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On the behaviour of natrolite under hydrostatic pressureContents
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Journal of Non-Crystalline Solids

j ourna l homepage: www.e lsev ie r.com/ locate / jnoncryso l

On the behaviour of natrolite under hydrostatic pressure

Joseph N. Grima , Ruben Gatt Department of Chemistry, University of Malta, Msida MSD 2080, Malta

Corresponding author. E-mail address: [email protected] (J.N. Grim

0022-3093/$ – see front matter © 2010 Published by E doi:10.1016/j.jnoncrysol.2010.05.069

a b s t r a c t

a r t i c l e i n f o

Article history: Received 21 May 2010 Available online 12 August 2010

Keywords: Auxetic; Mechanical properties; Negative Poisson's ratio; Silicates; Zeolites; Natrolite

Natrolite (NAT) is a zeolite which exhibits negative Poisson's ratios (auxetic) in its (001) plane (i.e. expands laterally when uniaxially stretched and becomes narrower when compressed), a property which has now been verified experimentally. It was recently shown that the empty SiO2 equivalent of the NAT framework is more auxetic if it is tested in conditions of positive external hydrostatic pressures. Here, we report results of additional simulations which show that the extent of auxetic behaviour of NAT having the standard composition (i.e. Na2(Al2Si3O10).2H2O) is also highly dependent on the magnitude of external pressure where it is tested. However, in contrast with the behaviour of its empty silica equivalent, NAT exhibits maximum auxetic behaviour at negative external hydrostatic pressures (i.e. partial vacuum conditions). This is very significant as it highlights the fact that auxetic behaviour may be more prominent in materials if they are tested at non-ambient pressure conditions. An attempt is made to explain this behaviour.

a).

1. Introduction

Auxetic materials (i.e. materials having a negative Poisson's ratio, NPR) exhibit the unusual property of expanding when stretched and getting thinner when compressed [1–5]. Although this type of behaviour is not commonly encountered in everyday materials, in recent years there has been a considerable effort aimed at identifying naturally occurring auxetics and to design new man-made ones [2– 59]. Examples of auxetics identified so far range frommicrostructured materials (e.g. microporous polymers [10–12] and foams [13–18]) to molecular-level auxetics such as polymers [1,19–23], metals [24], zeolites [29–31] and silicates [32–41]. These auxetic materials are characterised by very particular geometries (at the nano-, micro-, or macro-structural level) that permit NPR through specific deformation mechanisms. In recent years, a number of studies [13–18,25,27,29– 31,42–46] have proposed various two or three dimensional models which can explain the occurrence of negative Poisson's ratios.

A class of auxetic materials that has gathered much interest in recent years is that of zeolites [25–31,47]. Evidence of auxetic behaviour in the zeolite class of materials was first reported by Grima, Alderson and Evans in 1999 [35], where it was noted that simulations on the empty SiO2 equivalents of known zeolite frame- works had resulted in suggestions that various zeolite frameworks (including zeolites in the NAT group) may exhibit auxetic behaviour in particular crystallographic planes. These predictions were followed by further theoretical and experimental work which provided further evidence to this claim where it was also shown that even the actual

zeolites (i.e. in synthesised or natural form) can exhibit NPR. In particular, experimental work using a Brillouin light scattering technique, determined that NAT crystals exhibit NPR which is at a maximum in the (001) plane for loading at approximately 45° off-axis [26,28]. Modelling studies on NAT-type systems have also shown that this NPR in the (001)-plane may be explained in terms of 2D models based on semi-rigid rotating squares/parallelograms [27,29,31,43,46, 47,50–52].

Although it is well known that pressure and/or temperature can affect the magnitude of the Poisson's ratio [6,8,39,53–57], very little work has as yet been performed to assess the effect of external hydrostatic pressure on the elastic constants of zeolite-type frame- works. In fact, studies have only so far considered the SiO2 equivalent of the zeolites NAT (natrolite), THO (Thomsonite) and EDI (Edinto- nite) rather the actual crystals where it was shown that these hypothetical frameworks in the natrolite group exhibit maximum NPR when they are subjected to a positive external hydrostatic pressures which were predicted to be approximately 2–8% of the bulk modulus, rather than at ambient pressure conditions [59]. Although this study highlights the fact that the extent of auxetic behaviour exhibited by materials may be highly dependent on the external pressure conditions that thematerial is being subjected to, it is limited in scope due to the fact that the study was performed on hypothetical SiO2 frameworks rather than crystals having the standard composi- tion (i.e. in the presence of intra-framework cations and water molecules).

In view of this, here we assess how NAT crystals having the standard composition respond to changes in the external hydrostatic pressure so as to establish whether the behaviour predicted for the SiO2 equivalent (NAT_SI) is likely to be manifested by the actual zeolite crystals.

2. Method

Simulations were performed on NAT crystals having the compo- sition Na2(Al2Si3O10).2H2O, Fdd2 symmetry (i.e. the aluminosilicate framework in the presence of cations or water molecules), henceforth referred to as NAT_CW using the Cerius2 V4.1 molecular modelling package running on an SGI Octane 2 workstation. The NAT framework was aligned in space in such a way that the [001] direction of the crystal was fixed parallel to the Z-direction whilst the [010] direction was fixed to lie in the YZ plane. Energy expressions were set using the force-fields based on the Burchart and the CVFF 300 force-fields modified as discussed in [51]. All non-bond terms were summed up using the Ewald summation technique [60]. For both force-fields, a series of energyminimisations using the SMART compoundminimiser were carried out to the default Cerius2 high convergence criteria whilst the system was subjected to both positive and negative external hydrostatic pressures P, i.e. whilst the system was subjected to stresses defined by:

σij = − P 0 0 0 P 0 0 0 P

0 @

1 A:

Fig. 1. The cell parameters, Young's moduli and Poisson's ratio for NAT_CW and NAT_Si unde cell parameters, the cell angles α, β and γ were found to remain approximately constant at

For both sets of simulations, the mechanical properties, in particular the on-axis stiffness and compliance matrices, the on-axis Young's moduli and the on-axis Poisson's ratio for each of these minimised systems were then calculated using the second derivative method in accordance with the procedure followed by Alderson et al. [39]. The off-axis Young'smoduli and the on-axis Poisson's ratios in the (001) plane were also calculated using standard axis-transformation techniques [61]. These simulationswere then repeated for the all silica zeolite versions using the same force-fields since the earlier study [59] was carried out using the COMPASS force-field rather than the Burchart-Universal force-field.

