a force-field based analysis of the deformation mechanism in α … · 2017-11-05 · a...

A force-field based analysisof the deformation mechanismin a-cristobalite

Ruben Gatt*,1, Luke Mizzi1, Keith M. Azzopardi1, and Joseph N. Grima1,2

1Metamaterials Unit, Faculty of Science, University of Malta, Msida MSD2080, Malta2 Department of Chemistry, Faculty of Science, University of Malta, Msida MSD2080, Malta

Received 9 March 2015, revised 13 April 2015, accepted 17 April 2015Published online 17 June 2015

Keywords auxetic, a-cristobalite, force-field based simulations, mechanical properties

* Corresponding author: e-mail [email protected], Phone: þ356 2340 2840

a-cristobalite is a metastable polymorph of silica which has theremarkable property of exhibiting a negative Poisson’s ratio(auxetic). In this study, the mechanical properties anddeformation mechanisms of this system were investigatedthrough a force-field based approach. Besides simulations onthe actual structure of a-cristobalite, a number of simulationswhere the rigidity of SiO2 tetrahedra was increased, were alsoconducted. This was done in order to obtain a clearer picture of

the role which these tetrahedra play in determining themechanical properties of this system. These systems wereloaded in an off-axis direction in a number of planes about the[001] direction and the deformations observed were comparedwith those of type II rotating rectangle systems. It wasdiscovered that the mechanical properties of a-cristobalite canbe accurately described through the semi-rigid version of thismodel, even for cases where the tetrahedra were fully rigid.

� 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction The metastable polymorph of silica,a-cristobalite, has received a significant amount of interestin recent years due to its auxetic behaviour [1–10]. The term‘auxetic’ is used to describe materials with a negativePoisson’s ratio [11–14], i.e., systems which expand laterallyon application of a uniaxial load. It is generally accepted thatthis property arises primarily from the specific geometry ofthe system along with its deformation mechanism. Thusauxeticity is thought to be scale independent, as observedfrom a variety of macro-scale [15–35], micro-scale [36–39]and nano-scale [40–48] auxetic systems.

One popular approach employed by researchers for thedesign of novel auxetic systems is to study existent realsystems known to have a negative Poisson’s ratio and thenextrapolate their deformation mechanisms to design newmetamaterials with similar properties at similar or differentscales. a-Cristobalite is of particular interest in this regardsince both experimental and molecular modelling techni-ques show that it exhibits auxeticity in multiple on-axis andoff-axis planes [1–10] as well as in its polycrystallineaggregate form [1].

Numerous attempts have been made to explain thisphenomenon in a-cristobalite in terms of its molecular level

deformations. In particular, Alderson et al. have proposed amechanism based on concurrent rotations and dilations ofthe SiO4 tetrahedra which can successfully explain the on-axis Poisson’s ratios of these systems [4–7]. Grima and co-authors [9, 10] have also proposed an alternative hypothesisin which they explained the auxetic behaviour in the (100)and (010) planes of a-cristobalite in terms of a rotatingrectangles model. The rectangles in this model represent thetwo-dimensional projection of the three-dimensional frame-work in these two planes [9, 10]. Even though these modelselucidate the types of deformations that are taking place atthe molecular level, there is still a need for further studies inorder to obtain a more comprehensive picture of thebehaviour of a-cristobalite. In particular, it should be notedthat Grima’s ‘rotating rectangles’ model [9], which wasdesigned to explain the auxetic behaviour in the (100) and(010) planes, needs to be analysed in more detail withrespect to off-axis deformations since auxeticity is notlimited to these two planes, but is in fact found in any planeabout the [001] direction, with maximum auxeticityoccurring at 458 to the [001] direction.

