other sites and promote the further development of SBs.
Fig. 7(c), where Al and N atoms move in the opposite directions along zaxis, forming the GL phase. However, phase transformation is not found in the samples of nc-AlNr=70 Å, nc-AlNr=50 Å, and nc-AlNr=30 Å, which is attributed to that the hydrostatic stress in these samples is not sufficiently large to trigger phase transformation.
3.1.4. Effect of fv on phase transformation After work hardening, σ drops from Point e4 to Point e5 in the σ-ε curve of a-AlNr=90 Å, as shown in Fig. 2, corresponding to the phase transformation from ε = 0.220 to ε = 0.234, as illustrated in Fig. 7 (a), where CN is used to distinguish the wurtzite structure (B4) from graphene-like (GL). Some atoms with the space group of P63mc (B4) transform to the space group of P63/mmc (GL) (indicated by red) at ε = 0.220, as shown in Fig. 7(a), where a “carrot” region of GL forms at ε = 0.234, which leads to a drop of stress. When the phase transformation from B4 to GL occurs, the volume of the transformation parts would decrease (shrink), which would lead to the relaxation of the stress in the sample subjected to compressive deformation, and results in the stress drop in the stress-strain curve. With further loading, structural transformation continues along the maximum resolved shear stress plane, leading to a rectangle zone at ε = 0.30, as shown in Fig. 7(a). It is known that pressure can induce B4-GL-B1 (rocksalt structure) phase transformation in AlN where GL is the intermediate phase. However, it is worth to note that B1 structure is not found at ε = 0.30 only existing GL. Meanwhile, the intermediate GL structure has been proven stable using the first principle calculation [44]. To gain an insight into the phase transformation, the distribution of hydrostatic stress in nc-AlNr=90 Å at ε = 0.234 is shown in Fig. 7(b). Comparing the distributions of CN and hydrostatic stress at ε = 0.234 in Fig. 7(a) and (b), a high stress region can be found between the B4 and GL structure. For better observation, the lateral of the slice marked with a rectangle in Fig. 7(b) is enlarged and shown on right of Fig. 7(b), where the three parts from top to bottom are indicated by GL, Tr (transition), and B4, respectively. The Tr zone is in high hydrostatic stress with Al and N atoms under tensile and compressive stress, respectively. The side and top views of the B4-to-GL phase transformation zone are shown in
Interfaces play a significant role in the mechanical performance of nc-AlN, and surface-to-volume ratio should be a suitable parameter to describe the effect of the interfaces. Four models based on the cells with different numbers of nanoparticles (n = 9, 16, 25 and 36, respectively) but the same fv are used, as shown in the inset of Fig. 8, of which the compressive σ-ε curves are shown in Fig. 8. It can be seen that the four curves show insignificant differences. In the initial elastic stage, effective Young's moduli (Enc) are in line with each other. The
3.2. Effect of surface-to-volume ratio
Fig. 8. Compressive stress-strain curves for nc-AlNs with different number of crystalline nanoparticles. 7
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Fig. 9. Distributions of local shear strain, rotation angle and atomic displacement vectors in deformed nc-AlNn=9. (a) Local shear strain at ε = 0.076, 0.120, and 0.216, respectively; (b) local shear strain (up) and rotation angle (radian, down) in the dotted zone in (a) at ε = 0.089; (c) Atomic displacement vectors and local shear strain at ε = 0.120. The yellow arrows denote the atomic displacement vectors, and the atoms are coloured by their local shear strain. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
the IDS method), as manifested in Fig. 10(b). Since GL structure cannot be recognized by IDS method, it will be indicated in white. It is noticed that nearly no white atoms can be found in the nanoparticles at ε = 0.300, indicating the extinction of GL structure. To better understand this phenomenon, the distribution of CN in the dashed rectangle in Fig. 10(a) is enlarged and shown in Fig. 10(c), in which, at ε = 0.2251, the upper-left side and the lower-right side in the crystalline nanoparticle slip along the direction of the maximum resolved shear stress after the formation of the GL phase. The bonds of the upperleft side reconnect with that of the lower-right side, and the GL phase turns back into the wurtzite one at ε = 0.2254, which can still bear the load.