An attempt was also made to identify the underlying cause for the particular values of the predicted Poisson's ratios at the various pressures P by simulating the atomic level deformations of the NAT systems at various stresses σ at 45° to the X and Y axis (a direction of maximum auxetic behaviour) whilst still being subjected to a hydrostatic pressure P by performing energy minimisations whilst the system is being subjected to the following loading conditions:

σij = − P 0 0 0 P 0 0 0 P

0 @

0 @

1 A:

r various pressures as simulated by the different force-fields. Note that in the case of the 90°.

1883J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

3. Results and discussion

The cell parameters, mechanical properties and various geomet- rical features relating to the (001) projections for the various simulated NAT systems as they vary with pressure P are displayed graphically in Fig. 1 (cell parameters, Young's moduli for loading in the [001] direction and (001) plane at 45° off-axis and the Poisson's ratio in the (001) plane for loading in the (001) plane at 45° off-axis) and Fig. 2 (geometric parameters related to the (001) plane, see Fig. 3 for definition of parameters measured). Also shown in Fig. 4 are some images of the projection of the NAT_CW and NAT_SI systems in the (001) plane at various pressures.

Referring to Fig. 1, the results clearly suggest that for both NAT_CW and NAT_SI [59] irrespective of the force-field used at moderate pressures, the applied pressure results in significant changes in the a and b dimensions of the unit cell and a much lower change in c. This is to be expected given that atmoderate pressures, the Young'smoduli Ex and Ey are significantly lower than Ez. This difference in the modulus can be easily explained by looking at the nanostructure of the NAT

Fig. 2. The absolute values of various (i) dimensions and (ii) angles projected in the (001) pla force-fields. The parameters are defined in Fig. 3.

frameworks which are characterised by rigid columnar shaped units aligned down the [001] directions (hence the high Ez) which are connected together though highly flexible X–O–X (X=Si, Al) bonds which permit deformations when loaded in directions orthogonal to the [001] directions (hence the low Ex and Ey). This arrangements gives rise to the characteristic projected ‘rotating squares’ pattern in the (001) plane where the ‘rigid squares’ are the projections of the columnar units in the (001) plane whilst the flexible X–O–X represent the hinges. The fact that the ‘rotating squares’ mechanism is indeed operating when the systems are placed under moderate hydrostatic pressure is supported not only by the images in Fig. 4, but also from measurements of various projected dimensions and angles. In fact, irrespective of the force-field and/or system used, when various geometrical parameters related to the projected ‘rotating squares’ in the (001) plane were measured, it was found that for NAT_SI (as reported in [59]) and NAT_CW, the application of moderate amounts of pressure resulted in significant changes in the angles between the projected squares (caused by significant changes in the X–O–X hinge angles)when compared to the lengths of the sides and of the diagonals

ne for NAT_CW and NAT_Si as a result of pressure changes as simulated by the different

Fig. 3. The parameters measured and plotted in Fig. 2 and listed in Tables 1 and 2.

1884 J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

of the projected squares, the internal angles of the squares and the angles between the diagonals of the projected squares (see Fig. 2).

However, at higher pressures, other deformations such as dilation and deformation of the squares started to become more dominant. In fact, at the higher values of negative hydrostatic pressures, the projected squares are nearly in their ‘fully open conformation’, (see Fig 4) meaning that the ‘rotating squares’ mechanism which at moderate pressures was giving rise to low moduli in the (001) plane cannot operate. This permits the onset of other deformation mechanisms which result in greater deformations of the ‘projected squares’ as opposed to changes in the angles between the squares.

At this point, it is of interest to analyse the dependency of the mechanical properties (Young's moduli and Poisson's ratios) on pressure. Both For NAT_CW and NAT_SI, the lowest value for the Poisson's ratio in the (001) plane is not observed at ambient conditions (which here corresponds to P=0 GPa) but occurs at a positive hydrostatic pressure in the case of NAT_SI [59] and at a negative hydrostatic pressure in the case of NAT_CW.We also observe that conditions of maximum auxetic behaviour generally correspond to those of minimum values of Ex45 since we observe that in the case of NAT_SI system, the lowest values of the moduli occur at positive hydrostatic pressure, whilst for NAT_CW, the lowest value of the moduli are observed at a negative hydrostatic pressure. All this can be easily explained through the fact that the lowering Young's modulus in the direction of loading is an indication that the ‘rotating squares mechanism’ is operating more efficiently which in turn is responsible for more negative values of the Poisson's ratios. Also it is important to note that both force-field predict that the drop in Young's modulus of NAT_CW is much larger than the drop in modulus of NAT_SI, as is the increase in the extent of auxetic behaviour (there is a bigger increase in the auxetic behaviour as a result of the change in the external pressure conditions in the case of NAT_CW than in the case of NAT_SI). However, it should be noted that the range of pressures where negative Poisson's ratios are observed is generally smaller in the case of NAT_CW than in the case of NAT_SI.

Our hypothesis that the results obtained here may be interpreted through a ‘rotating squares’ analysis is further supported by the analysis of the minimised systems which were subjected to a combined hydrostatic pressure and uniaxial stress (see Tables 1 and 2). In fact, herewe found that themost auxetic systems analysed (i.e. in the case of NAT_CW the systems at a pressure P of−3.5 GPa when using the CVFF force-field and−2.0 GPawhen using the Burchart-Universal force-field and in the case of NAT_SI the systems at a pressure P of +1.0 GPawhen

using the CVFF force-field and +0.5 GPa when using the Burchart- Universal force-field) were always characterised with a deformation profile where the ‘rotating squares’ mechanism operates most effec- tively. In fact, in these systemswhere maximum auxetic behaviour was observed, we found that the angles between the squares change to the greatest extent when compared to the other geometrical parameters measured. In all of the other systems we observed that as the auxetic behaviour becomes less pronounced, the squares themselves start to deform to a greater extent when compared to the extent of rotation of the squares. It is also important to note that Tables 1 and 2 suggests that the role of the ‘rotating squares’mechanism ismuchmore evident in the most auxetic NAT_CW systems when compared to NAT_CW at 0PGa than in the case of NAT_SI.