In view of this, the current work provides a morecomprehensive picture of the behaviour of a-cristobalite by

Phys. Status Solidi B 252, No. 7, 1479–1485 (2015) / DOI 10.1002/pssb.201552133

basic solid state physics

statu

s

soli

di

www.pss-b.comph

ysi

ca

� 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

investigating the deformations that occur when it is loadedin the off-axis planes of maximum auxeticity. In addition, anattempt is also made to explain these deformations in termsof the rotating rectangles model, whilst relating its merits tothe more traditional model based on concurrent rotationsand dilations of the SiO4 tetrahedra.

2 Method All simulations were performed using theCerius2 V4.1 molecular modelling package. The crystalframework was aligned such that the [001] direction wasparallel to the z-global axis while the [010] direction wasoriented in the (100) plane. These alignments were set asconstraints during all simulations. For each force-field used,geometry optimisations were performed to the defaultCerius2 high convergence criteria. To minimise symmetryconstraints, the space group of the system was set to P1. Theelastic constants were calculated using the second derivativeof the energy expressions.

2.1 Simulations of the deformations for loadingin the non-major planes Force-field based energyexpressions were first set up using the Compass and CVFF300 force-fields modified with all non-bond terms beingsummed up using the Ewald summation technique [49].These force-fields were chosen since a previous study hasshown that these force-fields can accurately reproduce theexperimentally measured mechanical properties of a-cristobalite [9].

The Poisson’s ratios in the (100) plane and other planeswhich pass through the [001] direction were then calculatedusing standard axis transformation techniques [50]. Inaddition, a series of geometry optimisations were conductedin which the system was subjected to both positive andnegative uniaxial stress in particular directions in planespassing through the [001] direction.

In particular, the stresses applied were in the form:

cosðz0 Þ sinðz0 Þ 0

�sinðz0 Þ cosðz0 Þ 0

0 0 1

0B@

1CA

1 0 0

0 cosð458Þ sinð458Þ0 �sinð458Þ cosð458Þ

0B@

1CA

0 0 0

0 0 0

0 0 s

0B@

1CA

�cosðz0 Þ sinðz0 Þ 0

�sinðz0 Þ cosðz0 Þ 0

0 0 1

0B@

1CA

T1 0 0

0 cosð458Þ sinð458Þ0 �sinð458Þ cosð458Þ

0B@

1CA

T

ð1Þ

where z, the in-plane rotation, was set to 458 so as to load inthe directions of maximum auxeticity. z0 was set to 08(corresponding to stresses being applied in the (100) plane),158, 308, 458, 608 and 908 (corresponding to stresses beingapplied in the (010) plane).

Referring to Fig. 1, for each system minimised under anapplied load, various lengths and angles corresponding tothe quadrilaterals ABCD in the (100) plane and A0B0C0D0 onthe (010) plane were measured. These measurements wereused to extrapolate the changes of these parameters (either

as absolute values in the case of angles, or as percentage

changes in the case of lengths) per EE45Z ðCVFFÞ

h iGPa applied

stress, where E45Z ðCVFFÞ is the Young’s modulus for

loading at 458 off-axis in the (100) plane, as simulated by theCVFF force-field. This was done so as to modulate theresults.

2.2 Simulations of the deformations for hypo-thetical a-cristobalite systemswhere the tetrahedrahave different extents of rigidity Additional energyexpressions for each crystal were set up using modifiedversions of the CVFF force-field with non-bond terms addedusing the Ewald summation technique [49] in which theoriginal terms in the CVFF force-field for the Si–O bondstretching (VSTR) and O–Si–O angle bending terms (VBEND)were modified to:

VSTRM ¼ VSTR þ 12k�STR l� lminð Þ2; ð2Þ

VBENDM ¼ VBEND þ 12k�HING u � uminð Þ2; ð3Þ

where lmin and umin are the respective values of l and u asminimised by the original CVFF force-field whilst k

�str and

k�hing, the added stiffness constants, were given values of 0,100, 500, 5000 and 999,999 kcal A�1 and kcal rad�2,respectively. The magnitudes of k

