nano networks exhibiting negative linear compressibility

Nano networks exhibiting negativelinear compressibility

Joseph N. Grima*,1,2, Edera P. Degabriele1, and Daphne Attard2

1 Faculty of Science, Department of Chemistry, University of Malta, Msida, MSD 2080, Malta2Metamaterials Unit, Faculty of Science, University of Malta, Msida, MSD 2080, Malta

Received 2 May 2016, revised 23 May 2016, accepted 25 May 2016Published online 20 June 2016

Keywords auxetic, mechanical properties, molecular mechanics, molecular networks, negative linear compressibility

* Corresponding author: e-mail [email protected], Phone: þ356 2340 2274

Molecular mechanics simulations were used to analyse the on-axis mechanical properties of a series of nanonetworks. Theseconsisted of naphthacene-like units linked together throughacetylene chains, and designed to mimic the geometry of awinerack-like structure in one of the major planes, a geometrywhich is closely associated with negative linear compressibil-ity. It was found that these networks have a potential to exhibita negative linear compressibility in one of the major axis. An

analysis of the deformation mechanism as well as themagnitudes of the compressibility in the major directionssuggest that this effect arises from relative rotations of the rigidpolyphenyl chains and is comparable to that predicted by anidealised winerack model deforming through hinging. Thisconfirms that there is a very strong link between negative linearcompressibility and geometry, as is the case with auxeticmaterials.

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1 Introduction Polyphenylacetylene networks are aclass of materials made from phenyl rings connectedtogether through acetylene chains. In the last decades, thesenetworks attracted considerable attention from the scientificcommunity, particularly since the report by Evans et al. [1]who proposed a family of poly(phenylacetylene) networksexhibiting ‘auxetic’ behaviour, i.e. get wider when pulledrather than thinner (negative Poisson’s ratio).

From a materials design point of view, these systems areextremelyversatile andcan result invarious2D/3Dconstructsdepending on how the different building blocks are connectedtogether. Inparticular, Evanset al. has shown that it is possibleto design 2D networks which may either be auxetic or non-auxetic, depending on the manner of substitution on thephenyl rings which could result in a re-entrant or non-re-entrant network [1–3]. More importantly, this and similarwork [1–13] has suggested that a viable methodology fordesigning nanoscale auxetics was one which downscaled amacroscale auxetic mechanism to the atomic level, as it wasshown that, provided that the material has the appropriatenanostructural features and the forms deform via the requiredmechanism, then auxeticity would be retained.

It is now becoming more widely recognised thatauxeticity is not the only ‘negative’ mechanical propertywhich can be explained in terms of the geometry and

deformation profile of the internal structure of the material.For example, it has been shown that particular geometriesand deformation mechanisms can result in the anomalousproperty of negative linear compressibility (NLC), i.e. theability of a material to expand rather than contract in at leastone direction when subjected to a positive hydrostaticpressure [14–38].

One of the geometries/deformation mechanismswhich isknown to result inNLC, is thatwhichmimics the behaviour ofawine rackwhich, as a result of this geometry and constraints,can expand in one direction when being subjected to a biaxialcompressive force (similar to hydrostatic pressure), i.e.exhibit NLC [27–38], see Fig. 1a. This mechanism hasbeen used by Fortes et al. to explain NLC in methanolmonohydrate [28]. These systems have provided a significantinsight into the field of NLC. In particular, they haveconfirmed that NLC can indeed exist, and reconfirmed thata mechanism can be implemented in various classes ofmaterials and lead to the samemacroscopic properties, in thiscase NLC. Here it should be noted that all work carried out sofar in this field has always confirmed the requirement for apositive volume compressibility unless the system isunstable [17] or the system is porous and pressure is beingexerted on both from the ‘outside’ and ‘inside’. In such cases,although the system appears to expand in ‘volume’ under

Phys. Status Solidi B, 1–9 (2016) / DOI 10.1002/pssb.201600276

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positive hydrostatic pressure, the ‘solid portion’ of thesystem is in fact still shrinking thus not violating anythermodynamic requirement [14, 16].

From a more practical point of view, it has also beenshown that negative compressibility should not be regardedas a mere scientific curiosity, as there are a number of realapplications which would benefit from negative compress-ibility. These include the use of NLC materials in pressuresensors, artificial muscles and actuators [18, 27, 35].

