J. CHEM. SOC. FARADAY TRANS., 1995, 91(16), 2671-2680 267 1

Auxetic Two-dimensional Polymer Networks An Example of Tailoring Geometry for Specific Mechanical Properties

Ken E. Evans,* Andrew Alderson? and Frances R. Christian School of Engineering, University of Exeter, North Park Road, Exeter, UK EX4 4QF

The Poisson's ratios and Young's moduli of 2D molecular networks having conventional and re-entrant honey- comb forms have been modelled using molecular modelling. Three principle deformation mechanisms were observed : bond hinging, flexure and stretching. Analytical models have also been developed that can be used to describe each of these modes of deformation acting either independently or concurrently. A parametric fit of the force constants in the concurrent analytical model calculations to the molecular model calculations yields good agreement in the mechanical properties for all the structures studied. Specific trends in the force constants required to fit the data are observed. Consequently, a force constants library can be compiled and has been used to predict accurately the properties of more complex variants of the networks. This semi-analytical sub-unit approach enables a more efficient use of computer-intensive molecular modelling programs.

1. Introduction This paper considers one example of how the geometric arrangement of a polymer network may be tailored to achieve specific macroscopic mechanical properties. The example considered here is the development of auxetic molec- ular structures ; polymer networks that exhibit a negative Poisson's ratio. At the moment this work is in its early stages. Some research has been published on the design of novel polymeric networks that exhibit this unusual effect '-' and some work has been progressed on the synthesis of such net- w o r k ~ . ~ , ~ Applications for such materials have also been iden- tified.7-9 As yet, no molecular structure has been modelled, synthesised and characterized, but we expect that this will not be long in coming. The development of a molecular network with a negative Poisson's ratio is an excellent example of understanding form to obtain a particular function. The first examples of structures (as opposed to materials) with auxetic functionality (the ability to expand laterally when stretched) are to be found in the analysis of the mechanical properties of honeycombs.".' ' Here, one further attribute other than form is required to demonstrate a function and that is the mode of deformation. So, in the first analysis," the form, a hexagonal honeycomb, was assumed to deform in a particular manner, by flexure, leading to the mechanical property, a positive Poisson's ratio. By modifying the form, using a re-entrant honeycomb (see Fig. l), but maintaining the same mode of deformation, a negative Poisson's ratio results.

Classical elasticity is not a scale-dependent formulation and none of the simple analyses of structure that have been used to predict negative Poisson's ratios contain any scaling factors. One may therefore assume, naively, that geometric structures and deformation mechanisms that apply at the macrostructural level may also apply at the microstructural and molecular level.

This assumption has been proved, experimentally, in the microstructural case by the synthesis and testing of a novel form of auxetic polyethylene.' 2*1 This paper considers the further use of this approach at the molecular level. Of course, at some length-scale classical mechanics will break down and quantum mechanics take over. However, for the polymer net- works considered here, classical mechanics is still an ade-

f Current address : British Nuclear Fuels plc, Company Research Laboratory, B5 16, Springfields Works, Preston, Lancashire, UK PR4 OXJ.

quate approximation, as embodied in many commonly used molecular mechanics programs, where atomic forces and long-range interactions are modelled by spatial and angular- dependent, classical force constants. These issues have been thoroughly covered e l s e ~ h e r e . ' ~ * ' ~

In the next section we introduce and develop the molecular and analytical models used to describe the deformation of polymer networks having conventional and re-entrant honey-

(a 1

H Fig. 1 Honeycomb cell geometry and co-ordinate system used in the analytical models for (a) conventional and (b) reentrant molecu- lar networks

2672 J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

comb forms. The analytical models are based on the net- works deforming by hinging, flexure and stretching of the honeycomb cell walls, with each mode acting independently of the others in the simplest (single-mode) models. A more complex (multi-mode) analytical model where all three modes act concurrently is then developed. Comparison between the analytical and molecular model predictions of the mechanical properties of the networks is then made and the results dis- cussed accordingly.

2. Models In this section we first describe the molecular model used to predict the Poisson’s ratios and Young’s moduli of the molec- ular network structures and then the analytical models. In all cases expressions for the Poisson’s ratios and Young’s moduli are derived.

2.1 Molecular Model

Examples of the earliest conventional and re-entrant molecu- lar network sub-units designed’,’ are shown in Fig. 2(a) and (b), respectively. The networks consist of branches of acety- lene groups joined by benzene rings at the junctions. Each benzene ring has three polyacetylene arms connected to it. The connectivity of the polyacetylene arms to the benzene rings determines the honeycomb geometry. When the arms are connected to alternate benzene ring carbon atoms the conventional honeycomb geometry is realised, whereas the re-entrant structure is produced when the arms are connected to three adjacent benzene ring carbon atoms. A naming con- vention has been adopted:’ (n,m)-flexyne refers to the conven- tional honeycomb structure, where n and M are the number of acetylene links on the diagonal and vertical branches, respectively. (n,m)-reflexyne refers, similarly, to the re-entrant structure. Hence the sub-units shown in Fig. 2(a) and (b) are (1,4)-flexyne and (1,4)-reflexyne, respectively.

The mechanical properties of these molecular networks were determined using the POLYGRAF molecular modelling programI6 (version 2.20) employing the DREIDING force field,17 which is a well established package for predicting a range of properties for polymeric materials. An infinite system was approximated by periodically extending the repeat unit in the x and y directions. The energy mini- misation process (using the method of conjugate gradients) described in ref. 2 was adopted in this work. For each struc- ture, minimisations were performed for the undeformed struc- ture and for uniaxial applied loads of kO.5 GPa in each of the x and y directions. Previous calculations on a wider range of stress increments have shown that this is sufficient to give accurate results.’ Values of Poisson’s ratio ( v i j ) and Young’s modulus (Ei ) were then evaluated from the stress and strain data with

v i j = - E j / E i

Ei = b i / E i

where bi is the stress applied in the i (=x or y ) direction and E~ and cj are the true strains calculated using

ci = ln(I/Z,) (3)

where I (= X or Y) and I, (= X, and Yo) are the deformed and undeformed repeat unit-cell lengths in the i direction.

