Atomistic and mean-field estimates of effective stiffness tensor of nanocrystalline materials of hexagonal symmetry
aa r X i v : . [ phy s i c s . c o m p - ph ] S e p Atomistic and mean-field estimates of effective stiffnesstensor of nanocrystalline materials of hexagonalsymmetry
Katarzyna Kowalczyk-Gajewska ∗ , Marcin Ma´zdziarz Institute of Fundamental Technological Research Polish Academy of Sciences, Warsaw,Poland
Abstract
Anisotropic core-shell model of a nano-grained polycrystal is extended to esti-mate the effective elastic stiffness of several metals of hexagonal crystal latticesymmetry. In the approach the bulk nanocrystalline material is described as atwo-phase medium with different properties for a grain boundary zone and agrain core. While the grain core is anisotropic, the boundary zone is isotropicand has a thickness defined by the cutoff radius of a corresponding atomisticpotential for the considered metal. The predictions of the proposed mean-field model are verified with respect to simulations performed with the use ofthe Large-scale Atomic/Molecular Massively Parallel Simulator, the EmbeddedAtom Model, and the molecular statics method. The effect of the grain size onthe overall elastic moduli of nanocrystalline material with random distributionof orientations is analysed.
Keywords:
Molecular statics, Elasticity, Polycrystal, Effective medium,Hexagonal symmetry
1. Introduction
In nanocrystalline materials, usually defined as those polcyrystalline mediafor which the average grain size is less than 100 nm [1, 2], a significant numberof atoms occupies the grain boundary zone or the grain boundary affected zone[3]. Therefore such materials can be treated as composed of two main phases.The effect of grain boundaries on the effective properties of a bulk nanocrys-talline material is the more pronounced the smaller is a grain size [4, 2]. Animpact of the atom arrangements at the nanoscale on the effective properties ofsuch materials has been studied mainly by means of atomistic simulations [5],although some experimental data, in majority related to fcc materials, can be ∗ Corresponding author.
E-mail address: [email protected] the Mori-Tanaka(MT) and self-consistent (SC) core-shell model , respectively. In view of the pro-posed geometrical idealization of nanocrystalline medium an additional phasethat forms an uniform isotropic coating around the anisotropic grain core is in-troduced. Let us mention that a more sophisticated treatment of a grain bound-ary zone can be found in [19, 20] – studies dedicated to metal-matrix compositereinforced by nanosized inclusions. Following [21], authors assumed that theinterphase layer between the inclusion and the matrix has isotropic propertieswhich vary smoothly with ”upward convexity”. Alternatively, a step-wise gra-dation of interphase properties has been assumed by [22]. As demonstrated in[23] also in the frame of the core-shell model inhomogeneous shell propertiescan be assumed, though, on the cost of a more complicated formulation andnecessity to identify additional material parameters.Most often to identify those parameters and validate the proposed estimatesthe molecular dynamics/statics simulations are used [24, 25, 26, 2, 27, 28, 8].Finite element calculations are scarce because they require a non-standard con-stitutive models accounting for size effects [29]. The common trend observed inthe majority of such simulations is reduction of elastic stiffness with a decreasinggrain size [30, 2, 31]. Such variation of elastic moduli with a grain size wouldbe predicted by the core-shell models when, on average, the boundary zone iselastically less stiff than a grain core [14, 18, 2, 8]. It is worth mentioning thata reverse trend was found in atomistic simulation by [23] for two (i.e. vanadiumand niobium) out of eight metals of cubic symmetry studied therein. Interest-ingly, these two crystals have a Zener anisotropy factor lower than one, contraryto remaining six metals.The challenging issue for those multi-phase concepts is to propose an ap-propriate description of a grain boundary zone (or zones), namely its volumefraction, morphology and local properties. To this end, likewise, molecularstatic/dynamic simulations are employed, commonly in a bi-crystal configura-tion, e.g. [32, 33]. Results depend on the disorientation axis and angle betweentwo grains, see also [34, 35]. For a mean-field model of random nanocrystallinemedium the average properties representative for all types of boundaries are ofinterest, therefore we apply the procedure adopted in [23]. Elastic propertiesof a grain boundary zone are identified on generated polycrystal samples for2hich the fraction of transient shell atoms encompasses the whole volume. Athickness of this zone is assumed as equal to the cutoff radius of a respectiveatomistic potential.The present paper reports a follow-up to the recent studies by [8, 23]. Thegoal of this research is to evaluate applicability of the core-shell model proposedtherein for describing the effective elastic stiffness of nanocrystalline metals ofhexagonal lattice symmetry. In particular, the assumptions concerning the de-scription of a grain boundary zone are verified.The paper is constructed as follows. The successive section presents detailsof spectral decomposition of elasticity tensor for crystals of hexagonal symme-try, which due to the properties of the fourth order tensors is equivalent to atransverse isotropy case. The possible anisotropy measures for such tensor arealso discussed. Moreover, this section reminds the formulation of a core-shellmodel and shows how its different variants can be obtained from the generalformula. Section 3 is devoted to fundamentals of atomistic simulations. Com-parison of the results of atomistic simulations and core-shell model predictionsis performed in Section 4 (detailed results of molecular simulations are collectedin Appendix B.). The last section contains summary and conclusions.
2. Two-phase core-shell model for bulk nanocrystals of hexagonalsymmetry
The anisotropic linear law between the stress σ and strain ε in the grain isassumed, namely σ = C ( φ c ) · ε , ε = S ( φ c ) · σ , S ( φ c ) C ( φ c ) = I , (1)where C ( φ c ) and S ( φ c ) are the fourth order elastic stiffness and compliancetensors of a given symmetry. Argument φ c denotes symbolically an orientationof local axes { a k } with respect to some macroscopic frame { i k } . I is a fourthorder symmetrized identity tensor.A unit cell of crystal lattice with a hexagonal closed packed (hcp) spatialdistribution of atoms has a six-fold rotational symmetry axis c . Therefore forhcp crystals the local elastic stiffness tensor C ( φ c ) exhibits transverse isotropy.It means that, from the point of view of hcp unit cell geometry [36], as concernselastic properties only orientation of c axis matters, while orientation of a i axes(e.g. the so-called armchair or zigzag one) does not influence the form of C ( φ c ).The spectral form of the fourth order tensor of transverse isotropy is [37, 38, 39] C ( φ c ) = h P ti ( ξ, φ c ) + h P ti ( ξ, φ c ) + 2 G P ti ( φ c ) + 2 G P ti ( φ c ) , (2)where P tii are fourth order orthogonal projectors of the form P ti ( ξ, φ c ) + P ti ( ξ, φ c ) = I P + 16 (3 N − I ) ⊗ (3 N − I ) , (3) P ti ( φ c ) = 12 (cid:16) [( I − N ) ⊗ ( I − N )] T (23)+ T (24) − ( I − N ) ⊗ ( I − N ) (cid:17) , (4) P ti ( φ c ) = 12 [ N ⊗ ( I − N ) + ( I − N ) ⊗ N ] T (23)+ T (24) (5)3 c cc r qp qu -u-u v vpf( ) x ppp 2s/f( ) x -s-s -s P (1D) P (1D) P (2D) P (2D) pf( ) x