Unfortunately, despite all these recent advances, it iswidely recognised that this field is still in its infancy. Forexample, nanoscale systems exhibitingNLCwhichhave beendiscovered or designed so far have limited tunability, in thesense that the nanoscale building blocks which constitute thesystems permit little variability, if any. The present work willattempt to address this lacuna by proposing a novel and moretuneable polyphenylacetylene-based nanonetwork which ismeant tomimic thewine rackmodel structure [29].Anattemptwill be made to assess how slight modifications in thenetwork’s nanostructure could result in a different set ofmacroscopic properties, in particular different compressibil-ity characteristics. The behaviour of these systems at ambientpressure conditions and also at extreme negative pressureconditions are also investigated.

2 Methodology2.1 Design concept The concept behind this work is

to design and study nanolevel systems which, as much aspossible, have the nanostructural characteristics that maypermit them to behave and respond to pressure in a ‘wine-rack’-like manner. This requires: (i) components whichwould be straight and rigid to be used as the ‘rib’ elements ofthe wine-rack, and (ii) components which may behave as‘hinges’ which connect the ‘ribs’ together. These character-istics may be achieved through the use of long naphthacene-like units as the ‘rib’ elements which are connected togethervia acetylene chains which would act as hinges, seeFig. 1b and c. This construction is possible since theacetylene chains that are meant to act as hinges would beorthogonal to the polyphenyl naphthacene-like chain units,i.e. the different polyphenyl chains should be able to, at leastin theory, rotate freely relative to each other. Through thismethod of design, different variants of the networks may beachieved through variation of the length of ribs by changingthe number of rings between each ‘hinge’ in the polyphenylchain and/or the length of the acetylene chains. Note that intheory, it is possible to either have ‘standard’ trellis-likesystems, if the lengths of the polyphenyl linkages betweenthe ‘hinges’ are all equal (see Fig. 1b-i), or, more complexsystems where some variations are used, e.g. two differentsets of lengths are used (see Fig. 1b-ii).

In the case of the system with equal polyphenyllinkages, the system may be described by a representativecuboidal unit cell (not the smallest unit cell) which, asillustrated in Fig. 1a projects as a rectangular unit cell in theplane orthogonal to the acetylene chains, i.e. plane of the‘wine-rack’motif. For convenience, these types of networks

Figure 1 (a) The wine-rack mechanism which results in negativelinear compressibility. (b) Typical polyphenylacetylene networksconsisting of polyphenyl naphthacene-like chains linked togetherby acetylene chains that may mimic the system in (a). Thepolyphenylacetylene chains can be either (b-i) equal or (b-ii)different in length. (c) The naming convention used to refer tonetworks where the polyphenyl chains are of equal length:m refersto the number of phenyl rings and n refers to the number ofacetylene linkages, hence the name PmAn.

2 J. N. Grima et al.: Nano networks exhibiting negative linear compressibility

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are henceforth referred to as PmAn networks, where ‘m’refers to the number of phenyl rings denoted by ‘P’ and ‘n’refers to the number of acetylene links denoted by ‘A’. Thiswine-rack motif displayed by such systems, see Fig. 1, hasbeen extensively modelled and it was shown that for asystem with unit thickness in the third direction, thecompressibility properties, if deformations are limited tochanges in u (the angle between the ribs in the ‘wine-rack’)should be estimated by [29]:

bhx ¼ � 1

khcos uð Þtan u

2

� �; ð1Þ

bhy ¼

1kh

cos uð Þcot u

2

� �; ð2Þ

where bhx and b

hy are the linear compressibilities in the x- and

y- directions respectively, assuming that the wine-rackdeforms only through hinging, and kh is the hinging stiffnessconstant. These equations suggest that these systems arecapable of exhibiting negative compressibility properties, inparticular, negative bx for 0

o < u < 90o and a negative byfor 90o < u < 180o. Moreover, they suggest that the ratio ofthe compressibility ratios should be:

bhy

bhx

¼ �cot2u

2

� �: ð3Þ

Obviously, this idealised uni-mode behaviour isdifficult to achieve in real systems. This makes it necessaryto perform more detailed simulations so as to estimatebetter the true properties that could be achievable by suchsystems.