2.2 Single-mode Analytical Models

In the derivations that follow we assume unit thickness in the z-direction (perpendicular to the x-y plane).

E

Fig. 2 MolecuIar network sub-units: (a) (1,4)-flexyne, a positive Poisson’s ratio molecular network; (b) (1,4)-reflexyne, a negative Poisson’s ratio molecular network

on conventional and re-entrant honeycomb networks. Hence, if we neglect the detailed molecular structure, it is possible to derive expressions for the elastic properies of these networks using conventional honeycomb theory.’, In this case the network sub-units are represented by the conventional and re-entrant honeycomb cells shown in Fig. l(a) and (b), respec- tively, and the deformation is due to flexure of the diagonal honeycomb arms. The repeat unit-cell lengths in the x and y directions are given by

(4)

Y = 2(h + I sin 0) (5 )

where h and I are the lengths of the vertical and diagonal arms, respectively. 8 is the honeycomb angle which is positive for the conventional cell, and negative for the re-entrant cell, see Fig. 1. The Poisson’s ratios and Young’s moduli associ- ated with these networks have been developed elsewhere” and are quoted directly here:

x = 21 cos e

2.2.1 Flexure Model The molecular networks described in Section 2.1 are based

cos e x V X Y = vy;’ = - -

sin 0 Y

J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

(7)

2673

For an x directed load we have

and

K , is the flexure force constant governing the flexure of the ribs. For ribs of length I, thickness t , depth w and intrinsic Young’s modulus E,, K , is found from standard beam theory” to be

K , = E, wt3/l (9) From eqn. (6) we note that positive Poisson’s ratios are rea- lised owing to flexure when 8 is positive (conventional honeycomb) whereas a negative value of 0 (re-entrant honeycomb) yields negative Poisson’s ratios.

For an orthotropic material to have a symmetric stiffness matrix and a positive definite strain energy for static equi- librium we require’

vxy Ey = vyx Ex and

I v x y I d (EJE,)’”

From eqn. (6)-(8) we have

vxy E, = vyx Ex = K,/lz sin 8 cos 0 (12) and

I vxy I = (EJEy)”2 (13)

Hence the flexure model satisfies the requirements of a sym- metric stiffness matrix [eqn. (lo)] and a positive strain energy Ceqn. (1 113.

2.2.2 Hinging Model We now consider the conventional and re-entrant honey- combs shown in Fig. 1 deforming by hinging of the honey- comb cell walls, i.e. by varying the honeycomb angle 0. From eqn. (4) and (5) we have

dX = -21 sin 0 d8 (14)

dY = 21 cos 0 d8 (15) The increment of true strain in the x direction is defined by

dE, = dX/X (16) with a similar expression for the y direction. In the case of an x-directed load the Poisson’s ratio is defined by

vXy = -d~,,/d&, (17) which holds for non-linear as well as linear elastic behaviour. Giving, from eqn. (4), (5), (16) and (17):

cos 0 x vxy = - -

sin 8 Y To derive the Young’s moduli due to hinging we introduce

a hinging force constant Kh defined in the usual manner by

M = Kha (19) where M is the moment applied to a diagonal honeycomb arm and a is the angular displacement of the arm due to M. M i s given by

M = IF (20) where F is the force applied perpendicular to the diagonal arm of length 1. Hence for an infinitesimal increment in applied force the change in angular displacement is

da = 1 dF/Kh (21)

(23) F = F, sin 8/2 where F, is the force applied in the x direction which, for unit thickness in the z direction, is given by

(24)

(25) d0 = -(IY sin 8/2K,) dux Given the possibility of non-linear elastic behaviour we use the tangent Young’s modulus20

F, = 0, Y

From eqn. (21)-(24) we have

Ex = dadds, (26) From eqn. (14), (16), (25) and (26) the Young’s modulus for unit thickness in the z direction is

Similarly,

sin 0 Y vyx = - -

cos e x and

Note that the hinging-model Poisson’s ratio expressions [eqn. (18) and (28)] are identical to the flexure model expres- sions [eqn. (6)]. Furthermore, the hinging-model Young’s moduli [eqn. (27) and (2911 differ only by the nature of the force constant from those for the flexure model [eqn. (7) and (8), respectively]. Hence eqn. (10) and (1 1) will apply.

2.2.3 Stretching Model The final single-mode analytical model we consider is defor- mation due to stretching of the arms of the honeycomb network shown in Fig. 1, i.e. by varying 1 and h. From eqn. (4) and (5) we have

a x p i = 2 cos e (30)

aY/ah = 2 (3 1)

aY/al= 2 sin 8 (32) The changes in the unit-cell lengths due to infinitesimal increments ds, and ds, in the lengths 1 and h, respectively are then

d x = (ax/azyds, (33)

dY = (aY/dl) ds, + (aY/ah) ds, (34) Consider a load applied in the x direction. In this case the

vertical ribs (length h) remain at constant length since there is no resultant component of the applied force along the length of these ribs, i.e.

ds, = 0 (35)

Therefore, from eqn. (30)-(35) we have

dX = 2 cos 8 ds, (36) dY = 2 sin 8 ds,

and hence the Poisson’s ratio vXy is

-sin 0 X VXY = - -

cos e Y

(37)

2674 J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

The stretching force constant is defined by

F = K , s (39) where F is the force applied along the length of an honey- comb arm of length s. K, can be related to the intrinsic material Young’s modulus and dimensions of the arm. For example, consider the increment of extension ds of an arm of length s, thickness t, depth w and Young’s modulus E, due to an infinitesimal increase d F in the force applied along the length of the arm. From Hooke’s law

Therefore, from eqn. (39) and (40)