�str and k

�hing were the same

at all times, so the term k� will be used to represent themcollectively. Thus, when k� represents the force constant of abond length it bears the units kcal A�1, while when itrepresents the force constant of a bond angle it bears the unitskcal rad�2. These modified equations permit the control ofthe extent of rigidity of the SiO4 tetrahedra through the valueof the k� parameters. The resulting systems are such thatwhen k� ¼ 0, the system will be equivalent to that obtainedusing the original CVFF force-field, while the systems withk� ¼ 100, 500 and 5000 have tetrahedra which areincreasingly more rigid than predicted by the CVFF force-

Figure 1 The parameters relating to the type II rotating rectanglesmeasured for a-cristobalite in the (100) and (010) planes,respectively.

1480 R. Gatt et al.: Force-field based analysis of deformation in a-cristobalite

� 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

ph

ysic

a ssp stat

us

solid

i b

field proper. The system when k� � 1,000,000, this being themaximum value permitted by the program, represents asystem in which the tetrahedra may be assumed to be fullyrigid. This system will be henceforth referred to as the fullyrigid system.

3 Results and discussion3.1 Simulations using original CVFF and

COMPASS force-field The mechanical properties assimulated from the original CVFF and COMPASS force-fields confirm that a-cristobalite exhibits negative Poisson’sratio with maximum auxeticity of equal magnitudes beingexhibited for loading at 458 to the [001] directions, as shownin Fig. 2. Such properties result from the symmetry of a-cristobalite (P41212), and are also observed in the results ofthe experimental work [1]. A similar deformation profilewas also observed for loading in off-axis directions in theoff-axis planes.

In an attempt to obtain a quantitative picture of thedeformation in the (100) and (010) plane for loading inthe (100), (010) and off-axis planes, the measurements ofthe various lengths and angles (Fig. 1) corresponding to thequadrilaterals ABCD in the (100) plane and A0B0C0D0 in the(010) plane were compared. Two rectangles were studiedsince each unit cell projects as two ‘rectangles’ in the (100)and in the (010) planes each.

Figure 3 clearly shows that irrespective of the plane ofloading, there is a rotation of the projected rectangles in the(100) and (010) planes. From this analysis it may easily benoted that in all systems:

(1) The change in the angles u1 and u2 between thequadrilaterals ABCD in the (100) plane and that of u10

and u20 between the quadrilaterals A0B0C0D0 in the (010)plane is larger than that observed in the internal anglesof the individual quadrilaterals.

(2) In both the (100) and (010) planes, irrespective of theplane in which a-cristobalite is being loaded (i.e.independently of the value of z0), one of the projectedrectangles in each plane changes its dimensions to much

lesser extent than the other rectangle. In other words, thetwo projected rectangles in each of the (100) and (010)planes are not equivalent to each other.

All this confirms that the molecular level deformationsin the systems considered here can be explained in terms of‘rotating semi-rigid rectangles’ which are projected in the(100) and (010) planes, and that this model can beconstructed from two rectangles per unit cell, where therigidity of one of the rectangles needs to be less than that ofthe other. It should be noted that a load in the (100) planewill not only cause the projected rectangles in the (100)planes to rotate relative to each other, but also cause arelative rotation of the projected rectangles in the (010)plane. This effect is very likely to be due to the three-dimensional connectivity of the system.

3.2 Simulations using the modified CVFF force-field In an attempt to explain the observation that a load inthe (100) and off-axis planes will also cause rotation of theprojected rectangles in the (010) plane and vice-versa,additional simulations were performed using modifiedforce-fields as discussed in the Methodology section. TheO–Si–O angles were made increasingly more rigid, with thesystem with maximum rigidity (k� � 1,000,000) corre-sponding to a system which deforms solely via rotation ofthe tetrahedra, as described by the rotating tetrahedral modelproposed Alderson et al. [4–7].

