Smaller is Plastic: Polymorphic Structures and ... - ACS Publications

Apr 30, 2015 - planes has been seen to contribute to plastic deformation in Mg ... result in changes in the fault vectors, upon structural relaxation...
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Smaller is Plastic: Polymorphic Structures and Mechanism of Deformation in Nanoscale hcp Metals Meha Bhogra,† U. Ramamurty,‡,§ and Umesh V. Waghmare*,† †

Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560 064, India Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India § Center of Excellence for Advanced Materials Research, King Abdulaziz University, Jeddah 21589, Saudi Arabia ‡

S Supporting Information *

ABSTRACT: Using first-principles calculations, we establish the existence of highly stable polymorphs of hcp metals (Ti, Mg, Be, La and Y) with nanoscale structural periodicity. They arise from heterogeneous deformation of the hcp structure occurring in response to large shear stresses localized at the basal planes separated by a few nanometers. Through Landau theoretical analysis, we show that their stability derives from nonlinear coupling between strains at different length scales. Such multiscale hyperelasticity and long-period structures constitute a new mechanism of size-dependent plasticity and its enhancement in nanoscale hcp metals. KEYWORDS: Structural transitions, hcp metals, polymorphism, hyperelasticity, planar faults

M

shear stresses, particularly those leading to a high density of planar faults on the basal plane, are still unexplored. In this work, we use first-principles calculations to study planar faults on basal (0001) plane of Ti and show that localized shear stresses associated with them lead to formation of long-period polymorphs which have essentially the same energies as of the hcp structure. The relevance of c/a ratio in determining the nature of deformation prompted us to investigate similar polymorphic structures in other hcp metals, viz. Mg, La, Be, Y and Zn. Indeed, the extraordinary stability of these polymorphs is found to be quite general in all the hcp metals, except in Zn which has an anomalously large c/a ratio. Originating from the release of shear stresses resulting from large localized strains at the planar faults, these polymorphs involve nonlinearly coupled strains at different length scales and provide a new mechanism of enhanced plasticity in nanoscale hcp metals. Our calculations are based on first-principles density functional theory (DFT) and plane-wave pseudopotential method as implemented in Quantum ESPRESSO20 (QE). The ionic core−valence electron interactions are represented with ultrasoft pseudopotentials21 and electronic exchange− correlation energy is approximated with a generalized gradient approximation (GGA) as parametrized by Perdew−Wang22 for Ti, Perdew−Burke−Ernzerhof23 for Be, Y, and La, and with a local density approximation (LDA) as parametrized by

any modern materials exhibiting a combination of high strength and improved ductility are nanostructured.1−3 This has stimulated several efforts to understand the deformation mechanisms in materials at submicrometer length scales.4−10 Specifically, hcp metals have been shown to exhibit significant crystal size effects. While the bulk hcp structure permits fewer number of active slip systems than those in bcc and fcc metals11 and exhibits twinning12 as the geometrically favored mode of deformation, ordinary dislocation-mediated plasticity (ODP) becomes dominant13−15 when the sample size decreases below 1 μm. For instance, in situ TEM studies on nanopillars of hcp metals like Mg and Mg-0.2 wt % Ce,16 and Ti-5 at% Al10 have shown dislocation slip on the basal plane at very high-stress fields in the necked region, which coincides with their softening and enhanced plasticity. Activity of dislocations on both basal and nonbasal (⟨c⟩ and ⟨c + a⟩) planes has been seen to contribute to plastic deformation in Mg nanoparticles (∼100 nm).7 This indicates a significant reduction in the anisotropy of critical resolved shear stress (τCRSS) and a more homogeneous plastic flow at submicrometer-scale. Recently, twinning-induced plasticity (TWIP) has been reported in nanosized [0001]-oriented single-crystal Mg through {1011̅} “contraction” twinning and {1012̅} “extension” twinning at very high stresses.17,18 Mg nanopillars are seen to exhibit twinning-like lattice reorientation, and coordinated motion of interfaces between basal and prismatic planes under compressive stresses.19 These studies suggest that there are interactions between planar faults that may be responsible for occurrence of a coherent shearing process and enhanced plasticity in hcp metals at nanoscale. However, consequences of © XXXX American Chemical Society