nanostructures of the nc-AlN cell with 9 nanoparticles under uniaxial compression is shown in Fig. 9, and the others are given in Fig. S1 in the supplementary material. It can be seen in Fig. 9(a) that nc-AlNn=9 does not yield until ε = 0.076 when STZs initiate. Different from the response of nc-AlNr=90 Å mentioned above, for nc-AlNn=9, there is no sharp drop after the yield (Fig. 8), even if fv ≈ 60%, which can be ascribed to that no mature SB forms until ε = 0.120, as shown in Fig. 9(a) nc-AlNn=9 enters the hardening stage at ε = 0.216 due to the interaction between immature SBs, as manifested in Fig. 9(a). It is intriguing to note that the regions between immature SBs are of high shear strain. In order to understand the formation of SB in the nc-AlN, the distributions of the local shear strain and rotation angle in the dotted rectangle in Fig. 9(a) at ε = 0.089 are shown in Fig. 9(b). Sopu et al. [45] proposed a mechanism for STZ percolation to illustrate the formation of SB in metallic glasses. It is remarkable that the vortexes develop in the vicinity of the nanoparticles and gather together rather than be activated one by one, which is different from that in the amorphous structure (like metallic glasses). Since the vortexes could be activated under relatively high stress [45] and the stress concentration may exist at the interfaces between the amorphous matrix and the crystalline nanoparticles, the vortexes would be easily activated at interface, as shown in Fig. 9(b). Fig. 9(c) shows that the atoms in the immature SBs near the particle Ⅱ slip in the direction along 135°, which is the same as that far away from the particle Ⅲ (the arrow in the upper right corner), and vice versa. As the shear modulus of the crystalline nanoparticle is larger than that of the amorphous matrix, it should be more difficult for the atoms in the crystalline particle to be activated, which would hinder the formation of mature SBs. Phase transformation occurs at ε = 0.225, as shown in Fig. 10(a), leading to a drop in the σ- ε curve (Fig. 8). We can also find that some crystalline nanoparticles are cut into half along the direction of the maximum resolved shear stress direction at ε = 0.300 (recognized with
3.3. Effect of crystal nanoparticle distribution The distribution of particles plays a crucial role in the mechanical behaviour of composite [46,47], which is to be investigated in this subsection. As mentioned above, the number (or the size) of the nanoparticles in the amorphous matrix has little influence on the mechanical properties, thus, three kinds of distributions of particles in a cell are adopted, i.e., square distribution, (SD, inset of Fig. 8), triangle distribution (TD), and random distribution (RD, inset of Fig. 11), each of which contains 9 identical particles. The compressive σ-ε curves of these patterns are shown in Fig. 11. It can be seen that the effective Young's modulus, Enc, are almost identical, which is reasonable since the models have the same fv. It is noteworthy that the model with TD has larger yield strength than the other two cases. At ε = 0.076, the initial STZs corresponding to TD (Fig. 12(a)) is smaller than those corresponding to SD (Fig. 9(a)) and RD (Fig. 12(b)), which indicates that in the case of TD, it is more difficult for STZ to be activated. Hardening occurs in the model with TD during ε = 0.130 and ε = 0.195, due to the interaction between the immature SBs, as shown 8
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Fig. 10. (a) Distributions of CN at ε = 0.225; (b) Atomic structure recognized with IDS method, the white represents the amorphous structure, the other colours represent the crystal structure; (c) Distributions of CN corresponding to the dashed rectangle box in (a) at ε = 0.2250, 0.2251, and 0.2254, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
crucial role in the mechanical performance of nc-AlN. For instance, in the model with TD, the immature SB develops near Particle I indicated with white dotted rectangle at ε = 0.130 (Fig. 12(a)) and meets the particle Ⅱ at ε = 0.195. The immature SB bypasses Particle Ⅱ and keeps moving since the strength of the crystalline particle is larger than that of the amorphous matrix, which is difficult for immature SB to move across the whole sample. No hardening is observed in the RD, attributed to that the immature SB could go across the whole sample without intersection. Although the strain hardening stage of the model with TD is a little smaller than that with SD, the former should be the best choice. 4. Conclusions The mechanical properties of dual-phase AlN materials (nc-AlN) composed of amorphous AlN matrix and crystalline AlN nanoparticle were investigated with molecular dynamics simulations for the purpose to uncover the mechanism of the enhancement in strength and ductility. The effects of volume fraction (fv), surface-to-volume ratio, and distribution of crystalline AlN particles were studied, from which the following conclusions can be drawn.
Fig. 11. Compressive stress-strain curves for nc-AlNs with different distributions of crystalline nanoparticles — Square, Triangle, and random, respectively.
in Fig. 12(a), and this duration is smaller than that of SD, because of the phase transformation that occurs previously, as shown in Fig. 13(a1). Similarly, some particles in the model with TD are cut into half at ε = 0.300, as shown in Fig. 13(a2). Different from the models with SD and TD, there is no hardening in the model with RD, which resembles the a-AlN model, as shown in Fig. 11. An immature SB develops at ε = 0.130 and the next one appears at ε = 0.195, as shown in Fig. 12(b). No drop of stress can be found in the model with RD, in which no phase transformation occurs, as shown in Fig. 13(b1). Meanwhile, the particle is still of perfect wurtzite structure without any fracture, as shown in Fig. 13(b2). The distribution of particle plays a
(1) Stress concentration exists in the nc-AlN samples, where crystalline particles and amorphous matrix are subjected to tensile and compressive hydrostatic stresses, respectively. (2) The elastic modulus of the nc-AlN increases with the increase of fv, but the magnitude is larger than that evaluated by Mori-Tanaka approach, and Voigt and Reuss approximations, attributed to the effect of interfaces that take strong effect at nanoscale. Hardening Would occur in the nc-AlN of fv ≥ 40.9%, as a result of the intersection between shear bands. The transformation from the wurtzite structure (B4) to graphene-like (GL) occurs layer by layer in the ncAlN of fv = 40.9%, ascribed to the high hydrostatic stress. 9
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Fig. 12. Distributions of local shear strain corresponding to (a) triangle distribution, and (b) random distribution at ε = 0.076, 0.130, and 0.195, respectively.
Fig. 13. (a1) Distributions of CN at ε = 0.200, (a2) atomic structure recognized with IDS method at ε = 0.300, corresponding to triangle distribution; (b1) Distributions of CN at ε = 0.300, (b2) atomic structure recognized with IDS method at ε = 0.300, corresponding to random distribution.
fv ≥ 40.9% has a better performance.
(3) The plastic performance of nc-AlN is less dependent on surface-tovolume ratio because the extension of the immature shear band is restricted by the crystalline particles. The vortexes develop in the vicinity of nanoparticles and gather together, rather than be activated one by one. (4) The distribution pattern of crystalline nanoparticle plays an important role in the mechanical performance of nc-AlN. Triangle distribution (TD) of may be the best choice compared with square or random distributions (SD and RD). The nc-AlN with TD and
The results obtained in this article would enrich the research of the mechanical properties of ceramics and provide available information for the design of high-performance AlN based ceramics.