x Figure 1: Illustration of eigen-subspaces of the elasticity tensor of hexagonal symmetry with ( A T (23)+ T (24) ) ijkl ≡ ( A ) ikjl + ( A ) ilkj and N ( φ c ) = ¯ c ( φ c ) ⊗ ¯ c ( φ c ). Unitvector ¯ c is a normalized axis of a hcp unit cell: c / | c | . Two single Kelvin moduli h and h are two single eigenvalues of the 2 × (cid:20) K L L G (cid:21) (6)where: 3 K = (2 C + C + 2 C + 4 C ) / , (7)2 G = ( C + 2 C + C − C ) / , (8) L = √ C − C + C − C ) / , (9)while in-plane G and out-of-plane G shear moduli are specified as: G = ( C − C ) / , G = C (10) C ijkl are the components of the elasticity tensor C in the orthonormal basis forwhich i = c .Four strictly positive Kelvin moduli: h K ( K = 1 , G and 2 G correspondto four eigen-subspaces of strain or stress states established by the elasticitytensor, which are respectively: • two one-dimensional subspaces of axially symmetric stretching along c .The specification of these two subspaces depends on the value of stiffnessdistributor ξ (more details can be found in Appendix A), • the two-dimensional subspace of in-plane pure shears (i.e. pure shears inthe isotropy plane which is a plane perpendicular to c axis), • the two-dimensional subspace of out-of-plane pure shears (i.e pure shearsin the plane containing c axis).This subspaces are schematically illustrated in Fig. 1. For the states belongingto the respective subspaces the proportionality is observed between stress andstrain tensors. It should be mentioned that if L equals zero then the space P is the space of hydrostatic states, P the space of isochoric axially symmetricstretching and h = 3 K , h = 2 G . 4s discussed by [23] in the case of cubic crystal the Zener anisotropy factor ζ enables the assessment of an anisotropy degree but also distinction betweenanisotropy types. Cubic crystal is elastically anisotropic if ζ = 1 and crystalscan be classified as those for which ζ < ζ >
1. In the caseof hexagonal (transverse isotropic) crystals definition of a unique parameter ofsuch property is not possible. Instead, a set of three parameters is proposed,which play a similar role as the Zener parameter, namely: ζ = { L , G /G , G /G } . (11)Hexagonal crystal is in fact isotropic if and only if ζ = { , , } . Six subclasses oftransverse isotropy may be distinguished depending if the ratios ζ II = G /G , ζ III = G /G are larger or smaller than 1 (note that they are always positive)and on their relative value so if ζ II > ζ III or reversely. A subclass of materialsfor which L = 0 is called volumetrically isotropic. Note that for such materialshydrostatic state is an eigenstate in the spectral decomposition (2), similarly tothe case of isotropic material. It should be stressed that anisotropy degreeas such can be also assessed using a single scalar, for example the universalanisotropy factor [40] or the anisotropy measure ζ (Eq. 21), which is based onthe closest isotropic approximation. However, two latter anisotropy factors donot enable us to distinguish between transverse isotropy subclasses.The standard micromechanical theories treat coarse-grained polycrystals asone-phase heterogeneous materials. In the elastic regime heterogeneity of strainand stress fields results from the varying orientation of crystal axis c in thepolycrystalline representative volume element (RVE). Estimates of effective re-sponse of the hcp grain aggregate are obtained on the basis of knowledge of thelocal elastic properties and the assumed micro-macro transition scheme. Theformulas for the standard estimates, such as the Voigt, Reuss, Hashin-Shtrikmanor self-consistent one, can be found in Appendix A. These estimates are notsensitive to the grain size. A fundamental difference as compared to cubic poly-crystals studied within similar framework by [8, 23] is that, as long as L = 0,the overall bulk modulus for random polycrystal is different from the local oneand varies between the schemes.As discussed in the Introduction, for nanocrystalline materials the commonway to assess the effective properties of the bulk material is to use a two-phasemodel. In the present research the core-shell model developed in [8] is used withdifferent properties for a grain boundary zone and a grain core. While the graincore is anisotropic, the boundary zone surrounding the core is isotropic. Themodel enables estimation of the effective stiffness tensor ¯ C for an arbitrary orien-tation distribution. By fundamental theories of micromechanics [41] such tensorrelates the averaged strain E = h ε i and stress Σ = h σ i in the polycrystallineRVE, namely: Σ = ¯ C · E (12)where h . i = V R V ( . ) dV denotes averaging performed over the representativematerial volume. 5n idea behind the core-shell model is to calculate effective stiffness byexploiting the double inclusion scheme of [42]. Accordingly the coated grain isembedded in the infinite medium of the stiffness C m taken equal to C s or ¯ C CS forMori-Tanaka (MT) or self-consistent (SC) variants of the model, respectively.As a result it is obtained¯ C CS = [ f C s A s + (1 − f ) h C ( φ c ) A ( φ c ) i O ] [ f A s + (1 − f ) h A ( φ c ) i O ] − (13)where A ( φ c ) = ( C ( φ c ) + C ∗ ( C m )) − ( C m + C ∗ ( C m )) , (14) A s = ( C s + C ∗ ( C m )) − ( C m + C ∗ ( C m )) (15)and C ∗ ( C m ) is the Hill tensor [43]. Quantity f is the volume fraction of thegrain boundary zone. It is calculated by the formula f = 1 − (cid:18) − d (cid:19) , (16)where d is an averaged grain diameter and ∆ – the coating thickness. Theformula is found assuming the spherical shape of grain cores and the coating.Previous studies indicated [8, 23] that ∆ can be assumed as equal to the cut-off radius of the atomistic potential valid for the considered metal. Presenceof the ratio 2∆ /d makes the estimate ¯ C CS sensitive to the grain size. Moredetails on the model formulation can be found in the mentioned papers. Theisotropic shell properties need to be identified separately. In the present work,following [23], they are established by means of atomistic simulations by ana-lyzing polycrystalline aggregates with a very small grains, in which the grainboundary zone encompasses whole grains. Note that Eq. (13) can be under-stood in a generalized fashion enabling one to encompass also another two-phaseschemes applicable to nanocrystalline media known in the literature. For exam-ple, a simple mixture rule-based model (Voigt’s iso-strain scheme) is obtainedassuming A ( φ c ) = A s = I , while Reuss’ iso-stress scheme is recovered when A ( φ c ) = C ( φ c ) − and A s = C − s .A limit of a coarse-grained polycrystal is obtained when f →
0, so whenthe volume fraction of grain boundary zones approaches zero. In such limit theeffective properties ¯ C CS / SC approach the self-consistent estimate of [44] for aone-phase polycrystal. Respective limit estimates of the bulk and shear modu-lus related to the effective stiffness ¯ C CS / MT , and perfectly random orientationdistribution, approach the following values:¯ K ∞ CS / MT = K − L G + 9 K ∗ (17)¯ G ∞ CS / MT = 5 G + G ∗ − L K + K ∗ ) + 2 G + G ∗ + 2 G + G ∗ − − G ∗ (18)6here K ∗ = 4 G s , G ∗ = G s G s + 9 K s G s + K s ) (19)These values are some lower (resp. upper) bound estimates of ¯ C if the difference C s − C ( φ c ) is negative (resp. positive) definite for any φ c . Those bounds liewithin less rigorous Reuss and Voigt bounds, which are approached if G s tends to0 and ∞ , respectively. Evidently, for another limit value: f → C CS / MT and ¯ C CS / SC are equal and coincide with C s , so with the shell properties.