Received: December 26, 2014 Revised: April 18, 2015

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Nano Letters Perdew−Zunger24 for Mg and Zn. The choice of these functionals is based on the availability of reliable pseudopotentials with QE, and the physical results obtained are not sensitive to these choices. Kohn−Sham wave functions are expanded in a plane-wave basis set truncated with energy cutoff of 30 Ry (and a corresponding cutoff of 240 Ry for charge density). Integrations over Brillouin Zone (BZ) are sampled with a uniform (20 × 20 × 12) k-grid for the bulk hcp structures, and the atomic structures are relaxed using Broyden, Fetcher, Goldfarb, Shanno (BFGS)-based algorithm until the Hellman−Feynman forces on each atom are lower than 0.001 Ry/bohr. Energies of the new structures of Ti are confirmed using ABINIT25 implementation of DFT with a normconserving pseudopotential26 and LDA as parametrized by Perdew−Wang.27 Dynamical matrices (DMs) and phonon frequencies of polymorphs are obtained using the DFT linear response28 method (QE implementation) at q-points on a uniform (2 × 2 × 1) mesh in the BZ of supercells. We simulate planar faults on the basal (0001) plane of Ti using periodic supercells containing six unit cells (12 atomic planes) of hcp Ti stacked along the c-axis. A localized shear strain is introduced through a generalized stacking fault (GSF) vector f ⃗ = xa⃗1 + ya⃗2, where a1⃗ , a2⃗ are periodic cell vectors in the basal plane, and (x,y ∈ [0,1]) are the fractional coordinates, by changing a3⃗ to a3⃗ + f. In practice, (x,y) are chosen points on a uniform (6 × 6) grid on (0001) plane. To reduce the in-plane stresses built up in faulted configurations, we optimize these structures with respect to all the degrees of freedom (atomic positions and overall strain). Because of the small (nanoscale) size of supercell, with c = 6c0 ≈ 2.7 nm, the strain-mediated interactions between faults propagate across the supercell and result in changes in the fault vectors, upon structural relaxation. The variable cell relaxation of the faulted nanoscale structures has revealed three inequivalent structures with energies that are same as that of pristine hcp Ti, within the computational errors (see Table 1). These energies (Table 1) are quite small in

To establish further credibility in these surprising results of stability, we repeated these calculations using another DFT code, ABINIT with norm-conserving pseudopotentials.25 The final relaxed structures are indeed equivalent to those obtained using QE, within the typical DFT errors, and have energies essentially the same as that of pristine hcp Ti (Table S1). Increase in the separation between adjacent planar faults (c = 12c0 ≈ 5.4 nm) results in a slight increment in energy, as exemplified by the structure II with ΔE of 1.9 meV/atom, which indicates that such polymorphs are stabilized only when the density of planar faults is high. Hence, we call these novel structures as the ‘deformation-induced polymorphs’ of Ti. We now identify the structural and vibrational signatures of these polymorphs to establish their uniqueness and structural distinction with respect to hcp Ti (see Figure S1 and Table S2). These polymorphs show significant changes in their out-ofplane bond lengths and negligible increase in their in-plane bond lengths (Figure 1b), while preserving the coordination number of the hcp structure. Calculated vibrational spectra of these polymorphs establish their structural stability (see Figure 1c for phonon density of states) and highlight significant softening of longitudinal acoustic (LA) mode along Γ− A(0,0,1/2) segment (Figure 1d),16 which originates from the increase in the out-of-plane bond lengths in these polymorphs. The decrease in the out-of-plane bond lengths leads to the hardening of optical modes with eigen displacements along the c-direction from 204 cm−1 in hcp Ti to 215, 211, and 216 cm−1 in polymorphs I, II, and III respectively. Frequencies of LA mode along Γ−M(1/3,1/3,0) for q → 0 show no change in the polymorphs relative to hcp Ti (Figure 1d), thus confirming the uniformity in their in-plane bond lengths. Phonon DoS of the polymorphs (Figure 1c) also show an increase in the density of low-frequency modes which dominate in vibrational contribution to free energy and play an important role in their hightemperature stability. Neglecting phonon−phonon anharmonic interactions over the temperature range from 0 to 1000 K (∼0.5 Tm), we estimate temperature-dependent vibrational free energies within a harmonic approximation30,31 as

Table 1. Energies and Cell Vectors of New Structures with Reference to Pristine hcp Ti Using QE

