Development of novelpoly(phenylacetylene) network polymersand their mechanical behaviour

Joseph N. Grima*,1,2, Christine Zerafa1, and Jean-Pierre Brincat2

1 Faculty of Science, Department of Chemistry, University of Malta, Msida MSD 2080, Malta2 Faculty of Science, Metamaterials Unit, University of Malta, Msida MSD 2080, Malta

Received 30 December 2013, revised 9 January 2014, accepted 10 January 2014Published online 29 January 2014

Keywords auxetic, deformation, novel structures, simulations

* Corresponding author: e-mail joseph.grima@um.edu.mt, Phone: þ356 2340 2274

Auxetics exhibit the anomalous property of getting wider ratherthan thinner when uniaxially stretched, i.e. have a negativePoisson’s ratio. This work presents the results of simulationsperformed on a number of hypothetical poly(phenylacetylene)networks which are meant to mimic the behaviour of rotating

scalene triangles. It was shown that such systems do indeedexhibit a negative Poisson’s ratio being as low as c. �1.2, i.e.lower than the c. �1 found in earlier systems. This negativePoisson’s ratio is due to the relative rotation of nanoscopicunits.

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1 Introduction The first auxetic system (i.e. amaterial which expands laterally when uniaxially stretched)was reported by Voigt [1], when single crystals of ironpyrites were described as having a negative Poisson’s ratio.However, at that time this phenomenon was regarded asan anomaly, and the behaviour was attributed to twinningdefects [2]. It was only in the 1980s that interest in suchauxetic behaviour started to grow, especially through theimportant contributions made by pioneers such as Lakes [3],Evans [4] and Wojciechowski [5].

Since then, a wide range of auxetic systems have beendiscovered, including partially auxetic metals [6], cubicmetallic alloys of extremely negative (n

inspired by the generalised polytriangles networks studiedby Grima et al. [32] and Chetcuti et al. [33] and are similar tomolecular-level systems that already exist in the litera-ture [34]. However, these new systems possess additionalversatility imparted to them through some chemical structuralvariations that give rise to interesting systems with differentmechanical properties from the parent systems.

2 The proposed new nano-networks In thesystems being proposed here, the triangular units arerepresented through a ‘1,3,5-triacetylenylbenzene’-like unit,connected together through cis-ethylene junctions as inFig. 2a. Such a construction would not only avoid the1,2,3,4-tetrasubstituted phenyl rings in the original nano-polytriangles network proposed by Grima [8], but moreimportantly, permit the triangles to assume a non-equilateralshape, meaning that it would become possible to construct awider variety of triangular networks of the form as discussedin Grima, Chetcuti et al.’s work, giving significantly moreversatility.

In fact in such systems, one is not constrained tohaving the triangles equilateral, an imposition previouslyimparted through the 608 internal angles at the verticeswhich result from the substitution pattern of the benzenerings in Grima’s polytriangles [8]. In the novel systems,assuming that the triangles are of side lengths a, b, cdefined as in Fig. 2b, then these lengths a, b, c can assume

a much wider range of distinct values which may beapproximated by

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL21 þ L22 � 2L1L2 cos

2p3

� �s

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL21 þ L22 þ L1L2

q;

ð1Þ

b ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL22 þ L23 � 2L2L3 cos

2p3

� �s

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL22 þ L23 þ L2L3

q;

ð2Þ

c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL21 þ L23 � 2L1L3 cos

2p3

� �s

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL21 þ L23 þ L1L3

q;

ð3Þ

where Li is the length indicated in Fig. 2b, which may berelated to the length of the acetylene chain through

Li ¼ jCR � CRj þ jCR � C1jþ jC2 � C1j þ jC2 � C2jþ ð2ni � 1ÞjC1 � C1j;

ð4Þ

Figure 1 (a) The chemical structure of polytriangles-2-yne [8]; (b) the ‘rotating triangles’ model which depicts the idealised deformationmechanism of polytriangles-2-yne.

Figure 2 (a) The 1,3,5-triacetylenylbenzene group which makes up the triangular units that are connected together through cis-ethylenejunctions; (b) the triangular unit, indicating triangle side lengths a, b and c, as well as the acetylene-chain-related lengths L1, L2 and L3;(c) the 3,2,1-polytriangles-Type II system, indicating the location and assigned values of n1, n2 and n3 which form the basis of the suggestednomenclature. (Note that the location of ni is determined by referring to the ‘generic rigid triangles’ model proposed by Grima et al. [32].)