Acknowledgements The authors gratefully acknowledge the financial support from 10
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National Natural Science Foundation of China (Grant No. 11332013), ARC Discovery Project (180101955 and 180102003), the program of China Scholarships Council (No. 201606050043), The High Performance Computing (HPC) and CARF at QUT have kindly provided access to their facilities.
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Ceramics International journal homepage: www.elsevier.com/locate/ceramint
Dual phase nano-particulate AlN composite — A kind of ceramics with high strength and ductility Yinbo Zhaoa,b, Xianghe Penga,∗, Bo Yanga, Cheng Huanga, Ning Hua, Cheng Yanb,∗∗ a
College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China School of Chemistry, Physics and Mechanical Engineering, Science and Engineering Faculty, Queensland University of Technology, 2 George Street, G.P.O. Box 2434, Brisbane, Australia
b
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanocomposite AlN Shear band Phase transformation Interface Molecular dynamics simulations
Ceramics are widely used in many fields due to their excellent properties. However, the brittle fracture is a short board restricting their applications. To understand their deformation mechanism and explore a way to enhance both the strength and ductility, we investigated the mechanical behaviour of dual-phase AlNs composed of amorphous AlN matrix and crystalline nanoparticles under compression via molecular dynamics simulations. The stress concentration exists at the interface of nanocomposite AlN, where the particles and matrix are in the tensile and compressive states of stress, respectively. Strain hardening occurs when crystalline nanoparticle fraction fv ≥ 40.9%, attributed to the intersection between shear bands. The phase transformation from wurtzite structure (B4) to graphene-like structure (GL) is observed in the crystalline phase, as a result of high hydrostatic stress. After phase transformation, the particle might be cut into half during further compression along with the recovery of the GL structure to the wurtzite structure that could still bear load. The investigation of the effects of the volume fraction, surface-to-volume ratio, distribution pattern of the crystalline nanoparticles indicates that the dual-phase AlN nanocomposite with fv ≥ 40.9% and triangle distribution of particles would possess both higher strength and ductility.
1. Introduction Ceramics own many excellent physical and mechanical properties like low density, high melting point, high hardness, etc., however, its brittle nature strongly restricts their applications [1]. A lot of effort has been devoted to preparing ceramics of high toughness [2], among which the introduction of second ductile phases seems promising, such as fibers [3], graphene platelets [4], polymer [5], carbon nanotubes (CNT) [6], and metal infiltration [7], etc. However, these methods can toughen the ceramics by triggering crack defects instead of suppressing crack initiation with less ductility [8]. Similar issue also occurs in metallic glasses, which are a class of amorphous materials that possess high strengths and high elastic strain limits (∼2%) [9]. Dual-phase metallic glass was also developed to achieve better ductility [10]. Recently, Wu et al. [11] designed a dual-phase metallic glass which exhibits near-ideal strength as the localized shear bands are impeded by the crystalline phase. Different from the metallic glass, the amorphous ceramics can endure notable plastic deformation [12] but have lower strength
∗
compared with their crystalline counterparts [13,14]. Based on the concept of dual-phase metals, a dual-phase amorphous carbon ceramic was synthesized, which exhibits high strength and ductility under compression [15]. Although experimental work has shown the important role of second phases in the mechanical performance of the dual-phase ceramics, it is difficult to observe interactions and evolutions of the internal structures, such as shear band and inclusions, and to uncover the mechanism for the high strength and ductility. Molecular dynamics simulation (MD) is a suitable means that has been widely used to investigate the responses and the corresponding atomic structures (including the shear bands [14,16], dislocations [17,18], and phase transformation [19], etc.) as well as their evolutions during the deformations at nanoscale. Aluminium nitrides (AlNs) have attracted considerable attention due to their distinctive properties - high hardness [20], wide band gap [21], high piezoelectric coefficients [22], etc., which have been extensively applied in the fabrication of different kinds of devices in micro electromechanical system (MEMS) [23], high power electronics [24], and optoelectronic materials [25], etc. To achieve higher ductility and
Corresponding author. Corresponding author. E-mail addresses: [email protected] (X. Peng), [email protected] (C. Yan).
∗∗
https://doi.org/10.1016/j.ceramint.2019.06.239 Received 10 May 2019; Received in revised form 14 June 2019; Accepted 22 June 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.