2.2 Simulations2.2.1 Construction of the system Simulations

were performed on six variants of the polyphenylacetylenenetworks of the simpler ‘standard’ system shown in Fig. 1b,where m¼ 1,2 and n¼ 1,2,3. These systems wereconstructed within Materials Studio (version 7.0) asmolecular crystals through the use of periodic boundaryconditions. The networks were oriented in such a way thatthe acetylene chains are aligned parallel to the c crystalaxis, while the polyphenyl chains are aligned along thediagonals of the crystal plane orthogonal to the acetylenechains. This method of construction results in a unit cellwhich, if the system has a geometry where the acetyleneand polyphenyl chains are perfectly straight, the cellparameters a, b and g would be all equal to 908. The crystallattice itself was aligned with the global x-, y- andz-directions in such a way that the c crystal axis is alwaysparallel to the global z-direction, the b crystal axis lies in theyz-plane whilst the a crystal axis is free to orient itselfin any direction. With this orientation, the a, b, c latticevectors would be in the form:

a ¼ ax ay azð Þ;b ¼ 0 by bzð Þ;c ¼ 0 0 czð Þ;

ð4Þ

where ax, by and cz are the projected lengths of the unit cellin the x-, y- and z-directions, respectively (hereafterreferred to as X, Y and Z, respectively) and may beexpressed in terms of the cell parameters as follows:

ax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � acos bð Þð Þ2 � abcos gð Þ � abcos að Þcos bð Þ

bsin gð Þ� �2

s;

ð5Þ

by ¼ bsina; ð6Þ

cz ¼ c; ð7Þ

where a, b and c are the lengths of the lattice vectors a, b andc, respectively whilst a, b and g are the angles betweenthe lattice vectors ‘b and c’, ‘a and c’ and ‘a and b’,respectively.

It should be emphasised that the systems so constructedare not the smallest possible representations of thesemolecular crystals. In fact, an even smaller rhombic unitcell could have been chosen with the polyphenyl chainsbeing placed parallel to the crystallographic axis a and b. Itshould also be noted that throughout this work, allsimulations were always performed on a 2� 2� 2 superlattice of the smallest rectangular unit cell. This unit cellsize was found to be large enough to adequately representthe expected true properties of the system. In fact, evensimulations performed on the smaller 1� 1� 1 rectangularunit cells were found to give comparable results and,therefore, even the smaller unit cell could, in theory, havebeen used. No symmetry constraints were applied, i.e. a P1symmetry was used.

2.2.2 Simulations For each of the systems con-structed, the energy expression was set up using theDreiding force-field [39] as implemented in the Forcitemodule within Materials Studio. The default settings wereused with the exception of the summation method ofthe non-bond interactions, which were summed up using theEwald summation technique [40]. Atomic charges werecalculated as point charges using the charge equilibrationprocedure as developed by Rapp�e et al. [41]. This force-fieldhas been used in earlier studies to study other polypheny-lacetylene systems, in particular their mechanical propertiesand auxetic potential [2–4]. Once the energy expression wasset up, the potential energy of the structure was thenminimised as a function of the atomic coordinates andsix cell parameters at an external pressure p¼ 0.00GPausing a compound minimiser (the SMART minimiser asimplemented in Materials Studio Forcite) which starts off

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with the steepest descent algorithm and then proceeds withother more efficient Adopted Basis Newton-Raphsonmethod, and quasi-Newton methods. Minimization wasconsidered as complete when the RMS gradient was lessthan 0.04 kcalmol�1 A�1.

The elastic constants of the systems were thensimulated through a constant-strain approach using theForcite module within Materials Studio which computesthe 6� 6 compliance matrix S relating an applied stress sto the resultant strain e. For these simulations, fourdistorted structures were generated for each strain patternand the maximum strain amplitude was set to 0.003, i.e.0.3% strain. This low value of maximum applied strain wasused so as to ensure that the stress–strain behaviourremains as much as possible approximated by a linearrelationship. From this computed compliance matrix,the on-axis linear compressibility coefficients bx, by andbz in the x-, y- and z-directions, as well as the areacompressibility bA in the xy-plane and the volumecompressibility bV were calculated as follows:

bx ¼ � 1X

@X@p

� �T

¼ � 1ax

@ax@p

� �T

¼ s11 þ s12 þ s13;

