Elastic constants of polycrystals with generally ...

JOURNAL OF APPLIED PHYSICS 120, 165105 (2016)

Elastic constants of polycrystals with generally anisotropic crystals Christopher M. Kube1,a) and Maarten de Jong2,b) 1

Vehicle Technology Directorate, U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005-5069, USA 2 Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA

(Received 26 August 2016; accepted 8 October 2016; published online 26 October 2016) A homogenization model is developed that describes the effective elastic constants of polycrystalline materials with constituent crystallites of general anisotropy (triclinic symmetry). The model is solved through an iterative technique where successive iterations improve the estimates of the polycrystal’s elastic constants. Convergence of the solution provides the self-consistent elastic constants, which are the polycrystal’s elastic constants resulting from continuity between local and far-field stress and strains. Iterative solutions prior to convergence are the bounds on the elastic constants including the Voigt-Reuss and Hashin-Shtrikman bounds. The second part of the article establishes a formal link between the present model and single-crystal elastic anisotropy. An analysis from a dataset containing 2176 inorganic crystalline compounds, spanning all crystallographic symmetries, is provided. The role of elastic anisotropy and related properties such as crystalline structure and elastic stability are discussed as it relates to the model. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4965867] I. INTRODUCTION

In nature, crystalline materials are often found as constituents of expansive, multicomponent systems. Some examples are metallurgical systems that are polycrystalline, which are composed of an assemblage of crystallites (grains). In many instances, the physical properties of crystalline materials are well-characterized from either laboratory studies conducted on a single crystal or modeling of single-crystal properties. It is practical and highly desirable to use this data to predict the response of systems containing an assemblage of the single crystallites. For polycrystalline media, the well-known properties of a representative constituent crystallite can be homogenized to describe the response of the larger system. Statistical homogenization techniques treat the constituent crystallite properties as stochastic variables, where, upon averaging, the bulk properties of the polycrystal are obtained. Statistical homogenization methods have been used to model a number of properties pertaining to polycrystalline media (see Markov1 for a review). The elastic case has been investigated most frequently and is the subject of this work. The primary obstacle to the elastic homogenization problem is relating the local stress and strain within a crystallite to that of the polycrystal. Full compatibility of the stresses and strains throughout the polycrystal lead to homogenized elastic properties that are known as self-consistent (SC). Failure to meet compatibility results in nonunique estimates depending on a strain- or stress-based homogenization. Voigt2 assumed the strain in the crystallite to be uniform and equal to the strain in the polycrystal. Such an assumption fulfills kinematic compatibility while losing compatibility of the stress fields. The effective elastic moduli of the a)

Electronic mail: [email protected]. Present address: Space Exploration Technologies, 1 Rocket Rd, Hawthorne, CA 90250, USA.

b)

0021-8979/2016/120(16)/165105/14/$30.00

polycrystal following Voigt’s assumption are obtained by averaging the crystallite’s elastic moduli over all possible crystallographic orientations. Conversely, Reuss3 ensured static compatibility of the stress fields, which results in a loss of kinematic compatibility. This assumption leads to estimates of the polycrystal’s compliance as an orientation average of the crystallite’s elastic compliance tensor. Elastic moduli in the Reuss formulation follow by inverting the result of the averaged compliance. The uniform strain assumption of Voigt and uniform stress assumption of Reuss were proven to be extremum mechanical states by Hill.4 This led to the proof that the Voigt and Reuss homogenization techniques form bounds on the actual properties.4 Improved estimates of the elastic constants were proposed by finding the arithmetic average of the Voigt and Reuss bounds, known as the Hillestimate.4 Such a procedure is purely empirical and has no theoretical basis. Thomsen5 compared experimental data spanning a large collection of polycrystalline materials and concluded that Hill’s estimate has little experimental support. Despite these deficiencies, the Hill estimate (or sometimes referred to as the Voigt-Reuss-Hill estimate) remains in use, primarily because of its simplicity. Another approach to obtain more accurate estimates of elastic constants of polycrystals is to seek a homogenization that produces tighter bounds. Based on variational principles, Hashin and Shtrikman6,7 (HS) derived the next order of bounds or the second-order bounds, which are tighter than the first-order Voigt and Reuss bounds. The bound theorems were generalized further by Dederichs and Zeller8 (DZ) who derived formulas to produce all odd-order bounds. Similarly, Kr€oner9 developed formulas for the even-order bounds. As the order of the bounds increase, the bounds become closer together. Due to complexity, the closed-form evaluation of the various bounds6–9 was not performed except for the simplest case of statistically isotropic polycrystals with crystallites of

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C. M. Kube and M. de Jong

cubic crystallographic symmetry. In a series of papers, Watt10–13 developed the Hashin-Shtrikman bounds further by considering polycrystalline materials with hexagonal, trigonal, tetragonal, orthorhombic, and monoclinic crystallite symmetries. Watt14 later provided a computational scheme based on this work10–13 to make the Voigt, Reuss, and HS bounds accessible to the geophysics community. Each symmetry class was handled individually and the complexity of the algorithms likely inhibited widespread adoption. Recently, Brown15 developed a new computational algorithm for the calculation of the HS bounds. This method is computationally more efficient as it does not partition the symmetry classes.15 The SC estimates are obtained when full kinematic and static compatibility is met. This condition is met for the order of bounds that effectively coincide. Hershey16 was the first to derive a quartic expression for the SC shear modulus for a statistically isotropic polycrystal containing cubic crystallites. An equivalent cubic expression was later independently derived by Kr€ oner17 for the same case. A remarkable and important unification between the SC estimates and the bounds was achieved by Gairola and Kr€oner.18 This connection was found using a simple iterative process, in which each iteration gave the next order of tighter bounds and the self-consistent result, upon convergence. Subsequently, Berryman19 developed an iterative method starting from the HS bounds and produced the SC values for polycrystals containing crystallites of hexagonal, trigonal, and tetragonal symmetries upon convergence. Berryman20 extended this approach to polycrystals having crystallites of orthorhombic symmetry. In this work, the SC shear modulus was used as an input to a search routine to produce the HS bounds.20 In this article, the bounds and SC estimates are determined for polycrystalline materials that are statistically isotropic and homogeneous, and contain a microsctructure consisting of a single-phase of anisotropic crystallites. The theory is developed in Sec. II. General expressions for the effective Lame and shear constants are obtained. These expressions are similar in spirit to those in Gairola and Kr€oner18 in which repeated iteration produce the various orders of bounds including the HS bounds, and upon convergence, the SC estimates. Unlike previous work on this topic, the present model is applicable to crystallites belonging to any of the seven crystal classes (cubic, hexagonal, tetragonal, trigonal, orthorhombic, monoclinic, and triclinic). Closed-form expressions for the Lame and shear modulus of polycrystals having crystallites of cubic symmetry are derived from the general case of triclinic symmetry. These results produce closed form expressions for the Lame constant and the shear modulus.17,18 In Sec. II C, a theoretical connection between the bounds and the single-crystal elastic anisotropy is developed. The theoretical model is valid for polycrystals with crystallites of any symmetry. This feature permits a systematic analysis of the Voigt, Reuss, Hill, and SC estimates of the bulk and shear modulus, which is given in Sec. III. The bounds and SC estimates are obtained for a set of 2176 inorganic crystalline materials, which includes calculations spanning all symmetry groups. The crystallite elastic constant data

