Macroscopic uncertainty of the effective properties of random media and polycrystalsPham Duc Chinh Citation: Journal of Applied Physics 101, 023525 (2007); doi: 10.1063/1.2426378 View online: http://dx.doi.org/10.1063/1.2426378 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Estimating the dynamic effective mass density of random composites J. Acoust. Soc. Am. 128, 571 (2010); 10.1121/1.3458849 Wave-induced fluid flow in random porous media: Attenuation and dispersion of elastic waves J. Acoust. Soc. Am. 117, 2732 (2005); 10.1121/1.1894792 Uncertainty limits for the macroscopic elastic moduli of random polycrystalline aggregates J. Appl. Phys. 88, 1346 (2000); 10.1063/1.373823 Percolation for a model of statistically inhomogeneous random media J. Chem. Phys. 111, 5947 (1999); 10.1063/1.479890 High-resolution finite-volume methods for acoustic waves in periodic and random media J. Acoust. Soc. Am. 106, 17 (1999); 10.1121/1.428038

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Macroscopic uncertainty of the effective properties of random mediaand polycrystals

Pham Duc Chinha�

Vietnamese Academy of Science and Technology, Vien Co hoc, 264 Doi Can, Hanoi, Vietnam

�Received 25 July 2006; accepted 3 November 2006; published online 26 January 2007�

The concept of macroscopic properties �conductivity, elasticity,...� of heterogeneous media isreexamined and defined, with the assessment that the effective properties of randomlyinhomogeneous materials �in the large representative volume element limit� generally are notunique, but scatter within some uncertainty limits; hence, the statistical homogeneity, statisticalisotropy, and ergodicity hypotheses often attributed to them may be considered only asapproximations, and random irregular systems may not have definite percolation thresholds. Ourformal bounds on the elastic moduli of random polycrystals are used to derive explicit estimates ofthe uncertainty of the moduli with numerical results for the aggregates of hexagonal and tetragonalcrystals of all classes. The results indicate that the macroscopic moduli of many polycrystallinematerials are determined within just two or three significant digits—similar to the respective resultsfor the conductivity properties. © 2007 American Institute of Physics. �DOI: 10.1063/1.2426378�

I. INTRODUCTION

Most practical materials are inhomogeneous at one ormore scales �look at the polycrystals, cellular solids, porousmedia, colloid, gels, composites,...�. To study the macro-scopic behavior of the materials, one needs the concept ofrepresentative volume element, on which the materials’ ef-fective properties are defined. The macroscopic �effective�properties depend on the material-components’ properties aswell as their often-complex microgeometry. The theory ofeffective properties �or homogenization� started since theclassical works1–3 has been well established in the broadliterature.4–13 Mathematical tools are well developed for pe-riodic structures, but rigorous treatment of the broader classof randomly inhomogeneous systems requires additional as-sumptions such as statistical homogeneity, statistical isot-ropy, and ergodicity hypotheses, or even an equivalence tosome periodic one. All the assumptions are plausible �atleast, they are very good approximations to the real worldobservations�, serve their purposes well, and help us to ob-tain many important results on the effective properties ofheterogeneous media. Still, recent studies14–18 assess thatmacroscopic properties of randomly inhomogeneous materi-als generally may be not unique, but scatter over some un-certainty intervals, though small. This aspect requires us toreconsider the concept of the representative volume element�RVE� and effective properties, and to start with as few as-sumptions as possible to set a framework to study the objec-tive.

There is a large literature on the macroscopic behavior ofvarious random systems, including random networks, perco-lation threshold problems, symmetric cell geometries, ran-dom Voronoi and Delaunay cellular materials, disorderedmatrix mixtures, random polycrystals,..., some results ofwhich related to the subject of this article are discussed. Our

main attention then turns to the random polycrystals, thegrains’ shapes, and crystalline orientations of which are un-correlated. Our formal upper and lower bounds on the elasticmoduli of the aggregates are used to derive explicit estimatesfor the uncertainty of the moduli of some polycrystallinematerials.

II. RVE AND THE EFFECTIVE CONDUCTIVITY

We will examine specifically the conductivity problem�which represents transport properties of inhomogeneousmedia, such as electrical and thermal conductivities, dielec-tric and magnetic permeabilities, diffusion coefficients�,while the whole procedure also applies to the more complexproblem on the elastic constants. Let us consider a body Bmade of an inhomogeneous material. The actions of externalagencies over B create the local gradient intensity field E�x�,

E�x� = − ���x�, x � B , �1�

where ��x� is the potential field �electric potential, tempera-ture,...�. The resulting �electric, thermal,...� flux J�x� understeady-state conditions with no source terms should satisfythe equilibrium equation

� · J�x� = 0, x � B . �2�

The solenoidal flux J and irrotational intensity field Eare related through a constitutive relation J�x�=C�x� ·E�x�,where C�x� is the second order symmetric tensor, which rep-resents the conductivities of the constituent inhomogeneities.Within the framework of continuum physics, Eqs. �1� and �2�applied anywhere in the heterogeneous material imply thatthe potential � and the normal component of the flux J arecontinuous across the inhomogeneities’ interface. We take anin situ �representative� volume element occupying a region Vinside the body B. RVE is often presumed to be sufficientlysmall compared to B and other macroscopic dimensions ofinterest, while being sufficiently large compared to the mi-croscopic sizes of the inhomogeneities. The particular valuesa�Electronic mail: pdchinh@imexh.ac.vn