K , = E,wt/s (41) For honeycombs consisting of homogeneous or uniform

material then s = I (and ds = ds,) in the case of the diagonal arms at angle 8. However, as will be shown, in the stretching model the molecular structure is more important than it was in the hinging and flexure models. It becomes necessary to consider which parts of the honeycomb network undergo stretching. Molecular model calculations (see Section 2.1) were performed for (2,2)-flexyne for stresses of -2.0, 0 and +2.5 GPa applied along the y direction. It was found that ca. 62.5% of the total extension of the vertical arms was due to extension of the bonds connecting the acetylene arms to the benzene rings. A further ca. 19% of the extension was due to distortion of the benzene rings themselves, with only ca. 19% of the extension due to deformation of the acetylene branches. In other words, ca. 80% of the extension of the honeycomb arms was found to be related to the benzene ring junctions (benzene ring and connecting bond). Hence, the length s in eqn. (41) may not be directly related to either I or h since the molecular structure introduces inhomogeneity into the material forming the honeycomb networks on this length-scale. As a first approximation then, we neglect acety- lene branch elongation and consider all the extension in the stretching model to be due to the benzene ring and connect- ing bond. Hence, since there are equal numbers of benzene ring junctions for the vertical and diagonal arms (two) in all cases, then s is the same in both arms and, therefore, from eqn. (41) the stretching force constant for both types of arm is equal.

In the case of an x-directed load, the force applied along one of the diagonal honeycomb arms is given by

and we have

s = s, (43) where s, is the active length (i.e. benzene ring junction) of the diagonal arm. From eqn. (39), (42) and (43) we have

ds, = (Y cos 8/2K,) do, (44) Hence the increment of strain in the x direction is, from eqn. (W, (301, (33) and (44)

ds, = ( Y COS’ 8/K,X) do, (45) and from eqn. (26) and (45) the Young’s modulus in this case is

Now consider a load applied in the y direction. In this case the vertical ribs also extend, i.e.

dSh # 0 (47)

For a y-directed load the force F, applied along the length of the diagonal ribs is

(48)

F h = Xo, (49)

ds, = (X sin 8/2K,) do, (50)

dsh = (X/K$ do, (51)

F, = (X sin 8/2)o,

and that along the length of the vertical ribs (Fh) is

Therefore, from eqn. (39), (48) and (49) we have

Hence, from eqn. (30)-(34), (50) and (51) the changes in the unit-cell lengths due to an infinitesimal incremental change do, in a y-directed load are

dX = (X cos 8 sin 8/KJ do,

dY = [X(2 + sin’ 8)/K,] do, (52)

(53) and the Poisson’s ratio vyx is, therefore,

cos 8 sin 8 Y (2 + sin’ e) X Vyx = - (54)

The increment of strain in the loading direction is in this case

ds, = [X(2 + sin’ 8)/K, y3 do, (55) and hence the Young’s modulus is

K , y E, = (2 + sin’ 8) x

From eqn. (38) and (54) we notice that the stretching model predicts negative Poisson’s ratios when 8 is positive, whereas positive Poisson’s ratios are realised for negative values of 8. In other words, the stretching model predicts the opposite sign of Poisson’s ratio to the flexure and hinging models. We also note from eqn. (38), (46), (54) and (56)

vXy E , = v,, E , = - K , sin 8/[cos 8(2 + sin’ 8)] (57)

(EJEy)’l’ = [(2 + sin’ 8)1/2X]/(cos 8Y) > I vXy I (58)

and hence the conditions for a symmetric stiffness matrix [eqn. (lo)] and a positive definite strain energy [eqn. (ll)] are satisfied by the stretching model.

and

2.3 Concurrent Analytical Model

In a real molecular structure all three modes of deformation may operate. Hence a multi-mode analytical model for the networks where hinging, flexure and stretching act concur- rently is developed.

In the derivations that follow we once again assume unit- thickness in the z direction.

The total incremental change dX in the unit-cell length X due to an incremental change in the applied load for the con- current model is simply given by the sum of the incremental changes due to each of the individual modes of deformation, i.e.

(59) dX = dXh + dX, + dX, where dX,, dX, and dX, are the incremental changes in X in the hinging, stretching and flexure models, respectively. Simi- larly, the change in Y is given by

dY = dYh + dY, + dY, (60) Consider the case of an x-directed load. We have for the

flexure model :’ dX, = (YZ’ sin’ 8/K,) do,

dY, = -(YZ2 sin 8 cos B/K,) do, (61)

(62)

J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91 2675

In the case of the hinging model we find from eqn. (14), (15) and (25)

(63)

(64)

(65)

(66)

dxh = (Y1’ Sin’ 8/Kh) da,

dY, = -(Y1’ Sin 8 COS 8/Kh) da,

For the stretching model eqn. (36), (37) and (44) yield

dX, = ( Y COS’ 8/Ks) do,

dY, = (Y sin 8 cos O/KJ da,

Substituting eqn. (61), (63) and (65) into (59) gives the total change in X :

dX = Y[(sin’ e/Khf) + (cos’ 8/K,)] do, (67)

where Khf is a function of the hinging and flexure force con- stants, defined by

Khfl = 12(K, + K , ’) (68)

(69)

Eqn. (16) (and the de, equivalent), (17), (26), (68) and (69) give a Poisson’s ratio v,, of

Similarly, from eqn. (60), (62), (64) and (66) we have dY = Y sin 6 cos 0(Ks-’ - Khfl) da,

sin 8 cos O[(Ks/Khf) - 11 X [(K$KM)sin2 8 + 60s’ 01 Y (70) -

vxy =

and a Young’s modulus E x of

Eqn. (70) illustrates that the sign of vxy is now dependent on the geometry (0) and the force constants ratio (KJKhf ) , e.g. negative and positive values of v,, are realised for 8 positive with KJK,, < 1 and >1, respectively. That is, the sign and magnitude of the Poisson’s ratio is both geometry- and mechanism-dependent .