Fvib = Etotal +

c vector (in units of a0) hcp Ti structure I structure II structure III

ΔE (meV/atom)

cx

cy

cz

0.0 0.78 1.06 1.15

0.0 0.834 0.916 0.501

0.0 0.0 0.001 0.685

1.580 9.499 9.492 9.499

kBT Nq

⎡

⎛ ℏwiq ⎞⎤ ⎟⎥ ⎝ 2kBT ⎠⎥⎦

∑ log⎢⎢2sinh⎜ q,i

⎣

(1)

where Nq is the number of q-points on (20 × 20 × 12) mesh in the BZ, and ωiq is the frequency of ith normal mode at wavevector q. Since the relative stability of polymorphs at T > 0 is dominated by slightly softer acoustic phonons, resulting from the strain in these structures, anharmonicity does not bring in significant changes in the free energies up to 0.8 of Debye temperature. We find negative free energies of the polymorphs II and III (see inset of Figure 1b) at elevated temperatures relative to hcp Ti, indicating that an enhanced plastic flow during the high-temperature deformation of nanostructured Ti is expected to involve these highly stable long-period polymorphic structures. Landau theoretical description of polymorphs: Landau theory is an effective tool in modeling a structural phase transition in which the low-temperature phase typically is of a lower symmetry and can be derived as a structural distortion of the parent higher symmetry phase stable above the transition temperature T0. The latter structure is used as a reference, its symmetry lowering distortion defines an order parameter which is nonzero at T < T0. The order parameter is typically a normal mode of the reference structure and has a definite symmetry.

comparison to the cohesive energy of Ti at 0 K (4.87 eV/ atom). The morphology of these structures shows nonvanishing atomic displacements in each of the atomic planes, in contrast to the localized nature of fcc-stacking faults (SF) involving the formation of a shear step (f ⃗ = 2/3a1⃗ + 2/3a2⃗ ) (Figure 1a) and thus cannot be directly compared to the polytypic structures with well-defined stacking sequences occurring during the hcp-fcc transition.29 This structural feature of the polymorphs showing delocalization of shear strain over the entire supercell suggests an improved capacity of the nanostructured Ti to sustain large shear stresses developed during plastic deformation without the formation of any faults or dislocations. This is analogous to mechanism of formation of nanotwins in Mg at the crack tip, the site of localized high tensile stresses, leading to deflection of the crack and toughening of the sample.17 B

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Figure 1. Structural fingerprints of Ti polymorphs shown with (a) atomic displacements (dfaulted = |rn⃗ |faulted − |rn⃗ |ideal) in faulted configurations after n relaxation with all degrees of freedom, clearly highlighting the structural dissimilarity between fcc-stacking fault and the polymorph II, (b) distribution of normalized bond lengths in the three polymorphs (bo = 2.87 Å is the out-of-plane bond length in hcp Ti), (c) stability of polymorphs evident from absence of any unstable mode in phonon density of states (p-DoS), and (d) softening of dilational mode involving out-of-plane (left) and shear mode for q →0 involving in-plane atomic displacements (right). The inset of (c) shows enhanced temperature-dependent stability of polymorphs II and III from vibrational free energy analysis.

Figure 2. Structural analysis of deformation-induced polymorphs. (a) Rectangle (B) represents the structure of polymorph I and rectangle (A) is the region of localized strain. The color gradation from red to white shows the degrees of inhomogeneous deformation in polymorph I, the values of which are shown in the horizontal bar as fractions of a0. Periodic displacement fields associated with acoustic and optical modes in (b) real space and (c) Fourier space that describes polymorphs as distortions of the hcp structure.

Free energy of the relevant low-symmetry structures can be expressed as a Taylor expansion that is invariant under the symmetry of the reference structure. Identification of the order parameter in Landau analysis uncovers the precise relationship between the two structures in terms of phonons or normal modes, and the form of free energy expansion reveals the mechanism, i.e., coupling between various degrees of freedom that drives the structural transition. In the present case, many degrees of freedom are involved in shear-induced transition of hcp structure to the polymorphs, and the coupling between them defines multiscale hyperelasticity.