Please cite this article as: Yinbo Zhao, et al., Ceramics International, https://doi.org/10.1016/j.ceramint.2019.06.239
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where it can be seen that there exist several peaks in the curve of nanoparticles, which indicates that the nanoparticle is of crystalline structure. In contrast, the curve corresponding to the matrix does not show the characteristic of long range order, indicating that the matrix is amorphous. Using the melt-quench-duplicate approach, an a-AlN sample is also prepared for the comparative MD simulations [14]. It is crucial to select an appropriate interatomic potential in MD simulations. Xiang et al. [33] compared different kinds of interatomic potentials for AlN and found that Vashishta potential [34] is more suitable to describe the mechanical properties of AlN. This potential has been used to explore the deformation mechanism of crystalline AlN during indentation [35] and shock loading [36], and the mechanical responses of a-AlN under uniaxial loading [14,28]. In this work, the mechanical responses of nc-AlN samples subjected to uniaxial compressions are investigated. The compression is applied in z-direction at a strain rate of 108 s−1, PBCs are maintained in the yand z-directions while free surface condition is imposed in the x-direction. A microcanonical (NVE) ensemble is adopted with the temperature kept at 10 K using Langevin thermostat. Atomic stresses is calculated through the virial stress [37]. The local strain shear invariant
toughness, Heard et al. [26] introduced high hydrostatic pressure condition and Guo et al. [27] decreased the sample size to promote the plastic flow in AlN. We found that amorphous AlN (a-AlN) possesses high ductility, which could be ascribed to the self-repairing capability [14] and less sensitivity to fissures [28] compared with wurtzite structured AlN (w-AlN). However, a-AlN has low strength due to its amorphous structure, which is different from crystalline materials, in which there are grain boundaries, twin boundaries and dislocations, which may induce strengthening. In this work, we introduce crystalline AlN nanoparticles into amorphous AlN matrix for the purpose to obtain dual-phase AlN composites with both high strength and ductility. Although much effort has been made in the research of the interaction between crystalline nanoparticle and shear band in metallic glass composites to uncover the mechanisms [16,29,30], the mechanism of high strength and ductility in the dual-phase AlN ceramic remain unclear. How do shear bands develop and how do they react with crystalline nanoparticle in dual-phase ceramic composite? How do the crystalline particles in the amorphous matrix deform during loading? What kind of roles do interfaces play in the enhancement of the mechanical performance of dual-phase composites? To solve the problems mentioned above, the mechanical response of the dual phase AlN nanocomposites under compression is simulated using MD simulations. The effects of volume fraction, surface-to-volume ratio, and distribution patterns of the crystalline nanoparticles are also taken into consideration for better performance of the dual-phase nanocomposites (DPNC). The content of this article is arranged as follows: In Section 2, the simulation details are briefly introduced. In Section 3, the simulated results are exhibited and the corresponding discussion is made. The work in this article is summarized and some conclusions are drawn and given in Section 4.
1
ηMises [38] is computed with ηMises = 2 (ηij − ηm δij )(ηij − ηm δij ) , where ηij and ηm represent local Lagrangian strain and its hydrostatic component at an atom, respectively. For clarity, the atoms with ηMises > 0.2 are considered as those in a shear transformation zone (STZ). Identify diamond structure (IDS) method [39] is used to distinguish different atomic structures in AlN. The coordination number (CN) is utilized to identify the phase structure in crystalline AlN. All the post-processing analyses and graphical visualization are performed by using the software Ovito [40]. 3. Results and discussion
2. Computational methods 3.1. Effect of crystal nanoparticle volume fraction Nanocomposite AlN (nc-AlN) can be prepared by two ways. For the first one, amorphous AlN (a-AlN) is prepared firstly using a meltquench-duplicate method [14], then, nc-AlN is constructed by replacing some areas in the a-AlN with crystalline AlN nanoparticles. This method can hardly guarantee the charge balance at the interface between amorphous matrix and crystalline nanoparticle. In the second method, a crystalline wurtzite AlN (w-AlN) is prepared at first, then it is heated to molten state, followed by quenching it at an extremely large cooling rate. The material obtained with this method could have crystalline cores and the amorphous surroundings (i.e., particulate nc-AlN), due to the non-uniform temperature field during the quench. To obtain a nc-AlN sample for MD simulations, a cuboidal cell of wAlN is built firstly, as shown in Fig. 1(a). It has the size of 295.45 Å (x) × 59.25 Å (y) × 298.7 Å (z), containing 501,600 atoms and with x, y and z along [1¯21¯0], [101¯0] and [0001], respectively. Then, the w-AlN cell except the central region (with radius of 9 nm) is heated to 3500 K and homogenized for 500 ps (to reach a uniform molten state) in isothermal-isobaric (NPT) ensemble by a Nose-Hoover thermostat, followed by cooling it down to 10 K in 17.45 ps [14]. During this time interval, the atoms in the central area (nanoparticle) are kept to be crystalline structure. Finally, the cell is relaxed at 10 K for 500 ps to make it stable and eliminate residual stress. During the heat treatment, periodic boundary conditions (PBCs) are applied in all the three directions. The shape of the nanoparticle is set to be circular (Fig. 1(a)) which is similar to the experimental results [31] where spherical crystalline AlN nanoparticles are randomly embedded in the amorphous matrix as shown in Fig. 1(b). The radius of the nanoparticle can be changed by controlling the number of the atoms in the area frozen in crystalline state during the melting and cooling processes. All the MD simulations are performed using the large-scale atomic/molecular massively parallel simulator (LAMMPS) [32]. The time step of 1.0 fs is used throughout our MD simulations. The nc-AlN model is validated with the radial distribution functions (RDFs), as shown in Fig. 1(c),
3.1.1. Stress concentration in nc-AlNs Four different nc-AlN samples are adopted for MD simulations, of which the radii of the nanoparticle of are r = 30 Å, 50 Å, 70 Å and 90 Å, corresponding to particle volume fractions of fv = 6%, 14.4%, 26.4% and 40.9% (after relaxation), respectively, as shown in the inset of Fig. 2. It can be found that after relaxation stress concentration exists in these nc-AlN samples, as shown in the inset in Fig. 3. The hydrostatic stress is adopted for analysis, which can be expressed as:
σh =
1 (σx + σy + σz ) 3
(1)
where σx , σz , σz are the normal components of the stress at an atom in the Cartesian coordinate system. The distributions of σh in the samples along z-axis are shown in Fig. 3, where the values of σh are the mean of that over a small region (10(x) × 59(y) × 6(z) Å3). It can be seen that the crystalline nanoparticle is in the tensile stress state, while the amorphous matrix is in compressive stress state, and at the interface σh varies from compressive to tensile states, as a result of strain incompatibility of these two phases. Such effect is stronger at the interface than at the centre, so the tensile hydrostatic stress at the interface is relatively larger compared with that at the centre of the crystalline particle. It is noteworthy that the stress concentration tends to decrease with the increase of fv, as shown in Fig. 3. 3.1.2. Effect of fv on elastic response The compressive stress-strain (σ-ε) curves of these nc-AlNs are shown in Fig. 2, where the σ-ε curve of a-AlN is also provided for comparison. In the initial stage, the slope of the curves, i.e., the Young's modulus (E) increases with the increase of fv. The effective Young's modulus of each nc-AlN is calculated as shown in Fig. 4(a). The effective Young's modulus of a dual-phase composite, Enc, can be estimated with Mori-Tanaka approach [41]: 2
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Fig. 1. (a) Flow chart of nc-AlN procedure; (b) TEM micrograph of crystalline AlN particles embedded in an amorphous matrix (reproduced from Ref. [31]); (c) RDFs corresponding to nanoparticle and matrix in nc-AlN, respectively.