JOURNAL OF APPLIED PHYSICS 101, 023525 �2007�

0021-8979/2007/101�2�/023525/9/$23.00 © 2007 American Institute of Physics101, 023525-1

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of the in situ fields on V will be designated specifically asE*�x� , �*�x� , J*�x�, while we reserve the usual notationsE�x� , ��x� , J�x� for the general purpose. The volume aver-age on V of any object, e.g., E�x�, is defined as �E�= 1

V�VE�x�dx, where V is also used to represent the volumeof the region V. The in situ apparent conductivities�CEJ , CE , CJ� of the in situ volume element V are expressedvia the constitutive definition as

�J*� = CEJ · �E*� , �3�

and via the energetic definitions as

�E*� · CE · �E*� = �E* · C · E*�

= �E* · J*�

= �J* · C−1 · J*�

= �J*� · �CJ�−1 · �J*� . �4�

Note that for the in situ definitions stated, we use thewords “in situ apparent conductivity” and “in situ volumeelement” instead of “effective conductivity” and “RVE,”which are reserved for the latter use. Generally, CEJ , CE , CJ

differ; however, for those particular fields that satisfy Hill’scondition5

�E* · J*� = �E*� · �J*� , �5�

they are equal and may be referred to as the unique in situapparent conductivity CA,

CA = CEJ = CE = CJ. �6�

The definitions of the apparent �effective� conductivity�CD ,CN� frequently used in the literature are those with ide-alistic Dirichlet uniform boundary conditions on �V,

E0 · CD · E0 = inf����V=E0·x

�E · C · E� , �7�

and Neumann uniform boundary conditions

J0 · �CN�−1 · J0 = inf�J · n��V=�J0 · n��V

�J · C−1 · J� , �8�

where n is the unit vector normal to the boundary �V of V;E0 and J0 are constant vectors; trial fields E in Eq. �7� and Jin Eq. �8� are required to satisfy Eqs. �1� and �2�, respec-tively. Optimal solutions of both problems �7� and �8� satisfyHill’s condition for every one to yield a unique apparentconductivity from constitutive �3� and energetic �4� view-points. It has been established19 that CD�CN �in the sensethat E0 ·CD ·E0�E0 ·CN ·E0, ∀E0�. Moreover, Hill’s condi-tion holds also for the problems on V with orthogonal uni-form mixed boundary conditions,20 which form the base forusual experimental measurements of the macroscopic prop-erties, and the resulting apparent conductivities fall betweenCN and CD. The question is how our in situ apparent con-ductivities CEJ , CE , CJ from Eqs. �3� and �4�, which reportthe material’s real behavior in a particular problem, are com-pared with the uniform boundary ones CD, CN from Eqs. �7�and �8� often used as the benchmarks for the effective con-ductivity without solid mathematical justification?

Other definitions of the apparent conductivities �CE , CJ�used in the literature include that with average intensity fieldconstraint

E0 · CE · E0 = inf�E�=E0

�E · C · E� , �9�

and that with average flux constraint

J0 · �CJ�−1 · J0 = inf�J�=J0

�J · C−1 · J� , �10�

where E0 and J0 are constant vectors; Trial fields E in Eq.�9� and J in Eq. �10� are irrotational and solenoidal, respec-tively. Comparing Eqs. �4� and �9�, and taking E0= �E*�, onesees that CE�CE, since the in situ field E* from Eq. �4� isjust a trial field for Eq. �9�. One may check that the optimalsolution J =C ·E of Eq. �9� satisfies some uniform boundary

condition �J ·n��V= �J0 ·n��V �where J0=const�, hence, �J�= J0=CE ·E0. If we take J0= J0, then the solution of Eq. �8�coincides with that of Eq. �9� and we have CN=CE�CE.Alternatively, we can also prove CN�CE directly, using theprinciple of minimum of the potential energy applied to theproblem Eq. �8�,

UE�E� =1

2

V

E · C · Edx − �V

J0 · n�dsx . �11�

Indeed, putting the in situ field �* , E* from Eq. �4� as atrial field into Eq. �11�, we obtain

UE�E*� =1

2

V

E* · C · E*dx − V

� · �J0�*�dx

=V

2�E�� · CE · �E�� − VJ0 · �E�� . �12�

On the other hand, with the exact solution E , J of Eq.�8� �which naturally satisfies Eqs. �1� and �2��, from Eq. �11�one obtains

UE�E� =1

2

V

E · C · Edx − �V

J0 · n�dsx

= −1

2

V

E · C · Edx = −1

2

V

J · C−1 · Jdx

= −V

2J0 · �CN�−1 · J0. �13�

Then, taking J0=CN · �E*�, from the statement of theminimum potential energy principle that UE�E��UE�E*�,we get CN�CE, which is required.