Similarly, for a y-directed load we have

Eqn. (70)-(73) yield

Vxy Ey = VyxEx

(74) K , sin 8 cos 8[(:(K$Khf) - 11 - -

[(K$Khf)sin’ 8 + COS’ el x [(K,/K&OS’ 8 + 2 + sin’ el

and

I v,, I -= (EX/E,)”’ (75) The expressions for the elastic moduli involve the unit-cell

dimensions X and Y which are given by eqn. (4) and (9, respectively, and are dependent on the honeycomb angle 8 and the cell-wall lengths h and 1. In order to relate X and Y to the molecular networks it is necessary to determine h and 1 in terms of the molecular dimensions. We define A and B to be the acetylene and benzene junction ‘lengths’ defined in Fig. 2(a). A is simply the length of one acetylene link. B/2 is the sum of the ‘radius’ of the benzene ring and :he length of that part of the connecting bond not associated with an acetylene link [i.e. where most of the stretching is found to occur, see Section 2.2.3 and Fig. 2(a)]. Hence

l = B + n A (76)

h = B + r n A (77)

and, therefore

3. Results 3.1 Molecular Model Data

Molecular model calculations were performed on a total of 12 structures with n = 1 or 2, 2 < rn d 8 and 8 = +3W. For each structure, energy minimisations were performed for the undeformed (zero stress) configuration and with uniaxial stresses of k0.5 GPa applied in each of the x and y direc- tions. Previous calculations on a wider range of stress increments have shown this to give accurate results.’ The Poisson’s ratio and Young’s modulus data were then calcu- lated following the procedure outlined in Section 2.1. In most cases different values of each elastic constant are realised for tension and compression. This has been discussed elsewhere,’ where strain-dependent effects have been considered. However, in this paper we are concerned only with the elastic moduli of the undeformed state (although the molecular and analytical models can be used to calculate strain-dependent mechanical properties) and so the average of the values due to tension and compression will be compared to the analyti- cal model calculations.

In Table 1 we present the average values of the elastic moduli calculated from the molecular model for each struc- ture, with an error estimated from the difference between tension and compression. We also show the calculated unde- formed unit-cell lengths for each structure. For reasons to become apparent later, the data due to loading in the x direc- tion for the (2,rn)-reflexyne structures were not averaged (i.e. tension and compression in the x direction were considered separately for these structures). The v,, E , us. vyx E x data, evaluated from the average elastic moduli in Table 1, are plotted in Fig. 3. With the exception of the data point corre- sponding to compressive loading in the x-direction for (2,5)- reflexyne, the data are scattered about the equality line within the calculated uncertainties. Hence the molecular model data satisfy the requirement of a symmetric stiffness matrix [eqn. (lo)]. Furthermore, the data are grouped into four clusters with each cluster characterised by the value of n and 8 [i.e. (n, 8) = (1, +30”), (1, -30”), (2, +30”) and (2, -30”)]. This will be discussed later.

It is readily shown that the data in Table 1 also satisfy the condition of a positive definite strain energy for static equi- librium [eqn. (1 l)].

3.2 Comparison of Analytical Models with Molecular Model

The concurrent analytical model Poisson’s ratio expressions can be rearranged to give the force constants ratio K J K , required to fit the analytical model to the molecular model data. From eqn. (70) and (72) we have

Ks/Khf = [(vxy/tan exy/x) -k l1/C1 - ‘xy tan e(y/x)l (79) and

(80) [(2 + sin’ 8)v,, + (Y/x)sin 8 cos 81

[(Y/X)sin e cos e - cosz ev,,] KJKhf =

For each structure four values of KJK,, , required to fit the concurrent analytical model Poisson7s ratio data to the molecular model data, were evaluated using eqn. (79) and (SO), corresponding to tensile and compressive uniaxial loading in each of the x and y directions. With the notable exception of tensile loading in the x direction for the (2,m)- reflexyne structures, the values of K J K , were found to be

2676 J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

TaMe 1 Undeformed unit-cell lengths Xo and Yo (A) are also presented

Average Poisson's ratios (v,, and vyx) and Young's moduli [Ex and E, (GPa)] calculated from the molecular model calculations.

structure vxY VYX EX EY XO YO

(12)-flexyne ( 1,4)-flexyne (1,6)-flexyne (2,2)-flexyne (2,4)-flexyne (2,6)-flex yne ( 1,4)-reflexyne (1,5)-reflexyne ( 1,6)-reflexyne (2,5)-reflexyne

(2,6)-reflexyne

(2,8)-reflexyne

0.46 [6] 0.32 [3] 0.24 [3] 0.85 [2] 0.60 [4] 0.47 [2]

-0.29 [8] -0.29 [S] -0.22 [S] - 0.69" - 1.04' - 0.72" -0.88' - 0.50" - 0.Mb

0.696 16)

0.99 [4] 0.84 [S] 1.11 [S] 1.33 [4]

0.9 [l]

-0.29 [2] -0.386 [2] -0.42 [3] -0.53 [4]

-0.70 [3]

-0.902 [2]

75 c41 56.2 [4] 45 c13 30 C5l 23 c21

124 [63 95 c11

57.4" 44.7' 35.8" 35.8* 3 1.4" 31.5'

19 c41

84 c41

120 [lo] 160 [lo] 220 [40] 30 c31 42 c41 55 c41

21 c41

31 c31

110 [lo]

140 [lo] 116 [6]

40 ClOl

1 1.4709 1 1.4735 11.4671 15.63 14 15.5975 15.6375 11.7017 11.7102 11.7151 15.8858

15.9271

15.9098

24.6685 34.2045 43.7892 27.0193 36.6377 46.1 108 21.3666 26.1698 30.9691 23.8009

28.6706

38.1895

Numbers in square brackets are uncertainties in least significant figures. In the case of the (2,m)-reflexyne structures under loading in the x direction actual values calculated due to compression' and tensionb are quoted (see text).

dependent on the number of acetylene links in the diagonal branches (n) and on the sign of the honeycomb angle (O), i.e. conventional or re-entrant honeycomb, but independent of the number of acetylene links in the vertical branches (m). For the (2,mkreflexyne structures the values of K J K , obtained due to tensile loading in the x direction were found to be significantly different to the values obtained from the other loading conditions.