B1 = fv D1 + D2 + (1 − fv )(S2211 + S3311) B2 = fv + D3 + (1 − fv )(S2222 + S3322) B3 = fv + D3 + (1 − fv )(S2233 + S3333) B4 = fv + D3 + (1 − fv )(D1 S2211 + S3311) B5 = fv D1 + D2 + (1 − fv )(D1 S2222 + S3322) B6 = fv + D3 + (1 − fv )(D1 S2233 + S3333) B7 = fv + D3 + (1 − fv )(D1 S3311 + S2211) B8 = fv + D3 + (1 − fv )(D1 S3322 + S2222) B9 = fv D1 + D2 + (1 − fv )(D1 S3333 + S2233)
(2c)
D1 = 1 + 2(μp − μm )/(λp − λm) D2 = (λm − 2μm )/(λp − λm) D3 = λm /(λp − λm)
with λm , μm and λp , μp the Lamé constants of the matrix and nanoparticle, respectively. The components of the Eshelby Tensor Sijkl are shown in the supplementary material. The variation of Enc against fv determined by Equation (2a) is shown in Fig. 4(a), where it can be seen that Enc develops almost linearly with the increase of fv and the value is smaller than the result by MD simulations. To verify this result, we also calculate the Enc using the Voigt approximation, Enc = Ep fv + Em (1 − fv ) (Mixture rule 1), and the Reuss approximation,
Fig. 2. Compressive stress-strain curves for a-AlN and nc-AlNs with various radii of crystalline nanoparticles.
Enc =
Em 1 + fv [A5 − νm (A 4 + A6 )]/ A
(2a)
where E m and v m are the Young's modulus and the Possion's ratio of the matrix, fv the volume fraction of the particle, and
A 4 = D1 (B4 B9 − B6 B7) + B1 (B6 − B9) + B3 (B7 − B4 ) A5 = D1 (B3 B7 − B1 B9) + B4 (B9 − B3) + B6 (B1 − B7) A6 = D1 (B1 B6 − B3 B4 ) + B9 (B4 − B1) + B7 (B3 − B6) A = B1 (B5 B9 − B6 B8) + B2 (B6 B7 − B4 B9) + B3 (B4 B8 − B5 B7)
(2d)
Ep Em
(Mixture rule 2), respectively, as shown in Fig. 4(a). It can be seen that Enc by the Mori-Tanaka approach is located between the two mixing rules. As fv < 0.34, the Enc by MD simulation is even larger than that by Voigt approximation, which is regarded as the upper bound of Enc. As these theoretical approaches are obtained based on the concept of continuum mechanics, the effects of the interface are not taken into account, which might take effect at nanoscale and lead to the enhancement elastic properties.
Enc =
(2b)
in which 3
(1 − fv ) Em + fv Em
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Fig. 3. Hydrostatic stress σh distributions corresponding to (a) nc-AlN (r = 30 Å); (b) nc-AlN (r = 50 Å); (c) nc-AlN (r = 70 Å); (d) nc-AlN (r = 90 Å) after relaxation. Regions for the amorphous matrix, the interface, and the crystalline particle are highlighted in light red, green and blue, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
A criterion of the effective yield stress of dual-phase composite was proposed to identify the start of plastic deformation: it depends on the value of σym/cm and σyp/cp, where σym and σyp are the yield stress of the matrix and nanoparticle, respectively, and cm and cp the average stress concentration factors of the matrix and the nanoparticles, respectively [42]. If σym/cm < σyp/cp, the matrix would yield first compared with the particle [42]. The values of cp and cm are nearly the same, as the average stress concentration only depends on the chemical composition [43]. Therefore, the ductile a-AlN matrix in the nc-AlN would yield first, which agrees with our simulation results, as illustrated in Figs. 5(a1)-(e1). As was mentioned above, the yield strength of the ncAlNs increases with the increase of fv, i.e., the larger the fv, the sooner the nc-AlN yields, as shown in Fig. 2. To gain an insight into the difference between the distributions of PSTZs in nc-AlNs and that in a-AlN, the polar coordinate frame (ρ, θ) is introduced, in which the shear stress component σρθ can be expressed
3.1.3. Effect of fv on plastic deformation The yield strength of the nc-AlNs increases with the increase of fv, as shown in Fig. 4(b), which can be ascribed to that, as the bond energy of the crystalline AlN is larger than that of the amorphous AlN, the deformation of the former needs more energy than the latter under the same strain. The nc-AlNs with fv = 0, 6%, 14.4%, 26.4% and 40.9% begin to yield (Fig. 2) at ε = 0.134, 0.110, 0.086, 0.074 and 0.069, with the corresponding distributions of ηMises shown in Figs. 5(a1)-(e1), respectively. It can be found that the preliminary shear transformation zones (PSTZs) develop randomly in pure a-AlN, which is in stark contrast to the case of nc-AlNs, where most PSTZs form at the interface between the crystalline nanoparticle and amorphous matrix. This phenomenon becomes more obviously with the increase of fv. In pure aAlN, as there is no pre-existing defect that would result in stress concentration, the flow units would distribute randomly in a-AlN [14], STZs would be activated randomly.