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where ni is the number of triple bonds in the chain whilstjCR � CRj is the average length of a C–C bond in a benzenering, jCR � C1j is the average bond length between a Catom in a benzene ring and a C atom in an acetylene chain,jC2 � C1j is the average length between bonded sp2 andsp hybridised C atoms, jC2 � C2j is the average lengthbetween two bonded sp2 hybridised C atoms, whilst jC1 �C1j is the average length between two bonded C atoms inan acetylene chain. Taking parameters from standardforce-fields, such as the Dreiding force-field [35], thislength becomes approximately equal to:

Li ¼ ð1:39A� Þ þ ð1:29A� Þ þ ð1:26A� Þ þ ð1:33A� Þþ ð2n� 1Þð1:19A� Þ ¼ ½4:08þ 2:38ni�A� :

ð5Þ

For example, a triangle made from systems havingn1¼ 1, n2¼ 2 and n3¼ 3 would have side lengths of:

½a� b� c� � ½9:86� 13:96� 12:08�: ð6Þ

Once again, to simplify the discussion, a simplenomenclature system will be used for these systems,which will be termed (n1,n2,n3)-polytriangles-Type II, where(n1,n2,n3)- refer to the number of triple bonds per acetylenechain, as indicated in Fig. 2c. Note that the Type II suffix isbeing used to distinguish these novel systems from the onesdescribed originally by Grima [8].

Given the large number of systems that may beconstructed in the manner described above, modellingwas only performed on a small sub-section of the possiblesystems. This was also done in order to identify any issueswhich could potentially reduce the auxetic character of thesesystems.

3 Simulations A number of (n1,n2,n3)-polytriangles-Type II organic networks having the connectivity illustratedin Fig. 2 were constructed with the number of carbon–carbontriple bonds in each of the three acetylene chains (n1,n2,n3)being (1,1,1), (2,2,2), (3,3,3), which systems where meant toresult in equilateral triangles, or (2,1,1), (1,2,2), (2,2,3),(2,3,3), (3,2,1), which systems where meant to result innon-equilateral triangles.

For all systems, the starting unit-cell structures for the2D networks listed above were constructed using the Buildertools in Materials Studio V.6.0 (Accelrys, Inc. USA) andplaced in the global YZ plane, with the alignment shown inFig. 3. This alignment ensured that the networks remainedaligned parallel to the YZ plane as much as possible in thesimulations and is not dissimilar to that employed by Grimaet al. in the analytical model of the general polytriangularsystems.

Periodic boundary conditions were applied with the[001] cell vector being aligned parallel to the global Zdirection, while the [010] direction was aligned to lie in theYZ plane. An energy expression was set up for each of the

constructed systems using three different force-fields:Dreiding [35], PCFF [36, 37] and COMPASS [38–40].These force-fields were chosen since they have all beenadequately parameterised for all atom types used in thisstudy, and they have in fact already been used to successfullymodel the properties of the related molecular-level reflexyneand flexyne systems [41–43]. In all cases the default force-field settings were applied, with the exception of non-bondinteractions (Van der Waals and Coulombic interactions)summation method, for which the Ewald procedure [44] wasused. Note that charge equilibration [45] was performedprior to using the Dreiding force-field, since this force-fielddoes not include a list of partial charges to be used informulating the energy expression.

The energy of the systems was minimised as a functionof the atomic coordinates and six cell parameters to theearlier of reaching the default fine convergence criteria(including an energy change threshold of 0.0001 kcal mol�1

and maximum force of 0.005 kcal mol�1 Å�1) or 15,000minimisation steps, whichever occurs first. The SMARTminimiser implemented in Materials Studio was used toperform the energy minimisation calculation, which is acascade of the steepest descent, ABNR (Adjusted basisset Newton–Raphson), and quasi-Newton methods. Thisstep was repeated three times to enable re-calculation ofthe non-bond list and to ensure complete minimisation ofthe structure.

The resulting systems were visually analysed and themechanical properties were calculated using the ‘constant-strain’ approach (using the Forcite module within MaterialsStudio). The maximum strain amplitude was set to 0.010 i.e.1.0% strain and seven distorted structures were generatedfor each strain pattern. The on-axis Young’s moduli and

Figure 3 A representative example of the newly designednetworked polymer, the (1,1,1)-polytriangles-Type II system.