3. Computational methods
The molecular statics (MS) method (i.e. at 0 K temperature) [45, 46, 47]simulations were performed with the use of the Large-scale Atomic/MolecularMassively Parallel Simulator (LAMMPS) [48]. As an approximation describingthe energy between atoms the Embedded Atom Model (EAM) [45, 49] was used.The Open Visualization Tool OVITO [50] was used to analyse and visualize theresults of the simulations. The methodology for preparing polycrystal sam-ples by the Voronoi tessellation algorithm implemented in the Atomsk program[51], their pre-relaxation and atomic simulations was adapted almost straight-forwardly from [8, 23]. All calculation samples were approximately cubes. Thesize of the samples was chosen so that: small sample contained only an amor-phous structure representing the grain boundaries, an medium sample of about0.5 million atoms and a large sample of about 4 million atoms. To get the com-ponents of stiffness tensor, ¯ C ijkl , for all pre-relaxed structures, the stress-strainmethod with the maximum strain amplitude of 10 − was utilised [48, 52].In order to study the effect of the anisotropy degree as well as the number andsize of grains on mechanical properties of polycrystalline material, six metals ofhcp lattice symmetry with seven grain sizes each were considered in this work,see the following enumeration I–VI, Tabs. B.5–B.15 and Fig. 2.The stiffness parameters of a grain boundary zone used in the core-shellmodel should be representative for an averaged stiffness of an interphase layersbetween any pair of grain orientations. In [23] it was proposed to identify suchparameters by performing atomistic simulations on samples for which the sizewas reduced so that the fraction f of transient shell atoms approaches unity.The name of these samples starts with a letter S in Table 1.I. Ruten (Ru)
The ruten EAM potential parametrized by [53] was used. This poten-tial reproduces the hcp-ruten monocrystal equivalent orthogonal cell (butthat still respects the hexagonal lattice) lattice constants a hcp =2.704 ˚A, b hcp =4.684 ˚A, c hcp =4.288 ˚A, the cohesive energy E c =-6.86 eV, and the elas-tic constants in crystallographic axes coinciding with Cartesian coordinatesystem axes: C =546.54 GPa, C =619.07 GPa, C =169.87 GPa, C =170.85 GPa, and C =199.58 GPa. The characteristics of compu-tational ruten samples are listed in the Tab.B.5.7a) (b) ζ I =G /G v ζ II =G /G v ζ I / ζ II =G /G Ru Ti Co Zr Mg Re0.00.20.40.60.81.0
Ru Ti Co Zr Mg Re - (cid:1) [ D eg r ee ] (c) (d) Ru Ti Co Zr Mg Re0123456 E rr o r - Lg ( ζ )[ % ] Ru Ti Co Zr Mg Re0.00.20.40.60.8 U n i v e r s a l A n i s o t r op y F a c t o r Figure 2: Anisotropy measures for considered hcp metals: (a) Zener-like anisotropy factors ζ ,(b) Non-caxiality ratio Φ (see Appendix A), (c) universal anisotropy measure ζ , (d) universalelastic anisotropy index. II.
Titanium (Ti)
The titanium EAM potential parametrized by [54] was used. This potentialreproduces the hcp-titanium monocrystal equivalent orthogonal cell (butthat still respects the hexagonal lattice) lattice constants a hcp =2.953 ˚A, b hcp =5.114 ˚A, c hcp =4.681 ˚A, the cohesive energy E c =-4.85 eV, and the elas-tic constants in crystallographic axes coinciding with Cartesian coordinatesystem axes: C =171.47 GPa, C =189.96 GPa, C =84.23 GPa, C =77.07 GPa,and C =52.79 GPa. The characteristics of computational titanium sam-ples are listed in the Tab.B.7.III. Cobalt (Co)
The cobalt EAM potential parametrized by [55] was used. This poten-tial reproduces the hcp-cobalt monocrystal equivalent orthogonal cell (butthat still respects the hexagonal lattice) lattice constants a hcp =2.519 ˚A, b hcp =4.362 ˚A, c hcp =4.056 ˚A, the cohesive energy E c =-4.39 eV, and the elas-tic constants in crystallographic axes coinciding with Cartesian coordinatesystem axes: C =310.01 GPa, C =357.51 GPa, C =145.67 GPa, C =119.48 GPa, and C =92.54 GPa. The characteristics of compu-tational cobalt samples are listed in the Tab.B.9.IV. Zirconium (Zr)
The zirconium EAM potential parametrized by [56] was used. This poten-tial reproduces the hcp-zirconium monocrystal equivalent orthogonal cell(but that still respects the hexagonal lattice) lattice constants a hcp =3.230 ˚A,8 hcp =5.596 ˚A, c hcp =5.186 ˚A, the cohesive energy E c =-6.02 eV, and the elas-tic constants in crystallographic axes coinciding with Cartesian coordinatesystem axes: C =174.27 GPa, C =211.40 GPa, C =109.70 GPa, C =80.54 GPa, and C =46.45 GPa. The characteristics of computa-tional zirconium samples are listed in the Tab.B.11.V. Magnesium (Mg)
The magnesium EAM potential parametrized by [57] was used. Thispotential reproduces the hcp-magnesium monocrystal equivalent orthog-onal cell (but that still respects the hexagonal lattice) lattice constants a hcp =3.199 ˚A, b hcp =5.541 ˚A, c hcp =5.210 ˚A, the cohesive energy E c =-1.55 eV,and the elastic constants in crystallographic axes coinciding with Cartesiancoordinate system axes: C =55.88 GPa, C =69.40 GPa, C =28.70 GPa, C =20.19 GPa, and C =13.86 GPa. The characteristics of computa-tional magnesium samples are listed in the Tab.B.13.VI. Rhenium (Re)
The rhenium EAM potential parametrized by [58] was used. This potentialreproduces the hcp-rhenium monocrystal equivalent orthogonal cell (butthat still respects the hexagonal lattice) lattice constants a hcp =2.761 ˚A, b hcp =4.782 ˚A, c hcp =4.477 ˚A, the cohesive energy E c =-8.03 eV, and the elas-tic constants in crystallographic axes coinciding with Cartesian coordinatesystem axes: C =340.24 GPa, C =448.68 GPa, C =259.96 GPa, C =217.92 GPa, and C =52.51 GPa. The characteristics of compu-tational rhenium samples are listed in the Tab.B.15.
4. Results
The following notation for computational samples of nanocrystalline hcpmaterial subjected to the atomistic simulations is usedSIZE − N g − SYSwhere SIZE is a relative size of sample (S – small, M – medium or L – large) as-sessed by the number of unit cells in the sample, N g - a number of orientationsof crystal axes (here 16, 54, 125, 128 or 250 randomly selected orientations),while SYS denotes the geometry of grain distribution, i.e.: BCC or random, seeTables B.5–B.15 presented in Appendix B. As in [8, 23] orientations are definedin terms of Euler angles. Detailed results, in the form of full elasticity tensors¯ C , derived from molecular simulations of analysed samples for six hcp metalsare collected in the Tables B.6, B.8, B.10, B.12, B.14 and B.16, respectively.Consistently with the previous studies on cubic nanocrystalline metals men-tioned above, it is found that the number of orientations and the morphologicaldistribution of grains have much smaller impact on the value of elastic stiffnessthan a number of atoms per grain, which in the present context is equivalent tothe grain size. 9 henium (HCP) Figure 3: Visualization of selected atomistic computational samples and cohesive energy Ec(eV/atom) for rhenium.