J. Appl. Phys. 120, 165105 (2016)

used in the analysis were obtained from the Materials Project (MP),21–25 which was originally generated from calculations based on the first-principle density functional theory (DFT). The DFT calculations are reviewed first in Sec. III A. In Sec. III B, previously defined anisotropy indexes26–30 are linked to the various orders of bounds. The universal log-Euclidean index30 is highlighted and employed to help analyze the set of inorganic crystalline materials. In Sec. III C, we first compare the SC estimate to the Hill average, HS bounds, and experimental measurements in the context of ultrasound experiments, which clearly shows the significance of the SC scheme for calculating elastic moduli. Next, the SC estimates are compared to the Hill average for all 2176 compounds from the Materials Project dataset and the anomalous cases are analyzed. Subsequently, we demonstrate the relation between how well the different schemes perform with respect to each other and the degree of elastic anisotropy. Section III D discusses the origins of discrepancies between the SC estimate and the Hill average in terms of elastic anisotropy and mechanical instabilities. Our findings are illustrated with several examples. Finally, Sec. IV presents the conclusions of this work. II. THEORY

The theory is organized by first introducing notation and tensor definitions in Sec. II A. A derivation of the effective elastic moduli C and compliances S is given in Sec. II B, which is based on the compatibility delivered by the concentration tensors G for stress and H for strain. All necessary tensors and operations needed to construct C and S are given. A universal log-Euclidean anisotropy index, valid for all crystallite symmetries, is given in Sec. II C based on the first-order bounds on C and S .30 A. Preliminaries

The following notation and tensor conventions are applied throughout Sec. II. Boldface Latin characters denote fourthrank tensors, e.g., F. The equivalent index form of F is Fijkl. If F is invertible and D ¼ F1 is its inverse, then D is obtained by solving for its components through F : D ¼ I with I as the fourth-rank identity tensor, Iijkl ¼ ðdik djl þ dil djk Þ=2. The inner product operator is given by a boldface colon, which contracts the last two indices and the first two indices of the product of two fourth-rank tensors. For example, Fijmn Dmnkl ¼ Iijkl is the equivalent index form to F : D ¼ I where the Einstein summation convention over repeated indices from 1 to 3 is assumed. The components of all fourth-rank tensors in this article are given in lowercase Latin characters with two indices by making use of Voigt’s index notation where the pairs of indices obtain the following single values: 11 ! 1; 22 ! 2; 33 ! 3; 23 or 32 ! 4; 13 or 31 ! 5, and 12 or 21 ! 6. Thus, all fourthrank tensors in this paper have the minor symmetries Fijkl ¼ Fjikl ¼ Fijlk ¼ Fjilk . For example, F2213 ¼ F2231 ¼ f25 . If F is an isotropic tensor, it can be written as F ¼ f12 1 1 þ2f44 I or in index form, Fijkl ¼ f12 dij dkl þ 2f44 ðdik djl þ dil djk Þ, where we have defined 1 1 ¼ dij dkl .


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hGi ¼ hðS þ QÞ1 i : ðS þ QÞ ¼ I:

B. Expressions for the effective elastic constants

The generalized Hooke’s law for a linearly elastic polycrystalline medium that is statistically isotropic and homogeneous is r ¼ C : ;

¼S :r ;

(7b)

Applying inverses and rearranging Eq. (7) leads to expressions for the effective elastic moduli C and compliances S for the polycrystal

(1a)

C ¼ hðC þ RÞ1 i1 R;

(8a)

S ¼ hðS þ QÞ1 i1 Q:

(8b)

(1b)

where is the infinitesimal strain tensor and C is the effective fourth-rank elastic modulus tensor for the medium. The inverse of C is the compliance S ¼ ðC Þ1 . Similarly, Hooke’s law for an individual ellipsoidal crystallite embedded within the polycrystal is

These expressions can be written in the form of the isotropic tensors C ¼ c12 1 1 þ 2c44 I;

(9a)

r ¼ C : ;

(2a)

S ¼ s12 1 1 þ 2s44 I:

(9b)

¼ S : r;

(2b)

S is the inverse of C ; Sijmn Cmnkl ¼ Iijkl , which can be used to obtain the stiffness/compliance relations c12 ¼ s12 = ½2s44 ð3s12 þ 2s44 Þ; c44 ¼ 1=ð4s44 Þ. The evaluation of R and Q in Eq. (5) require the inner products between 1 1 and I, which are 1 1 : 1 1 ¼ 31 1; 1 1 : I ¼ I : 1 1 ¼ 1 1, and I : I ¼ I. This procedure allows R and Q to be written in the forms R ¼ r12 1 1 þ 2r44 I and Q ¼ q12 1 1 þ 2q44 I, which has isotropic symmetry. C; S; Q, and R can each be expanded in the form

where C and S ¼ C1 being the elastic modulus and compliance of the inclusion, respectively. The strain and stress fields throughout the material including the interior of the crystallite are uniform.31 In general, r 6¼ r and 6¼ unless C ¼ C. The connection between the stress and strain of the crystallite and polycrystal are the concentration tensors H and G where r ¼ G : r ;

¼H: :

(3a) (3b)

Bijkl ¼ aia ajb akc ald Babcd ¼ ai1 aj1 ak1 al1 b11 þ ai1 aj1 ak1 al2 b16

(10)

þ ai1 aj1 ak1 al3 b15 þ ai1 aj1 ak2 al1 b16 þ ;

The concentration tensors are32,33 H ¼ ðC þ RÞ1 : ðC þ RÞ;

(4a)

G ¼ ðS þ QÞ1 : ðS þ QÞ

(4b)

where a transforms the tensor into an alternate coordinate system. The form of Eq. (10) allows the sums C þ R and S þ Q to be constructed. Constructing C þ R, for example, leads to Cijkl þ Rijkl ¼ aia ajb akc ald ðCabcd þ Rabcd Þ

where

¼ ai1 aj1 ak1 al1 ðc11 þ r12 þ 2r44 Þ R ¼ C : ½E1 I;

(5a)

Q ¼ S : ½ðI EÞ1 I;

(5b)

31

with E being the Eshelby tensor. The Eshelby tensor for a spherical crystallite embedded in the polycrystal is E¼

2ð3c12 þ 8c44 Þ 3c12 2c44 11þ I; 15ðc12 þ 2c44 Þ 15ðc12 þ 2c44 Þ

(7a)

(11)

C þ R has the same symmetry as C while S þ Q has the same symmetry as S. Thus, the inverses of C þ R and S þ Q are constructed by solving for components using the system of equations generated by

(6)

where c12 and c44 are the effective Lame and shear constants of the polycrystal, respectfully. r and of the crystallite are dependent on the crystallographic orientation of the crystallite with respect to the principal directions of r and strain . However, the volume average of r and are equal to r and , respectively. Letting h i denote the volume average and applying this to Eq. (3) gives hri ¼ r ¼ hGi : r and hi ¼ ¼ hHi : , which leads to the conditions hGi ¼ hHi ¼ I. Applying the orientation average to Eq. (4) gives hHi ¼ hðC þ RÞ1 i : ðC þ RÞ ¼ I;

þ ai1 aj1 ak1 al2 c16 þ ai1 aj1 ak1 al3 c15 þ ai1 aj1 ak2 al1 c16 þ

ðC þ RÞ : L ¼ I;

(12a)

ðS þ QÞ : P ¼ I;

(12b)

where we have let L ¼ ðC þ RÞ1 and P ¼ ðS þ QÞ1 . The orientation averages hLi¼hðCþRÞ1 i and hPi¼hðSþQÞ1 i can be written as hLi ¼ l12 1 1 þ 2l44 I;