Fig. 4. Volume fraction effect of nanoparticle on E and σs of nc-AlN. 4
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Fig. 5. Evolution of von-Mises strain in the a-AlN and nc-AlNs with different crystalline volume fraction at different strain levels.
comparing the σρθ curve in Fig. 6(a) and those in Figs. 6(b)-(d). In Fig. 6(a), σρθ in the matrix is comparable to that in the nanoparticle, which could render the PSTZs to develop not only at the interface, but also in the amorphous matrix, as illustrated in Fig. 5(b1). In Figs. 6(b)(d), σρθ in the matrix is distinctly smaller than that at the interface, which can account for why the PSTZs mainly develop at the interface in these samples, as illustrated in Figs. 5(c1)-(e1). PSTZs are activated at the interface in the direction of 45° and 135° (Figs. 5(c1)-(e1)), which can be ascribed to the high shear stress (maximum resolved shear stress) along these two directions, as shown in Fig. 6. After yield, the stress decreases sharply from Point e1 to Point e2 in the σ-ε curve of nc-AlNr=90 Å as manifested in Fig. 2, which can be attributed to the formation of four embryonic shear bands (SBs) as shown in Fig. 5(e2). Analogously, σ also drops from Point d1 to Point d2 in the σ-ε curve of nc-AlNr=70 Å (Fig. 2) due to the formation of two mature
as:
σρθ = −sinθcosθσx + sinθcosθσz + [(cosθ)2 − (sinθ)2] σxz
(3)
z−b
where θ = tan−1 x − a , with (x,z) indicating the coordinate of each atom, and a and b are the half edge lengths of box along x and z directions, respectively. The distributions of shear stress σρθ of the nc-AlNs at the yield point (Fig. 2) are shown in Fig. 6. It is rather remarkable that the distribution of σρθ in the nanoparticle are symmetrical about the lines of θ = 135° and θ = 45°, respectively, and zero stress zones exist between them. The radial distributions of σρθ along the while line (135°) in the samples containing particles of r = 30, 50, 70 and 90 Å are given respectively on the right column in Fig. 6, in which each σρθ curve takes the shape of a swallow, i.e., it increases rapidly in the nanoparticle from its centre to outer radius and then decreases at the interface. Difference can be found in the distributions of σρθ in the amorphous matrix by 5
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Fig. 6. Radial distributions of shear stress σρθ corresponding to (a) nc-AlN (r = 30 Å) at ε = 0.110; (b) nc-AlN (r = 50 Å) at ε = 0.086; (c) nc-AlN (r = 70 Å) at ε = 0.074; (d) nc-AlN (r = 90 Å) at ε = 0.069. Regions for the amorphous matrix, the interface, and the crystalline particle are highlighted in light red, green and blue, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
SBs, as illustrated in Fig. 5(d2). For nc-AlNr=50 Å, σ declines gradually from Point c1 to Point c2 along with the development of SB, as indicated in Fig. 5(c2). There is no significant drop of stress in the σ-ε curve of ncAlNr=30 Å and that of pure a-AlN. The embryonic SBs take shape at the interface in nc-AlNs, but not in a-AlN until ε = 0.20, as shown in Figs. 5(b2) and (a2). It is evident in the σ-ε curve shown in Fig. 2 that σ increases to Point e3, i.e., work hardening occurs to nc-AlNr=90 Å,
attributed to the intersection between SBs, which leads to an “ear-like” extrusion of atoms, as shown in Fig. 5(e3), where high level of strain appear at the four corners. The shear band offset occurs for ncAlNr=70 Å and nc-AlNr=50 Å at ε = 0.19, as shown in Figs. 5(d3) and (c3), respectively. At ε = 0.30, there is no formation of mature SB in ncAlNr=30 Å and a-AlN, as shown in Figs. 5(b3) and (a3). In the previous part, we found that there is a concentration of σρθ at the interface in nc6
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Fig. 7. Distributions of (a) CN at ε = 0.220, 0.234, and 0.300, respectively; (b) hydrostatic stress at ε = 0.234 in nc-AlNr=90 Å; (c) The schematic of the wurtzite-tographene like (GL) phase transformation. Arrows represent the relative motion of atoms. The colour red and blue denote the atoms in tensile and compressive stress state, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
AlN samples along both 45° and 135° directions, however, different number of SBs form eventually for nc-AlNs samples containing nanoparticles of r = 50 Å, 70 Å and 90 Å, respectively, as manifested in Figs. 5(c3)-(e3). As the space for the extension of SB (in 45° and 135° directions) decreases with the increase of fv, free surface may suppress the progress of SB, which may trigger the STZs in other sites and promote the further development of SBs.
Fig. 7(c), where Al and N atoms move in the opposite directions along zaxis, forming the GL phase. However, phase transformation is not found in the samples of nc-AlNr=70 Å, nc-AlNr=50 Å, and nc-AlNr=30 Å, which is attributed to that the hydrostatic stress in these samples is not sufficiently large to trigger phase transformation.