The obvious reason for such correlations can be deduced from Fig. 3 whereselected atomistic computational samples and cohesive energy E c (eV/atom)are visualized. As it is seen, with a decreasing average grain size, the frac-tion of transient shell atoms in the sample rises, increasing the impact of thegrain boundary zone on the overall response. The samples with a smallest ratioSIZE / N g have almost all atoms belonging to this zone. As SIZE / N g increasessamples can be described as a two-phase medium composed of crystalline coresof well-ordered atoms surrounded by amorphous wrapping. It is consistent withthe idea of a core-shell model recalled in Sec.2. Present results indicated thatthe assumption concerning the assessment of the shell thickness ∆ taken for cu-bic nanocrystals can be extended to hcp metals, so that ∆ is assumed as equalto the cutoff radius of the atomistic potential. The respective values of d and f obtained using Eq. (16) are collected in Tables B.5, B.7, B.9, B.11, B.13and B.15 in Appendix B. Because the assumed orientation distribution withinthe samples is random the closest isotropic approximation of the calculatedelasticity tensors, collected in Tables B.6-B.12 in Appendix B, is established.Following previous studies by [8, 23] the closest isotropic approximation ¯ C L iso of anisotropic ¯ C is defined employing the Log-Euclidean metric as proposed by[59]. Using this method two scalars, approximated isotropic bulk modulus ¯ K L iso and shear ¯ G L iso are obtained and next compared with the respective estimatesfound using the core-shell model. As an universal (i.e. applicable to any materialsymmetry) anisotropy measures the error ζ resulting from the applied isotropicapproximation is used in this work. It is calculated as a normalized differencebetween ¯ C L iso , ¯ C L iso = 3 ¯ K L iso I P + 2 ¯ G L iso ( I − I P ) , (20)and the actual ¯ C . It is defined as [60] ζ = || Log ¯ C − Log ¯ C L iso |||| Log ¯ C || × ≥ , (21)where || A || = √ A · A = p A ijkl A ijkl and Log A = P K log λ L P K ( λ K - eigen-values of A , P K - eigenprojectors of A obtained by its spectral decomposition).10ore on the approximation and detailed formulas can be found in [8, 23]. An-other universal anisotropy measures have been discussed in [40]. In particular,the non-dimensional quantity defined in terms of Voigt and Reuss estimates ofthe overall bulk and shear modulus for random polycrystal (see Appendix A)has been recommended in that paper. This so-called universal anisotropy index,equal zero for isotropy, is defined as: A U = 5 ¯ G V ¯ G R + ¯ K V ¯ K R − ≥ K s and G s ,were established for the sample with f approaching unity, namely S − − BCC.The identified values, together with values of K , G , L and two shear moduli G and G , that is constants defining the Kelvin moduli and stiffness distributorof monocrystals, are collected in Table 2. The cutoff radius ∆ of the appliedatomistic potential is also placed there. Note that this is a set of necessaryinput data to obtain the predictions of a core-shell model in the next subsection.Metals in this table are ordered according to the increasing value of anisotropydegree measured by ζ . For comparison purpose universal anisotropy index isalso included. Table 1: The overall isotropized bulk and shear moduli ¯ K L iso [GPa] and ¯ G L iso [GPa] andanisotropy measure ζ [%] calculated for the effective stiffness tensors resulting from the atom-istic simulations for metals of hcp lattice geometry. Samples are ordered according to theincreasing average grain size d , while metals according to the decreasing anisotropy measure ζ of single crystal (see Table 2). Sample ¯ K L iso ¯ G L iso ζ ¯ K L iso ¯ G L iso ζ ¯ K L iso ¯ G L iso ζ Ru Ti CoS-128-BCC 157.38 80.37 0.63 93.68 21.46 0.70 196.86 34.85 1.08M-250-BCC 162.94 102.96 1.09 96.29 23.25 5.42 196.43 49.17 1.48M-128-BCC 179.71 113.93 0.93 100.70 28.48 3.99 197.56 55.10 1.23M-125-Random 181.50 96.53 2.34 98.76 28.14 2.29 197.05 48.25 2.93M-54-BCC 218.25 114.07 2.30 100.35 28.22 5.01 196.63 60.10 1.14M-16-BCC 221.48 134.40 0.91 103.81 35.73 1.57 195.61 65.59 1.61L-16-BCC 251.27 155.36 0.65 107.36 41.33 0.61 195.06 75.21 0.86Zr Mg ReS-128-BCC 85.55 17.34 1.38 33.14 6.29 3.27 341.05 88.11 0.37M-250-BCC 96.98 21.51 3.03 33.59 7.81 5.35 297.46 62.10 1.84M-128-BCC 99.12 23.23 2.77 33.88 9.22 2.61 297.04 66.01 1.08M-125-Random 99.02 23.76 2.55 33.94 9.27 1.91 301.03 67.43 0.41M-54-BCC 103.51 26.31 3.16 34.15 10.05 2.32 293.37 62.29 0.80M-16-BCC 108.51 30.67 3.15 34.50 11.68 1.53 291.08 60.16 1.51L-16-BCC 114.89 35.00 1.36 34.76 12.48 2.05 284.48 56.67 1.66 able 2: Constants K , G , G , G and L of monocrystal samples defining four Kelvinmoduli and the stiffness distributor, identified shell elastic moduli K s and G s and cutoffradius ∆ of the applied atomistic potential for analysed metals. Metals are ordered with anincreasing anisotropy parameter ζ . Respective universal elastic anisotropy index A U is alsoincluded.Metal K G G G L ζ A U K s G s ∆[GPa] [GPa] [GPa] [GPa] [GPa] [%] [GPa] [GPa] [˚ A ]Ru 303.9 211.9 188.3 199.6 34.65 0.83 0.016 157.4 80.37 7.6Ti 112.2 54.57 43.63 52.79 5.346 1.91 0.051 93.68 21.46 6.72Co 194.1 115.5 82.17 92.54 10.05 2.11 0.079 196.9 34.85 6.5Zr 122.4 64.10 32.28 46.45 3.757 5.00 0.345 85.55 17.34 7.6Mg 35.48 23.77 13.59 13.86 2.362 5.50 0.250 33.15 6.290 7.15Re 280.1 104.3 40.15 52.51 31.30 6.29 0.656 341.1 88.11 5.5 The estimates of effective bulk and shear moduli obtained by the core-shellmodel for nanocrystalline hcp metals are now compared with the results ofatomistic simulations reported in Table 1. All analytical estimates are calculatedfor perfectly random distribution of orientation, so the overall stiffness specifiedby Eq. (13) is isotropic.First, let us discuss the classical bounds and mean-field estimates for coarsegrained polycrystals of six hcp metals with local properties specified in Table2. Their values for each metal are collected in Table 3. As it is seen, due tosmall non-coaxiality angle Φ, the Reuss and Voigt bounds on the bulk modulusare very close. A larger difference between those bounds exists as concerns theshear modulus. Nevertheless when one compares the value of the self-consistentand CS/MT estimates they are again close to each other. This observationleads to the conclusion that the estimates delivered by two variants of a core-shell model for nanocrystalline medium will not be far from each other as well.Therefore, since the CS/MT estimate is specified by an explicit and closedform equation, contrary to the implicit CS/SC estimate, the analysis of themodel validity is focused on this variant. For this range of grain sizes atomisticsimulations are not applicable due to hardware limitations related to excessivelylarge number of atoms required to represent polycrystal. Instead, in Table 3for a purpose of comparison, results of computational FE homogenization [61]are included. Effective properties have been obtained using RVE geometries andperiodic boundary conditions described in [62]. For each hcp metal 5 realizationsof RVE composed of 125 grains with randomly selected orientations and 6 elements per grain were analyzed to find the effective elasticity tensor ¯ C F E .Isotropized bulk and shear moduli of such tensor are reported in Table 3. Itis seen that the obtained values are close to the SC estimate. This result is inagreement with other literature studies, e.g. [61, 63].Figure 4 compares the CS/MT model predictions with the correspondingresults of atomistic simulations for nano-grained polycrystals. Since there is a12 able 3: The overall bulk and shear modulus ¯ K ∞ iso and ¯ G ∞ iso [GPa] of