As explained elsewhere [59], this variation of the Poisson's ratio with hydrostatic pressure can be explained in terms of deviations from the idealised ‘rotating rigid squares’ model which predicts Poisson's ratios of νijrot-sqr=−1. Bearing in mind that in real systems the idealised scenario where projected squares behave as perfectly rigid bodies is difficult to accomplish, and that in reality the ‘rotating rigid squares’ having Poisson's ratios νij

rot-sqr=−1 and Young's moduli Ei

rot-sqr are replaced by ‘rotating semi-rigid squares’ where the additional modes of deformations have Poisson's ratio and Young's moduli given by νijother and Ei

other, then the observed ‘total’ Young's moduli and Poisson's ratio νijtotal and Ei

total are given by:

Eotheri

: ð1Þ

All this clearly suggests that in all the systems where the ‘rotating squares’ model is operating, the contribution to the overall Poisson's ratio of the ‘−1 Poisson's ratio’ which arises from the idealised ‘rotating squares’ mechanism decreases as the angle θ between the squares increases and starts to approach 90° (i.e. as the squares approach their most open conformation). In such systems, the total Poisson's ratio is not expected to remain dominated by the auxetic contribution since the Young's moduli of the idealised ‘rotating squares’ structure increases to the extent that, when θ=90°, the Young's moduli for the idealised ‘rotating squares’ mode of deforma- tion would have been infinite (structure is locked) and thus the observed Poisson's ratio and Young's moduli would become equal to νijother and Ei

other respectively (i.e. the rotating squares mechanism

% change in lengths Changes in angles (angles in degrees)

P (GPa) n AnBn BnCn CnDn DnAn AnCn BnDn AnBnCn BnCnDn CnDnAn DnAnBn n Squares θ θ′

Compass −9 1 0.34 0.21 0.34 0.21 0.72 −0.13 0.49 −0.49 0.49 −0.49 −0.08 1–3 0.13 −1.1 3 0.21 0.41 0.21 0.41 0.72 −0.12 0.48 −0.48 0.48 −0.48 0.11 2–3 1.1 −0.13

−3.5 1 0.64 −0.34 0.64 −0.34 0.47 −0.14 0.35 −0.35 0.35 −0.35 −0.56 1–3 5.38 −6.14 3 0.21 0.16 0.21 0.16 0.52 −0.18 0.4 −0.4 0.4 −0.4 −0.03 2–3 6.14 −5.38

0 1 0.48 −0.17 0.48 −0.17 0.4 −0.08 0.27 −0.27 0.27 −0.27 −0.37 1–3 0.45 −1.02 3 −0.09 0.43 −0.09 0.43 0.42 −0.1 0.3 −0.3 0.3 −0.3 0.3 2–3 1.02 −0.45

4 1 0.55 −0.22 0.55 −0.22 0.38 −0.05 0.24 −0.24 0.24 −0.24 −0.44 1–3 0.15 −0.67 3 −0.07 0.4 −0.07 0.4 0.4 −0.08 0.27 −0.27 0.27 −0.27 0.27 2–3 0.67 −0.15

BurchartUniversal −5 1 0.41 0.17 0.41 0.17 0.9 −0.27 0.67 −0.67 0.67 −0.67 −0.14 1–3 −0.2 −1.31 3 0.43 0.48 0.43 0.48 1.16 −0.31 0.85 −0.85 0.85 −0.85 0.03 2–3 1.31 0.2

−2 1 −0.32 0.67 −0.32 0.67 0.94 −0.55 0.85 −0.85 0.85 −0.85 0.57 1–3 5.79 −7.42 3 −1.05 1.85 −1.05 1.85 1.02 −0.33 0.78 −0.78 0.78 −0.78 1.66 2–3 7.42 −5.79

0 1 0.72 −0.43 0.72 −0.43 0.9 −0.5 0.81 −0.81 0.81 −0.81 −0.66 1–3 1.42 −2.99 3 0.05 0.75 0.05 0.75 1.03 −0.32 0.77 −0.77 0.77 −0.77 0.4 2–3 2.99 −1.42

0.5 1 0.67 −0.42 0.67 −0.42 0.92 −0.55 0.85 −0.85 0.85 −0.85 −0.62 1–3 1.2 −2.88 3 −0.03 0.78 −0.03 0.78 1.05 −0.4 0.83 −0.83 0.83 −0.83 0.46 2–3 2.88 −1.2

Fig. 4. Images of NAT_CW and NAT_Si in the (001) plane at various pressures as simulated by the Burchart-Universal and CVFF force-fields.