Structurally, the deformation-induced polymorphs of Ti (see illustration of polymorph I in Figure 2a) involve a combination of (a) an overall (homogeneous) supercell deformation, originating from the initial slip f ⃗ on the basal plane at c = 6c0, (b) inhomogeneous strains given by spatial variations in average displacements of A and B sublattice sites in each primitive cell, and (c) internal strains associated with relative displacements of sublattice atomic planes A and B within a primitive cell. Homogeneous shear strain in the cell, ϵxz is ϵxz = C

cx 6c0

(2) DOI: 10.1021/nl504978t Nano Lett. XXXX, XXX, XXX−XXX

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Nano Letters where cx is the x component of c vector of a polymorph and c0 is the cell parameter of hcp Ti (Table 1). The ϵxz values for polymorphs I, II, and III are 0.087, 0.096, and 0.088 (8−10% strain), respectively (the strain has both xz and yz components in polymorph III). To separate the inhomogeneous deformation into acoustic phonons (inhomogeneous strain) and optical phonons (internal strain), we define two degrees of freedom for each unit cell, i (i = 1−6, in the present study) belonging to the supercell: ai =

uAi + uBi u − uBi and oi = Ai 2 2

analysis has revealed that the specific acoustic phonons (q = 0.6π/c0) (see Figure 2c) are involved predominantly in addition to homogeneous strain ϵ in the formation of these polymorphs starting from hcp as a reference structure. Since the wavevector q of a phonon is a Bloch vector belonging to reciprocal space, an expression of the Landau free energy function can be more readily written in reciprocal space spanned by q. A generic symmetry allowed term representing the lowest order coupling between these degrees of freedom is (aq’s are Fourier transform of ai’s): F∼

(3)

(5)

where f(q,ϵ) is an even function of q, and periodic in reciprocal space (q), and limq→0 f(q,ϵ) = 0. For small q, f(q,ϵ) ∼ gϵq2, where g is the third-order coupling that gives the leading order change in acoustic phonons (or elastic modulus) with strain ϵ. From Figure S2, it is clear that the acoustic modes at q in the range (0.4,0.6)π/c0 couple most strongly and anharmonically with strain ϵ. Hence, the associated length-scale is 1/Δq is 6c0 ∼ 3 nm. The multiscale hyperelastic model (eq 5) has been written in terms of displacement field given by the acoustic phonons ai’s, rather than strain, and it does not correspond to a simple strain energy density function in the continuum limit. Noting that ϵ = ∂a/∂r ∼ iqaq, a reasonable continuum limit can be taken assuming that the strain fields vary on long lengthscales, and then this model becomes

where uαi is the displacement of atomic plane α (A or B), ai describes the average displacement of adjacent A and B sublattice planes, and oi describes the relative displacement of adjacent A and B sublattice planes in a polymorph with reference to the pristine hcp structure. We point out that internal strains, oi’s are unique to the hcp metals because of their crystal structure with a unit cell containing two-atom basis (similar to shape-memory alloys). To compute ai and oi, we first filter out the contribution of homogeneous strain to changes in the atomic positions. This is done by representing atomic positions in a polymorph in reduced (crystal) coordinates and obtaining unp = dpn − don, dpn and don being atomic positions of nth atom (n = 1−12, in the present study) in a polymorph and the hcp structure, respectively. The ai’s, obtained using eq 3, represent a displacement field of inhomogeneous strains (acoustic modes) in the symbol and contain linear and periodic components of atomic displacements. We isolate the periodic part of atomic displacements: ∼ ai = ai − i*(ϵ′xz)(c0) − a ̅

∫ dq|a(q)|2 f (q , ϵ)

F∼

∫ drE(ϵ(r), ∇ϵ(r), ∂ 2ϵ(r), ...)

(6)

where E is the energy density of local strain ϵ(r)⃗ and its spatial derivatives, and there exists a nonlinear coupling between them. Generality of polymorphs in hcp metals: To establish the possible existence and stability of similar polymorphic structures in other hcp metals, viz. Y, Be, La, Mg, and Zn, we obtained the minimum-energy structure of the polymorph II of each of these metals and identified structural features that can be rationalized through correlation with their c/a ratio. In each of these metals, the structure of the polymorph II is generated by shear deformation on basal plane with the same fault vector as that of polymorph II of Ti and then optimized with respect to all the degrees of freedom, resulting in the minimum energy structure. The energy of polymorph II of each of these metals is then estimated relative to the respective hcp structure (Table 2). Our results uncover a striking existence of highly stable