(13a)

hPi ¼ p12 1 1 þ 2p44 I;

(13b)

where


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15l12 ¼ l11 þ l22 þ l33 þ 4ðl12 þ l13 þ l23 Þ 2ðl44 þ l55 þ l66 Þ;

J. Appl. Phys. 120, 165105 (2016)

(14a)

15l44 ¼ l11 þ l22 þ l33 ðl12 þ l13 þ l23 Þ þ3ðl44 þ l55 þ l66 Þ;

(14b)

15p12 ¼ p11 þ p22 þ p33 þ 4ðp12 þ p13 þ p23 Þ 2ðp44 þ p55 þ p66 Þ; 15p44 ¼ p11 þ p22 þ p33 ðp12 þ p13 þ p23 Þ þ3ðp44 þ p55 þ p66 Þ:

(14c)

(14d)

Because L and P have isotropic symmetry, their inverses are easily obtained as ðl12 Þ1 ¼ l12 =½2l44 ð3l12 þ 2l44 Þ; ðl44 Þ1 ¼ 1=ð4l44 Þ; ðp12 Þ1 ¼ p12 =½2p44 ð3p12 þ2p44 Þ, and ðp44 Þ1 ¼ 1=ð4p44 Þ. Now, all tensor quantities contained in Eq. (8) are defined. However, solutions to Eq. (8) are difficult to obtain because Eq. (8) are transcendental because C is present on both sides in Eq. (8a), while S is present on both sides in Eq. (8b). In order to solve for C and S we employ an iterative method where the right hand sides of Eq. (8) receive suitable guesses for C and S . The right hand sides are then evaluated in order to give the updated expressions for C and S . Self-consistency is observed when C becomes equal to ðS Þ1 . As an example, the first-order bound of Voigt CV is obtained if c12 ¼ 0 and c44 ¼ 1 is placed in the right hand side of Eq. (8a) while the first-order bound of Reuss CR is obtained when c12 ¼ 1 and c44 ¼ 0 is placed in the right hand side. Iterating again, using the pairs cV12 ; cV44 or cR12 ; cR44 in the right hand side produces the third-order bounds CþDZ and CDZ first derived by Dederichs and Zeller (the next tighter bounds following the second-order Hashin-Shtrikman bounds).8 Subsequent iterations produce tighter odd-order bounds that quickly converge to the true SC estimates of c12 and c44 . Figure 1 illustrates this procedure for the layered ternary chalcogenide TlSbS2, which belongs to the triclinic symmetry group. Suitable convergence is observed after 4

FIG. 1. Odd-order bounds for the layered crystallite TlSbS2 calculated using the iterative solutions of Eq. (8a).

iterations, which are the 7th-order bounds. In Fig. 1, the effective bulk modulus is obtained from j ¼ c12 þ 2c44 =3, which is valid for all crystallite symmetries (this form for j is easily found from the definition of the bulk modulus j ¼ Ciijj =9 and applying isotropic symmetry conditions). Alternatively, the even-order bounds are obtained if the iteration procedure begins with the pairs c12 ¼ min ½a1i a1j a2k a2l Cijkl ; c44 ¼ max½a2i a3j a2k a3l Cijkl and c12 ¼ max ½a1i a1j a2k a2l Cijkl ; c44 ¼ min½a2i a3j a2k a3l Cijkl in the right hand side, which are the extremum values of c12 and c44 over all possible crystallographic orientations (aij are rotation matrices used to define all possible crystallite orientations). The first iteration leads to the second-order Hashin-Shtrikman bounds CþHS and CHS . Subsequent iterations converge to the SC estimates. Figure 2 illustrates this procedure for TlSbS2. For this case, the sixth-order bounds are effectively coincident after 3 iterations. The iterative solution procedure is included, see supplementary material for a Matlab resource, STRAIN_BOUND.m, which calculates the upper bounds and self-consistent estimates of the bulk and shear modulus. This function can be used with either numerical inputs to obtain estimates of a polycrystal’s elastic constants for a specific material or symbolic inputs to arrive at analytical expressions. The iterative solutions are required for crystals having crystallographic symmetry lower than cubic. For cubic crystals, equating the two elastic moduli invariants Cijij ¼ Cijij and Ciijj ¼ Ciijj leads to the relation 3c12 þ 2c44 ¼ c11 þ 2c12 , which allows either c12 or c44 to be written in terms of the other. Using this relation, the components of the tensor C from Eq. (8a) are found to be c12 ¼

ðc12 Þ2 a2 þ c12 a1 þ a0 ; ðc12 Þ2 þ c12 b1 þ b0

(15a)

c44 ¼

ðc44 Þ2 c2 þ c44 c1 þ c0 ; ðc44 Þ2 þ c44 d1 þ d0

(15b)

FIG. 2. Even-order bounds for the layered crystallite TlSbS2 calculated using the iterative solutions of Eq. (8a).


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c12, c44, and the anisotropy factor ¼ c11 c12 2c44 , which yields zero when the isotropic condition c11 ¼ c12 þ 2c44 is satisfied. When ! 0, Eq. (15) give c12 ¼ c12 and c44 ¼ c44 . In this case, the polycrystal exhibits the same elastic properties as the constituent crystallites (the polycrystal is effectively a single crystal). As a result, all orders of bounds coincide when ! 0. However, when the crystallite anisotropy (for any crystallite symmetry) is not zero, the converged iterative solution to Eq. (8) effectively overcomes the crystallite anisotropy to arrive at the homogenized or self-consistent values of c12 and c44 . Many anisotropy indexes have been used to quantify the degree of crystallite anisotropy.26–30 The present model helps unify these different anisotropy definitions because they are all based on the bounds on the elastic constants coinciding when the crystallite is isotropic. Zener’s anisotropy index27 for cubic crystals is the ratio of the maximum and minimum values of the crystallite’s shear modulus

where 1 3 2 25 þ c12 þ 4c44 a0 ¼ 12 3 80 þ 23c212 þ c12 c44 þ 4c244 3 2 þ c12 21c12 þ 44c12 c44 þ 20c244 ; 17 2 2 þ ð53c12 þ 17c44 Þ 60 17 5 þ c12 ð33c12 þ 34c44 Þ ; 17

a1 ¼

1 a2 ¼ þ c12 ; 5 b0 ¼

1 59 2 þ 2 ð141c12 þ 134c44 Þ 180 þ15ð3c12 þ 2c44 Þð7c12 þ 10c44 Þ;

A¼

1 b1 ¼ ð73 þ 165c12 þ 170c44 Þ; 60 1 2 c44 þ c44 ð3c12 þ 4c44 Þ þ 2c244 ð3c12 þ 2c44 Þ ; 8 3 2 c1 ¼ þ ð3c12 þ 17c44 Þ þ 15c44 ðc12 þ 2c44 Þ ; 40 1 c2 ¼ þ c44 ; 5 3 2 1 10 þ ð9c12 þ 16c44 Þ þ c44 ð3c12 þ 2c44 Þ ; d0 ¼ 40 3 3 c0 ¼

d1 ¼

3 ð11 þ 15c12 þ 30c44 Þ; 40

and ¼ c11 c12 2c44 is an unnormalized measure of the elastic anisotropy of crystallites of cubic symmetry. Equation (15b) is exactly the expression derived by Gairola and Kr€ oner, while the expression for Lame’s constant c12 in Eq. (15a) is believed to be reported for the first time.18 Analogous expressions can be written for S , but are omitted for brevity. The bounds on c12 and c44 can be realized by using the same iteration procedure as described previously. Alternatively, algebraic manipulation leads to the two cubic equations ðc12 Þ3 þ ðc12 Þ2 ðb1 a2 Þ þ c12 ðb0 a1 Þ a0 ¼ 0;