Similarly, comparing Eqs. �4� and �10�, and taking J0

= �J*�, one sees that CJ�CJ, since J* from Eq. �4� is just atrial field for Eq. �10�. The optimal solution J , E , � fromEq. �10� corresponds to some uniform boundary condition

���V= E0 ·x �where E0=const�, hence, J0=CJ · E0. If we take

E0= E0, then the solution of Eq. �7� coincides with that ofEq. �10� and we have CD=CJ�CJ. Alternatively, we canalso prove CD�CJ directly, using the principle of minimumof the complementary energy applied to the problem Eq. �7�,

023525-2 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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UJ�J� =1

2

V

J · C−1 · Jdx − �V

�J · n��E0 · x�dsx . �14�

For that, putting the in situ field J* from Eq. �4� as a trialfield into Eq. �14�, we obtain

UJ�J*� =1

2

V

J* · C−1 · J*dx − V

� · �J*�E0 · x��dx

=1

2

V

J* · C−1 · J*dx

− V

�E0 · J* + E0 · x � · J*�dx

=V

2�J*� · �CJ�−1 · �J*� − VE0 · �J*� , �15�

since J* satisfies Eq. �2�. On the other hand, with the exactsolution E , J of Eq. �7� �which naturally satisfies Eqs. �1�and �2��, from Eq. �14� one obtains

UJ�J� =1

2

V

J · C−1 · Jdx − �V

�J · n��E0 · x�dsx

= −1

2

V

J · C−1 · Jdx

= −1

2

V

E · C · Edx

= −V

2E0 · CD · E0. �16�

Then, taking �J*�=CJ ·E0, from the statement of theminimum complementary energy principle that UJ�J��UJ�J*�, we get CD�CJ, which is required. Summing theresults we have partial estimations

CN�=CE� � CE, CD�=CJ� � CJ. �17�

The question remaining is how CE compared with CD

and CJ with CN, and CEJ with both CD and CN? The bound-ary values �on �V� of the in situ fields �* , J* generally maynot fall into the category of uniform ones. One can take thevolume element V as a selection of exotic islands from thoseparts of the material space where the in situ fields E* , J* arestrong in the high-conducting phase and weak in the low-conducting phase �or vice versa� to make the apparent con-ductivities as unlikely high �low� as possible. One may alsoconnect those islands via thin tunnels of negligible materialvolume to make V a simply connected region. Even if V istaken as a simply connected region with all the sizes beinglarge compared to the microscopic dimensions, one still can-not control the microgeometry and the boundary conditionsof V. Though the corresponding fields should converge to thesmooth ones of Eqs. �7� and �8� types over the boundary ofV, when V is small compared with B and other macroscopicdimensions and if the real heterogeneous body B should bereplaced by some imagined homogeneous one with an effec-tive property, the real in situ fields E* , J* can fluctuatestrongly on �V near the sharp corners and edges of the con-

stituent inhomogeneities. Those strong fluctuations on theboundary �V can also spread their influence deep inside Valong crack-like structures started from �V and filled withsome component material of extreme conductivity. It is hardto count all possible odd situations to classify the fieldsE* , J* qualitatively and quantitatively—the task may alsomake the approach complicated and impractical. Our practi-cal way is to formulate a concept—simple but reasonable,broad, and effective enough to describe our real world obser-vations. Whenever one encounters a new awkward situationof practical significance not covered by our old concept, oneshould come back to reexamine the assumptions and to refinethe approach accordingly. Here, in particular, we choosethose simply connected large volume elements V with re-spective in situ fields E* , J* that satisfy Hill’s consistencycondition �5� to represent the macroscopic behavior of theheterogeneous materials. Because of Eq. �5�, the in situ ap-parent conductivity CA is defined unanimously as stated inEq. �6�. That is plausible since we do not expect the apparent�effective� conductivities to be different in the constitutive�3� and energetic �4� expressions—at odds with our percep-tion that they should be the same for us to tackle the effectivemedium with the usual apparatus of continuum physics. Ouradmissible set of in situ fields, which are required to satisfyHill’s condition, is broad enough to include all the solutionsof the problems �7�–�10� as well as those with orthogonaluniform mixed boundary conditions as special cases. More-over, Eqs. �6� and �17� are combined to yield

CN � CA � CD, �18�

which means that our in situ apparent conductivity CA is wellcontrolled by the benchmark uniform boundary ones CN andCD. As the size of the volume element V increases,CN , CA , CD are expected to converge toward each other�but that does not necessary mean that they should also con-verge to those of the other or larger volume elements!�. Spe-cifically, if CN and CD �hence, CA� of all admissible volumeelements V, as the sizes of them increase to infinity, shouldconverge to some unique value, then that value would becalled the effective conductivity of the composite, and a suf-ficiently large V from the set may be called a RVE of thematerial. More generally, if CN , CA , CD should convergejust to fall within some definite uncertainty limits, then anyvalue within the limits would be called a (possible) effectiveconductivity �Ce� of the randomly inhomogeneous medium,and the macroscopic characterization of the material requiresthe determination of the macroscopic uncertainty, i.e., thewhole uncertainty interval, in the sense

E0 · Cl · E0 � E0 · Ce · E0 � E0 · Cu · E0, ∀ E0. �19�

Even if Cl and Cu should be isotropic �Cl=ClI, Cu

=CuI, I is the unit tensor�, as for statistically isotropic com-posites, Ce may be accidently slightly anisotropic as allowedby the bounds. Statistical homogeneity and isotropy hypoth-eses should be understood in the broader sense that allowsfor the possible small uncertainty �19� specific for every par-ticular random system. Ergodic assumption, equating the en-semble and volume averages, appears not exact, e.g., if theeffective conductivity �defined over the volume average of