coarse-grained polycrystal,obtained by the Voigt (V), Reuss (R), self-consistent (SC) estimate, the limit value obtainedby MT core-shell (CS/MT) model (Eq. (17) and (18)) and computational FE homogenizationfor polycrystals with random orientation distributions and six metals of hexagonal symmetry.Local properties of single crystal are collected in Table 2. Metal R SC V CS/MT FE R SC V CS/MT FE¯ K ∞ iso[GPa] ¯ G ∞ iso[GPa]Ru 302.98 303.44 303.92 303.42 303.44 197.04 197.28 197.54 197.18 197.29Ti 112.095 112.14 112.18 112.12 112.14 48.98 49.24 49.47 49.16 49.25Co 193.94 194.01 194.09 193.98 194.01 91.54 92.25 92.98 91.97 92.28Zr 122.35 122.365 122.39 122.36 122.37 41.455 42.92 44.31 42.36 42.98Mg 35.44 35.45 35.48 35.45 35.46 14.99 15.32 15.74 15.19 15.33Re 278.52 279.14 280.085 279.31 279.16 51.255 54.24 57.92 54.96 54.41 (a) (b) (c) (d) Figure 4: The isotropic bulk ¯ K L iso (a) and shear ¯ G L iso (b) moduli as functions of the averagegrain diameter d for 5 hcp metals: Ru, Ti, Co, Zr and Mg - the results of atomistic simula-tions reported in Tables B.6, B.8, B.10, B.12 and B.14 as well as the Mori-Tanaka core-shell(CS/MT) model predictions (dashed line); (c) and (d) contain analogical plots for Re (TableB.16). For the purpose of comparison the grain diameter is scaled by the double cutoff radius of atomistic potential 2∆ for the given metal (the last column in Table 2), while moduli arescaled by the respective estimates of CS/MT scheme for the coarse-grained random polycrystal(Table 3). K ∞ CS/MT and ¯ G ∞ CS/MT (Table 3),while the grain diameter by a double cutoff radius of corresponding atomisticpotential. As it is seen in Fig. 4b the shear modulus of five out of six hcpmetals (Ru, Ti, Co, Zr, Mg) follows the common qualitative and quantitativetrend – with a decreasing grain diameter the value drops from ¯ G ∞ CS/MT to thevalue of approximately 0 . G ∞ CS/MT when the grain boundary zone encompassesthe whole volume. For a grain diameter of 30 × ∆ the value of 0 . G ∞ CS/MT isattained. As concerns bulk modulus the qualitative trend is similar, however,quantitatively the relative value attained when d/ (2∆) → E L iso = 9 ¯ K L iso ¯ G L iso K L iso + ¯ G L iso , ¯ ν L iso = 3 ¯ K L iso − G L iso K L iso + 2 ¯ G L iso . (23)The Young modulus follows qualitatively the trend observed for the shear mod-ulus. As concerns Poisson’s ratio for Ru, Ti, Co, Zr and Mg, it decreases witha grain size, while an opposite relation is found for Re. Additionally, presentedresults confirm the observation that for analyzed hcp metals the CS/MT andCS/SC estimates are close to each other. In spite of these two mean-field modelsfigures contain also predictions obtained using the mixture rule-based iso-strain(Voigt) scheme and its counterpart – an iso-stress Reuss scheme. Those twoare upper and lower bounds for stiffness moduli (but not Poisson’s ratio) of atwo-phase random polycrystalline medium. Comparing the predictions of allpresented averaging models with the atomistic simulations it is seen that onoverall the CS/MT scheme can be recommended as delivering reasonable pre-dictions for all hcp metals. Moreover, consistency of model estimates with theresults of atomistic simulations proves validity of the assumed procedure for theassessment of size and average properties for the grain boundary zone.In [23] the correlation between the Zener parameter and the character of therelation between effective moduli and the grain size has been found for cubicnanocrystalline metals. However, in the case of analyzed six hcp metals it is14oung’s modulus Poisson’s ratioa) Ru ● ● b) Ti ● ● c) Co ● ● (cid:4)(cid:5)(cid:6)(cid:7) Figure 5: The isotropic Young modulus ¯ E L iso and Poisson’s ratio ¯ ν L iso as a function of theaverage grain diameter d by the two variants of the core-shell model (CS/MT, CS/SC) andthe two-phase iso-strain Voigt (V) and iso-stress Reuss (R) schemes - comparison with resultsof atomistic simulations, calculated using Eq. 23: (a) Ru, (b) Ti (c) Co (d) Zr (e) Mg (f) Re.The horizontal dashed black lines in left figures indicate the limit value of CS/MT estimateof Young’s modulus for a coarse-grained polycrystal. Zr ● ● (cid:8)(cid:9)(cid:10)(cid:11) e) Mg ● ● (cid:12)(cid:13)(cid:14)(cid:15) f) Re ● ● (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !" Figure 6: continue... ξ and A U (see Table 2), it is not that much different from other five metalsunder study. From this point of view, it would be interesting to verify if theidentified behaviour is an artefact consequence of the inherent features of theapplied atomistic potential or is also observed in reality. Unfortunately, authorswere not able to find any experimental data in the literature to confirm eitherof hypotheses.
5. Summary and conclusions
Different variants of a mean-field core-shell model [8, 23] for estimation ofelastic properties of bulk nanocrystalline metals have been validated for hcpcrystal lattice symmetry. Because there is not enough experimental data, vali-dation has been conducted by comparing the estimates with the results of atom-istic simulations. Six metals of hexagonal (hcp) lattice geometry were selectedfor which the verified EAM potentials are available in the literature. All of themare characterized by relatively low non-coaxiality angle Φ and the same relationbetween Zener-like anisotropy factors (11), see also the collective figures 2.Following previous research [8, 23], for each hcp metal atomistic simulationshave been conducted on seven generated samples of polycrystalline materialswith randomly selected orientations. Samples vary as concerns the average grainsize, so that the averaged grain diameter takes values between ca. 1 nm to 20 nm.In the simulations all 21 components of the anisotropic elastic stiffness tensor areidentified. The smallest sample served to identify the average properties of thegrain boundary zone. For further analysis of the grain-size effect on the elasticmoduli the closest isotropic approximation is found using the Log-Euclideannorm [59].It has been observed that for five out of six studied metals (Ru,Ti,Co,Zr,Mg)the elastic bulk and shear moduli increase with a grain size. The reverse trend isobserved for rhenium (Re). This metal exhibits the strongest anisotropy amongconsidered metals, although the correlation between the anisotropy degree andthe character of grain size dependence is not clear. It would be interesting toconfirm experimentally this qualitative difference in the grain size effect for thismetal, since the present observations strongly relays on validity of the appliedatomistic potential.Among the considered variants of core-shell model the estimates of elasticmoduli obtained by the Mori-Tanaka scheme are on overall in the most sat-isfactory qualitative and quantitative agreement with the results of atomisticsimulations for all considered hexagonal metals, independently of the charac-ter of the grain size effect. The study demonstrated also the validity of theassumptions concerning the shell thickness and properties.The applied variants of mean-field core-shell model can be extended to es-timate a non-linear response of a nanocrystalline material and specifically theyield strength [14, 16]. Atomistic simulations may serve to validate such anextension. 17
CKNOWLEDGMENTS
The research was partially supported by the project No. 2016/23/B/ST8/03418of the National Science Centre, Poland. Additional assistance was grantedthrough the computing cluster GRAFEN at Biocentrum Ochota, the Interdis-ciplinary Centre for Mathematical and Computational Modelling of WarsawUniversity (ICM UW) and Pozna´n Supercomputing and Networking Center(PSNC).