(16a)

ðc44 Þ3 þ ðc44 Þ2 ðd1 c2 Þ þ c44 ðd0 c1 Þ c0 ¼ 0;

(16b)

which can be solved exactly for c12 and c44 . Equation (16b) was first derived by Hershey16 and then, independently, by Kr€ oner.17 C. Anisotropy indexes

Equations (15) are implicit formulas for the effective Lame and shear elastic constants of polycrystals having crystallites of cubic crystallographic symmetry. These two formulas are written in terms of the single-crystal elastic constants

2c44 : c11 c12

(17)

Zener’s index is the ratio of the zeroth-order iterative solutions for c44 (for cubic crystals). The Ledbetter-Migliori index was defined through the squared ratio of the maximum and minimum acoustic transverse wave velocities.29 The Chung-Buessem index is28 AC ¼

cV44 cR44 ; 2cH 44

(18)

which is based on the first-order Voigt and Reuss bounds coinciding for the case of crystallite isotropy. Similarly, Ranganathan and Ostoja-Starzewski26 defined a universal anisotropy index AU, valid for all crystallite symmetries, based on the closeness of the tensors CV ¼ hCi (Voigt’s average or the orientation average of C) and hSi (the orientation average of S), which is given by26 AU ¼ CVijkl hSijkl i 6 ¼ 5

cV44 jV þ 6: cR44 jR

(19)

AU yields zero when the first-order Voigt and Reuss bounds coincide. Thus, the zeroth and first-order iterative solutions to the present model can be used to calculate any of the previously defined anisotropy indexes.26–29 Additional anisotropy indexes based on the present model can be conceived by formulating ratios of higher-order bounds or by observing the rate of convergence to the SC values. 1. Universal log-Euclidean anisotropy index

In Sec. III, the influence of crystallite anisotropy on the present model is investigated for 2174 crystalline materials. To facilitate the analysis, an alternative anisotropy index is employed based on a Euclidean distance measure between CV and CR , which is zero if CV ¼ CR when the crystallite is isotropic. The standard Euclidean distance dE between CV and CR is dE ¼ ðCVijkl CRijkl Þ1=2 (summation is over all four of the


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repeated indices).30 From Eq. (19), the universal anisotropy index AU is equal to the squared standard Euclidean distance between CVijkl ¼ hCijkl i and hSijkl i (minus a normalization constant of 6). Similarly, a log-Euclidean formula30,34 is used to define a universal log-Euclidean index as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " !#2ffi u 2 u V V j c þ 5 ln 44 ; (20) AL ¼ t ln R j cR44 where lnð Þ is the standard natural base-e logarithm. This index, based on a distance measure between the Voigt and Reuss bounds, is scaled correctly for perfect isotropy, i.e., AL ¼ 0. A derivation of Eq. (20) can be located elsewhere.30 Equation (20) has a structure similar to AU and retains the universal features of AU, i.e., it is valid for all crystallographic point symmetry groups. However, AL is found to be more appropriate for the present analysis because it is less sparse than AU when considering extremely anisotropic crystallites.

J. Appl. Phys. 120, 165105 (2016)

elements and compounds. This procedure produces sets of elastic constants that have been verified to be within 5% of published values for over 95% of a dataset consisting of 1181 materials.22 The present work considers an expanded data set of 2176 crystalline materials spanning all crystallographic point group symmetries and over 40 different atomic species. This yields a large variety of chemistries that includes metals, metallic alloys, semiconductors, and insulators. The model given in Sec. II B and the universal anisotropy distance defined in Eq. (20) are evaluated for each of the materials in the dataset. B. Universal log-Euclidean anisotropy index

A. First-principles calculations of elastic tensors

Figure 3 shows the distribution of the calculated anisotropy parameter AL for the set of 2176 inorganic crystalline compounds. The value of AL ranges between 0 and 10.26 with 90% of the compounds having an anisotropy AL < 1:0. It is of particular interest to examine materials in the dataset that are either highly anisotropic (AL 1) or those that are nearly isotropic. See supplementary material for an overview of the elastically most anisotropic and isotropic systems considered in this work. In particular, Table I gives the 60 most anisotropic materials in the dataset. The Materials Project (MP) ID is presented, together with the crystal system and the chemical formula. We also show the three elastic constants c11, c12, and c44, which are used to aid the discussion in Sec. III D. Finally, for each compound it is indicated whether the underlying structure exhibits a layered/lamellar type of configuration. It is difficult to provide a precise definition of what is meant by layered compounds. In general, these materials have relatively short bond distances, high stiffness, and isotropic properties in-plane. However, the properties out of plane are vastly different (often by at least a factor 5). In this case, the bond distances between atoms are much larger and the stiffness is much smaller. Often, the primary bonds between layers are weak Van der Waals bonds. Graphite would be the prototypical example of a layered material. Many other example materials considered in this work are similar, e.g., PtO2, TiNCl, BN, and MnSe.

The high-throughput calculations of elastic tensors in the MP are briefly reviewed here. Further details have been described elsewhere.22,35–39 Complete sets of elastic constants for a number of crystalline materials are obtained using first-principles calculations based on DFT. The process begins by relaxing the unit cell structure and then applying a number of small deformations. The resulting stress tensor is calculated and the components of the elastic tensor are obtained as fitting parameters to the stress data. The DFT calculations are based on the projector augmented wave (PAW) method,40,41 which is implemented in the Vienna Ab Initio Simulation Package (VASP).42,43 The Perdew, Becke, and Ernzerhof (PBE) Generalized Gradient Approximation (GGA) for the exchange-correlation functional44 is used. For metals and metallic compounds, a cut-off energy of 700 eV for the plane waves is used. Additionally, a uniform k-point density of approximately 7000 per reciprocal atom (pra) is used. A k-point density of 1000 pra is used for nonmetallic

FIG. 3. The universal log-Euclidean anisotropy inxex AL calculated for the MP dataset.

III. ANALYSIS

The present model was formulated to be applicable to general crystal symmetries, which allows input of the full elastic modulus tensor. In this section, we take advantage of this feature by analyzing polycrystals based on full elastic tensor inputs generated from first-principle DFT. The inputs span a set of 2176 inorganic crystallite compounds found within the Materials Project (MP).21–25 This dataset is used in Sec. III B to survey the elastic anisotropy (AL) of each material, which is then used in Sec. III C to analyze the influence of AL on the polycrystal’s elastic constants. Finally, the SC estimates of the bulk modulus j and the shear modulus c44 for the 2176 compounds are compared with the various orders of bounds and the Hill estimate. Discrepancies between the various estimates are rationalized by considering the degree of elastic anisotropy and the underlying structure of the compounds.