The mean values of KJK, for each particular combination of n and 8 are given in Table 2. From the molecular model calculations 8 was found to be ca. + 30 and cu. - 30" for the conventional and re-entrant honeycomb geometries, respec- tively, hence these are the values used in Table 2. With the exception of the data for the (2,m)-reflexyne structures under tensile loading in the x direction, the errors associated with each KJK, value in Table 2 were taken to be the standard deviation in the mean of the 12 values for each (n, 0) com- bination since this dominates over the errors inherent in the molecular modelling package. In the case of the data for the (2,m)-reflexyne structures under tension in the x direction, the uncertainty associated with KJK, was calculated by adding the fractional errors in K , and K, (see later) in quad-

80

-60 " I I I I I I I -60 -40 -20 0 20 40 60 80

vwEV IGPa F'ig. 3 v,, E , 0s. vyx Ex from the molecular model calculations. Uncertainties are calculated from the deviation of the elastic moduli in tension and compression from the average value. 0, (1,m)-flexyne; A, (2,m)-flexyne, 0, (2,mbreflexyne (tension in x direction); V,(2,m)- reflexyne (compression in x direction) and 0, (1,m)-reflexyne.

rature since the smaller sample of K$K, values involved in this case (three values instead of 12) make a statistical treat- ment inappropriate.

In all structures K J K , > 1 in Table 2, indicating that hinging and/or flexure dominate over stretching. Stretching is most important for the (1,m)-reflexyne structures since they have the lowest values of K J K , .

The concurrent analytical model Poisson's ratio calcu- lations for the structures modelled in the molecular model calculations are shown in Table 3. The calculations used the library of mean K J K , values in Table 2 and the actual unit- cell lengths determined for the undeformed structures in the molecular model calculations (see Table 1). Also shown in Table 3 are the hinging, flexure and stretching analytical model Poisson's ratio calculations, employing the molecular model unit-cell lengths. As noted in Section 2.2.2 the hinging and flexure model Poisson's ratio expressions are identical and so these models yield the same numerical values.

It is clear from Tables 1 and 3 that the stretching model consistently predicts the incorrect sign of Poisson's ratio. The hinging and flexure models predict the correct sign but con- sistently overestimate the magnitude of the Poisson's ratio (by as much as a factor of three in some cases), indicating that some other mechanism (e.g. stretching) is necessary to reproduce the molecular model data. The concurrent analyti- cal model us. molecular model Poisson's ratios for loading in the x and y directions are shown in Fig. 4(a) and (b), respec- tively. The excellent scatter of the concurrent model calcu- lations about the equality line in Fig. +a) and (b) confirms that the deformation of these structures is principally due to a combination of bond hinging, flexure and stretching. It also indicates that the concurrent model and associated force con- stants library may be used to predict the Poisson's ratios of other (n,m)-flexyne and (n,m)-reflexyne structures, and other related structures built from similar benzene ring and acety- lene link subunits (see later).

The values of K , in the concurrent model can be found by fitting the Young's moduli expressions [eqn. (71) and (7311 to the molecular model Young's moduli in Table 1, using the corresponding K J K M values from the fit to the Poisson's ratio data. K, can then simply be evaluated from the values of K$K, and K, thus obtained. This was done for each loading condition for all the structures considered in the molecular model calculations. Following the same procedure used for the KJK, data, the mean values of K, and K , for each (no) combination were evaluated and are also presented in Table 2.

J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91 2677

Table 2 Mean force constants and B/A [see text and Fig. %a)] data library for concurrent analytical model evaluated from parametric fits to molecular model Poisson's ratio and Young's modulus data for (n,m)-flexyne and (n,m)-reflexyne

1 1 2 2" 2b

+ 30

+ 30 - 30

- 30 - 30

5.6 C1.21 358 [56] 65.1 C9.11 1.776 [0.017] 2.83 C0.57) 281 [57] 101 [20] 1.776 [0.017]

20.7 [6.8] 320 [llo] 15.6 C1.91 1.776 [0.017] 8.9 [1.9] 173 [58] 19.1 C4.81 1.776 [0.017]

37.7 C9.41 697 [41] 19.1 C4.8) 1.776 [0.017]

" Values for loads applied in y direction or compressive loads in x direction. brackets are the estimated standard deviation in the mean value.

Values for tensile load applied in x direction. Numbers in square

Table 3 Analytical model Poisson's ratio calculations for (n,m)-flexyne and (n,m)-reflexyne structures

structure

concurrent stretching hingindflexure

VXY VYX VXY vYx VXY VYX

(1,2)-flexyne (1,4)-flexyne (1,6)-flexyne (2,2)-flexyne (2,4)-flexyne (2,6)-flexyne ( 1,4)-reflexyne ( 1,5)-reflexyne (1,6)-reflexyne (2,5)-reflexyne

(2,6)-reflexy ne

(2,8)-reflexyne

0.43 [ 5 ] 0.31 [4] 0.24 [3] 0.83 [5] 0.61 [4] 0.49 [3]

-0.30 [6] -0.24 [5] -0.21 [4] -0.77 [6]" -1.04 [31b -0.64 [5]" -0.87 [2]' -0.48 [4]" -0.65 [21b

0.66 [8] 0.9 [l] 1.2 [2] 0.83 [5] 1.13 [7] 1.42 [9]

-0.33 [7] -0.41 [9]

-0.58 [5]

-0.69 [6]

-0.92 [8]

-0.5 [l]

- 0.27 -0.19 -0.15 -0.33 - 0.25 - 0.20

0.32 0.26 0.22 0.39

0.32

0.24

-0.41 -0.57 -0.73 -0.33 - 0.45 -0.57 0.35 0.43 0.5 1 0.29

0.35

0.46

0.81 0.58 0.45 1 .oo 0.74 0.59

- 0.95 - 0.78 - 0.66 - 1.16

- 0.96

- 0.72

1.24 1.72 2.20 1 .oo 1.36 1.70

- 1.05 - 1.29 - 1.53 - 0.87

- 1.04

- 1.39

Concurrent model calculations employ K J K , library of Table 2. Molecular model Y / X values were employed in all analytical model calcu- lations. " Compression in x direction; ' tension in x direction. Numbers in square brackets are the estimated uncertainty in the least significant figures.