ð8Þ

by ¼ � 1Y

@Y@p

� �T

¼ � 1by

@by@p

� �T

¼ s21 þ s22 þ s23;

ð9Þ

bz ¼ � 1Z

@Z@p

� �T

¼ � 1cz

@cz@p

� �T

¼ s31 þ s32 þ s33;

ð10Þ

bA ¼ 1XY

@XY@p

� �T

¼ 1axby

@axby@p

� �T

¼ s11 þ s22 þ 2s12 þ s13 þ s23;

ð11Þ

bV ¼ 1XYZ

@XYZ@p

� �T

¼ 1axbycz

@axbycz@p

� �T

¼ s11 þ s22 þ s33 þ 2s12 þ 2s13 þ 2s23;

ð12Þ

where p is the pressure, T is the temperature and sij are theelements of the 6� 6 on-axis compliance matrix S. To aidthe discussion, the elastic constants in the xy-plane, i.e. theplane where the system projects in the ‘wine-rack’ motif,were also computed as follows:

Ex ¼ 1s11

; Ey ¼ 1s22

; Gxy ¼ 1s66

;

nxy ¼ � s21s11

; nyx ¼ � s12s22

;ð13Þ

where Ex and Ey are the Young’s moduli in the x- andy-directions respectively, Gxy is the shear modulus in thexy-plane whilst nxy and nyx are the Poisson’s ratio in thexy-plane for loading in the x- and y-directions, respectively.

Furthermore, in an attempt to understand better theeffect of pressure on the crystal structure and compressibil-ity properties of these systems, in particular their potentialto exhibit NLC, the systems so obtained were then re-minimised at various values of externally hydrostaticallyapplied pressure to a maximum of 0.50GPa in incrementsof 0.05GPa. Also, the system at 0.00GPa was then re-minimised at negative pressures to a minimum of �0.50GPa in increments of �0.05GPa, then to a minimum of�3.00GPa in increments of �0.25GPa and then to aminimum of �5.00GPa in increments of �0.50GPa.

Finally, in attempt to confirm that the results obtainedby the Dreiding force-field were not an artefact of the force-field used, additional simulations were also performed onthese systems using other potentials. In particular, thesimulations were repeated using COMPASS and Universalforce-fields using their default settings with the exception ofthe summation method for non-bond interactions whichwere once again summed up using the Ewald summationtechnique. For simulations using COMPASS force-field, theatomic charges were calculated using force-field assignedvalues whilst with the Universal force-field (UFF), thecharges were calculated using the charge equilibration byRapp�e et al. [41].

3 Results and discussion All the simulations werecarried out successfully in the sense that all systemsreached the set convergence criteria. Images of the xy-planeprojected cross-sections of a typical PmAn system atdifferent hydrostatic pressures, as simulated by the Dreidingforce-field, are shown in Fig. 2 whilst plots of X, Y and Z; theprojections of the systems in the x-, y- and z-directions areshown in Fig. 3. (The equivalent plots for the projectionsas obtained by the other force-fields are shown in thesupplementary information.)

These images suggest that for all systems, the networksproject in the xy-plane as a wine-rack like geometry where,at near ambient hydrostatic pressures, the ‘wine-rack’ issemi-closed whilst at high negative pressure, the ‘wine-rack’ appears to be fully-open. This indicates that there is asignificant shrinkage in the y-direction relative to the systemat 0.00GPa as the system opens up under the conditionsof high negative pressure. Although a prima facie thesestructural changes are indicative of a giant negative linearcompressibility, the term can only be loosely applied sincethe changes in the linear dimensions occur very suddenly asevident from Fig. 3, suggesting that this “giant negativelinear compressibility” is in fact resulting from a phasechange. Large changes in linear dimensions associated withpressure-induced phase transitions are not unusual andhave also been observed in naturally occurring crystals. Inthis particular case, it is worth noting that the changesin dimensions, i.e. the contraction in the y-direction



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accompanied by a sudden expansion in the x-direction,result in an overall increase in area (in the xy-plane) andvolume (see Fig. 4) at negative pressures, and is in line withwhat is expected to happen when a system undergoes a‘wine-rack’-like deformation (see Fig. 4).