1885J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

% change in lengths Changes in angles (angles in degrees)

P (GPa) n AnBn BnCn CnDn DnAn AnCn BnDn AnBnCn BnCnDn CnDnAn DnAnBn n Squares θ θ′

CVFF −9.0 1 0.35 0.20 0.35 0.20 0.67 −0.11 0.45 −0.45 0.45 −0.45 −0.08 1–3 0.01 −0.91 3 0.23 0.33 0.23 0.33 0.67 −0.11 0.45 −0.45 0.45 −0.45 0.06 2–3 0.91 −0.01

0.0 1 0.40 −0.07 0.40 −0.07 0.51 −0.17 0.39 −0.39 0.39 −0.39 −0.27 1–3 4.64 −5.42 3 −0.05 0.39 −0.05 0.39 0.52 −0.18 0.40 −0.39 0.40 −0.40 0.25 2–3 5.13 −4.33

1.0 1 0.37 −0.15 0.37 −0.15 0.41 −0.19 0.35 −0.35 0.35 −0.35 −0.30 1–3 5.00 −5.69 3 −0.13 0.37 −0.13 0.37 −0.19 −0.43 0.36 −0.36 0.36 −0.36 0.29 2–3 5.92 −5.20

4.0 1 0.28 −0.19 0.28 −0.19 0.27 −0.18 0.26 −0.26 0.26 −0.26 −0.27 1–3 1.09 −1.61 3 −0.17 0.26 −0.17 0.26 0.27 −0.19 0.26 −0.26 0.26 −0.26 0.25 2–3 1.61 −1.09

Burchart-Universal −4 1 0.28 0.23 0.28 0.23 0.71 −0.20 0.53 −0.53 0.53 −0.53 −0.02 1–3 −0.53 −0.53 3 0.28 0.23 0.28 0.23 0.71 −0.20 0.53 −0.53 0.53 −0.53 −0.02 2–3 0.53 0.53

0 1 0.48 −0.13 0.48 −0.13 0.62 −0.27 0.50 −0.51 0.51 −0.51 −0.35 1–3 8.80 −9.81 3 −0.12 0.47 −0.12 0.47 0.62 −0.27 0.51 −0.51 0.51 −0.51 0.34 2–3 9.81 −8.80

0.5 1 0.55 −0.06 0.55 −0.06 0.66 −0.17 0.47 −0.47 0.47 −0.47 −0.35 1–3 7.34 −8.29 3 −0.05 0.53 −0.05 0.53 0.66 −0.17 0.48 −0.48 0.48 −0.48 0.33 2–3 8.29 −7.34

2.5 1 0.47 −0.08 0.47 −0.08 0.53 −0.14 0.38 −0.38 0.38 −0.38 −0.31 1–3 1.92 −2.69 3 −0.05 0.44 −0.05 0.44 0.53 −0.14 0.39 −0.39 0.39 −0.39 0.28 2–3 2.69 −1.92

1886 J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

plays no role), thus explaining the behaviour of the NAT_CW and NAT_SI systems at higher negative pressures.

The behaviour of the NAT_CW and NAT_SI systems under high positive pressures which are predicted to exhibit a lower extent of auxetic behaviour may be explained by the fact that our molecular systems are much more complex than simple mechanical structures as our systems are in reality composed of atoms which interact with each other, which interactions becomemuchmore pronounced as the separation between the atoms decreases, something which occurs at higher positive pressures. This increase is obviously much more significant in systems where the intra-framework cations and water molecules are present since the distance between the density of NAT_CW is much higher than NAT_SI which makes the NAT_CW systems much more sensitive to non-bond interactions which are stronger in NAT_CW than in NAT_SI as a result of the increased density.

The result of all this is that whilst in the case of NAT_SI this reduction of auxetic behaviour starts to bemanifested at higher values of positive pressure, in the case of NAT_CW, this reduction in auxetic behaviour starts to occur at near ambient pressure since the repulsive interactions which restrict the auxetic behaviour are much more pronounced in the case of NAT_CW. As a consequence, whilst in the case of NAT_SI, maximum auxetic behaviour is predicted to occur at a positive hydrostatic pressure, in the case of NAT_CW, maximum auxetic behaviour is predicted to occur at a negative hydrostatic pressure. This is due to the fact that such conditions (i.e. moderate negative pressure in the case of NAT_CW and moderate positive pressure in the case of NAT_SI) represent optimal conditions for auxetic behaviour, i.e., the best balance between the ‘pro-auxetic’ nature that is a result of the proposed mechanism, and ‘against- auxetic’ nature due to atomic level effects. (The extra framework species present additional hindrance to the ‘rotating squares’ mechanism with the result that moderate values of negative pressure create the void spaces between the projected squares which are required to provide the necessary freedom for the ‘rotating squares’ mechanism to operate.)

Before we conclude we note that although our earlier results for NAT_SI [59] appeared to provide an exceptional case to the hypothesis proposed byWojciechowski et al. [57,58], which states that in general, negative hydrostatic pressure can transform conventional materials into auxetics, our new results for NAT_CW (which correspond to the system which exists in nature) are now in agreement with Wojciechowski's predictions. It is important to highlight the fact that the work presented in this paper is based on predictions made

using force-field based simulations which may not necessarily be mirrored in a quantitative manner in experimental work performed on real zeolite. In fact, it should be pointed out that in general, the systems simulated by the CVFF force-field appear to be stiffer than those predicted by the Burchart-Universal force-field. Nevertheless, the fact that the results obtained are qualitatively consistent, we are confident that what is being observed here bears good resemblance with what is likely to be measured experimentally.

4. Conclusion

This paper has shown that the external hydrostatic pressure has a significant effect on the crystal structure andmechanical properties of natrolite with its standard composition. In particular we have shown that auxetic behaviour is at a maximum at a negative hydrostatic pressure something which contrasts with earlier predictions that showed that the empty SiO2 framework equivalent to NAT exhibits maximum auxetic behaviour at a positive hydrostatic pressure. All this is very significant as it highlights the fact that the lowest values of the Poisson's ratio a system can exhibit may occur at non-ambient conditions and that one should look at both the positive and negative range of hydrostatic pressures if one wants to obtain maximum auxetic behaviour. In otherwords, this work suggests that if one has to analyse the whole pressure–temperature domain, one is likely to find auxetic behaviour in systems which are conventional at ambient conditions. Furthermore, this work has highlighted the importance that extra framework species may have on the observed mechanical properties of crystalline systems since their presence could offer significant hindrance to the mechanism which results in negative Poisson's ratio.

Acknowledgements

The financial support of the Malta Council for Science and Technology through their national RTDI programme is gratefully acknowledged. We also acknowledge the contribution of Ms. Daphne Attard and Mr Victor Zammit of the University of Malta and Mr. Richard N. Cassar.