Appendix A. Spectral decomposition of elasticity tensor for hcp crys-tal and standard estimates of effective stiffness for ran-dom polycrystals of hexagonal symmetry
Spectral decomposition of C for a crystal of hcp symmetry is given by Eq.(2), where the projectors P ( φ c ) and P ( φ c ) for two 2D eigen-subspaces aregiven by Eq. (4) and (5). These two projectors are common for all crystalof this symmetry. Two remaining projectors depend also on the distributor ξ = 1 / P ( ξ, φ c ) = (cos Φ) I P + 12 √ I ⊗ D n + D n ⊗ I ) + (sin Φ) D n ⊗ D n (A.1) P ( ξ, φ c ) = (sin Φ) I P − √ I ⊗ D n + D n ⊗ I ) + (cos Φ) D n ⊗ D n , (A.2)Angle Φ = Φ( ξ ) is calculated using the components of 2 × (cid:18) L K − G (cid:19) ∈ h− π/ , π/ i (A.3)This angle is a measure of non-coaxiality between the given anisotropic stiffnessof hcp crystal and any isotropic tensor. If Φ = 0 then they are coaxial and C ( φ c ) ¯ C iso − ¯ C iso C ( φ c ) = O for any ¯ C specified by Eq. (20). It is worth notingthat the stiffness distributor ξ can be expressed by the invariants of orthogonalprojector P [64].The formulas for standard estimates of effective elastic stiffness of one-phasepolycrystals of any anisotropy has been provided in Appendix A of [8]. Theirspecification for materials of random texture composed of grains of hexagonalsymmetry are collected in Table A.4 (for details see [65] and [66]). It is seenthat if L = 0, which is equivalent to Φ = 0, then all estimates of the overallbulk modulus coincide and are equal to K . Such crystals belong to the class ofvolumetrically isotropic materials [39]. It is worth to note that for hexagonalcrystals and random orientation distribution the specification of two equationsenabling to find the self-consistent estimate, proposed by [44], was first givenby [67]. It can be verified that the respective two equations in Table A.4 areequivalent to Kneer’s formulas. 18 able A.4: Classical mean-field estimates of the overall bulk and shear moduli ( ¯ K , ¯ G ) for aone phase random polycrystal of hexagonal symmetry. V – Voigt, R – Reuss, H-S – Hashin-Shtrikman (U – upper, L – lower), SC – self-consistent (equiaxial, spherical shape of grainsis assumed). K U/L and G U/L for the Hashin-Shtrikman bounds are established from theoptimality conditions (for details see [65] or [66]) Estimate ¯ K ¯ G V K ( G + 2 G + 2 G )R K − L G (cid:16) (cid:16) G − L / (6 K ) + G + G (cid:17)(cid:17) − H-S (U/L) K − L G +9 K U/L ∗ o G + G ∗ o − L K K ∗ o ) + G + G ∗ o + G + G ∗ o !! − − G ∗ o K U/L ∗ o = 4 G U/Lo G U/L ∗ o = G U/Lo K U/L +8 G U/Lo K U/Lo +2 G U/Lo ) positive solutions of the set of two equations:SC K − ¯ K − L G +9 ¯ K ∗ = 0 G + ¯ G ∗ − L K
1+ ¯ K ∗ ) + G + ¯ G ∗ + G + ¯ G ∗ !! − − ¯ G ∗ − ¯ G = 0¯ K ∗ = 4 ¯ G ¯ G ∗ = ¯ G K +8 ¯ G
6( ¯ K +2 ¯ G ) Appendix B. Detailed results of atomistic simulations
In this Appendix detailed results of atomistic simulations for eight samplesof six metals of hcp symmetry are collected is subsequent subsections. For eachmetal the following convention is used,[ C KL ] = C C C C C C C C C C C C C C C C C C Sym. C C C . (B.1)The quantitative data describing analysed samples are collected in the first table,while the calculated 21 components of the anisotropic elasticity tensor for eachsample (the Voigt notation (B.1) is used) are given in the second table.19 ppendix B.1. Nanocrystalline ruten Table B.5: Ruten: Volume (˚A ), box lengths: a,b,c (˚A), number of atoms, average graindiameter d (˚A), fraction of transient shell atoms f (16), average cohesive energy E c (eV/atom)of analysed computational samples. Used EAM potential [53] with cutoff radius =6.5(˚A). Sample V a b c No.of atoms d f c Monocrystal 44.56 2.70 4.68 4.29 4 -6.86S-128-BCC 143764.4 52.39 52.36 52.41 10324 12.90 1.00 -6.60M-128-BCC 6841505.5 189.85 189.81 189.85 496965 46.74 0.59 -6.75M-16-BCC 6824700.3 189.63 189.73 189.69 497225 93.39 0.34 -6.80M-54-BCC 6839237.3 189.81 189.85 189.79 497232 62.31 0.47 -6.77M-250-BCC 6877358.3 190.13 190.18 190.19 497416 37.45 0.69 -6.72M-125-Random 6867930 190.11 190.10 190.04 497109 47.17 0.59 -6.74L-16-BCC 54324813 378.74 378.74 378.72 3976847 186.48 0.18 -6.83
Table B.6: Ruten: Elasticity tensors ¯ C [GPa] of analysed samples (for notation used see Eq.(B.1)). Monocrystal small sample, 128 grains in BCC system, 10324 atoms(S-128-BCC) .
54 169 .
87 170 .
852 0 0 0546 .
54 170 .
85 0 0 0619 .
068 0 0 0199 .
58 0 0
Sym. .
58 0188 . .
98 106 .
61 102 .
56 2 .
76 1 .
52 0 . .
14 101 .
94 4 . − . − . .
35 0 . − . − . . − . − . Sym. .
738 0 . . medium sample, 128 grains in BCC system, 496965 atoms medium sample, 16 grains in BCC system, 497225 atoms(M-128-BCC) (M-16-BCC) .
49 154 .
52 171 .
95 11 . − .
72 5 . .
65 159 .
56 13 .
67 10 .
03 3 . . − . − . − . .
53 10 . − . Sym. .
75 8 . . .
47 134 .
67 133 .
33 1 . − . − . .
65 138 .
81 3 . − . − . .
52 2 . − . − . . − . − . Sym. . − . . medium sample, 54 grains in BCC system, 497232 atoms medium sample, 250 grains in BCC system, 497416 atoms(M-54-BCC) (M-250-BCC) .
82 148 .
29 149 . − . − .
08 1 . .
59 158 . − .
83 0 .
51 2 . . − .
13 6 . − . . − . − . Sym. .
47 0 . . .
66 137 .
14 126 . − .
11 3 . − . .
96 120 .
63 7 .
93 4 . − . . − .
78 10 .
04 4 . .
92 11 .
25 3 . Sym. .
70 7 . . medium sample, 125 grains in random system, 497109 atoms large sample, 16 grains in BCC system, 3976847 atoms(M-125-Random) (L-16-BCC) .
38 110 .
49 101 .
51 4 .
24 1 . − . .
70 127 .
41 2 .
97 6 . − . .
57 7 . − . − . .
16 7 .
38 2 . Sym. . − . . .
11 155 .
18 149 . − . − . − . .
08 151 .
72 1 .
57 1 . − . .
07 1 . − . − . . − .
79 0 . Sym. .
78 0 . . Appendix B.2. Nanocrystalline titanium
Table B.7: Titanium: Volume (˚A ), box lengths: a,b,c (˚A), number of atoms, average graindiameter d (˚A), fraction of transient shell atoms f (16), average cohesive energy E c (eV/atom)of analysed computational samples. Sample V a b c No.of atoms d f c Monocrystal 70.69 2.95 5.11 4.68 4 -4.85small-128-BCC 194855.95 58.06 58.01 57.85 11024 14.27 1.00 -4.74M-128-BCC 8782331.5 206.34 206.35 206.26 494687 50.79 0.55 -4.79M-16-BCC 8768351 206.16 206.21 206.25 494702 101.53 0.31 -4.82M-54-BCC 8779701.6 206.28 206.35 206.25 494820 67.72 0.44 -4.80M-250-BCC 8786488 206.34 206.29 206.42 494813 40.64 0.65 -4.78M-125-Random 8781498.7 206.38 206.28 206.27 494670 51.19 0.55 -4.79L-16-BCC 70048712 412.25 412.16 412.26 3957154 202.97 0.17 -4.83 able B.8: Titanium: Elasticity tensors ¯ C [GPa] of analysed samples (for notation used seeEq. (B.1)). Monocrystal small sample, 128 grains in BCC system, 11024 atoms(S-128-BCC) .