Considering, first, the data for (1, m)-flexyne, (1,m)-reflexyne and (2,m)-flexyne we see that, within the calculated standard deviation from the mean values, K, remains constant at K, x 320 N m-'. However, the value of K,, varies greatly between these three (no) combinations, leading to the trends observed in the KJK, data. Hence, in these cases changing the geometry primarily affects the hinging/flexure force coeffi- cient K, . This will be discussed later.

In the case of the (2,m)-reflexyne data, K , showed similar variation about a mean value for all loading conditions, as observed for the other structures. Hence in Table 2 we show the mean K, value (with associated standard deviation) for all loading conditions. K , , however, showed a significant increase when a tensile load was applied in the x direction compared with the other loading conditions, for the (2,m)- reflexyne networks. Hence two values of K, are given in Table 2 for (no) = (2, -30"); these being the mean of the K, values for a tensile load in the x direction, and the mean of the K, values for all other loading conditions.

In Table 4 we present the concurrent analytical model Young's modulus calculations for the networks considered in this paper, using eqn. (71) and (73) employing the Y/X ratio from the molecular model calculations and KJK, and K, values from Table 2. These are plotted against the molecular model calculations (Table 1) in Fig. 5(a) and (b) for loading in the x and y directions, respectively. Once again, excellent agreement is achieved in both cases. One might expect rea- sonable agreement since the force constants used in the con- current model calculations were the mean of those derived from a fit to the molecular model data for each structure. However, the fact that good agreement is achieved in all the Poisson's ratio and Young's modulus data for uniaxial loading in both principal directions indicates that the concur-

rent model contains the essential features of the deformation of these structures. (The Young's modulus expressions for the single-mode analytical models cannot be fitted to the molecu- lar model data using the same value of the appropriate force constant for loading in both the x and y directions.) Further- more, the agreement in the data indicates that more complex networks can be assembled and their properties predicted to reasonable accuracy using the concurrent model and the associated force constants library (Table 2), see later.

The force constants trends also explain why the molecular

Table 4 Concurrent analytical model Young's modulus calculations for (n,m)-flexyne and (n,m)-reflexyne, employing the K$KM and K , library of Table 2

structure EJGPa Ey/GPa

(1,2)-flexyne ( 1,4)-flexyne (1,6)-flexyne (2,2)-flexyne (2,4)-flex yne (2,6)-flex yne (1,4)-reflexyne (1,5)-reflexyne (1,6)-reflexyne (2,5)-reflexyne

(2,6)-reflexyne

(2,8)-reflexyne

77 c11 56 c11 43.6 [0.8] 31 c21 23 c11 18 c11

110 [lo]

73 C81 39 171" 46 [8Ib 32 [6]" 38 [7Ib 24 [4]" 29 [5Ib

86 c91

119 [2] 166 [3] 212 [4]

31 C2I

120 [lo] 140 [20]

42 c31 53 c31

29 c51 170 [20]

35 C61

47 C81

Molecular model Y / X values were used in the calculations for all structures. " Compression in x direction; ' tension in x direction. Numbers in square brackets are the estimated uncertainty in each value.

2678 J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

1.2

0.8

0.4 n - a ._ ; 0.0 - a s -0.4 2

-0.8

-1 .2

- 1.6 -

1.6

1.2

0.8 n - .- 0.4

a - s 5 0.0 Si

v

-0.4

-0.8

1.6 -1 .2 -0.8 -0.4 0.0 0.4 0.8 1.2 vxy ( m o I ec u I a r m ode I I i n g )

-0.8 -0.4 0.0 0.4 0.8 1.2 1 .6 vyx (molecular modelling)

Fig. 4 (a) Concurrent analytical model vxy us. molecular model vxy for (n,m)-flexyne and (n,m)-reflexyne structures. (b) Concurrent ana- lytical model vyx us. molecular model vyx for (n,rn)-flexyne and (n,m)- reflex yne.

model vxy E , us. vyx E x data are grouped into four sets accord- ing to their (n,8) combination (Fig. 3). From eqn. (74) we see that ~ , , E , ( = v , ~ E ~ ) is dependent on K,/K, , K, and 8 only, and is independent of Y / X . Therefore, the concurrent analyti- cal model predicts the value of v x y E y to be constant for a given (n,O) combination, irrespective of m, as observed in the molecular model.

3.3 Model Predictions for More Complex Networks

Attempts to synthesise networks of the type so far discussed indicate that one or more benzene rings need to be incorpor- ated into the vertical branches in order for the networks to remain in a stable configuration.2'*22 The force constants

trends established in this paper indicate that the mechanical properties of these more complex networks can be modelled to a reasonable degree of accuracy by the concurrent analyti- cal model. For example, we consider here the addition of an extra benzene ring, midway along the vertical arms [referred to here as stabilised (n,m)-flexyne and stabilised (n,m)- reflexyne, where m is now the number of acetylene links between a junction benzene ring and the non-junction benzene ring in the vertical branches].