An analysis of the measurements pertaining to theprojected ‘rectangles’ (Fig. 4) for loading in the on-axisplanes suggests that, as the tetrahedral units become morerigid the behaviour of the two projected rectangles in each ofthese planes becomes more similar, until it is almostidentical when the system is fully rigid. The extent of thedeformation observed for the same amount of loading isnaturally reduced as the system becomes more rigid. Here itshould be emphasised that the projected rectangles in thefully rigid system are not perfectly rigid as shown by themeasurements displayed in Fig. 4. In fact, the simulated off-axis plots (Fig. 5) in the (100) and (010) planes for thissystem is not as expected for the type II rotating rectanglesmodel, i.e. an isotropic Poisson’s ratio of �1 is notobserved.

The increased rigidity of the tetrahedra affects the off-axis Poisson’s ratio plots in the (100) and (010) planes. Withincreased tetrahedral rigidity, the Poisson’s ratio is observedto:

(1) become less anisotropic in the sense that it tends to �1in both planes,

(2) lose its characteristic rotational symmetry of orderapproximately 4 that is observed in the polar plots fora-cristobalite proper, and gain one of order 2.

It is also important to point out that as the tetrahedrabecome more rigid, the internal angles of the projected‘rectangles’ are observed to be closer to the value of 908.

Figure 2 The experimental [1] and simulated Poisson’s ratios(nyz) confirm that a-cristobalite exhibits negative Poisson’s ratios,with a maximum auxeticity of equal magnitudes being exhibitedfor loading at 458 to the [001] directions. In each of these cases, forall directions about the [001] direction, the profile of the off-axisplot is identical.

Phys. Status Solidi B 252, No. 7 (2015) 1481

www.pss-b.com � 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original

Paper

Despite all this, one may still observe that in the (100)and (010) planes, maximum auxeticity is measured in an off-axis direction (Fig. 5). Here one should mention that the linewhich joins the corners of two opposite connectedrectangles is also in an off-axis direction. This directionis a direction of maximum auxeticity in unmodified

a-cristobalite. On increasing the rigidity of the tetrahedra,the direction of maximum auxeticity is shifted.

Also, in all simulations, a load in the (100) plane wasobserved to cause a rotation (and other deformations) of theprojected rectangles in both the (100) and (010) planes. Infact, the simulations suggest that as the extent of rigidity

Figure 3 Quantitative measures of the deformation of the quadrilaterals ABCD in the (100) and A0B0C0D0 in the (010) plane where eachunit cell projects as two ‘rectangles’ labelled 1 and 2, as detailed in Fig. 1. These measurements were obtained using the CVFF force-field.

Figure 4 Quantitative measures of the deformation of the quadrilaterals ABCD in the (100) plane and A0B0C0D0 in the (010) plane whereeach unit cell projects as two ‘rectangles’ labelled 1 and 2 as detailed in Fig. 1, for the modified CVFF force-fields with k* having a valueof 999,999.



ph

ysic

a ssp stat

us

solid

i b

increases when loading in the (100) plane, the projectedrectangles in the (100) plane always behave more like ‘rigidrectangles’ than the projected rectangles in the (010) plane,and vice versa. This trend shifts gradually as z0 increasesfrom 08 to 908, i.e. moving the load from the (100) to the(010) plane. It is also interesting to note that as z0 approaches458, the deformations of the rectangles in both the (100) and(010) planes tend to become more similar (something whichis best observed for the rigid tetrahedra).