023525-3 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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the large volume elements� should scatter uniformly over�Cl , Cu�, or distribute symmetrically about the middle pointof the interval, the ensemble average �for those volume ele-ments� appears to converge to that middle point. Still, thepractical spacial “randomness,” which avoids any ordernessof the inhomogeneities and makes the material macroscopicbehavior so close to homogeneous and isotropic, is muchmore restrictive than the theoretical “arbitrariness,” whichinvolves all possible extreme locally ordered configurationsand can make the uncertainty intervals much larger.15,21,22

Here we do not consider the effect of finite size RVE relativeto the sizes of the inhomogeneities13,19,23 and possible imper-fect contact between the inhomogeneities, which clearly con-tribute to the larger uncertainties.

Beside the earlier definitions, the effective properties arealso defined on a periodic cell with periodic boundary con-ditions and the constraint on the average field �E�=E0 �or theconstraint on the average flux �J�=J0�. If one compares therespective energetic definitions of the effective properties de-

signed here as CE �and CJ� with those of Eq. �9� �and Eq.

�10��, it is immediately clear that CE�CE�=CN�, and CJ

�CJ�=CD�.

III. DISCUSSION

Most technical materials have irregular random structureat some levels �e.g., the polycrystalline level�. The result isthat their macroscopic properties could only be determinedwithin limited accuracy and the partial differential equationsdescribing their macroscopic behavior could not yield thesolution more accurate than that allowed by the uncertaintyof the equations’ coefficients �the effective properties�. Themacroscopic conductivity of random irregular mixtures in-volving superconductor or insulator, generally, may fluctuatestrongly without definite percolation thresholds. One canimagine theoretically all possible composites with differentdegrees of randomness. However, only those theoretical es-timates or computer-generated models that approximate theobserved scatter ranges for the macroscopic properties of thenaturally occurring randomly inhomogeneous materials inthe real world should have major practical significance.

Many works are devoted to the practical class of randompolycrystals.16,18,24–29 In Refs. 16 and 18 we predict that thescatter deviation S= �Cu−Cl� / �Cu+Cl� of the macroscopicconductivity and elastic moduli of many polycrystalline ma-terials may range from a few thousandths to a few hun-dredths �see also the additional results in the next section�. Itis imperative to make accurate experimental measurementsto check the assessment. Note that the measured macroscopicproperties of practical polycrystalline materials tabulated intechnical literature are also given with only two or threesignificant digits. Computer simulations to assess the pos-sible uncertainty should be easier to perform on symmetriccell materials,30 such as those based on the random Voronoior Delaunay tessellations of the material space. For practicalnonsymmetric randomly inhomogeneous materials, accurateexperiments should give some information on the uncertaintysizes of the macroscopic properties. Numerical simulationsfor them are difficult because it is hard to characterize their

particular microgeometry via a few simple rules, while theresults for some idealistic systems such as random overlap-ping or hard monosized spheres,13 or polydispersed multi-coated spheres31 indicate that their effective properties con-verge to unique values. The random networks based onvarious regular skeletons with the bonds or sites being givendifferent conductivity values randomly appear to haveunique macroscopic conductivities and even well definedpercolation thresholds for those mixtures involving compo-nents of extreme conductivities, with a very high degree ofaccuracy �up to six significant digits�.13,32,33 However, ran-dom networks based on random irregular skeletons may nothave unique conductivities and percolation thresholds. Thenumerical simulations for the two-dimensional random cellVoronoi models reveal that the effective elastic moduli scat-ter deviations may be as high as 5%−10%—qualitatively inagreement with experimental observations on practical cellu-lar materials.14,17 More work is needed in this interestingdirection. In the mentioned references, the cell walls of thecellular materials are approximated by bending elements,which is realistic for the real world application purposes. Fortheoretical investigations, one may start with simpler prob-lems on elasticity of two-dimensional random Delaunay pin-connected truss systems and on conductivity of two-dimensional random Voronoi and Delaunay networks. Highaccuracy computer simulations of three-dimensional randomnetworks and cellular materials should be much more sophis-ticated. Note that, in the framework set earlier, the uncer-tainty should be determined in the sense of Eq. �19�, and theeffective property Ce should not be treated as the isotropictensor even for statistically isotropic media, as already ex-plained in the context.

The uniqueness of the effective properties of randomlyinhomogeneous materials is too idealistic to be true and adegree of uncertainty is more realistic. It is common forpracticing engineers to see the scatter in the strength charac-teristics of materials. The same can be said about the con-ductivity and elastic properties, the difference may be onlythat the sizes of scatter appear smaller there. Even in themicroscopic world, where the laws of physics are very accu-rate, the accuracy is still limited by Heisenberg’s uncertaintyprinciple. In the macroscopic world, naturally we also facevarious kinds of macroscopic uncertainties because of thecomplexity of interactions within random systems. Observa-tions indicate that the matter and radiation distributions inour universe appear macroscopically homogeneous and iso-tropic and the cosmic microwave background radiation isextremely uniform across the sky at the temperature of about2.725 K, but with small temperature fluctuations on the orderof one part in 105.