Here it must be noted that it is also evident that at nearambient pressures, the attractive p–p interactions [42]dominate and result in a conformation where the polyphenyl

chains tend to stack in a manner which optimizes theseinteractions, thus explaining why at such pressures thenetworks mimic a ‘semi-closed wine-rack’. In fact, with noexternally applied pressures, the inter-phenyl layer separa-tion d (see Fig. 1a) as simulated by the Dreiding force-fieldwas approx. 3.45� 0.03 A (see Supporting Information forvariation of the inter-phenyl layer separation with pressure),a separation which is suggestive of p–p stacking andcomparable to that in graphite, 3.35 A [43]. Also, at lowpressures, both positive and negative, such stacking waspreserved. In particular, the inter-phenyl separation in thecase of the Dreiding force-field ranged from ca. 3.39� 0.02Aatp¼ 0.50GPa to3.51� 0.04 A atp¼�0.50GPa.Althoughthese changes are non-negligible, they are insignificantwhen compared to the changes observed at higher valuesof negative pressure, when the networks are in their fullyopen conformation. More specifically, the inter-phenylseparation for the P1An systems was ca. 5.03� 0.01 A at

Figure 2 Projections in the xy-plane ofthe P1A1 network simulated using theDreiding force-field [39] at (i) �5.00GPa,(ii) �2.00GPa, (iii) �0.50GPa, (iv) 0.00GPaand (v) þ0.50GPa (see also animations,Supporting Information).

Figure 3 Variation of the (a) X, (b) Y and (c) Z projections withpressure in the range �5GPa� p�þ0.5GPa for the variousPmAn modelled using the Dreiding force-field.

Figure 4 Variation of the (a) area and (b) volume with pressurein the range �5GPa� p�þ0.5GPa for the various PmAnmodelled with the Dreiding force-field.



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�2.00GPa, a pressure at which the systems had alreadyopened up but not yet in their fully open form, whilst that ofP2An systems was 7.56� 0.03 A. These separations con-tinue to increase slightly further to ca. 5.11� 0.04 A for theP1An systems and 7.74� 0.09 A for the P2An systems at�5.00GPa. At this point, the systems are virtually fullyopened. Beyond this point, the negative compressibilitywould not be observed and instead the system behaves in aconventional manner vis-�a-vis its compressibility. This canbe confirmed by the negative slopes in plots of the dimensionof the unit cell vs. pressure. In fact, the abrupt changes in thecell projections that are occurring at negative pressures, andwhich indicate a pressure-induced phase change, are in linewith the behaviour in inter-phenyl separation. This behaviouris indeed to be expected since, as the extent of negativepressure increases, the forces being applied on the systemwill eventually be greater than those holding the differentlayers together resulting in sudden opening of the system.

It is also interesting to note that the negative pressure atwhich the phase change occurred was different for each ofthe systems modelled, where for the P1An series, thepressure required to open up P1A3 was lower than thatrequired by the P1A2, which in turn was lower than thatrequired by the P1A1. This behaviour was also observed forthe P2An series. This may be explained by the fact that uponincreasing the length of the acetylene chain, the density ofthe systems decreases, as does the density of the p–pinteractions, thus permitting opening of the system at a lessnegative value of the hydrostatic pressure.

Furthermore, it should be highlighted that systemswithin each of the series considered (P1An and P2An),deformed in a similar manner, where it seems that the lengthof the acetylene chain does not markedly affect the extent ofdeformation but only the pressure at which the phase changeoccurs. This is in sharp contrast to what was observed whencomparing systems from the two different series but withthe same n-value. As evident from Fig. 3, the phase changeresulted in very different extent of expansions in thex-directions, with almost doubling in size for the P1Ansystems and a tripling in size for the P2An systems. In fact,comparing the systems at �2.00GPa to those at 0GPa, a(93� 1)% increase in the x-projection was observed for theP1An systems and a (199� 1)% increase for the P2Ansystems. Once again, this type of behaviour was to beexpected and may be explained by looking at the maximumpermitted dimensions of the systems and bearing in mindthat at 0.00GPa, the P1An systems had a geometry wherethe angle u ¼ 136:74o � 0:04o whilst the P2An systems hada geometry where the angle u ¼ 152:19o � 0:04o. For thesesystems, at a pressure of �2.00GPa, u ¼ 90:00o � 0:00o.Comparing this to the idealised model in Fig. 1a, for a givenangle u, the x-projection of the ‘wine-rack’ is given byX ¼ 2lcos u