Here, it is appropriate to compare and contrast theseresults with those obtained by Grima et al. for the silicateswhich are equivalent to NAT, THO and EDI [51]. In thisanalysis, one should note that in the case of the NAT-type,THO-type and EDI-type frameworks, the fully rigid systemsexhibit behaviour where the projected squares in the (001)plane behave in a quasi-rigid manner, and exhibit therequired in-plane isotropic Poisson’s ratio of �1. In otherwords, in the case of zeolites with the NAT-type framework,the rotations of the tetrahedra become fully equivalent to the‘rotating rigid squares’model projected in the (100) plane ofthe zeolites (the projected squares in the (100) planeremained perfectly rigid as the systems were loaded in adirections of maximum auxeticity). However, in the case ofa-cristobalite the type II rotating rigid rectangles model isnot sufficient to describe the deformations in all planes. Thisdeviation is due to the three-dimensional nanostructure of a-cristobalite, which differs significantly from that of theframeworks of zeolites in the NAT group. In fact, in the caseof a-cristobalite, it should be emphasised that the columnarstructure which projects as one set of rectangles in the (100)plane will in fact incorporate the Si–O–Si hinges whichcorrespond to the soft hinges between the projectedrectangles in the (010) plane. As highlighted above, aload in the (100) plane will cause a rotation of the projectedrectangles in both the (100) and (010) planes, as well as adeformation of the internal angles of the projectedrectangles in both planes (although not to the same extent).The rectangles in a-cristobalite cannot be perfectly rigidsince the same atoms which constitute the Si–O–Si hinges ineither plane form the internal angles in the other plane. Thismakes the observation of the ‘rotating semi-rigid’ rectanglesmechanism in a-cristobalite even more remarkable.

If one now looks at the Poisson’s ratio profiles in thenon-major planes, particularly the planes which passthrough the [001] direction but are not the (100) or (010)planes, one may note that the loss of the characteristicrotational symmetry of order approximately 4 to one ofrotational symmetry of order 2 is much more evident inthe non-major planes than in the major planes. Thismeans that the planes passing through the [001] directionare no longer equal as in the case of naturally occurringa-cristobalite.

Here, it should be noted that an approximate rotationalsymmetry of order 4 in unrestrained a-cristobalite meansthat the properties for loading in any direction in a particularplane are roughly similar to those observed for loading in anequivalent direction in a second plane which is at 908 to theoriginal direction (Fig. 5). The loss of order of suchrotational symmetry was also predicted by the rotatingtetrahedra models by Alderson et al. [5], who showed thatnzy ¼ nyz

� ��1 6¼ nyz, a property which is only observedslightly in a-cristobalite.

Before concluding, it is important to note that the resultsobtained in this study are consistent with previous studies onlow-temperature cristobalite [1]. It is likely that thesefindings could also be extended to cristobalite at highertemperatures since it has been suggested that cristobalite hasthe potential to exhibit on-axis auxetic behaviour over awide temperature range [52].

It is important to highlight that the importance of thework reported here lies not only in relation to a-cristobalite,but also in relation to other systems which, like a-cristobalite, can be described in terms of rotating two-dimensional or three-dimensional units. For example, it hasbeen confirmed that the behaviour of auxetic polyurethanefoams can be described in terms of rotating semi-rigid joints,and it would be useful if their behaviour is compared to thatof the naturally occurring silicate a-cristobalite [1].Furthermore, the information reported here could be usedto guide scientists and engineers in their design of novelauxetic materials, which could be made to mimic thebehaviour of this naturally occurring material. In thisrespect, it should be highlighted that since the theory ofelasticity is scale independent, the observations made here at

Figure 5 Off-axis Poisson’s ratio polar plots fora-cristobalite as simulated by the CVFF force-field and the CVFF force-field with addedrestraints.



Original

Paper

the molecular level can be adopted at any size scale,including the macro level.

4 Conclusion This work has shown that the auxeticbehaviour of a-cristobalite in the planes passing through the[001] direction can be explained in terms of simple two-dimensional models involving semi-rigid rotating rectan-gles. These rectangles are the projections in the (100) and(010) planes of a more complex three-dimensional model. Itwas shown that in the case of the two-dimensional model,one needs to consider two sets of rectangles, with one setbeing more rigid than the other. It was also shown that thesetwo modelling approaches (i.e. the two-dimensional andthree-dimensional approaches) are complementary to eachother, and that as the rigidity of the tetrahedral unitsincreases, the projected rectangles behave more likeidealised ‘rotating rigid rectangles’, thus highlighting therelationship between these two modelling approaches. It ishoped that, given the many beneficial effects associated withauxetic behaviour, this work will provide an impetus tofurther studies not only in relation to a-cristobalite, but alsoon other systems which can be made to mimic the behaviourof this naturally occurring silicate.