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in-plane on-axis Poisson’s ratios and shear modulus for thesesystems were then calculated from the compliance matrix asshown below:

Ex ¼ 1s11 ; Ey ¼1s22

; Ez ¼ 1s33 ; ð7Þ

nyz ¼ � s32s22 ; nzy ¼ �s23s33

; Gyz ¼ 1s44 ; ð8Þ

where sij are the elements of the compliance matrix S, Ex, Eyand Ez are the Young’s moduli in the X, Y and Z directions,nyz and nzy are the Poisson’s ratios in the YZ plane for loadingin the Y and Z directions respectively and Gyz is the on-axisshear modulus in the YZ plane.

From the simulated on-axis stiffness matrix, the in-planeoff-axis properties were calculated using standard axistransformation techniques as described in Nye [46] andimplemented and coded for the Maple package byGrima [47].

Note that all simulations were performed using just twotriangular units per unit cell, thus resulting in a paral-lellogramic unit cell, since a preliminary convergence studysuggested that this system is sufficient to adequately modelthe properties of such system.

4 Results The simulated structural properties of theconstructed polytriangles systems are summarised in Fig. 4,while their simulated mechanical properties are given inFig. 5 together with plots of the off-axis Poisson’s ratio uponrotation around the X-axis. These results immediately showthat for all the studied systems, most combinations of ni¼ 1,2 or 3 in the structure can give a negative Poisson’s ratio inthe YZ plane, with n being more negative off-axis.

The structural properties were first analysed andcompared by obtaining information on the sizes of the unitcell and also by visually observing the YZ and XY projectionsof the minimised structures. It can be noted that ingeneral, each system minimised with different force-fieldsachieved similar conformations and characteristics, includ-ing volume and density. A few exceptions were notedthough, particularly in systems having longer acetylenechains such as (3,2,1)- and (3,3,3)-polytriangles systems. Inthese cases, slightly different packing was achieved withdifferent force-fields. The acetylene chains, for example,overlapped each other in some cases but did not in others.

It can also be noted from the XY projections that allthe systems adopted a slightly out-of-plane configuration,the extent of which seems to be dependent on the lengthof the acetylene chains, n. For example, this out-of-planebending was very significant in the (3,3,3)-polytriangles-Type II system, where long acetylene chains overlap in orderto be able to pack within a small space so as to maximisethe density. Other systems such as the (2,3,3)-polytriangles-Type II system, also packed in a highly dense mannerbut they tend to pack without the need of any chains

overlapping. Such packing is less efficient and gives riseto rather large pores. In contrast to this, the (1,1,1)-polytriangles-Type II system packs more efficiently due tothe short acetylene chains (having only one carbon–carbontriple bond), thus having relatively smaller pores in theminimised structure. Nevertheless, despite this out-of-planebending, the systems generally still retained their expectedgeometric features when viewed down their YZ plane, in thesense that one can still visualise the expected ‘connectedtriangles’ motif. One should also note that the properties ofthe (1,1,1)-polytriangles system showed significant consis-tency when simulated with different force-fields, somethingwhich to a certain extent was not entirely evident in most ofthe other systems. This may be attributed to the greaterdegree of freedom imparted on the systems by the longeracetylene chains in cases where ni >1.

On analysing the mechanical properties results (Fig. 5),it is immediately evident that various values of negativePoisson’s ratio were obtained, ranging up to �1.2 in the(3,2,1)-polytriangles system. Exceptionally low Poisson’sratios were also obtained in the (2,1,1)-system (whensimulated with the Dreiding force-field) and the (2,2,2)-system (when simulated with the PCFF force-field), whichexhibited n of up to �1.5 and �2, respectively. This is verysignificant as this high magnitude of auxeticty had notbeen demonstrated in the earlier work by Grima [8], asexpected since those systems had triangular units which wereequilateral, i.e. even in their most idealised scenario, are onlyexpected to achieve maximum auxeticity corresponding toPoisson’s ratios of �1. Such bounds in the Poisson’s ratioare relaxed for the more general models built fromscalene triangles, with the result that Poisson’s ratios whichare more negative than �1 may be obtained, as is the casereported here. It is however unfortunate that these moregeneral systems are no longer isotropic, as discussed indetail elsewhere [33]. Here it should be noted that in thestudy performed here, such anisotropy was very evident, asexpected. Anisotropy was also found in the systems havingn1¼ n2¼ n3, such as the (1,1,1)-, (2,2,2)- and (3,3,3)-polytriangles systems even if such systems were meant tomimic the isotropic �1 behaviour, however this was not thecase. This unexpected behaviour is probably because ofthe out-of-plane bending which did not allow the triangularunits to behave as ideal units in a single plane.