Incorporating an extra benzene ring into the vertical poly- acetylene branches introduces an extra component of stretch- ing into the vertical branches. Following the procedure outlined in Section 2.3, the elastic constants of stabilised net- works can be derived. It is found that the expressions for vXy

and E, remain the same as for the simpler structures, i.e. eqn. (70) and (71), respectively. However, the unit-cell lengths ratio to be employed in these equations for the stabilised structures is now

Y (2[(B/A) + rn]/[(B/A) + n] + sin 01 (81)

For the case of loading in the y direction it is easily shown that

-- - X cos e

sin 8 cos 8[(K$KM) - 13 Y vyx = [ ( K , / K ~ ) c o s ~ e + 4 + sin2 el X (82)

Y (83)

where Y / X is again given by eqn. (81). When n = rn the expression for the unit-cell lengths ratio [eqn. (8 l)] becomes trivial, depending only on 8. In order to predict the elastic constants for the more complex networks when n is not equal to rn, however, it is necessary to know the value of B/A to be used in eqn. (81). This can be found by returning to the simpler structures modelled earlier. For each structure mod- elled in Table 1, B/A was calculated by substituting the unde- formed unit-cell lengths, given in Table 1, from the molecular model calculations into eqn. (78). n, rn and 8 are known for each structure, enabling B / A to be determined. Unlike the force constants, no discernible trend in B / A with n or 8 was observed, the value remaining approximately constant for all structures. A mean value of B/A = 1.776 _+ 0.017 was found for the simpler structures modelled in this paper, with the calculated uncertainty being the estimated standard deviation in the mean value. This would be the value of B/A to use in the calculations for the stabilised structures when n is not equal to m. However, in this paper we will only consider the case of stabilised structures with n = rn, which leads to a can- cellation of the B / A terms in eqn. (81).

Table 5 contains the unit-cell lengths ratio, Poisson's ratio and Young's modulus calculations from the concurrent ana- lytical model for some stabilised structures. No uncertainty in Y / X is quoted since there is no dependence on B/A in these cases. The corresponding molecular model data are also shown in Table 5. It is important to note that the concurrent analytical model calculations use the force constants estab- lished from the earlier comparison between the concurrent analytical model and molecular model calculations on the simpler (n,rn)-flexyne and (n,m)-reflexyne structures (Table 2). There has been no fitting of the concurrent model data to the molecular model data for the stabilised structures in Table 5.

The values of Y / X from the molecular model are predicted to two decimal places by the concurrent analytical model. This level of agreement is easily acceptable when considering the errors associated with the force constants to be employed in the concurrent analytical model (Table 2). Excellent agree- ment is found in the elastic moduli for each structure, con- firming that the concurrent analytical model and its associ- ated force constants library can be used to model more complex molecular networks accurately. This enables more efficient and effective use of the computer intensive molecular modelling package for the modelling of molecular networks with specific mechanical properties.

In order to test the validity of using a fixed value of B/A to calculate the unit-cell lengths ratio from the concurrent ana- lytical model when n # rn, the undeformed structure for sta- bilised (2,l)-reflexyne was determined using the molecular model program. A value of Y / X = 1.112 was found for this structure. Employing the mean value of B / A = 1.776 (established from the earlier work on the simple structures- see above) in the concurrent analytical model yields

E , = Ks - [ ( K ~ / K ~ ) c o s ~ 8 + 4 + sin2 81 x

J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91 2679

L

I I I I I I I

0 20 40 60 80 100 120 140 Edmolecular modelling)/GPa

m

Q)

160

0

m m

-

v lui 80 -

40 -

0 40 80 120 160 200 240 280 Ej(molecular modelling)/GPa

Fig. 5 (a) Concurrent analytical model E , us. molecular model E x data for (n,rn)-flexyne and (n,rn)-reflexyne structures. (b) Concurrent analytical model Ey us, molecular model Ey data for (n,rn)-flexyne and (n,rn)-reflexyne structures.

Y/X = 1.120 f 0.003. Hence the level of agreement in Y / X for n # rn is of the same order as that for n = rn (see Table 5).

4. Discussion We have modelled 2D molecular networks based on conven- tional and re-entrant honeycomb cells, using molecular mod- elling and analytical modelling techniques. The mechanical properties of these molecular networks are determined by

their geometry and the behaviour of the sub-unit. The network can be tailored by modifying the sub-unit, or by altering the connectivity of the network. Most interestingly, the force constants required to fit the concurrent analytical model data to the molecular model data were found to be directly related to the particular geometry of the network (see Table 2). This enabled accurate predictions to be made of the mechanical properties of the more complex stabilised struc- tures using a concurrent analytical model, without recourse to a full molecular modelling computation. We now consider why these force constants trends are observed.

So far we have used the analytical model Young's modulus expressions for unit-thickness in the z direction. Hence one possible source of deviation in the force constants is that the value of the unit-cell length perpendicular to the x-y plane, i.e. Z, may be different for different (n,@ combinations. However, the molecular model calculations yield essentially planar structures [slight deviation from planarity being observed only for (2,5)-reflexyne], having similar values of 2 for all 12 simple structures, with a mean value of Z = 3.572 & 0.015 A. Hence the observed force constants trends cannot be attributed to changes in 2.

In the case of the (n,rn)-flexyne structures, increasing n from 1 to 2 reduces the mean value of the hingindflexure force constant, K,, from 65.1 & 9.1 to 15.6 & 1.9 N m-' (see Table 2), i.e. by a factor of 4.2 & 0.8. For (n,rn)-reflexyne K , decreases from 101 & 20 to 19.1 & 4.8 N m-l as n increases from 1 to 2, i.e. a reduction in K, by a factor of 5.3 & 1.7, which is consistent with the value obtained from the (n,rn)- flexyne data. From the definition of K,, [eqn. (68)] we see that K, is inversely proportional to the square of the length of the diagonal honeycomb arm (i.e. K, proportional to l-'). However, for the molecular structures considered here, this length should be modified to be the length of the diagonal acetylene branches since flexure of the benzene rings is unlikely to occur and hinging occurs about the benzene ring (rather than hinging of the benzene ring itself). Hence doub- ling the length of the diagonal acetylene branch by increasing n from 1 to 2 should yield a decrease in K, by a factor of 4 (= 2*). In fact a factor of greater than 4 would be expected since K , is also inversely proportional to the length of the diagonal acetylene branches [see eqn. (9) and (68)].

The effect of varying n on the value of K , appears to be negligible, within the accuracy of the data of Table 2. K, = 358 f 56 and 320 & 110 N m-' for (1,rn)-flexyne and (2,rn)- flexyne, respectively, and hence K , remains constant as n increases from 1 to 2 within the calculated uncertainties. A similar comparison cannot be made for the (n,rn)-reflexyne structures due to the anisotropy observed in K , for (2,rn)- reflexyne under tensile and compressive loading in the x direction.