46 84 .
23 77 .
07 0 0 0171 .
46 77 .
07 0 0 0189 .
96 0 0 052 .
79 0 0
Sym. .
79 043 . .
29 78 .
92 79 . − . − . − . .
30 79 .
23 0 . − .
44 0 . . − . − . − . . − .
56 0 . Sym. .
42 0 . . medium sample, 128 grains in BCC system, 494687 atoms medium sample, 16 grains in BCC system, 494702 atoms(M-128-BCC) (M-16-BCC) .
10 79 .
56 83 . − .
10 2 .
22 5 . .
40 85 .
37 0 .
59 0 .
37 1 . .
26 1 . − . − . .
24 2 .
23 0 . Sym. . − . . .
64 80 .
87 80 .
26 0 .
47 0 . − . .
80 80 . − . − .
23 1 . .
97 0 . − .
46 1 . . − . − . Sym. .
91 1 . . medium sample, 54 grains in BCC system, 494820 atoms medium sample, 250 grains in BCC system, 494813 atoms(M-54-BCC) (M-250-BCC) .
42 84 .
15 76 . − .
73 4 . − . .
70 85 .
13 1 . − . − . . − .
24 4 . − . .
53 5 . − . Sym. .
27 0 . . .
96 79 .
66 80 .
19 1 .
73 1 .
62 0 . .
26 79 .
16 3 .
12 1 . − . . − .
20 0 .
38 1 . .
26 0 .
50 6 . Sym. .
85 0 . . medium sample, 125 grains in random system, 494670 atoms large sample, 16 grains in BCC system, 3957154 atoms(M-125-Random) (L-16-BCC) .
88 80 .
55 78 . − . − . − . .
908 80 .
32 1 .
90 3 . − . .
98 0 .
25 0 .
72 0 . . − . − . Sym. .
74 0 . . .
47 79 .
46 80 .
93 0 . − . − . .
70 80 .
79 0 .
76 0 . − . . − . − . − . . − . − . Sym. .
57 0 . . Appendix B.3. Nanocrystalline cobalt
Table B.9: Cobalt: Volume (˚A ), box lengths: a,b,c (˚A), number of atoms, average graindiameter d (˚A), fraction of transient shell atoms f (16), average cohesive energy E c (eV/atom)of analysed computational samples. Used EAM potential [55] with cutoff radius =6.5 (˚A). Sample V a b c No.of atoms d f c Monocrystal 44.56 2.52 4.36 4.06 4 -4.39small-128-BCC 147622.4 52.74 52.84 52.97 12692 13.01 1.00 -4.27M-128-BCC 5775045 179.44 179.41 179.39 506233 44.17 0.61 -4.33M-16-BCC 5716172.5 178.82 178.78 178.81 506408 88.04 0.36 -4.36M-54-BCC 5745458.3 179.14 179.04 179.14 506071 58.79 0.50 -4.34M-250-BCC 5801318.1 179.75 179.59 179.72 506367 35.39 0.71 -4.32M-125-Random 5779664.5 179.49 179.42 179.46 506180 44.53 0.61 -4.33L-16-BCC 45430931 356.78 356.81 356.87 4049913 175.69 0.19 -4.37 able B.10: Cobalt: Elasticity tensors ¯ C [GPa] of analysed samples (for notation used see Eq.(B.1)). Monocrystal small sample, 128 grains in BCC system, 12692 atoms(S-128-BCC) .
01 145 .
67 119 .
48 0 0 0310 .
01 119 .
48 0 0 0357 .
51 0 0 092 .
54 0 0
Sym. .
54 082 . .
84 174 .
23 173 . − .
25 1 . − . .
30 175 .
27 1 . − .
39 0 . . − .
70 0 .
66 0 . .
80 1 .
08 0 . Sym. .
07 0 . . medium sample, 128 grains in BCC system, 506233 atoms medium sample, 16 grains in BCC system, 506408 atoms(M-128-BCC) (M-16-BCC) .
68 163 .
80 162 .
70 3 .
63 0 .
42 0 . .
47 163 .
30 1 .
70 1 .
38 2 . . − .
04 1 .
04 0 . . − .
18 0 . Sym. . − . . .
92 150 .
26 156 .
34 0 . − . − . .
34 157 . − .
48 1 . − . .
10 1 .
65 0 .
96 5 . . − .
27 0 . Sym. . − . . medium sample, 54 grains in BCC system, 506071 atoms medium sample, 250 grains in BCC system, 506367 atoms(M-54-BCC) (M-250-BCC) .
98 157 .
78 159 . − .
83 0 . − . .
10 159 . − .
45 1 .
46 1 . .
01 0 . − . − . .
139 1 .
11 0 . Sym. . − . . .
16 167 .
37 166 .
43 0 . − .
13 1 . .
61 164 .
01 2 .
30 2 .
01 0 . . − .
80 1 . − . .
01 0 .
04 0 . Sym. .
58 1 . . medium sample, 125 grains in random system, 506180 atoms large sample, 16 grains in BCC system, 4049913 atoms(M-125-Random) (L-16-BCC) .
50 165 .
27 160 . − .
85 1 .
54 5 . .
05 156 . − . − . − . .
88 2 . − . − . . − . − . Sym. . − . . .
95 143 .
53 149 . − . − . − . .
14 147 .
50 1 .
85 0 . − . . − .
41 0 .
13 1 . . − . − . Sym. . − . . Appendix B.4. Nanocrystalline zirconium
Table B.11: Zirconium: Volume (˚A ), box lengths: a,b,c (˚A), number of atoms, average graindiameter d (˚A), fraction of transient shell atoms f (16), average cohesive energy E c (eV/atom)of analysed computational samples. Used EAM potential [56] with cutoff radius =7.6 (˚A). Sample V a b c No.of atoms d f c Monocrystal 93.75 3.23 5.59 5.19 4 -6.02S-128-BCC 264763.83 64.23 64.11 64.29 11172 15.81 0.99 -5.90M-128-BCC 11630649 226.55 226.60 226.55 492603 55.78 0.52 -5.95M-16-BCC 11592882 226.34 226.32 226.31 492442 111.44 0.29 -5.98M-54-BCC 11611827 226.35 226.42 226.57 492413 74.33 0.41 -5.97M-250-BCC 11641724 226.64 226.64 226.64 492635 44.64 0.61 -5.94M-125-Random 11626043 226.59 226.51 226.52 492598 56.21 0.51 -5.95L-16-BCC 92583204 452.41 452.34 452.41 3940813 222.74 0.15 -6.00 able B.12: Zirconium: Elasticity tensors ¯ C [GPa] of analysed samples (for notation used seeEq. (B.1)). Monocrystal small sample, 128 grains in BCC system, 11172 atoms(S-128-BCC) .
26 109 .
69 80 .
54 0 0 0174 .
26 80 .
54 0 0 0211 .
40 0 0 046 .
45 0 0
Sym. .
45 032 . .
52 74 .
07 73 .
50 0 .
36 0 .
93 0 . .
11 74 . − .
56 0 .
05 0 . .
46 0 . − . − . .
87 0 .
30 0 . Sym. . − . . medium sample, 128 grains in BCC system, 492603 atoms medium sample, 16 grains in BCC system, 492442 atoms(M-128-BCC) (M-16-BCC) .
81 85 .
97 81 . − . − .
28 3 . .
15 85 .
26 0 . − .
09 2 . .
99 1 . − . − . . − . − . Sym. . − . . .