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TABLE I. The 60 most anisotropic systems from the MP dataset (Units in GPa, Y ¼ yes, N ¼ no). MP-ID

Crystal system Formula

mp-568286 mp-606949 mp-169 mp-629015 mp-984 mp-48 mp-559976 mp-25587 mp-543096 mp-7581 mp-5745 mp-540793 mp-34134 mp-27850 mp-27863 mp-7868 mp-12910 mp-568346 mp-32450 mp-27848 mp-505531 mp-18717 mp-694 mp-765892 mp-617 mp-2156 mp-850131 mp-25118 mp-625055 mp-9920 mp-20311 mp-3567 mp-22992 mp-774160 mp-9921 mp-604910 mp-2207 mp-771247 mp-2076 mp-552787 mp-753059 mp-30562 mp-557993 mp-2194 mp-776678 mp-604908 mp-571033 mp-764312 mp-23002 mp-4366 mp-755000 mp-1387 mp-570081 mp-30423 mp-10181 mp-2793 mp-11489 mp-505825 mp-10919 mp-313

Orthorhombic Triclinic Trigonal Hexagonal Hexagonal Hexagonal Trigonal Monoclinic Trigonal Trigonal Trigonal Orthorhombic Trigonal Orthorhombic Orthorhombic Hexagonal Orthorhombic Orthorhombic Orthorhombic Orthorhombic Tetragonal Cubic Trigonal Monoclinic Trigonal Trigonal Monoclinic Orthorhombic Monoclinic Monoclinic Tetragonal Trigonal Orthorhombic Monoclinic Monoclinic Tetragonal Hexagonal Monoclinic Trigonal Orthorhombic Orthorhombic Tetragonal Monoclinic Trigonal Tetragonal Tetragonal Tetragonal Orthorhombic Orthorhombic Trigonal Hexagonal Cubic Trigonal Orthorhombic Cubic Monoclinic Cubic Trigonal Trigonal Tetragonal

C C C BN BN C Ta2CS2 LiNiO2 NiO2 MoSe2 Nb2CS2 VCl2O MnO2 TiNCl AlClO PtO2 GeS HfBrN VBr2O TiIN FeS SrVO3 VSe2 MnCoO4 PtO2 TiS2 NiS2 VClO AlHO2 TiS3 FeSe NaVSe2 TiClO NiTeO4 ZrS3 MnSe NbSe2 CoTeO4 ZrSe2 FeClO TiOF Sc2Co BiO2 TiSe2 FeO CoSe NiSe VOF2 TiBrO Na2PtC2 LiTiO2 AlV3 CuI VAu2 LiSiNi2 AuSe Li3Pd Cs2PtC2 Rb2PtC2 YC2

AL

c11

c12

c44

Layered?

10.27 9.77 9.50 9.44 8.86 8.77 7.09 7.09 6.61 6.54 6.49 6.24 5.93 5.92 5.79 5.75 5.66 5.45 5.43 5.37 5.33 5.30 5.22 5.21 5.17 5.13 5.05 4.99 4.92 4.88 4.78 4.66 4.52 4.50 4.41 4.25 4.20 4.11 4.09 4.02 3.93 3.87 3.80 3.77 3.61 3.59 3.53 3.51 3.48 3.41 3.38 3.35 3.34 3.23 3.13 3.06 3.06 3.06 3.06 3.03

879.42 878.25 868.47 722.35 734.32 903.97 249.44 174.88 301.26 145.95 235.65 329.95 270.08 175.90 176.04 290.28 99.94 155.56 57.17 160.43 136.75 223.69 97.73 8.70 293.06 120.63 174.34 145.93 76.02 8.43 91.19 68.81 150.31 153.79 17.64 87.39 127.68 134.77 94.35 132.84 330.74 133.33 127.74 92.47 227.45 65.18 59.16 192.74 93.60 47.75 278.48 212.16 47.13 240.97 159.78 3.36 39.79 31.77 35.46 175.69

0.65 1.07 167.58 160.98 163.90 158.15 46.96 59.46 61.90 34.57 45.65 2.30 41.98 2.21 1.58 75.01 1.92 15.77 0.08 1.69 81.11 158.62 14.14 3.35 73.33 25.07 7.44 2.79 34.34 3.43 30.85 46.75 5.34 52.48 4.41 72.83 44.32 54.33 20.43 4.72 3.76 82.42 45.86 20.67 31.85 6.55 4.44 5.69 1.00 19.14 114.27 132.92 17.57 112.26 146.44 3.30 35.54 16.13 15.98 77.68

0.77 1.41 1.86 1.47 2.07 2.48 1.47 7.57 1.76 1.03 1.86 0.87 2.24 1.24 2.31 3.01 0.51 1.09 2.33 1.85 1.64 102.29 1.11 96.46 3.58 1.92 13.67 1.84 123.05 27.79 1.33 23.84 1.38 14.32 26.18 1.88 2.32 14.49 1.92 1.99 6.50 1.61 61.39 2.43 11.26 1.56 1.12 5.32 3.40 8.15 5.86 2.41 1.39 59.69 110.20 8.96 30.00 4.43 5.09 7.61

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N Y Y Y Y N Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N Y N Y Y Y Y N Y N Y N N Y N N N N

Here, the distinction between layered and nonlayered compounds is made visually. Table I shows that the majority (78%) of compounds that have high values of AL exhibit a layered structure. This leads to highly directional bonding and relatively high tensile and shear moduli in the layered planes but low stiffness perpendicular to these planes. In other words, these structurally anisotropic compounds naturally lead to elastic properties that are anisotropic. However, the converse does not necessarily follow the same logic. Some structurally isotropic, close-packed structures are elastically anisotropic, whereas other materials belonging to the same crystal system are not. For example, face-centered-cubic (FCC) aluminum is nearly isotropic, whereas some cubic intermetallics such as LiSiNi2 (see Table I) are highly anisotropic, even though none of these exhibits a layered/lamellar structure and both are structurally similar (cubic). In such cases, it is not the physical (crystal) structure that directly gives rise to elastically anisotropic behavior, but rather the electronic structure of the material. This will be discussed in more detail in Sec. III D. Table II shows the 60 least anisotropic systems. Interestingly, only compounds with either a cubic or hexagonal crystal system are present where 85% are cubic compounds. Not surprisingly, none of the compounds in Table II exhibit a layered structure. Rather, they are densely packed structures with similar linear atomic densities in all directions. C. Self-consistent and Hill estimates

The self-consistent estimates are obtained from the converged iterative solution to Eq. (8). Unlike the estimates provided by the bounds, they represent the set of elastic constants that satisfy continuity of stress and strain throughout the polycrystal. They are important for estimating the mechanical behavior of polycrystalline materials from the polycrystal’s microstructural constituents. This is exemplified in Sec. III C 1 by comparing the SC shear modulus calculated from Eq. (8) to independent experimental measurements compiled by Ledbetter.45 Additionally, the model and its iterative solutions contain important insights into the elastic anisotropy of monocrystalline materials, which is considered in Sec. III D. 1. Comparison of self-consistent shear modulus to experimental values

The shear modulus of a polycrystalline sample can be determined by measuring the transit time of an acoustic shear wave propagating through the sample and relating it to the shear wave phase velocity. Ledbetter45 compiled a number of independent experimental shear wave velocity measurements on nominally pure polycrystalline copper from previously reported values. The aim of the compilation was to benchmark the many elastic constant averaging schemes against the shear wave velocities. Here, the reported experimental shear wave phase velocities are used to calculate the shear modulus of polycrystalline copper through the relation exp 2 cexp 44 ¼ qðvs Þ . Figure 4 illustrates the average of the thirteen measurements along with the standard deviation, which


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TABLE II. The 60 least anisotropic systems from the MP dataset (Units in GPa, Y ¼ yes, N ¼ no). MP-ID mp-5229 mp-7070 mp-866165 mp-16719 mp-866154 mp-1431 mp-505297 mp-567276 mp-2121 mp-823 mp-865989 mp-11231 mp-1367 mp-11227 mp-30443 mp-1029 mp-9295 mp-2028 mp-867107 mp-550 mp-22598 mp-5318 mp-867171 mp-2404 mp-2605 mp-416 mp-866166 mp-754333 mp-4964 mp-712 mp-5778 mp-19963 mp-7576 mp-22120 mp-1677 mp-11275 mp-593 mp-1648 mp-2027 mp-30365 mp-10198 mp-31055 mp-31454 mp-8578 mp-16764 mp-10760 mp-3871 mp-510624 mp-30798 mp-11282 mp-30784 mp-317 mp-644264 mp-7577 mp-13503 mp-2472 mp-12558 mp-22040 mp-91 mp-1342

Crystal system

Formula

AL

c11

c12

c44

Layered?

Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Hexagonal Cubic Cubic Cubic Cubic Cubic Cubic Hexagonal Hexagonal Cubic Cubic Cubic Cubic Cubic Cubic Cubic Cubic Hexagonal Cubic Hexagonal Cubic Hexagonal Hexagonal Cubic Cubic Cubic Cubic Hexagonal Cubic Cubic Cubic Cubic Cubic

SrTiO3 Mn3PtN YMgRh2 Al12Tc TiBeIr2 MnSi NbSbRu TaV2 ScAl3 TiCo Al2RuIr YAl3 Mg2Si Al12W Be17Ru3 BaF2 TaCu3Te4 Be2Nb BeSiOs2 Al12Mo In2O3 MnCoSb SrLi2Sn CaAl2 CaO Cr3Os Y2RuPd TiO YSbPt YCu AlVFe2 TiFe2Sn CrSi YRh3C Be2Mo Be2Re Al4Cu9 Al12Re HfCo BeAu Mn3PdN Sc3InN TaSbRu Sc3InC RbYTe2 MgSe Ti2SnC SrFeO3 NbZn2 Be2W Mg2Zn11 SnRh Ca4Mg4Fe3H22 CoSi ScZn2 SrO LiMgAs Sn7Ir3 W BaO

0.000002 0.000008 0.000020 0.000025 0.000049 0.000072 0.000120 0.000137 0.000140 0.000142 0.000161 0.000189 0.000258 0.000297 0.000300 0.000333 0.000434 0.000560 0.000564 0.000581 0.000590 0.000625 0.000733 0.000735 0.000794 0.000868 0.000958 0.000961 0.000996 0.001138 0.001148 0.001200 0.001219 0.001322 0.001388 0.001531 0.001537 0.001552 0.001581 0.001724 0.001785 0.001788 0.001827 0.001866 0.001907 0.002152 0.002155 0.002173 0.002497 0.002564 0.002582 0.002664 0.002759 0.002834 0.002928 0.003060 0.003100 0.003205 0.003341 0.003457

319.08 275.24 174.09 192.65 402.54 399.34 255.72 253.33 177.41 244.27 386.86 160.78 113.29 168.40 313.19 86.94 55.30 294.20 469.81 168.94 223.34 191.98 61.43 110.13 201.65 423.02 154.97 433.13 158.46 115.94 405.18 318.42 346.22 305.87 402.64 474.25 224.87 197.36 230.02 200.78 253.26 208.02 265.93 210.08 48.10 111.86 263.10 238.41 249.55 449.92 109.59 208.85 116.15 358.67 147.14 165.56 93.21 186.09 509.82 121.17

99.57 121.18 78.55 45.88 160.20 114.75 124.30 175.09 37.73 122.63 135.47 34.09 22.72 53.31 74.75 38.16 15.01 91.41 194.53 52.66 106.00 79.12 17.56 29.56 56.69 211.28 75.68 116.94 66.07 47.39 123.78 112.45 107.40 142.03 92.98 117.99 81.24 45.39 114.27 121.51 110.95 51.14 130.24 41.13 16.87 32.31 79.86 77.87 78.44 94.67 45.69 104.49 33.18 136.42 52.81 48.36 23.30 60.42 201.00 41.95

109.53 76.74 48.05 73.92 121.53 140.76 66.70 39.75 70.98 61.82 124.39 62.17 44.20 56.23 115.90 23.79 19.68 104.71 142.18 56.30 56.88 56.00 21.13 41.97 75.32 101.70 38.01 154.63 48.23 35.89 134.33 107.79 114.00 77.95 147.31 172.98 68.28 80.58 54.82 41.64 75.38 83.16 71.92 79.64 15.14 42.37 90.79 85.54 81.81 164.94 34.27 55.95 45.05 119.67 46.77 54.34 37.71 58.16 142.69 36.55

N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N


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FIG. 5. Voigt versus the Reuss estimated bulk modulus for the MP dataset. FIG. 4. Comparison of the arithmetic average of the Voigt-Reuss bounds (Hill average), the arithmetic average of the Hashin-Shrtrikman bounds, and self-consistent estimate to experimental measurements of the shear modulus of polycrystalline copper.

is cexp 44 ¼ 47:6160:0013 GPa. The small experimental variation indicates the ability to determine the shear modulus to a high precision in well-controlled experimental settings and adequate sample preparation. The experimental precision in this case gives a situation where a theoretical homogenization scheme needs to have a high degree of precision in order to represent the phase velocity measurements accurately. To further illustrate this point, three homogenization schemes are illustrated in Fig. 4. The Hill average is the arithmetic average of the first-order Voigt and Reuss bounds; the H-S average is the arithmetic average of the second-order Hashin-Shtrikman bounds; and the SC value is the converged solution for the shear modulus c44 . Each of these homogenizations are obtained from the various orders of iterative solutions to Eq. (15b) using the single-crystal elastic constants and density reported by Ledbetter.45 For this case, the SC estimate of c44 is only 0.168% greater than the averaged experimental modulus. The Hill and Hashin-Shtrikman arithmetic averages are 1.55% and 0.84% smaller, respectively. This result highlights the improvement gained when using the SC homogenization compared to the Hill and HashinShtrikman averages and underlines the necessity to apply SC homogenization schemes.

this section, the MP dataset is used to form a comparison between the various estimates in order to quantify the expected error when using the Voigt, Reuss, or Hill average. Figures 5 and 6 compare the Voigt and Reuss bounds on the bulk and shear moduli, respectively. It is observed that the differences between the Voigt and Reuss bounds tend to be relatively large for compounds with small values of the bulk and shear modulus. The Voigt and Reuss bounds on the bulk modulus differ by over 10% for 9.8% of the compounds considered. The Voigt and Reuss estimates of the shear modulus differ by over 10% for 45% of the compounds. Hence, the Voigt and Reuss bounds on the bulk modulus generally appear to be closer than the corresponding bounds on the shear modulus. This is expected, since 34% of the compounds in this set are cubic, for which the Voigt and Reuss bounds on the bulk modulus are identically equal. This analysis clearly indicates that it is generally inappropriate to choose either the Voigt or Reuss bound on the bulk and shear moduli when modeling polycrystalline materials, because these bounds are not necessarily in good agreement. We next consider the comparison between the Hill and SC estimates, as calculated from the iterative solution to Eq. (8). The comparison of the bulk and shear moduli are depicted in Figs. 7 and 8, respectively. The distribution of

2. Comparison between self-consistent estimates and Hill’s average for the MP dataset

Hill’s estimate or average is defined as the arithmetic mean of the first-order Voigt and Reuss bounds on the bulk and shear modulus of a polycrystal. The Hill estimate is commonly employed because of its simplicity in applications where a knowledge of elastic constants is required.4,46–48 The Voigt and Reuss bounds are based on extremum assumptions regarding strain and stress fields in the polycrystal, respectively. Thus, the Hill estimate is entirely empirical and has weak fundamental justification from a theoretical perspective. Alternatively, the SC estimates are obtained without unrealistic assumptions to the stress and strain fields and are held to a higher regard. In

FIG. 6. Voigt versus the Reuss estimated shear modulus for the MP dataset.