Table 5 Concurrent analytical model and molecular model undeformed unit-cell lengths ratio, Poisson's ratio and Young's modulus data for stabilised (n,rn)-flexyne and (n,rn)-reflexyne structures

model structure y / x V X Y vYx EJGPa E,/GPa

concurrent stabilised (1,l)-flexyne analytical stabilised (1,l)-reflexyne

stabilised (2,2)-flexyne stabilised (2,2)-reflexyne

molecular stabilised (1,l)-flexyne stabilised (1,l)-reflexyne stabilised (2,2)-flexyne stabilised (2,2)-reflexyne

2.8868 1.7321 2.8868 1.7321

2.8896 1.7377 2.8933 1.7335

0.32 [4]

0.50 [3] -0.31 [7]

-0.66 [6]" -0.90 [2]b

0.34 [l]

0.49 [4] -0.36 [l]

- 0.85" -0.82'

0.7 [l]

1.3 [l] -0.22 [ S ]

-0.54 [6]

0.74 [l]

1.26 [S] -0.24 [l]

-0.67 [ S ]

58 c11 110 [lo] 19 c11 34 [6]" 40 [7ib

19 c31

59 c21 120 [20]

3 1" 43'

122 [6] 80 [lo] 47 c41 27 [GI

123 [4] 90 [lo]

27 PI 49 c71

" Compression in x direction; ' tension in x direction, Numbers in square brackets are estimated uncertainty in least significant figures.

2680 J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

Considering now the effect of changing 8 from positive to negative, we see that for n = 1 an increase in K, from 65.1 f 9.1 to 101 k 20 N m-l is observed for 0 = +30 and -3o", respectively, which is an increase by a factor of 1.55 k 0.38. A corresponding decrease in K, (from K , = 358

56 to 281 & 57 N m-') by a factor of 0.78 f 0.20 is also observed. In the case of the n = 2 structures K, increases from 15.6 & 1.9 to 19.1 +_ 4.8 N m-' as 8 changes from +30 to -3O", i.e. K , increases by a factor of 1.22 f 0.34 which agrees within the calculated uncertainties with the value obtained above for the n = 1 structures. Again the anisotropy in K, for the (2,m)-reflexyne structures renders a comparison of the K, values meaningless in this case.

The increase in K, as 8 changes from +30 to -30" indi- cates that hinging or flexure becomes more difficult to achieve for the re-entrant structure. It is reasonable to assume that the increase in K, is due to hinging since flexure occurs along the acetylene arms whereas hinging occurs at the benzene junction and it is the connectivity at the benzene junction that is being investigated here. The reason for the increase in K, is not clear. Naively, we may expect (n,m)- flexyne structures to hinge more easily (and hence have a lower K, value, as observed) than (n,m)-reflexyne structures since there is an intermediate benzene ring carbon atom about which to hinge in the conventional honeycomb geometry [see Fig. 2(a) and (b)].

The slight decrease observed in K, as 8 changes from +30 to -30" for n = 1 structures may be due to the influence of non-bonded interactions. It is known, for example, that the value of the stretching force constant of the C-C bond is reduced by a factor of ca. 5 for diamond owing to the pres- ence of non-bonded interaction^.^^ It is clear from Fig. 2(a) and (b) that the bonds connecting the polyacetylene branches to the junction benzene rings are substantially more crowded (leading to an increased number of non-bonded interactions) for the (n,m)-reflexyne structures than for the (n,m)-flexyne structures. These bonds have been found to make the largest contribution to the stretching mode of deformation (see earlier) and hence a reduction in K, would be expected for (n,m)-reflexyne structures.

In the case of (2,m)-reflexyne, however, the value of K, was found to be dependent on the loading condition (see Table 2). This is probably due to the concurrent model not containing all the deformation mechanisms in this case. It appears that when n = 2 the benzene rings at the re-entrant junctions become close enough to each other to interact in such a way as to introduce other modes of deformation in the molecular model calculations. This is consistent with the observation of the slight deviation from planarity of the (2,5)-reflexyne struc- ture.

5. Conclusions The mechanical properties of 2D molecular networks, having either positive or negative Poisson's ratios, have been calcu- lated using molecular modelling and analytical modelling techniques. The molecular networks are based on conven- tional and re-entrant honeycomb sub-units and consist of polyacetylene branches connected by benzene rings at the junctions. An analytical model employing flexure, hinging and stretching modes of deformation acting concurrently has been found to produce excellent agreement with the molecu- lar model Poisson's ratio and Young's modulus data when a parametric fit of the force constants governing these deforma- tion mechanisms is performed. The force constants required to fit the concurrent analytical model to the molecular model are found to follow certain trends, being dependent on the number of acetylene links in the diagonal branches (n) and on

whether the honeycomb angle (0) is positive or negative. The force constants trends enable a library of force constants to be compiled for different (n,0) combinations, which can then be used to predict the properties of other molecular networks of the type considered here. The concurrent analytical model can be extended to cover more complex variants of these net- works. We developed the concurrent analytical model to describe networks having an extra benzene ring in the verti- cal branches of the sub-units. The mechanical properties of these stabilised structures predicted from the concurrent ana- lytical model and the associated force constants library (established from the earlier work on the simpler structures) were found to be in good agreement with the molecular model calculations. In principle the properties of even more complex networks (2D and 3D) can be modelled, as a first approximation, using this concurrent analytical model approach. This would enable more efficient use of molecular modelling programs, providing first-order estimates of properties having first obtained the properties of the appro- priate sub-units. Hence, better selection of the network struc- tures to be modelled using full molecular modelling techniques should be possible.

The authors acknowledge the support of the Engineering and Physical Sciences Research Council of the United Kingdom and of Oxford Materials Ltd. K.E.E. currently holds an EPSRC Advanced Fellowship.

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Paper 5/013251; Received 3rd March, 1995