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On the behaviour of natrolite under hydrostatic pressure

Introduction

Method

Journal of Non-Crystalline Solids

j ourna l homepage: www.e lsev ie r.com/ locate / jnoncryso l

On the behaviour of natrolite under hydrostatic pressure

Joseph N. Grima , Ruben Gatt Department of Chemistry, University of Malta, Msida MSD 2080, Malta

Corresponding author. E-mail address: [email protected] (J.N. Grim

0022-3093/$ – see front matter © 2010 Published by E doi:10.1016/j.jnoncrysol.2010.05.069

a b s t r a c t

a r t i c l e i n f o

Article history: Received 21 May 2010 Available online 12 August 2010

Keywords: Auxetic; Mechanical properties; Negative Poisson's ratio; Silicates; Zeolites; Natrolite

Natrolite (NAT) is a zeolite which exhibits negative Poisson's ratios (auxetic) in its (001) plane (i.e. expands laterally when uniaxially stretched and becomes narrower when compressed), a property which has now been verified experimentally. It was recently shown that the empty SiO2 equivalent of the NAT framework is more auxetic if it is tested in conditions of positive external hydrostatic pressures. Here, we report results of additional simulations which show that the extent of auxetic behaviour of NAT having the standard composition (i.e. Na2(Al2Si3O10).2H2O) is also highly dependent on the magnitude of external pressure where it is tested. However, in contrast with the behaviour of its empty silica equivalent, NAT exhibits maximum auxetic behaviour at negative external hydrostatic pressures (i.e. partial vacuum conditions). This is very significant as it highlights the fact that auxetic behaviour may be more prominent in materials if they are tested at non-ambient pressure conditions. An attempt is made to explain this behaviour.

a).

1. Introduction

Auxetic materials (i.e. materials having a negative Poisson's ratio, NPR) exhibit the unusual property of expanding when stretched and getting thinner when compressed [1–5]. Although this type of behaviour is not commonly encountered in everyday materials, in recent years there has been a considerable effort aimed at identifying naturally occurring auxetics and to design new man-made ones [2– 59]. Examples of auxetics identified so far range frommicrostructured materials (e.g. microporous polymers [10–12] and foams [13–18]) to molecular-level auxetics such as polymers [1,19–23], metals [24], zeolites [29–31] and silicates [32–41]. These auxetic materials are characterised by very particular geometries (at the nano-, micro-, or macro-structural level) that permit NPR through specific deformation mechanisms. In recent years, a number of studies [13–18,25,27,29– 31,42–46] have proposed various two or three dimensional models which can explain the occurrence of negative Poisson's ratios.

A class of auxetic materials that has gathered much interest in recent years is that of zeolites [25–31,47]. Evidence of auxetic behaviour in the zeolite class of materials was first reported by Grima, Alderson and Evans in 1999 [35], where it was noted that simulations on the empty SiO2 equivalents of known zeolite frame- works had resulted in suggestions that various zeolite frameworks (including zeolites in the NAT group) may exhibit auxetic behaviour in particular crystallographic planes. These predictions were followed by further theoretical and experimental work which provided further evidence to this claim where it was also shown that even the actual

zeolites (i.e. in synthesised or natural form) can exhibit NPR. In particular, experimental work using a Brillouin light scattering technique, determined that NAT crystals exhibit NPR which is at a maximum in the (001) plane for loading at approximately 45° off-axis [26,28]. Modelling studies on NAT-type systems have also shown that this NPR in the (001)-plane may be explained in terms of 2D models based on semi-rigid rotating squares/parallelograms [27,29,31,43,46, 47,50–52].

Although it is well known that pressure and/or temperature can affect the magnitude of the Poisson's ratio [6,8,39,53–57], very little work has as yet been performed to assess the effect of external hydrostatic pressure on the elastic constants of zeolite-type frame- works. In fact, studies have only so far considered the SiO2 equivalent of the zeolites NAT (natrolite), THO (Thomsonite) and EDI (Edinto- nite) rather the actual crystals where it was shown that these hypothetical frameworks in the natrolite group exhibit maximum NPR when they are subjected to a positive external hydrostatic pressures which were predicted to be approximately 2–8% of the bulk modulus, rather than at ambient pressure conditions [59]. Although this study highlights the fact that the extent of auxetic behaviour exhibited by materials may be highly dependent on the external pressure conditions that thematerial is being subjected to, it is limited in scope due to the fact that the study was performed on hypothetical SiO2 frameworks rather than crystals having the standard composi- tion (i.e. in the presence of intra-framework cations and water molecules).

In view of this, here we assess how NAT crystals having the standard composition respond to changes in the external hydrostatic pressure so as to establish whether the behaviour predicted for the SiO2 equivalent (NAT_SI) is likely to be manifested by the actual zeolite crystals.

2. Method

Simulations were performed on NAT crystals having the compo- sition Na2(Al2Si3O10).2H2O, Fdd2 symmetry (i.e. the aluminosilicate framework in the presence of cations or water molecules), henceforth referred to as NAT_CW using the Cerius2 V4.1 molecular modelling package running on an SGI Octane 2 workstation. The NAT framework was aligned in space in such a way that the [001] direction of the crystal was fixed parallel to the Z-direction whilst the [010] direction was fixed to lie in the YZ plane. Energy expressions were set using the force-fields based on the Burchart and the CVFF 300 force-fields modified as discussed in [51]. All non-bond terms were summed up using the Ewald summation technique [60]. For both force-fields, a series of energyminimisations using the SMART compoundminimiser were carried out to the default Cerius2 high convergence criteria whilst the system was subjected to both positive and negative external hydrostatic pressures P, i.e. whilst the system was subjected to stresses defined by:

σij = − P 0 0 0 P 0 0 0 P

0 @

1 A:

Fig. 1. The cell parameters, Young's moduli and Poisson's ratio for NAT_CW and NAT_Si unde cell parameters, the cell angles α, β and γ were found to remain approximately constant at

For both sets of simulations, the mechanical properties, in particular the on-axis stiffness and compliance matrices, the on-axis Young's moduli and the on-axis Poisson's ratio for each of these minimised systems were then calculated using the second derivative method in accordance with the procedure followed by Alderson et al. [39]. The off-axis Young'smoduli and the on-axis Poisson's ratios in the (001) plane were also calculated using standard axis-transformation techniques [61]. These simulationswere then repeated for the all silica zeolite versions using the same force-fields since the earlier study [59] was carried out using the COMPASS force-field rather than the Burchart-Universal force-field.