2

� �so that one would have expected a DX

X of ca.92% as u changes from 136:74o at 0GPa to 90o at�2.00GPa in the case of the P1An systems and ca. 194% asu changes from 152:19o at 0.00GPa to 90o at 0.00GPa in

the case of the P2An systems. These values are well withinthe range of what was really observed, i.e. a (93� 1)% forthe P1An systems and (199� 1)% for the P2An systems.Similar arguments can also be made for the contractionsobserved in the y-projections where, based on the anglesmeasured in the molecular networks, the analyticalmodels [29] predicted a shrinkage of 24 and 27% whereasthe simulations with the Dreading force-field suggested agiant negative linear compressibility resulting in shrinkageof (23� 0) and (26� 0)% for the P1An and the P2Ansystems, respectively when comparing the systems at�2.00GPa to those at 0.00GPa. Similar overall trends,for example, giant contractions and expansions in the x-and y-directions, respectively, upon applications of negativepressure, were obtained by UFF and Compass force-fieldsthus confirming that these results are not artefacts of thepotential used (see Supplementary Information).

All this is very significant as it suggests that by varyingthe length of the acetylene chain and/or the number ofphenyl rings, one may alter and fine-tune the macroscopicproperties of the system, including the pressure at whichthe system undergoes the phase change, and the extent ofchanges in the linear dimensions resulting through thephase-change, thus providing a blueprint for producingmaterials with tailor-made properties. More importantly,the giant change in dimensions that have been predicted totake place once the applied pressure is large enough toovercome the p–p interactions holding the system together,suggests that systems such as these can find applications forthe design of pressure sensors. In this respect, it should benoted that these sudden changes in dimensions are likelyto be accompanied by other changes in the macroscopicproperties, e.g. the conductivity properties.

The behaviour of the systems at near ambient conditionsis also interesting. In fact, on closer inspection, it becomesclear that when low pressures are being applied, the P1Ansystems behaved rather differently from the P2An systems,with the P1An systems exhibiting much more pronouncednegative compressibility properties in the y-direction, asillustrated in Fig. 5. This enhanced NLC of P1An comparedto P2An may be explained, at least partially, throughgeometric arguments. In fact, the P1An systems project inthe xy-plane with a ‘wine-rack’ like geometry where theangle u ¼ 136:74o � 0:08o, giving idealised compressibility

values of bhx ¼ 1

kh1:84 and bhy ¼ � 1

kh0:29, i.e.

bhybhx¼ �0:16

whilst the P2An have a geometry where angle u ¼152:19o � 0:04o resulting in lower idealised compressibility

values of bhx ¼ 1kh3:57 and bhy ¼ � 1

kh0:22, i.e.

bhy

bhx¼ �0:06.

In other words, the idealised model by Grima et al. [29] issuggesting that from a geometry point of view, the P1Anshould exhibit more enhanced negative compressibilityproperties than the P2An systems, something which wasindeed also observed in the simulations (see Table 1 whichlists the simulated mechanical properties at p ¼ 0:00GPaobtained though the constant strain method with small



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strains of up to 0.3% being applied). However, thesimulations also suggest some additional reduction in theextent of negative compressibility and whilst the P1Anand P2A1 systems all exhibited a negative by, negativecompressibility in the y-direction as p ! 0:00GPa for theP2A2 and P2A3 systems could not be confirmed. In fact,the simulations on the P2A2 and P2A3 carried out atdifferent pressures with different force-fields predicteddifferent and more complex trends (see Fig 5) with thebehaviour under positive hydrostatic pressure sometimesfollowing a different trend than that observed at negativepressures. This may be due to the fact that the systems withlonger polyphenyl chains would have more pronounced p–pinteractions thus forcing these nanosystems to behave in amanner which is less dictated by the geometry of sub-units.