Supporting InformationAdditional supporting information may be found in theonline version of this article at the publisher’s website.

References

[1] A. Yeganeh-Haeri, D. J. Weidner, and J. B. Parise, Science257, 650 (1992).

[2] N. R. Keskar and J. R. Chelikowsky, Nature 358, 222 (1992).[3] N. R. Keskar and J. R. Chelikowsky, Phys. Rev. B 48(22),

227 (1993).[4] A. Alderson and K. E. Evans, Phys. Chem. Miner. 28, 711

(2001).[5] A. Alderson and K. E. Evans, Phys. Rev. Lett. 89, 225503

(2002).[6] A. Alderson, K. L. Alderson, K. E. Evans, J. N. Grima, and

M. S. Williams, J. Metastab. Nanocryst. 23, 55 (2005).[7] A. Alderson and K. E. Evans, J. Phys.: Condens. Matter 21, 1

(2009).[8] H. Kimizuka, H. Kaburaki, and Y. Kogure, Phys. Rev. Lett.

84, 5548 (2000).[9] J. N. Grima, R. Gatt, A. Alderson, and K. E. Evans, J. Mater.

Chem. 15, 4003 (2005).[10] J. N. Grima, R. Gatt, A. Alderson, and K. E. Evans, Mater.

Sci. Eng. A 423, 219 (2006).[11] K. E. Evans, M. A. Nkansah, I. J. Hutchinson, and S. C.

Rogers, Nature 353, 124 (1991).[12] B. D. Caddock and K. E. Evans, J. Phys. D, Appl. Phys. 22,

1877 (1989).[13] K. W. Wojciechowski, A. Alderson, A. Branka, and K. L.

Alderson, Phys. Status Solidi B 242, 497 (2005).[14] R. Gatt, J. N. Grima, J. W. Narojczyk, and K. W.

Wojciechowski, Phys. Status Solidi B 249, 1313 (2012).[15] K. W. Wojciechowski, Phys. Lett. A 137, 60 (1989).[16] L. J. Gibson, M. F. Ashby, G. S. Schajer, and C. I. Roberts,

Proc. R. Soc. A 382, 25 (1982).

[17] R. S. Lakes, J. Mater. Sci. 26, 2287 (1991).[18] D. Prall and R. S. Lakes, Int. J. Mech. Sci. 39, 305

(1997).[19] O. Sigmund, S. Torquato, and I. A. Askay, J. Mater. Res. 13,

1308 (1998).[20] J. N. Grima, R. Gatt, and P.-S. Farrugia, Phys. Status Solidi B

245, 551 (2008).[21] M. R. Hassan, F. Scarpa, M. Ruzzene, and N. A. Mohammed,

Mater. Sci. Eng. A 481, 654 (2008).[22] H. Abramovitch, M. Burgard, L. Edery-Azulay, K. E. Evans,

M. Hoffmeister, W. Miller, F. Scarpa, C. W. Smith, and K. F.Tee, Compos. Sci. Technol. 70, 1072 (2010).

[23] F. Scarpa, Compos. Sci. Technol. 70, 1033 (2010).[24] G. Cicala, G. Recca, L. Oliveri, Y. Perikleous, F. Scarpa, C.

Lira, A. Lorato, D. J. Grube, and G. Ziegmann, Compos.Struct. 94, 3556 (2012).

[25] J. N. Grima and R. Gatt, Adv. Eng. Mater. 12, 460–464(2010).

[26] A. Alderson, K. L. Alderson, D. Attard, K. E. Evans, R. Gatt,J. N. Grima, W. Miller, N. Ravirala, C. W. Smith, and K.Zied, Compos. Sci. Technol. 70, 1042 (2010).