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FIG. 7. Hill estimate of the bulk modulus versus the self-consistent bulk modulus for the MP dataset.

the relative differences between the Hill estimate and the SC bounds is illustrated in Figs. 9 and 10 for the bulk and shear modulus, respectively. The variation between the estimates is small for most materials, although several outliers occur for materials having bulk or shear moduli values under 120 GPa. The SC value for the bulk modulus differs from the Hill estimate by over 10% for 2.6% of the materials. The SC shear modulus differs by more than 10% from the Hill estimate for 3.5% of the materials. The compounds with significant deviations between the estimates generally exhibit a strong degree of elastic anisotropy AL. Among these are the majority of the compounds listed in Table I. The 20 systems for which the SC bulk modulus and Hill average differ most are C (orthorhombic, trigonal, triclinic, and hexagonal polymorphs), BN (hexagonal polymorphs), VBr2O (orthorhombic), MoSe2 (trigonal), MnCoO4 (monoclinic), Nb2Cs2 (trigonal), VCl2O (orthorhombic), AlClO (orthorhombic), LiNiO2 (monoclinic), PtO2 (hexagonal), NiO2 (trigonal), Ta2CS2 (trigonal), TiIN (orthorhombic), TiNCl (orthorhombic), FeS (tetragonal), and TiS2 (trigonal). Among these systems, the largest discrepancy between the SC estimate and Hill average is over 70%, whereas the smallest equals

FIG. 8. Hill estimate of the shear modulus versus the self-consistent shear modulus for the MP dataset.

J. Appl. Phys. 120, 165105 (2016)

FIG. 9. Relative differences between the self-consistent and Hill estimated bulk modulus for the MP dataset.

approximately 40%. For the shear modulus, similar compounds appear in the top 20 having the largest discrepancy between the SC estimate and Hill average: C (orthorhombic, trigonal, triclinic, and hexagonal polymorphs), BN (hexagonal polymorphs), MnCoO4 (monoclinic), VCl2O (orthorhombic), VBr2O (orthorhombic), Ta2CS2 (trigonal), MoSe2 (trigonal), Nb2Cs2 (trigonal), NiO2 (trigonal), MnSe (tetragonal), MnO2 (trigonal), FeS (tetragonal), TiNCl (orthorhombic), PtO2 (hexagonal), AlClO (orthorhombic), and Cs2PtC2 (trigonal). Similar to the bulk modulus, also for the top 20 systems with the largest discrepancy between the SC estimate and Hill average for the shear modulus, the differences range from 40% to 70%. Note that for both the bulk modulus and shear modulus, most of the listed systems in the top 20 are among the most elastically anisotropic compounds shown in Tables I and II. The influence of high elastic anisotropy on the variation in the estimates is discussed in Sec. III C 3. 3. Relation to the elastic anisotropy

In this section, the relation between the anisotropy index AL and the deviation between the Hill and self-consistent

FIG. 10. Relative differences between the self-consistent and Hill estimated shear modulus for the MP dataset.


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FIG. 11. The relative difference between the self-consistent and Hill estimated bulk modulus versus AL. The calculated Pearson (r) and Spearman (q) correlation coefficients are indicated.

FIG. 12. The relative difference between the self-consistent and Hill estimated shear modulus versus AL. The calculated Pearson (r) and Spearman (q) correlation coefficients are indicated.

estimates is investigated further. The cases where the Voigt and Reuss bounds are considerably different, i.e., large values of AL, are those where the most improvement can be expected when using the SC estimate over the Hill average. Figure 11 plots the quantity jj jH j=jH 100 versus L A . A general trend can be observed where the more elastically anisotropic materials tend to have a larger relative difference between j and jH, as expected. The Pearson and Spearman correlation coefficients (r and q, respectively) are calculated to gauge the relationship between jj jH j=jH and AL. We find that r 0:85 and q 0:34, indicating that a reasonably strong linear association exists between the quantities but a rather poor monotone association. We next examine for how many compounds j and jH differ by 1% or more. A conservative threshold of 1% is chosen based on the percentage difference from experimental measurements observed in Fig. 4. Table III shows a break-up that relates jj jH j=jH and AL. It is found that 187 out of 2176 compounds obey jj jH j=jH 100 > 1%. Of these, 83 exhibit AL > 1:5. On the other hand, 104 out of 187 compounds have jj jH j=jH > 1% and AL < 1:5. Note that 748 materials considered have cubic symmetry where j ¼ jH , which is independent of AL. In general, compounds for which the differences in the SC estimates and the Hill average for the bulk modulus are large tend to have strong anisotropy. H Figure 12 plots the quantity jc44 cH 44 j=c44 100 versus L A . The variation of the shear modulus given in Fig. 12 displays a trend similar to the bulk modulus. The Pearson and Spearman correlation coefficients are computed as r 0:90

and q 0:90. A stronger linear association and monotone association exists for the shear modulus as compared to the bulk modulus. A notable difference compared to the bulk modulus is that for the cubic compounds, the Voigt and Reuss values of the shear moduli are not identically equal (as opposed to the shear modulus). Thus, points near jc44 cH 44 j= cH 0 are not as common. Consequently, the relation 44 between jj jH j=jH 100 and AL is less (linearly) associL H ated than is the case for jc44 cH 44 j=c44 100 and A . Table IV shows the calculated break-up relating the magnitude of L H jc44 cH 44 j=c44 and A .

TABLE III. Comparison of the self-consistent and Hill estimated bulk modulus relative to the elastic anisotropy AL for the MP dataset.

TABLE IV. Comparison of the self-consistent and Hill estimated shear modulus relative to the elastic anisotropy AL for the MP dataset.

jj jH j=jH 100 > 1% jj jH j=jH 100 < 1% Total

AL > 1:5

AL < 1:5

Total

83 56 139

104 1933 2037

187 1989 2176

D. Discussion

For certain classes of compounds, the SC values offer a significant improvement compared to the Hill estimate. Such compounds also tend to exhibit strong elastic anisotropy, which is associated with elastic stability and structural anisotropy (e.g., layered crystals). These sources of elastic anisotropy are analyzed for the 2176 compounds and linked to the SC values of the elastic constants. The general results are further illustrated by a number of specific material examples. First, consider the hexagonal carbon polymorph in Table I, identified by the MP-ID mp-48. This compound is graphite, which consists of connected 6-rings of carbon atoms that are stacked in planes along the c-axis. Strong covalent bonds act within the carbon layers, whereas much weaker forces work in the direction perpendicular to the layers. In such cases, the anisotropic structure directly results

H jc44 cH 44 j=c44 100 > 1% H jc44 cH j=c 44 44 100 < 1% Total

AL > 1:5

AL < 1:5

Total

130 9 139

397 1640 2037

527 1649 2176


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in elastically anisotropic behavior via the elastic constants; this material exhibits high moduli c11 ¼ c22 and c66 since these correspond to deformations (tensile and shear, respectively) within the layers. However, the moduli c33 and c44 ¼ c55 are close to zero since they correspond to deformations (tensile and shear, respectively) with components acting out of the layers, where the bonding is very weak. An intuitive view is that such an anisotropic structure directly results in a large elastic anisotropy. The relationship between mechanical stability and elastic anisotropy has been discussed previously.49–52 However, the connection between local mechanical stability of a single crystal and homogenization models has received far less attention. This link is easily observed by looking at the mechanical stability conditions of a hexagonal crystallite53 8 c11 > jc12 j > > > > 2 > < 2c13 < c33 ðc11 þ c12 Þ c44 > 0 > > > > 1 > : c66 ¼ ðc11 c12 Þ > 0; 2