31 85 .
18 86 .
49 1 . − . − . .
80 90 . − .
66 0 . − . . − .
58 0 .
65 2 . . − . − . Sym. .
27 0 . . medium sample, 54 grains in BCC system, 492413 atoms medium sample, 250 grains in BCC system, 492635 atoms(M-54-BCC) (M-250-BCC) .
04 87 .
52 87 .
94 0 .
69 0 . − . .
54 84 .
56 0 . − . − . . − . − .
56 0 . . − . − . Sym. . − . . .
18 81 .
47 85 . − . − . − . .
63 87 .
33 0 .
99 2 .
22 0 . .
27 0 .
63 1 . − . .
61 1 . − . Sym. .
59 2 . . medium sample, 125 grains in random system, 492598 atoms large sample, 16 grains in BCC system, 3940813 atoms(M-125-Random) (L-16-BCC) .
24 82 .
33 83 . − .
23 0 .
58 1 . .
48 80 . − .
14 1 .
49 0 . .
34 0 . − .
45 0 . . − . − . Sym. .
98 0 . . .
54 90 .
40 93 .
31 0 . − . − . .
52 92 .
95 0 . − . − . . − .
01 0 .
98 1 . . − . − . Sym. .
17 0 . . Appendix B.5. Nanocrystalline magnesium
Table B.13: Magnesium: Volume (˚A ), box lengths: a,b,c (˚A), number of atoms, average graindiameter d (˚A), fraction of transient shell atoms f (16), average cohesive energy E c (eV/atom)of analysed computational samples. Used EAM potential [57] with cutoff radius =7.15 (˚A). Sample V a b c No.of atoms d f c Monocrystal 92.37 3.20 5.54 5.21 4 -1.55small-128-BCC 243204.72 62.38 62.43 62.46 10188 15.37 0.99 -1.51M-128-BCC 11683248 226.94 226.87 226.92 496137 55.86 0.52 -1.53M-16-BCC 11588292 226.30 226.32 226.26 496294 111.42 0.29 -1.54M-54-BCC 11644640 226.63 226.69 226.66 496382 74.40 0.41 -1.53M-250-BCC 11388363 224.96 225.02 224.98 496234 44.31 0.61 -1.49M-125-Random 11387887 224.98 224.96 225.00 496477 55.83 0.52 -1.50L-16-BCC 92228298 451.88 451.82 451.73 3970217 222.46 0.15 -1.54 able B.14: Magnesium: Elasticity tensors ¯ C [GPa] of analysed samples (for notation used seeEq. (B.1)). Monocrystal small sample, 128 grains in BCC system, 10188 atoms(S-128-BCC) .
88 28 .
70 20 .
19 0 0 055 .
88 20 .
19 0 0 069 .
40 0 0 013 .
86 0 0
Sym. .
86 013 . .
98 28 .
92 28 . − . − . − . .
60 28 . − . − .
62 0 . . − . − .
10 0 . . − . − . Sym. . − . . medium sample, 128 grains in BCC system, 496137 atoms medium sample, 16 grains in BCC system, 496294 atoms(M-128-BCC) (M-16-BCC) .
51 28 .
38 27 .
90 1 .
33 0 .
13 0 . .
38 27 .
37 0 .
12 0 .
38 0 . . − . − . − . . − .
05 0 . Sym. .
43 0 . . .
12 26 .
20 27 . − . − .
26 0 . .
62 26 .
74 0 .
62 0 . − . . − .
36 0 . − . . − . − . Sym. . − . . medium sample, 54 grains in BCC system, 496382 atoms medium sample, 250 grains in BCC system, 496234 atoms(M-54-BCC) (M-250-BCC) .
55 26 .
57 27 . − . − . − . .
23 27 .
92 0 .
27 0 .
43 0 . . − . − .
56 0 . . − . − . Sym. .
25 0 . . .
78 28 .
27 28 . − . − .
34 0 . .
90 27 .
43 0 .
54 0 . − . .
27 0 .
72 0 . − . . − .
18 1 . Sym. .
30 0 . . medium sample, 125 grains in random system, 496477 atoms large sample, 16 grains in BCC system, 3970217 atoms(M-125-Random) (L-16-BCC) .
77 26 .
96 27 . − .
72 0 . − . .
76 27 .
89 0 . − . − . .
53 0 . − . − . . − . − . Sym. .
48 0 . . .
39 25 .
82 27 . − . − .
14 0 . .
83 26 .
17 0 . − . − . . − . − . − . . − .
49 0 . Sym. . − . . Appendix B.6. Nanocrystalline Rhenium
Table B.15: Rhenium: Volume (˚A ), box lengths: a,b,c (˚A), number of atoms, average graindiameter d (˚A), fraction of transient shell atoms f (16), average cohesive energy E c (eV/atom)of analysed computational samples. Used EAM potential [58] with cutoff radius =5.5 (˚A). Sample V a b c No.of atoms d f c Monocrystal 59.12 2.76 4.78 4.48 4 -8.03small-128-BCC 98039.39 46.25 46.06 46.03 6556 11.35 1.00 -7.76M-128-BCC 7605402 196.68 196.63 196.66 509469 48.41 0.57 -7.91M-16-BCC 7663630.7 197.10 197.19 197.18 509494 97.07 0.33 -7.91M-54-BCC 7592673.7 196.53 196.51 196.60 509561 64.52 0.46 -7.93M-250-BCC 7617042.6 196.74 196.79 196.74 509655 38.75 0.67 -7.88M-125-Random 7607934.3 196.69 196.73 196.61 509357 48.80 0.57 -7.90L-16-BCC 60415965 392.37 392.39 392.41 4075464 193.20 0.176 -8.00 able B.16: Rhenium: Elasticity tensors ¯ C [GPa] of analysed samples (for notation used seeEq. (B.1)). Monocrystal small sample, 128 grains in BCC system, 6556 atoms(S-128-BCC) .
24 259 .
95 217 .
92 0 0 0340 .
24 217 .
92 0 0 0448 .
68 0 0 052 .
51 0 0
Sym. .
51 040 . .
71 282 .
82 281 . − .
51 0 . − . .
84 282 . − .
87 2 .
46 0 . . − . − . − . . − .
12 0 . Sym. .
80 0 . . medium sample, 128 grains in BCC system, 509469 atoms medium sample, 16 grains in BCC system, 509494 atoms(M-128-BCC) (M-16-BCC) .
27 250 .
57 251 .
55 0 . − . − . .
13 260 .
17 3 . − . − . .
78 0 . − .
04 0 . . − .
21 0 . Sym. . − . . .
32 226 .
52 236 . − . − .
20 2 . .
72 218 . − . − .
32 2 . . − . − . − . .
94 1 .
11 9 . Sym. . − . . medium sample, 54 grains in BCC system, 509561 atoms medium sample, 250 grains in BCC system, 509655 atoms(M-54-BCC) (M-250-BCC) .
22 246 .
90 255 . − . − . − . .
50 254 .
43 1 .
72 0 .
01 0 . . − . − .
83 2 . . − .
77 0 . Sym. .
71 1 . . .
02 261 .
51 257 .
40 0 . − .
98 5 . .
30 257 . − . − .
59 1 . . − . − .
77 1 . .
25 3 .
76 2 . Sym. .
60 2 . . medium sample, 125 grains in random system, 509357 atoms large sample, 16 grains in BCC system, 4075464 atoms(M-125-Random) (L-16-BCC) .
93 257 .
81 255 .
82 2 . − .
76 0 . .
17 253 .
97 2 .
51 0 . − . .
19 1 . − . − . . − . − . Sym. . − . . .
23 240 .
56 250 . − . − . − . .
84 245 .
76 2 .
45 1 . − . . − . − .
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