An attempt was also made to identify the underlying cause for the particular values of the predicted Poisson's ratios at the various pressures P by simulating the atomic level deformations of the NAT systems at various stresses σ at 45° to the X and Y axis (a direction of maximum auxetic behaviour) whilst still being subjected to a hydrostatic pressure P by performing energy minimisations whilst the system is being subjected to the following loading conditions:

σij = − P 0 0 0 P 0 0 0 P

0 @

0 @

1 A:

r various pressures as simulated by the different force-fields. Note that in the case of the 90°.

1883J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

3. Results and discussion

The cell parameters, mechanical properties and various geomet- rical features relating to the (001) projections for the various simulated NAT systems as they vary with pressure P are displayed graphically in Fig. 1 (cell parameters, Young's moduli for loading in the [001] direction and (001) plane at 45° off-axis and the Poisson's ratio in the (001) plane for loading in the (001) plane at 45° off-axis) and Fig. 2 (geometric parameters related to the (001) plane, see Fig. 3 for definition of parameters measured). Also shown in Fig. 4 are some images of the projection of the NAT_CW and NAT_SI systems in the (001) plane at various pressures.

Referring to Fig. 1, the results clearly suggest that for both NAT_CW and NAT_SI [59] irrespective of the force-field used at moderate pressures, the applied pressure results in significant changes in the a and b dimensions of the unit cell and a much lower change in c. This is to be expected given that atmoderate pressures, the Young'smoduli Ex and Ey are significantly lower than Ez. This difference in the modulus can be easily explained by looking at the nanostructure of the NAT

Fig. 2. The absolute values of various (i) dimensions and (ii) angles projected in the (001) pla force-fields. The parameters are defined in Fig. 3.

frameworks which are characterised by rigid columnar shaped units aligned down the [001] directions (hence the high Ez) which are connected together though highly flexible X–O–X (X=Si, Al) bonds which permit deformations when loaded in directions orthogonal to the [001] directions (hence the low Ex and Ey). This arrangements gives rise to the characteristic projected ‘rotating squares’ pattern in the (001) plane where the ‘rigid squares’ are the projections of the columnar units in the (001) plane whilst the flexible X–O–X represent the hinges. The fact that the ‘rotating squares’ mechanism is indeed operating when the systems are placed under moderate hydrostatic pressure is supported not only by the images in Fig. 4, but also from measurements of various projected dimensions and angles. In fact, irrespective of the force-field and/or system used, when various geometrical parameters related to the projected ‘rotating squares’ in the (001) plane were measured, it was found that for NAT_SI (as reported in [59]) and NAT_CW, the application of moderate amounts of pressure resulted in significant changes in the angles between the projected squares (caused by significant changes in the X–O–X hinge angles)when compared to the lengths of the sides and of the diagonals

ne for NAT_CW and NAT_Si as a result of pressure changes as simulated by the different

Fig. 3. The parameters measured and plotted in Fig. 2 and listed in Tables 1 and 2.

1884 J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

of the projected squares, the internal angles of the squares and the angles between the diagonals of the projected squares (see Fig. 2).

However, at higher pressures, other deformations such as dilation and deformation of the squares started to become more dominant. In fact, at the higher values of negative hydrostatic pressures, the projected squares are nearly in their ‘fully open conformation’, (see Fig 4) meaning that the ‘rotating squares’ mechanism which at moderate pressures was giving rise to low moduli in the (001) plane cannot operate. This permits the onset of other deformation mechanisms which result in greater deformations of the ‘projected squares’ as opposed to changes in the angles between the squares.

At this point, it is of interest to analyse the dependency of the mechanical properties (Young's moduli and Poisson's ratios) on pressure. Both For NAT_CW and NAT_SI, the lowest value for the Poisson's ratio in the (001) plane is not observed at ambient conditions (which here corresponds to P=0 GPa) but occurs at a positive hydrostatic pressure in the case of NAT_SI [59] and at a negative hydrostatic pressure in the case of NAT_CW.We also observe that conditions of maximum auxetic behaviour generally correspond to those of minimum values of Ex45 since we observe that in the case of NAT_SI system, the lowest values of the moduli occur at positive hydrostatic pressure, whilst for NAT_CW, the lowest value of the moduli are observed at a negative hydrostatic pressure. All this can be easily explained through the fact that the lowering Young's modulus in the direction of loading is an indication that the ‘rotating squares mechanism’ is operating more efficiently which in turn is responsible for more negative values of the Poisson's ratios. Also it is important to note that both force-field predict that the drop in Young's modulus of NAT_CW is much larger than the drop in modulus of NAT_SI, as is the increase in the extent of auxetic behaviour (there is a bigger increase in the auxetic behaviour as a result of the change in the external pressure conditions in the case of NAT_CW than in the case of NAT_SI). However, it should be noted that the range of pressures where negative Poisson's ratios are observed is generally smaller in the case of NAT_CW than in the case of NAT_SI.

Our hypothesis that the results obtained here may be interpreted through a ‘rotating squares’ analysis is further supported by the analysis of the minimised systems which were subjected to a combined hydrostatic pressure and uniaxial stress (see Tables 1 and 2). In fact, herewe found that themost auxetic systems analysed (i.e. in the case of NAT_CW the systems at a pressure P of−3.5 GPa when using the CVFF force-field and−2.0 GPawhen using the Burchart-Universal force-field and in the case of NAT_SI the systems at a pressure P of +1.0 GPawhen

using the CVFF force-field and +0.5 GPa when using the Burchart- Universal force-field) were always characterised with a deformation profile where the ‘rotating squares’ mechanism operates most effec- tively. In fact, in these systemswhere maximum auxetic behaviour was observed, we found that the angles between the squares change to the greatest extent when compared to the other geometrical parameters measured. In all of the other systems we observed that as the auxetic behaviour becomes less pronounced, the squares themselves start to deform to a greater extent when compared to the extent of rotation of the squares. It is also important to note that Tables 1 and 2 suggests that the role of the ‘rotating squares’mechanism ismuchmore evident in the most auxetic NAT_CW systems when compared to NAT_CW at 0PGa than in the case of NAT_SI.