3.1.4. Effect of fv on phase transformation After work hardening, σ drops from Point e4 to Point e5 in the σ-ε curve of a-AlNr=90 Å, as shown in Fig. 2, corresponding to the phase transformation from ε = 0.220 to ε = 0.234, as illustrated in Fig. 7 (a), where CN is used to distinguish the wurtzite structure (B4) from graphene-like (GL). Some atoms with the space group of P63mc (B4) transform to the space group of P63/mmc (GL) (indicated by red) at ε = 0.220, as shown in Fig. 7(a), where a “carrot” region of GL forms at ε = 0.234, which leads to a drop of stress. When the phase transformation from B4 to GL occurs, the volume of the transformation parts would decrease (shrink), which would lead to the relaxation of the stress in the sample subjected to compressive deformation, and results in the stress drop in the stress-strain curve. With further loading, structural transformation continues along the maximum resolved shear stress plane, leading to a rectangle zone at ε = 0.30, as shown in Fig. 7(a). It is known that pressure can induce B4-GL-B1 (rocksalt structure) phase transformation in AlN where GL is the intermediate phase. However, it is worth to note that B1 structure is not found at ε = 0.30 only existing GL. Meanwhile, the intermediate GL structure has been proven stable using the first principle calculation [44]. To gain an insight into the phase transformation, the distribution of hydrostatic stress in nc-AlNr=90 Å at ε = 0.234 is shown in Fig. 7(b). Comparing the distributions of CN and hydrostatic stress at ε = 0.234 in Fig. 7(a) and (b), a high stress region can be found between the B4 and GL structure. For better observation, the lateral of the slice marked with a rectangle in Fig. 7(b) is enlarged and shown on right of Fig. 7(b), where the three parts from top to bottom are indicated by GL, Tr (transition), and B4, respectively. The Tr zone is in high hydrostatic stress with Al and N atoms under tensile and compressive stress, respectively. The side and top views of the B4-to-GL phase transformation zone are shown in
Interfaces play a significant role in the mechanical performance of nc-AlN, and surface-to-volume ratio should be a suitable parameter to describe the effect of the interfaces. Four models based on the cells with different numbers of nanoparticles (n = 9, 16, 25 and 36, respectively) but the same fv are used, as shown in the inset of Fig. 8, of which the compressive σ-ε curves are shown in Fig. 8. It can be seen that the four curves show insignificant differences. In the initial elastic stage, effective Young's moduli (Enc) are in line with each other. The
3.2. Effect of surface-to-volume ratio
Fig. 8. Compressive stress-strain curves for nc-AlNs with different number of crystalline nanoparticles. 7
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Fig. 9. Distributions of local shear strain, rotation angle and atomic displacement vectors in deformed nc-AlNn=9. (a) Local shear strain at ε = 0.076, 0.120, and 0.216, respectively; (b) local shear strain (up) and rotation angle (radian, down) in the dotted zone in (a) at ε = 0.089; (c) Atomic displacement vectors and local shear strain at ε = 0.120. The yellow arrows denote the atomic displacement vectors, and the atoms are coloured by their local shear strain. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
the IDS method), as manifested in Fig. 10(b). Since GL structure cannot be recognized by IDS method, it will be indicated in white. It is noticed that nearly no white atoms can be found in the nanoparticles at ε = 0.300, indicating the extinction of GL structure. To better understand this phenomenon, the distribution of CN in the dashed rectangle in Fig. 10(a) is enlarged and shown in Fig. 10(c), in which, at ε = 0.2251, the upper-left side and the lower-right side in the crystalline nanoparticle slip along the direction of the maximum resolved shear stress after the formation of the GL phase. The bonds of the upperleft side reconnect with that of the lower-right side, and the GL phase turns back into the wurtzite one at ε = 0.2254, which can still bear the load.
nanostructures of the nc-AlN cell with 9 nanoparticles under uniaxial compression is shown in Fig. 9, and the others are given in Fig. S1 in the supplementary material. It can be seen in Fig. 9(a) that nc-AlNn=9 does not yield until ε = 0.076 when STZs initiate. Different from the response of nc-AlNr=90 Å mentioned above, for nc-AlNn=9, there is no sharp drop after the yield (Fig. 8), even if fv ≈ 60%, which can be ascribed to that no mature SB forms until ε = 0.120, as shown in Fig. 9(a) nc-AlNn=9 enters the hardening stage at ε = 0.216 due to the interaction between immature SBs, as manifested in Fig. 9(a). It is intriguing to note that the regions between immature SBs are of high shear strain. In order to understand the formation of SB in the nc-AlN, the distributions of the local shear strain and rotation angle in the dotted rectangle in Fig. 9(a) at ε = 0.089 are shown in Fig. 9(b). Sopu et al. [45] proposed a mechanism for STZ percolation to illustrate the formation of SB in metallic glasses. It is remarkable that the vortexes develop in the vicinity of the nanoparticles and gather together rather than be activated one by one, which is different from that in the amorphous structure (like metallic glasses). Since the vortexes could be activated under relatively high stress [45] and the stress concentration may exist at the interfaces between the amorphous matrix and the crystalline nanoparticles, the vortexes would be easily activated at interface, as shown in Fig. 9(b). Fig. 9(c) shows that the atoms in the immature SBs near the particle Ⅱ slip in the direction along 135°, which is the same as that far away from the particle Ⅲ (the arrow in the upper right corner), and vice versa. As the shear modulus of the crystalline nanoparticle is larger than that of the amorphous matrix, it should be more difficult for the atoms in the crystalline particle to be activated, which would hinder the formation of mature SBs. Phase transformation occurs at ε = 0.225, as shown in Fig. 10(a), leading to a drop in the σ- ε curve (Fig. 8). We can also find that some crystalline nanoparticles are cut into half along the direction of the maximum resolved shear stress direction at ε = 0.300 (recognized with