(21)

and the corresponding Voigt and Reuss estimates of the bulk and shear modulus 1 jV ¼ ð2c11 þ 2c12 þ 4c13 þ c33 Þ; 9 cV44 ¼

1 ð7c11 5c12 4c13 þ 2c33 þ 12c44 Þ; 30 jR ¼

c33 ðc11 þ c12 Þ 2c213 ; c11 þ c12 4c13 þ 2c33

(22) (23) (24)

cR44 ¼ 15c44 ðc11 c12 Þ½c33 ðc11 þ c12 Þ 2c213 f6ðc11 c12 Þ½c33 ðc11 þ c12 Þ 2c213 þ 2c44 ½2c211 2c212 4c13 ðc12 þ 3c13 Þ þ 5c12 c33 þ c11 ð4c13 þ 7c33 Þg1 ;

(25)

respectively. Clearly, jR ¼ 0 and cR44 ¼ 0 if 2c213 ¼ c33 ðc11 þ c12 Þ, which leads to AL ! 1 from Eq. (20). Equation (25) also shows that cR44 ! 0 as c44 ! 0 or c11 c12 ! 0. This clearly shows that a crystal near a mechanical instability exhibits a strong elastic anisotropy. Thus, the Hill estimate becomes a poorer estimate for the cases when jR ! 0 or cR44 ! 0. In such cases, the SC values are expected to be much more realistic. Table I also shows that several cubic compounds exhibit anomalously strong anisotropy. For such cases, the electronic structure rather than crystal structure causes the crystal to be near a mechanical instability. For transition-metal alloys, this can be further rationalized by considering d-band filling and its connection to mechanical stability, which can occur near structural transitions. As an example, consider the compounds AlV3 (MP-ID mp1387) with elastic constants c11 ¼ 212:2 GPa, c12 ¼ 132:9 GPa, c44 ¼ 2.4 GPa and LiSiNi2 (MP-ID mp-10181) with

elastic constants c11 ¼ 159:8 GPa, c12 ¼ 146:4 GPa, c44 ¼ 110.2 GPa, which are shown in Table I as being extremely anisotropic. For these crystals, the conditions for mechanical stability are53 8 > < c11 c12 > 0 (26) c11 þ 2c12 > 0 > : c44 > 0: The Reuss and Voigt estimates are

cV44 ¼ cR44 þ

cR44

jV ¼ jR ;

(27)

3 ðc11 c12 2c44 Þ2 ; 5 3ðc11 c12 Þ þ 4c44

(28)

1 jR ¼ ðc11 þ 2c12 Þ; 3 5 c44 ðc11 c12 Þ ¼ : 3 3ðc11 c12 Þ þ 4c44

(29) (30)

For either ðc11 c12 Þ ! 0 or c44 ! 0, a mechanical instability occurs leading to cR44 ! 0 and AL ! 1. Note that an instability of the type c11 þ 2c12 ! 0 results in jR ¼ jV , which indicates that not all instabilities result in strong anisotropy. Equation (26) shows that AlV3 is near a mechanical instability due to its anomalously low c44. On the other hand, LiSiNi2 is near an instability because c11 and c12 are similar in magnitude. Both these compounds exhibit similarly high values of AL. However, the Zener anisotropy index tends to zero for AlV3 and infinity for LiSiNi2. For transition-metal alloys, it has been established that clear trends exist between the d-band filling and elastic moduli.54–57 Previous work shows that to the right of Ti/Zr/Hf in the periodic table and to the left of the V/Nb/Ta, i.e., for d-band fillings near 2.20–2.24, c11 c12 ! 0.58,59 This is where the hexagonal-close-packed (HCP) to body-centeredcubic (BCC) transition occurs and it coincides with a mechanical instability. Thus, it is expected that all transitionmetal alloys with d-band fillings in that range exhibit an infinitely large elastic anisotropy. A similar situation is expected to occur for d-band fillings between 4 and 5, where the BCC to HCP transition between Cr/Mo/W and Mn/Tc/Re occurs (although the situation for Mn is more complicated, as its ground state at 0 K is not HCP).60 At such d-band fillings, strong elastic anisotropy exists and the SC values are a more appropriate measure for the polycrystal composed of these crystallites. AlV3 is a cubic compound, near a mechanical instability, with an anomalously low value for the modulus c44. This could be attributed to d-band filling although the connection is less clear than the c11 c12 instability.58 Additionally, AlV3 is not a true transition-metal alloy. c44 obtains values close to zero for d-band fillings between Ti and V, although it does not actually cross zero, as does c11 c12 . Approximating AlV3 as a transition-metal alloy with a total d-band filling of 2.25 (3d-electrons for each V-atom, no d-electrons from the Al-atom), low values of c44 can be expected,58 indicating a near-mechanical instability.


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IV. CONCLUSIONS

In this article, the elastic properties of polycrystalline materials and the influence of crystallite elastic anisotropy was investigated. A theoretical formalism was developed that allows the bounds on the elastic properties to be easily calculated. The self-consistent estimates are the set of elastic constants that enforce continuity of stress and strain fields throughout the polycrystal. The self-consistent estimates are obtained by solving Eq. (8) iteratively and observing the converged solution. Both the bounds and self-consistent estimates are able to be calculated using the single-crystal elastic constants of crystals belonging to any symmetry class. Homogenization estimates (self-consistent estimates, the Hill average, and the arithmetic average of the HashinShtrikman bounds) are compared to experimental measurements45 of the shear modulus of polycrystalline copper. This example highlights the benefit of using the self-consistent estimates. The general applicability to any crystallographic symmetry class allows the full elastic tensors to be input into the model. This feature permitted the broad analysis of the elastic properties of 2176 inorganic crystalline materials. The single-crystal elastic constants of the 2176 materials were obtained from the Material’s Project database and were used to perform a systematic study of the different estimates and bounds on the bulk and shear modulus. In particular, a comparison of the Voigt and Reuss bounds, the Hill estimate, and the SC value of the bulk and shear modulus are given. It is found that the Hill estimate and the SC values are in agreement to within 10% for 2.6% of the compounds for the bulk modulus and 3.5% of the compounds for the shear modulus. The compounds for which the discrepancy between the SC values and the Hill estimate is largest, also tend to have high values of the anisotropy index AL. This behavior is fundamentally caused by the material being near a mechanical instability, meaning that at least one of the eigenvalues of the elastic tensor is near zero. Two cases can be distinguished: (1) the physical anisotropic structure directly leads to certain elastic moduli being close to zero, which causes a near mechanical instability and a high elastic anisotropy or (2) the structure itself is not anisotropic, but rather the underlying electronic structure causes eigenvalues of the elastic tensor such as c11 c12 or c44 to tend to zero, which in turn leads to a near mechanical instability and a high elastic anisotropy. For such materials, SC estimates can lead to substantial improvements compared to the Voigt, Reuss, or Hill estimates. SUPPLEMENTARY MATERIAL

See supplementary material for a script to calculate the bounds and self-consistent estimates based on inputs of single-crystal elastic constant data for any crystallite symmetry. 1

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