Still, our notion of RVE and effective properties does notappear to be a closed concept. Our admissible in situ fieldsrequired to satisfy Hill’s condition �5� do not cover all thepossible ones. How far can the in situ apparent conductivitiesCEJ , CE , CJ released from the restriction differ from eachother and from the bench mark uniform boundary onesCN , CD? �Generally we have just the partial estimations �17�and the obvious relation CE=CEJ · �CJ�−1 ·CEJ.� How couldthat affect the macroscopic description of the materials and

023525-4 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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add to the macroscopic uncertainty? We have already knownsome composites with unusual properties. For example,though the isotropic composites with negative Poisson’s ra-tios are rare,34 theoretically they may have Poisson’s ratiosapproaching −1.35 Another question is, once the effectiveproperties should scatter, are they distributed uniformlywithin the interval, or concentrated toward certain middlevalues? Perhaps, we should wait until experiments and nu-merical simulations point to something clearly. Experimentalvalues of the macroscopic elastic moduli of about 60 poly-crystalline materials have been collected to check if themoduli should be concentrated toward the self-consistentvalues.27 Instead, they reveal that the moduli are distributedalmost uniformly over an interval comparable to the thirdorder �in the series expansion of the moduli in the orders ofcrystal anisotropy contrast� bounds—the limits qualitativelyagree with the predictions of our theoretical results.16,18

IV. RANDOM POLYCRYSTALS

Macroscopic properties of random polycrystals are thesubject of many studies.16,18,24–29 In polycrystal forming pro-cesses, the constituent crystals are often born independentlyat random places in a bath and grow until they meet eachother. The kinematic constraints, boundary interactions, andinertia would not allow the crystals to turn over to be fit as aconfiguration with minimal surface energy �such as that of asingle big crystal�, but leave the primary crystalline orienta-tions of the grains intact and accommodate them with thehelp of various two-dimensional defects �such as dislocationwalls...� on their common boundary. Hence, it appears thatthe shape and crystalline orientations of the constituentgrains in a random aggregate are uncorrelated. A computer-generated model of the random polycrystal may be made asa Voronoi tessellation of the space from a set of centralpoints thrown randomly into it, with the cells being assignedthe crystalline orientations randomly. Because of the ran-domness, the practical random polycrystalline materials ap-pear macroscopically homogeneous and isotropic and havesuch definite macroscopic properties that can be measuredand tabulated for applications, but with possible small uncer-tainty. Here the randomness implies certain geometric re-strictions of a statistical character. In particular, it has beenassumed that certain tensors characterizing the microgeom-etry of the random polycrystals are isotropic �statistical isot-ropy hypothesis� and an interchange of the places betweenany two sets of crystals of two different crystalline orienta-tions should not alter the overall characteristics of the aggre-gate �statistical symmetry hypothesis�; but otherwise the par-ticular shapes and arrangement of the grains in the irregularaggregate are not specified �see the Refs. 16 and 18 for theparticular mathematical expressions of the assumptions�. Ourformal upper and lower bounds on the effectiveproperties16,18 have been constructed from the minimum en-ergy and minimum complementary energy principles �of theforms �9� and �10�� and based on those statistical isotropyand symmetry hypotheses. Because third or higher orderbounds on the macroscopic properties would require addi-tional shape and other arrangement information about the

grains within an aggregate29—the information is unlikely tobe definite for the real-world irregular polycrystals, our gen-eral partly third order bounds derived in Refs. 16 and 18might be close to the best possible ones for the random poly-crystals. In Ref. 18 explicit estimates with numerical resultsfor the elasticity of the simplest cubic polycrystals have beenobtained. Here we apply the formal bounds of Ref. 18 toderive respective explicit estimates for the aggregates of cer-tain lower symmetry crystals.

The elastic tensor C of tetragonal crystals of classes4 , 4 , 4 /m �designated here as T7� in its base crystal refer-ence is characterized by seven elastic constants, which in thetwo-index notation are given as C11, C12, C13, C33, C44,C66, C16. The correspondence between the usual fourth-rankelasticity tensor components Cijkl �i , j , k , l=1, 2 , 3� andthose in the two-index notation C�� �� ,�=1, 2 , ... , 6� is

C11 = C1111 = C2222, C33 = C3333,

C44 = C1313 = C2323, C13 = C1133 = C2233, �20�

C12 = C1122, C66 = C1212, C16 = C1112 = − C2212,

while other independent elastic constants are equal to zero.The elastic tensor of tetragonal crystals of classes

4 /mmm , 42m , 4mm , 422 �designated here as T6� involvessix elastic constants C11, C12, C13, C33, C44, C66 and maybe considered as a specific case of Eq. �20� with C16=0.

The elasticity of hexagonal crystals �designated here asH6� expressed through five elastic constantsC11, C12, C13, C33, C44 can also be considered as a specificcase of Eq. �20� with C16=0 and C66= 1

2 �C11−C12�.At last, the elasticity of cubic crystals �designated here

as C3� determined by just three elastic constantsC11, C12, C44 is a specific case of Eq. �20� with C16=0,C12=C13, C44=C66, and C11=C33.