[27] X. N. Liu, G. L. Huang, and G. K. Hu, J. Mech. Phys. Solids60, 1907 (2012).

[28] A. Spadoni and M. Ruzzene, J. Mech. Phys. Solids 60, 156(2012).

[29] L. Rothenburg, A. A. Berlin, and R. J. Bathurst, Nature 354,470 (1991).

[30] J. N. Grima and K. E. Evans, J. Mater. Sci. Lett. 19, 1563(2000).

[31] K. L. Alderson, A. Alderson, and K. E. Evans, J. Strain Anal.32, 201 (1997).

[32] J. N. Grima, A. Alderson, and K. E. Evans, J. Phys. Soc. Jpn.74, 1341 (2005).

[33] R. Gatt, J. P. Brincat, K. M. Azzopardi, L. M.Mizzi, and J. N.Grima, Adv. Eng. Mater. 17(2), 189–198 (2015).

[34] R. Gatt, D. Attard, P.S. Farrugia, K.M. Azzopardi, L. Mizzi,J.P. Brincat, and J.N. Grima, Phys. Status Solidi B 250(10),2012 (2013).

[35] L. Mizzi, D. Attard, A. Casha, J.N. Grima, and R. Gatt, Phys.Status Solidi B 251(2), 328 (2014).

[36] R. Lakes, Science 235(4792), 1038–1040 (1987).[37] C. W. Smith, J. N. Grima, and K. E. Evans, Acta Mater. 48,

4349–4356 (2000).[38] J. N. Grima, D. Attard, R. Gatt, and R. Cassar, Adv. Eng.

Mater. 11(7), 533–535 (2009).[39] F. Scarpa, J. R. Yates, L. G. Ciffo, and S. Patsias, J. Mech.

Eng. Sci. 216(2), 1153–1156 (2002).[40] J. B. Baker, A. G. Douglass, and A. C. Griffin, Abstr. Pap.

Am. Chem. Soc. 210, 146 (1995).[41] V. V. Novikov and K. W. Wojciechowski, Phys. Solid State

41, 1970 (1999).[42] K. V. Tretiakov and K. W. Wojciechowski, Phys. Status

Solidi B 244, 1038 (2007).[43] R. H. Baughman, J. M. Shacklette, A. A. Zakhidov, and S.

Stafstrom, Nature 392, 362 (1998).[44] J. W. Narojczyk and K. W. Wojciechowski, Phys. Status

Solidi B 244, 943 (2007).[45] J. N. Grima, R. Gatt, V. Zammit, J. J. Williams, K. E. Evans,

A. Alderson, and R. I. Walton, J. Appl. Phys. 101, 86102(2007).

[46] R. Gatt, V. Zammit, C. Caruana, and J. N. Grima, Phys.Status Solidi B 245, 502 (2008).



ph

ysic

a ssp stat

us

solid

i b

[47] K. M. Azzopardi, J. P. Brincat, J. N. Grima, and R. Gatt, RSCAdv. 5(12), 8974–8980 (2015).

[48] J. N. Grima, S. Winczewski, L. Mizzi, M. C. Grech, R.Cauchi, R. Gatt, D. Attard, K. W. Wojciechowski, and J.Rybicki, Adv. Mater., DOI: 10.1002/201404106 (2014).

[49] P. P. Ewald, Ann. Phys. (Leipzig) 253, 64 (1921).

[50] J. F. Nye, Physical properties of crystals (Clarendon, Oxford,UK, 1957).

[51] J. N. Grima, V. Zammit, R. Gatt, D. Attard, C. Caruana, T. G.Chircop Bray, J. Non-Cryst. Solids 354, 35–39 (2008).

[52] W. Pabst and E. Gregorova, Ceramics—Silikaty 57(3), 167–184 (2013).



Original

Paper

a force-field based analysis of the deformation mechanism in α … · 2017-11-05 · a...

Documents