3.3. Effect of crystal nanoparticle distribution The distribution of particles plays a crucial role in the mechanical behaviour of composite [46,47], which is to be investigated in this subsection. As mentioned above, the number (or the size) of the nanoparticles in the amorphous matrix has little influence on the mechanical properties, thus, three kinds of distributions of particles in a cell are adopted, i.e., square distribution, (SD, inset of Fig. 8), triangle distribution (TD), and random distribution (RD, inset of Fig. 11), each of which contains 9 identical particles. The compressive σ-ε curves of these patterns are shown in Fig. 11. It can be seen that the effective Young's modulus, Enc, are almost identical, which is reasonable since the models have the same fv. It is noteworthy that the model with TD has larger yield strength than the other two cases. At ε = 0.076, the initial STZs corresponding to TD (Fig. 12(a)) is smaller than those corresponding to SD (Fig. 9(a)) and RD (Fig. 12(b)), which indicates that in the case of TD, it is more difficult for STZ to be activated. Hardening occurs in the model with TD during ε = 0.130 and ε = 0.195, due to the interaction between the immature SBs, as shown 8
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Fig. 10. (a) Distributions of CN at ε = 0.225; (b) Atomic structure recognized with IDS method, the white represents the amorphous structure, the other colours represent the crystal structure; (c) Distributions of CN corresponding to the dashed rectangle box in (a) at ε = 0.2250, 0.2251, and 0.2254, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
crucial role in the mechanical performance of nc-AlN. For instance, in the model with TD, the immature SB develops near Particle I indicated with white dotted rectangle at ε = 0.130 (Fig. 12(a)) and meets the particle Ⅱ at ε = 0.195. The immature SB bypasses Particle Ⅱ and keeps moving since the strength of the crystalline particle is larger than that of the amorphous matrix, which is difficult for immature SB to move across the whole sample. No hardening is observed in the RD, attributed to that the immature SB could go across the whole sample without intersection. Although the strain hardening stage of the model with TD is a little smaller than that with SD, the former should be the best choice. 4. Conclusions The mechanical properties of dual-phase AlN materials (nc-AlN) composed of amorphous AlN matrix and crystalline AlN nanoparticle were investigated with molecular dynamics simulations for the purpose to uncover the mechanism of the enhancement in strength and ductility. The effects of volume fraction (fv), surface-to-volume ratio, and distribution of crystalline AlN particles were studied, from which the following conclusions can be drawn.
Fig. 11. Compressive stress-strain curves for nc-AlNs with different distributions of crystalline nanoparticles — Square, Triangle, and random, respectively.
in Fig. 12(a), and this duration is smaller than that of SD, because of the phase transformation that occurs previously, as shown in Fig. 13(a1). Similarly, some particles in the model with TD are cut into half at ε = 0.300, as shown in Fig. 13(a2). Different from the models with SD and TD, there is no hardening in the model with RD, which resembles the a-AlN model, as shown in Fig. 11. An immature SB develops at ε = 0.130 and the next one appears at ε = 0.195, as shown in Fig. 12(b). No drop of stress can be found in the model with RD, in which no phase transformation occurs, as shown in Fig. 13(b1). Meanwhile, the particle is still of perfect wurtzite structure without any fracture, as shown in Fig. 13(b2). The distribution of particle plays a
(1) Stress concentration exists in the nc-AlN samples, where crystalline particles and amorphous matrix are subjected to tensile and compressive hydrostatic stresses, respectively. (2) The elastic modulus of the nc-AlN increases with the increase of fv, but the magnitude is larger than that evaluated by Mori-Tanaka approach, and Voigt and Reuss approximations, attributed to the effect of interfaces that take strong effect at nanoscale. Hardening Would occur in the nc-AlN of fv ≥ 40.9%, as a result of the intersection between shear bands. The transformation from the wurtzite structure (B4) to graphene-like (GL) occurs layer by layer in the ncAlN of fv = 40.9%, ascribed to the high hydrostatic stress. 9
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Fig. 12. Distributions of local shear strain corresponding to (a) triangle distribution, and (b) random distribution at ε = 0.076, 0.130, and 0.195, respectively.
Fig. 13. (a1) Distributions of CN at ε = 0.200, (a2) atomic structure recognized with IDS method at ε = 0.300, corresponding to triangle distribution; (b1) Distributions of CN at ε = 0.300, (b2) atomic structure recognized with IDS method at ε = 0.300, corresponding to random distribution.
fv ≥ 40.9% has a better performance.
(3) The plastic performance of nc-AlN is less dependent on surface-tovolume ratio because the extension of the immature shear band is restricted by the crystalline particles. The vortexes develop in the vicinity of nanoparticles and gather together, rather than be activated one by one. (4) The distribution pattern of crystalline nanoparticle plays an important role in the mechanical performance of nc-AlN. Triangle distribution (TD) of may be the best choice compared with square or random distributions (SD and RD). The nc-AlN with TD and
The results obtained in this article would enrich the research of the mechanical properties of ceramics and provide available information for the design of high-performance AlN based ceramics.
Acknowledgements The authors gratefully acknowledge the financial support from 10
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National Natural Science Foundation of China (Grant No. 11332013), ARC Discovery Project (180101955 and 180102003), the program of China Scholarships Council (No. 201606050043), The High Performance Computing (HPC) and CARF at QUT have kindly provided access to their facilities.
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