Hence, Eq. �20� covers the elasticity of all the classes ofcrystals mentioned. The property functions playing centralpart in our estimates have particular expressions

Pk�C, k0, �0� = ��C + C*�iij j−1 �−1 − k�

=�C11

+� + C12+��C33

+� − 2�C13+��2

C11+� + C12

+� − 4C13+� + 2C33

+� − k�,

P��C, k0, �0� = 2

5�C + C*�ijij

−1 −2

15�C + C*�iij j

−1 �−1

− ��

=15

2 3

C44+� +

2C11+� + 2C12

+� + 4C13+� + C33

+�

C33+��C11

+� + C12+�� − 2�C13

+��2

+3C66

+� + 12 �C11

+� − C12+��

C66+��C11

+� − C12+�� − 2C16

2 �−1

− ��, �21�

where

C11+� = C11 + k� +

4

3��, C33

+� = C33 + k� +4

3��,

C44+� = C44 + ��,

023525-5 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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C12+� = C12 + k� −

2

3��, C13

+� = C13 + k� −2

3��,

C66+� = C66 + ��,

C� = T�k�,���, k� =4

3�0, �� = �0

9k0 + 8�0

6k0 + 12�0, �22�

T�k ,�� is the isotropic fourth-rank tensor function with com-ponents

Tijkl�k,�� = k�ij�kl + ���ik� jl + �il� jk −2

3�ij�kl , �23�

and �ij is the usual Kronecker symbol; conventional summa-tion on repeating Latin indices is assumed. Not only our

general bounds given below, but also Hashin–Shtrikmanbounds, the bounds for spherical cell polycrystals, and theself-consistent approximation can be expressed comprehen-sively through these property functions.18,36

The formal bounds on the effective moduli Ce of therandom polycrystals are given as18

�0:T�kl,�l�:�0 � �0:Ce:�0 � �0:T�ku,�u�:�0, ∀ �0.

�24�

One should keep in mind that generally Ce may not bepresented simply as the isotropic tensor T�ke ,�e�, even forthe statistically isotropic aggregate, because the effectivemoduli Ce is not unique and it can be accidentally slightlyanisotropic as allowed by the bounds. The upper bound ku

from Eq. �24� has the expression

ku = infk0, �0

Pk�C,k0,�0��k0 � kV, �0 � �V, UKg1 � 0, UKf1 +6

7UKg1 � 0� , �25�

where kV and �V are Voigt averages

UKf1 = Ciikk−0 DjjppDllqq�−

22

315 + Ciikk

−0 DjlppDjlqq22

315+ Cijij

−0 DkkppDllqq22

315+ Cijij

−0 DklppDklqq�−2

63

+ Ciikl−0 DklppDjjqq� 4

105−

42

315 + Ciikl

−0 DkjppDljqq�162

315−

2

35 + Cijkj

−0 DikppDllqq�162

315−

2

35

+ Cijkj−0 DilppDklqq� 1

10−

42

315−

4

35 + Cijkl

−0 DijppDklqq� 8

105−

1

15−

132

315 + Cijkl

−0 DikppDjlqq� 1

10−

4

35+

162

315 ,

UKg1 = Ciikk−0 DjjppDllqq

2

945+ Ciikk

−0 DjlppDjlqq�−2

378 + Cijij

−0 DkkppDllqq�−2

378 + Cijij

−0 DklppDklqq2

54+ Ciikl

−0 DklppDjjqq�−2

945

+ Ciikl−0 DkjppDljqq�−

42

189 + Cijkj

−0 DikppDllqq�−42

189 + Cijkj

−0 DilppDklqq102

189

+ Cijkl−0 DijppDklqq

432

1890+ Cijkl

−0 DikppDjlqq�−42

189 , �26�

Dijkl =1

2��ik� jl + �il� jk� − �kl�k+� −

2

3�+� �C + C*�ijnn

−1 − 2�+��C + C*�ijkl−1 ,

k+� = Pk�C, k0 �0� + k�, �+� = P��C, k0 �0� + ��, �27�

C−0 = C − T�k0,�0�, =3k0 + �0

3k0 + 4�0. �28�

Note a typographical error in the indices of the last term of UKf1 given in Ref. 18, which should be the same as therespective ones of UKg1. Following Ref. 36, we use short hand notations C��

−0 and D�� instead of Cijkl−0 and Dijkl to represent the

results for our tetragonal and hexagonal crystals. For the mentioned crystals, we have simplifying relations ��,�=13 D��=0,

D11+D12+D13=− 12 �D33+2D31�, D11+D21+D31=− 1

2 �D33+2D13�, while D13�D31. Then, for our particular crystals �20�, theexpressions �22�–�27� are simplified to

UKg1 = UKg1

2�D33 + 2D31�22, �29�

UKg = C11−0 9

70+ C33

−0 4

35+ C44

−04

7+ C66

−01

7− C12

−0 1

70− C13

−0 8

35, �30�

023525-6 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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UKf1 +6

7UKg1 = UKfg

1

2�D33 + 2D31�2, �31�

UKfg = �C11−0 + C12

−0��262

735+

2

105−

1

15

+ C33−0�1212

735−

44

105+

4

15 + C13

−0�442

147−

62

105+

4

15

+ �2C11−0 + C44

−0 + 2C66−0�� 82

245−

4

35+

1

10 . �32�

With Eqs. �29�–�32�, the upper bound �6� is simplified to

ku = infk0, �0

�Pk�C, k0, �0��k0 � kV, �0 � �V, UKg

� 0, UKfg � 0� . �33�

The expression of �u, which involves D��, is more com-plicated �particular expressions of D�� , C��

−0 and Voigt aver-ages kV , �V for our crystals �20� are given in Ref. 36�

�u = infk0,�0

�P��C, k0, �0��k0 � kV, �0 � �V, UMg1

� 0, UMfg � 0� , �34�

where

UMg1 = UMg − UKg2

6�D33 − 2D31�2,

UMfg = UMf1 +6

7UMg1

= UMf +6

7UMg − UKfg

1

6�D33 + 2D31

2 �; �35�

UKg and UKfg are from Eqs. �30� and �32�; UMf and UMg

have the forms similar to those of UKf1 from Eq. �26� andUKg1 from Eq. �27�, respectively, with all tensor-componentsDijpp as well as Dijqq being substituted by Dijpq, and

Ciikk−0 DjjpqDllpq = �2C11

−0 + 2C12−0 + 4C13

−0 + C33−0�

3

2�D33 + 2D13�2,

Ciikk−0 DjlpqDjlpq = �2C11

−0 + 2C12−0 + 4C13

−0 + C33−0��2D11

2 + 2D122 + 4D66

2 + D332 + 2D13

2 + 2D312 + 8D44

2 + 8D162 � ,

Cijij−0 DkkpqDllpq = �2C11

−0 + 2C66−0 + C33

−0 + 4C44−0�

3

2�D33 + 2D13�2,

Cijij−0 DklpqDklpq = �2C11

−0 + 2C66−0 + C33

−0 + 4C44−0��2D11

2 + 2D122 + 4D66

2 + D332 + 2D13

2 + 2D312 + 8D44

2 + 8D162 � ,

Ciikl−0 DklpqDjjpq = ��C11

−0 + C12−0 + C13

−0��2D13 − D11 − D12� + �C33−0 + 2C13

−0��D33 − D31���D33 + 2D13� ,

�36�Ciikl

−0 DkjpqDljpq = �C11−0 + C12

−0 + C13−0�2�D11

2 + D122 + 2D66

2 + D132 + 2D44

2 + 4D162 � + �C33

−0 + 2C13−0��D33

2 + 2D312 + 4D44

2 � ,

Cijkj−0 DikpqDllpq = ��C11

−0 + C66−0 + C44

−0��2D13 − D11 − D12� + �C33−0 + 2C44

−0��D33 − D31�� + �D33 + 2D13� ,

Cijkj−0 DilpqDklpq = �C11

−0 + C66−0 + C44

−0�2�D112 + D12

2 + 2D662 + D13

2 + 2D442 + 4D16

2 � + �C33−0 + 2C44

−0��D332 + 2D31

2 + 4D442 � ,

Cijkl−0 DijpqDklpq = 2C11

−0�D112 + D12

2 + D132 + 2D16

2 � + C33−0�D33

2 + 2D312 � + 2C12

−0�D132 − 2D16

2 + 2D11D12�

+ 16C44−0D44

2 + 8C66−0�D66

2 + D162 � + 4C13

−0�D11D31 + D12D31 + D13D33� + 8C16−0D16�D11 − D12� ,

Cijkl−0 DikpqDjlpq = 2C11

−0�D112 + D12

2 + D132 + 2D16

2 � + C33−0�D33

2 + 2D312 � + 4C12

−0�D662 + D16

2 � + 2C66−0�2D11D12 + D13

2 + 2D662 �

+ 8C13−0D44

2 + 4C44−0�D11D31 + D12D31 + D13D33 + 2D44

2 � + 8C16−0D16�D11 − D12 + 2D66� .

Similarly the lower bounds from Eq. �24� can be derived

kl = supk0,�0

�Pk�C, k0, �0��k0−1 � kR

−1, �0−1 � �R

−1, UKg � 0, UKfg � 0� ,

�l = supk0,�0

�P��C, k0, �0��k0−1 � kR

−1, �0−1 � �R

−1, UMg1 � 0, UMfg � 0� , �37�

023525-7 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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where UKg , UKfg , UMg1 , UMfg have the similar forms asthose of UKg , UKfg, UMg1 , UMfg from Eqs. �30�, �32�, and�35�, with C��

−0 and D�� taking the places of C��−0 and D��,

respectively. Particular expressions of C��−0 , D�� and Reuss

averages kR , �R for the crystals considered here are pre-sented in Ref. 36. The simple bounds for the specific spheri-cal cell polycrystals, which approximate practical equiaxedparticulate aggregates, remain identical to those given in Ref.36.

Numerical results of the estimates �22�, �33�, �34�, and�37� for the effective elastic moduli of the random aggregatesof various hexagonal �H6� and tetragonal �T6 and T7� crys-tals, the elastic constants C�� of which are taken from Ref.37 and collected also in Ref. 36, are given in Table I. Thebounded intervals are considerably larger than the respectiveones of Ref. 36, because the latter bounds are based on asimplifying assumption on the form of certain isotropiceight-rank tensors, which implies a restriction on the aggre-gate microgeometry—as explained in Ref. 18. Comparisonswith the results of Ref. 36 also indicate that our partly thirdorder bounds �24� are considerably tighter than Hashin–

Shtrikman second order bounds and the first order Hillbounds, but less tight than the simple third order bounds forthe specific spherical cell polycrystals—intended for the par-ticulate aggregates.