(4)

where ϵ′xz is the slope of linear part of ai’s, and a¯ is the average of ai’s. The values of ϵ′xz are −0.098, −0.098, and −0.097 for polymorphs I, II, and III, respectively. We note that ϵxz ′ is negative, which means that local strain mostly compensate or screen the homogeneous deformation (eq 2), imparting ai ’s are periodic after structural stability to the polymorphs. ∼ 12-atomic planes (see Figure 2b) and show distinct inhomogeneous strain patterns for each of the three ai ’s (Figure 2c) gives the polymorphs. Fourier transform of ∼ relative contribution to strain of the acoustic phonons with different wavevectors in the units of π/c0. We find that the acoustic phonons at q ∼ 0.6(π/c0) dominate the inhomogeneous strains associated with these polymorphs, similar to that in martensitic phases of shape-memory alloys.32 The presence of these strain fields with varied Fourier components in the stable polymorphs reflects a nonlinear coupling between strains at different length scales. This defines a multiscale hyperelasticity, as opposed to simple local nonlinear elasticity in which elastic moduli depend on strains. The periodic displacement field of optical phonons, oi’s (eq 3), which indicates the inhomogeneous internal strain in the polymorphs relative to hcp Ti, has an average value of about 0.1 (in terms of lattice constant, ao) for the three polymorphs (see Figure 2b). Fourier analysis (Figure 2c) reveals the dominance of optical modes at wavevector q → 0 showing that the internal strains involve primarily long-wavelength component. Multiscale hyperelasticity: Among the acoustic and optical phonon degrees of freedom (ai and oi), our Landau theoretical

Table 2. Energies of Polymorph II of hcp Metals hcp metal

c/a value

ΔE (me V/atom)

Y Be Ti La Mg Zn

1.55 1.57 1.58 1.61 1.62 1.87

0.10 0.28 1.01 0.46 0.50 1.86

deformation-induced long-period polymorphs in all of these hcp metals, except for Zn, which has an unusually high c/a ratio of 1.87. Temperature-dependent free energies of the polymorph II (see Figure 3) of these hcp metals show existence of highly stable polymorphs of Mg and Y at elevated temperatures. Though vibrational contributions lower the free energy of D

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changes with reference to the hcp structure, resulting in the overall c/a ratio close to the ideal value (1.633), with a relatively large energy cost of ∼1.86 meV/atom. This reflects a weaker tendency of polymorphism in Zn, in contrast to that in other hcp metals. In conclusion, using first-principles simulations, we predict remarkably stable polymorphic structures of hcp metals that nucleate in response to large localized shear strains. We argue how they provide a new mechanism that is relevant to the enhanced plasticity in nanoscale hcp metals. Using Landau theoretical analysis, we show that (a) the atomic-scale structure of these polymorphs involves a combination of homogeneous and inhomogeneous strains and internal strain field similar to that in martensitic structures, with no resemblance to the structure of localized stacking faults or twins in crystals, and (b) their stability is governed by interactions between strain fields at different length-scales, that defines multiscale hyperelasticity. The mechanism of the long-period structures relevant to sizedependent plasticity is distinct from that at microscale where the resulting strain gradients lead to accumulation of geometrically necessary dislocations and formation of shear bands.33 Modification of the energetics of these polymorphs by substitutional alloying or engineering of their nanoscaled structure should facilitate the development of lightweight materials with high strength and ductility.

Figure 3. Free energies of polymorph II of hcp metals viz. Be, Mg, La, Y, and Zn at finite temperatures relative to the respective zero-point energies. Inset shows the free energies at temperatures close to 0 K.

polymorph II of Zn, the energy of this structure is significantly high ∼1.43 meV/atom at 0.8 Tm. Local changes in the atomic structure of polymorph II of each of these metals classify them into groups that correlate with their c/a ratio (Figure 4). For metals with c/a < 1.633, namely Y and Be, there are slight changes in the in-plane and out-of-plane bond lengths, while there is a significant spread in the out-of-plane bond lengths for La and Mg whose c/a ∼ 1.633. Interestingly, the new structure of Zn shows significant

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/nl504978t.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone:+91.80.2208.2842. Fax:+91.80.2208.2767. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. K K Sankaran and Prof. Vikram Jayaram for valuable suggestions, discussions, and comments. We acknowledge Boeing for funding and Centre for Computational Materials Science (CCMS), Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR) for providing access to computational facilities. U.V.W. and U.R. acknowledge support from the J. C. Bose National Fellowship.

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Figure 4. Distribution of normalized bond-lengths in polymorph II of hcp metals: (a) Be and Y with c/a < 1.633, (b) La and Mg with c/a ∼ 1.633, and (c) Zn with c/a ≫ 1.633, where bo is the out-of-plane bond-length of the respective pristine structure. In-plane bond lengths of pristine hcp structures of Be, Y, La, Mg, and Zn are 1.027, 1.033, 1.002, 1.003, and 0.908 respectively in units of bo. E

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