In view of the more pronounced negative compressibil-ity exhibited by the P1An systems, the discussion will nowfocus on these systems. For these systems, it was found that

when the P1An systems where subjected to pressures in therange �0:50GPa � p � þ0:50GPa, the systems deform insuch a way that the polyphenyl chains hinge about theacetylene linkages, so that the whole structure behaves onceagain as a nanolevel ‘wine-rack’-like structure. In fact, asillustrated in Fig. 6a, there are changes of c. 28 in the angle ubetween the polyphenyl chains as projected in the xy-plane,which changes are accompanied by less significantchanges in length l (see Fig. 6b), of the polyphenyl chainsthemselves, thus confirming a ‘wine-rack’-type behaviour.As evident from Table 1, there is an overall good agreementbetween the behaviour of these nanonetworks as simulatedby the Dreiding force-fields and the much simpler analyticalmodel based on geometry [29]. Particularly remarkable isthe fact that the magnitude of the compressibility alsoexhibit similar ratios. It is also interesting to note that in allcases, the Young’s moduli in the x-direction are much lowerthan the Young’s moduli in the y-direction, this being inaccordance with the wine-rack model presented else-where [44] as a possible model for non-auxetic foams,although this idealised hinging model had predicted highervalues for the positive Poisson’s ratio for stretching in they-direction, meaning that the deviations from the idealisedhinging ‘wine-rack’ model could be resulting from thisdeviation. This may be explained by the fact that, hadthe idealised behaviour been observed, stretching in they-direction, would have resulted in an excessive shrinkagein the x-direction, and ipso facto, a reduction in theinter-layer separation, something which would have beenopposed by the non-insignficant repulsive part of non-bondinteractions.

All this is very significant as it gives a clearconfirmation that, in analogy with earlier work on designedmaterials with a negative Poisson’s ratio [1–13, 45–49], itshould be possible to design materials that exhibit negativelinear compressibility by ensuring that the nanostructure of

Figure 5 Variation in strains along the (a) x, (b) y and (c) z directions at low pressure values of�0.5GPa� p�þ0.5GPa for the PmAnsystems modelled with the Dreiding force-field.

Table 1 Mechanical properties for the various PmAn polyphe-nylacetylene networks simulated by the Dreiding force-field atpressures close to 0.00GPa.

Young’smoduli(GPa)

Poisson’sratio

compressibility(TPa�1)

structure Ex Ey nxy nyx bx by bz by:bx

P1A1 28.6 380.6 0.16 2.18 29.14 �3.61 1.50 �0.12P1A2 24.3 295.1 0.16 1.99 34.45 �3.88 1.37 �0.11P1A3 21.6 242.1 0.16 1.84 38.82 �3.99 1.18 �0.10P2A1 30.5 604.0 0.06 1.21 30.34 �0.79 2.07 �0.03P2A2 25.4 450.4 0.06 1.11 53.10 2.97 2.33 0.06P2A3 19.0 305.5 0.02 0.39 52.64 1.82 3.77 0.03



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the material has the necessary geometrical features thatenable it to mimic known mechanisms that can achieve thiseffect.

Before concluding, it should be emphasised that thework presented here was based entirely on simulations, anda definite proof whether NLC is indeed exhibited by suchsystems will only be available once such materials aremanufactured and tested. The behaviour of such realmaterials may differ to some extent from what is predictedhere, since in this work we only considered defect freematerials, a level of perfection which may be difficult toachieve in practice. More importantly, it may be necessaryto modify the systems proposed to facilitate synthesis. Anysuch modification is likely to have some effect on themacroscopic properties of the systems and furtherinvestigations may need to be done to recalculate thepredicted compressibility properties. Nevertheless, giventhe excellent agreement between the analytical model andthe properties predicted here, it is more than likely thatthese, or related, materials which are meant to mimic the‘wine-rack’ model would indeed exhibit NLC to someextent or another.

4 Conclusions This work has shown that it is shouldbe possible to design polyphenylacetylene-based materialswhich may exhibit the anomalous property of negativelinear compressibility by mimicking the wine-rack model.It was shown that such materials are likely to be stableas they do not violate the requirement that the volumecompressibility is positive. These materials may also havesome of their macroscopic properties fine-tuned throughchanges in their design.

Supporting Information Additional supporting infor-mation may be found in the online version of this article atthe publisher’s web-site.

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nano networks exhibiting negative linear compressibility

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