Some experimental values of the Young modulus E �E=9k� / �3k+��� and shear modulus � of some technicalpolycrystalline materials taken from different sources in Ref.38, which are in agreement with the bounds calculated usingthe data of Ref. 37, are illustrated in Table II. However,much more effort is needed to make accurate measurementsof the properties of the base crystals as well as polycrystal-line aggregates, while ensuring their random nature withoutany texture. That would make it possible for quantitativecomparisons of the observed scatter ranges of the macro-scopic properties and the bounds derived theoretically. Themeasured macroscopic values for a set of samples should allbe considered to define the scatter interval, not to be aver-aged as is usually done. It is hopeful that the scatter rangesfor the macroscopic properties of other practical random sys-tems can also be constructed theoretically, experimentally,

TABLE I. The upper and lower estimates for the macroscopic elastic moduli of random hexagonal �H6� andtetragonal �T6, T7� polycrystals: kl , ku—the estimates for the bulk modulus; �l , �u—those for the shearmodulus �all in GPa�; Sk= �ku−kl� / �ku+kl�, S�= ��u−�l� / ��u+�l�—the respective scatter measures.

Crystal Class kl ku Sk�%� �l �u S��%�CaWO4 T7 76.44 76.47 0.017 37.35 37.46 0.15PbMoO4 72.34 72.42 0.057 24.59 24.81 0.45

NbO3 235.8 235.8 0.0027 97.60 97.99 0.19C�CH3OH�4 15.23 16.60 4.3 3.191 3.638 6.5

AgCIO3 35.31 35.33 0.019 8.271 8.277 0.039SrMoO4 69.86 69.88 0.016 34.56 34.66 0.15C14H8O4 4.114 4.199 1.0 8.413 8.429 0.098CaMoO4 80.92 80.94 0.010 39.85 39.90 0.068BaTiO3 T6 176.5 179.3 0.79 52.86 54.12 1.2ZrSiO4 19.74 19.82 0.19 19.71 19.87 0.41

Sn 60.63 60.63 0.0007 18.35 18.61 0.72TiO2 214.7 215.4 0.16 114.7 116.1 0.62

In 41.59 41.60 0.0020 4.758 4.990 2.4Hg2Cl2 18.77 20.59 4.16 6.132 8.057 13SnO2 212.2 212.9 0.18 101.6 103.3 0.87Urea 14.47 18.58 12 3.227 4.100 12

Sr4LiKNb10O30 118.5 119.5 0.42 61.51 61.57 0.043C�CH2OH�4 1.753 1.775 0.60 3.670 3.677 0.093KH2AsO4 13.53 13.54 0.053 14.30 14.56 0.87KH2PO4 32.68 32.68 0.0054 14.66 15.00 1.1

Cd H6 53.84 54.84 0.92 23.96 24.39 0.88Zn 69.15 71.15 1.4 39.15 40.79 2.0Co 190.4 190.4 0.0002 81.93 82.01 0.050

TiB2 418.3 422.3 0.48 168.1 169.1 0.31LiIO3 33.19 33.30 0.16 22.34 22.37 0.050GaSe 33.33 34.85 2.2 20.18 21.09 2.2MnAs 25.17 25.61 0.86 25.32 25.60 0.55ZnO 66.83 68.80 1.4 40.55 42.29 2.1Tl 35.62 35.63 0.017 5.358 5.488 1.2

CeF3 112.9 112.9 0.0044 45.29 45.42 0.14CuCl 42.00 42.00 0 7.440 7.477 0.25

Hf 1.023 1.023 0.0012 7.343 7.679 2.2Ho 2.711 2.711 0.0002 15.45 16.18 2.3

023525-8 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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and numerically. Numerical simulations are simpler �but stillvery hard� to be performed on elasticity of two-dimensionalrandom polycrystals for a comparison with the theoreticalresults of Ref. 39.

ACKNOWLEDGMENT

The work is supported by the Science Foundation ofVietnam.

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TABLE II. The estimates and experimental values for the macroscopic elastic moduli of some hexagonal �H6�,tetragonal �T6� and cubic �C3� polycrystalline materials: El , Eu—the estimates for the Young modulus;�l , �u—those for the shear modulus; Eexp , �exp—the experimental values �all in GPa�.

Crystal Class El Eexp Eu �l �exp �u

Fe C3 210.59 211.41, 212.0 212.24 81.66 82.0, 82.36, 82.38 82.41Cu 127.51 129.45, 129.8 130.22 47.38 48.33, 48.50 48.50Ag 81.12 81.1 82.42 29.65 29.6, 30.0 30.18Ge 132.24 132.3 132.36 54.82 54.88Nb 104.8 105.2 37.53 37.5, 37.7 37.66Zn H6 98.80 101.9 102.7 39.15 40.8Cd 62.60 63.1 63.73 23.96 24.03, 24.12, 24.13 24.39In T6 13.75 13.9 14.39 4.758 4.8 4.990

023525-9 Pham Duc Chinh J. Appl. Phys. 101, 023525 �2007�

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