As explained elsewhere [59], this variation of the Poisson's ratio with hydrostatic pressure can be explained in terms of deviations from the idealised ‘rotating rigid squares’ model which predicts Poisson's ratios of νijrot-sqr=−1. Bearing in mind that in real systems the idealised scenario where projected squares behave as perfectly rigid bodies is difficult to accomplish, and that in reality the ‘rotating rigid squares’ having Poisson's ratios νij

rot-sqr=−1 and Young's moduli Ei

rot-sqr are replaced by ‘rotating semi-rigid squares’ where the additional modes of deformations have Poisson's ratio and Young's moduli given by νijother and Ei

other, then the observed ‘total’ Young's moduli and Poisson's ratio νijtotal and Ei

total are given by:

Eotheri

: ð1Þ

All this clearly suggests that in all the systems where the ‘rotating squares’ model is operating, the contribution to the overall Poisson's ratio of the ‘−1 Poisson's ratio’ which arises from the idealised ‘rotating squares’ mechanism decreases as the angle θ between the squares increases and starts to approach 90° (i.e. as the squares approach their most open conformation). In such systems, the total Poisson's ratio is not expected to remain dominated by the auxetic contribution since the Young's moduli of the idealised ‘rotating squares’ structure increases to the extent that, when θ=90°, the Young's moduli for the idealised ‘rotating squares’ mode of deforma- tion would have been infinite (structure is locked) and thus the observed Poisson's ratio and Young's moduli would become equal to νijother and Ei

other respectively (i.e. the rotating squares mechanism

% change in lengths Changes in angles (angles in degrees)

P (GPa) n AnBn BnCn CnDn DnAn AnCn BnDn AnBnCn BnCnDn CnDnAn DnAnBn n Squares θ θ′

Compass −9 1 0.34 0.21 0.34 0.21 0.72 −0.13 0.49 −0.49 0.49 −0.49 −0.08 1–3 0.13 −1.1 3 0.21 0.41 0.21 0.41 0.72 −0.12 0.48 −0.48 0.48 −0.48 0.11 2–3 1.1 −0.13

−3.5 1 0.64 −0.34 0.64 −0.34 0.47 −0.14 0.35 −0.35 0.35 −0.35 −0.56 1–3 5.38 −6.14 3 0.21 0.16 0.21 0.16 0.52 −0.18 0.4 −0.4 0.4 −0.4 −0.03 2–3 6.14 −5.38

0 1 0.48 −0.17 0.48 −0.17 0.4 −0.08 0.27 −0.27 0.27 −0.27 −0.37 1–3 0.45 −1.02 3 −0.09 0.43 −0.09 0.43 0.42 −0.1 0.3 −0.3 0.3 −0.3 0.3 2–3 1.02 −0.45

4 1 0.55 −0.22 0.55 −0.22 0.38 −0.05 0.24 −0.24 0.24 −0.24 −0.44 1–3 0.15 −0.67 3 −0.07 0.4 −0.07 0.4 0.4 −0.08 0.27 −0.27 0.27 −0.27 0.27 2–3 0.67 −0.15

BurchartUniversal −5 1 0.41 0.17 0.41 0.17 0.9 −0.27 0.67 −0.67 0.67 −0.67 −0.14 1–3 −0.2 −1.31 3 0.43 0.48 0.43 0.48 1.16 −0.31 0.85 −0.85 0.85 −0.85 0.03 2–3 1.31 0.2

−2 1 −0.32 0.67 −0.32 0.67 0.94 −0.55 0.85 −0.85 0.85 −0.85 0.57 1–3 5.79 −7.42 3 −1.05 1.85 −1.05 1.85 1.02 −0.33 0.78 −0.78 0.78 −0.78 1.66 2–3 7.42 −5.79

0 1 0.72 −0.43 0.72 −0.43 0.9 −0.5 0.81 −0.81 0.81 −0.81 −0.66 1–3 1.42 −2.99 3 0.05 0.75 0.05 0.75 1.03 −0.32 0.77 −0.77 0.77 −0.77 0.4 2–3 2.99 −1.42

0.5 1 0.67 −0.42 0.67 −0.42 0.92 −0.55 0.85 −0.85 0.85 −0.85 −0.62 1–3 1.2 −2.88 3 −0.03 0.78 −0.03 0.78 1.05 −0.4 0.83 −0.83 0.83 −0.83 0.46 2–3 2.88 −1.2

Fig. 4. Images of NAT_CW and NAT_Si in the (001) plane at various pressures as simulated by the Burchart-Universal and CVFF force-fields.

1885J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

% change in lengths Changes in angles (angles in degrees)

P (GPa) n AnBn BnCn CnDn DnAn AnCn BnDn AnBnCn BnCnDn CnDnAn DnAnBn n Squares θ θ′

CVFF −9.0 1 0.35 0.20 0.35 0.20 0.67 −0.11 0.45 −0.45 0.45 −0.45 −0.08 1–3 0.01 −0.91 3 0.23 0.33 0.23 0.33 0.67 −0.11 0.45 −0.45 0.45 −0.45 0.06 2–3 0.91 −0.01

0.0 1 0.40 −0.07 0.40 −0.07 0.51 −0.17 0.39 −0.39 0.39 −0.39 −0.27 1–3 4.64 −5.42 3 −0.05 0.39 −0.05 0.39 0.52 −0.18 0.40 −0.39 0.40 −0.40 0.25 2–3 5.13 −4.33