5 Deformation mechanism In attempt to under-stand the reason behind this auxetic behaviour, additionalsimulations were performed where the systems wereuniaxially stretched in the Y and Z directions and theminimum energy configurations of the stretched structureswere obtained. These simulations were performed throughthe Cerius2 modelling environment (Accelrys, Inc.) usingthe previously minimised structures as starting points. Herewe report only a sample of these simulations, namely thoseperformed on the (1,1,1)-polytriangles-Type II system whenthis was stretched by application of a uniaxial stress of1.0GPa in the Y direction.

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Figure 4 Structural properties of the resulting minimised (n1,n2,n3)-polytriangles-Type II systems obtained with Dreiding, PCFF andCOMPASS force-fields, including views of the systems in the YX and ZY planes.

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Figure 6 depicts the resulting deformation whichultimately gave rise to auxetic behaviour. These imagesconfirm that the rotating triangles mechanism is indeedtaking place, thus explaining the observed negativePoisson’s ratio.

This finding is very significant because it confirms thatthe novel designs proposed here contribute to the regime ofmolecular-level auxetic systems and also provide a betterunderstanding of the behaviour of such systems through thecombined information of their structural and mechanical

Figure 5 Mechanical properties of (n1,n2,n3)-polytriangles-Type II systems, including the on-axis Young’s moduli and shear modulus, aswell as the in-plane on-axis Poisson’s ratios and off-axis plots of the Poisson’s ratio upon rotation around the X-axis.

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properties. The resulting auxetic potential of the (n1,n2,n3)-polytriangles-Type II systems arising from the ‘rotatingtriangles’ mode of deformation is particularly important,since it confirms that the ‘rotating rigid units mechanism’ isindeed one of the more superior deformation mechanismswhich drives the system towards a negative Poisson’s ratio.

On closer inspection of the results from thesesimulations, one may note that the deformation mechanismalso involves the flexure of the acetylene chains duringstretching, which not only bring about the rotation of thetriangular units, as highlighted in Fig. 6, while keeping theirconnections through the cis-ethylene bond, but also resultin some deformations of the triangles themselves. Thesedeformations of the triangular units themselves would nothave been accounted for in the derivation of the analyticalmodel, which only considers the highly idealised scenariowhere the triangles are perfectly rigid, and are the mostlikely cause for the deviations from the predictions of theanalytical model.

Before concluding, it is important to point out that thisstudy was based on static mechanical simulations, and itwould be very interesting to understand the behaviour ofsuch systems at various temperatures and pressures, sincethese factors may have a significant effect on the structuralproperties, and hence also on the mechanical behaviour, ofthese systems. Besides, such a modelling study also assumesthat the networks under consideration are perfect networks,with no random defects (such as lack of connectivity). Workwe and others performed on non-perfect systems suggeststhat in general, imperfections may result in significantchanges to the Poisson’s ratio [48–51]. Therefore, it wouldalso be interesting to study such systems from a morerealistic point of view in order to understand how suchdefects may ultimately affect their mechanical properties.

Finally it is also important to highlight that all thesystems studied here are hypothetical and it is still unknownwhether they can be successfully synthesised, which is aproblem that has already been encountered with reflexyneand flexyne systems. However, it is very important tofurther the research of such hypothetical networks sincethey provide a better understanding of the behaviour ofmolecular-level systems, while also leading to the design ofnew auxetic systems.

6 Conclusions This study has shown that the novelmolecular-level (n1,n2,n3)-polytriangles-Type II systemshave auxetic potential, with all the systems exhibitingnegative Poisson’s ratio through the ‘rotating triangles’deformation mechanism. These systems were found to beanisotropic, even in systems were n1¼ n2¼ n3, whichunexpected behaviour can be attributed to their lack ofplanarity. These poly(phenylacetylene) networks show verypromising auxetic properties, and these predictions encour-age further research into similar systems and their chemistry,with the hope of having them successfully synthesised andmechanically tested in the future. Bearing in mind that thesesystems are also likely to benefit from additional otherinteresting properties, such as high electrical connectivelydue to their extensivep-bonding, we hope that this work willprovide an impetus to such further studies.

Acknowledgements The research work disclosed in thispublication is partially funded by the Strategic Educational PathwaysScholarship (Malta). The scholarship is part-financed by theEuropean Union – European Social Fund (ESF) under OperationalProgramme II – Cohesion Policy 2007–2013, ‘Empowering Peoplefor More Jobs and a Better Quality of Life’. This research hasbeen carried out using computational facilities procured throughthe European Regional Development Fund, Project ERDF-080‘A Supercomputing Laboratory for the University of Malta’ (http://www.um.edu.mt/research/scienceeng/erdf_080).

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