1.0 1 0.37 −0.15 0.37 −0.15 0.41 −0.19 0.35 −0.35 0.35 −0.35 −0.30 1–3 5.00 −5.69 3 −0.13 0.37 −0.13 0.37 −0.19 −0.43 0.36 −0.36 0.36 −0.36 0.29 2–3 5.92 −5.20

4.0 1 0.28 −0.19 0.28 −0.19 0.27 −0.18 0.26 −0.26 0.26 −0.26 −0.27 1–3 1.09 −1.61 3 −0.17 0.26 −0.17 0.26 0.27 −0.19 0.26 −0.26 0.26 −0.26 0.25 2–3 1.61 −1.09

Burchart-Universal −4 1 0.28 0.23 0.28 0.23 0.71 −0.20 0.53 −0.53 0.53 −0.53 −0.02 1–3 −0.53 −0.53 3 0.28 0.23 0.28 0.23 0.71 −0.20 0.53 −0.53 0.53 −0.53 −0.02 2–3 0.53 0.53

0 1 0.48 −0.13 0.48 −0.13 0.62 −0.27 0.50 −0.51 0.51 −0.51 −0.35 1–3 8.80 −9.81 3 −0.12 0.47 −0.12 0.47 0.62 −0.27 0.51 −0.51 0.51 −0.51 0.34 2–3 9.81 −8.80

0.5 1 0.55 −0.06 0.55 −0.06 0.66 −0.17 0.47 −0.47 0.47 −0.47 −0.35 1–3 7.34 −8.29 3 −0.05 0.53 −0.05 0.53 0.66 −0.17 0.48 −0.48 0.48 −0.48 0.33 2–3 8.29 −7.34

2.5 1 0.47 −0.08 0.47 −0.08 0.53 −0.14 0.38 −0.38 0.38 −0.38 −0.31 1–3 1.92 −2.69 3 −0.05 0.44 −0.05 0.44 0.53 −0.14 0.39 −0.39 0.39 −0.39 0.28 2–3 2.69 −1.92

1886 J.N. Grima, R. Gatt / Journal of Non-Crystalline Solids 356 (2010) 1881–1887

plays no role), thus explaining the behaviour of the NAT_CW and NAT_SI systems at higher negative pressures.

The behaviour of the NAT_CW and NAT_SI systems under high positive pressures which are predicted to exhibit a lower extent of auxetic behaviour may be explained by the fact that our molecular systems are much more complex than simple mechanical structures as our systems are in reality composed of atoms which interact with each other, which interactions becomemuchmore pronounced as the separation between the atoms decreases, something which occurs at higher positive pressures. This increase is obviously much more significant in systems where the intra-framework cations and water molecules are present since the distance between the density of NAT_CW is much higher than NAT_SI which makes the NAT_CW systems much more sensitive to non-bond interactions which are stronger in NAT_CW than in NAT_SI as a result of the increased density.

The result of all this is that whilst in the case of NAT_SI this reduction of auxetic behaviour starts to bemanifested at higher values of positive pressure, in the case of NAT_CW, this reduction in auxetic behaviour starts to occur at near ambient pressure since the repulsive interactions which restrict the auxetic behaviour are much more pronounced in the case of NAT_CW. As a consequence, whilst in the case of NAT_SI, maximum auxetic behaviour is predicted to occur at a positive hydrostatic pressure, in the case of NAT_CW, maximum auxetic behaviour is predicted to occur at a negative hydrostatic pressure. This is due to the fact that such conditions (i.e. moderate negative pressure in the case of NAT_CW and moderate positive pressure in the case of NAT_SI) represent optimal conditions for auxetic behaviour, i.e., the best balance between the ‘pro-auxetic’ nature that is a result of the proposed mechanism, and ‘against- auxetic’ nature due to atomic level effects. (The extra framework species present additional hindrance to the ‘rotating squares’ mechanism with the result that moderate values of negative pressure create the void spaces between the projected squares which are required to provide the necessary freedom for the ‘rotating squares’ mechanism to operate.)

Before we conclude we note that although our earlier results for NAT_SI [59] appeared to provide an exceptional case to the hypothesis proposed byWojciechowski et al. [57,58], which states that in general, negative hydrostatic pressure can transform conventional materials into auxetics, our new results for NAT_CW (which correspond to the system which exists in nature) are now in agreement with Wojciechowski's predictions. It is important to highlight the fact that the work presented in this paper is based on predictions made

using force-field based simulations which may not necessarily be mirrored in a quantitative manner in experimental work performed on real zeolite. In fact, it should be pointed out that in general, the systems simulated by the CVFF force-field appear to be stiffer than those predicted by the Burchart-Universal force-field. Nevertheless, the fact that the results obtained are qualitatively consistent, we are confident that what is being observed here bears good resemblance with what is likely to be measured experimentally.

4. Conclusion

This paper has shown that the external hydrostatic pressure has a significant effect on the crystal structure andmechanical properties of natrolite with its standard composition. In particular we have shown that auxetic behaviour is at a maximum at a negative hydrostatic pressure something which contrasts with earlier predictions that showed that the empty SiO2 framework equivalent to NAT exhibits maximum auxetic behaviour at a positive hydrostatic pressure. All this is very significant as it highlights the fact that the lowest values of the Poisson's ratio a system can exhibit may occur at non-ambient conditions and that one should look at both the positive and negative range of hydrostatic pressures if one wants to obtain maximum auxetic behaviour. In otherwords, this work suggests that if one has to analyse the whole pressure–temperature domain, one is likely to find auxetic behaviour in systems which are conventional at ambient conditions. Furthermore, this work has highlighted the importance that extra framework species may have on the observed mechanical properties of crystalline systems since their presence could offer significant hindrance to the mechanism which results in negative Poisson's ratio.

Acknowledgements

The financial support of the Malta Council for Science and Technology through their national RTDI programme is gratefully acknowledged. We also acknowledge the contribution of Ms. Daphne Attard and Mr Victor Zammit of the University of Malta and Mr. Richard N. Cassar.

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On the behaviour of natrolite under hydrostatic pressure

Introduction

Method