NLCE properties and physical structure

The films of monodomain side-chain LCE were composed of 6-(4-cyano-biphenyl-4’-yloxy)hexyl acrylate (A6OCB, 34 mol%), 2-ethylhexyl acrylate (EHA, 49 mol%) and 1,4-bis-[4-(6-acryloyloxyhex-yloxy)benzoyloxy]-2-methylbenzene (RM82, 17 mol%) (Fig. 1a) produced as described in ref. 30, further details are given in the methods and Supplementary Fig. 1. This LCE is a modified version of a commonly used LCE, which does not display auxetic behaviour31,32. In addition to introducing auxeticity, the modifications reduced the glass transition temperature of the LCE from 50 °C to 14 °C and ensured a room temperature nematic phase prior to polymerisation30. Monodomain LCEs were produced by polymerising the monomer mixture inside liquid crystal (LC) devices coated with a uniformly oriented planar alignment agent. We have previously shown this material to be optically anisotropic (birefringence, Δn = 0.08) and also to have a highly anisotropic and non-linear stress–strain behaviour30. The LCE films used here were prepared with a thicknesses, d, in the range of 95–105 µm. For mechanical characterisation, strips of dimensions ~2 × 18 mm were cut with the nematic director at 89 ± 1° (s.d., n = 6) to the film long axis (Fig. 1b).

Fig. 1 Composition and macroscopic appearance of the LCE. a Chemical structures of 1,4-bis-[4-(6-acryloyloxyhex-yloxy)benzoyloxy]-2-methylbenzene (RM82), 6-(4-cyano-biphenyl-4’-yloxy)hexyl acrylate (A6OCB) and 2-ethylhexyl acrylate (EHA)—the acrylate monomers used to produce the auxetic side-chain LCE. b Diagram of undeformed sample geometry with liquid crystal director parallel to the y-axis and perpendicular to the film long (x) axis. In this work strains (ε x )/deformations (λ x ) are applied along the x-axis with the auxetic response observed along the z-axis. c Photograph of the final LCE showing its flexibility and high optical quality. d Polarising microscopy images of a unstrained LCE. Scale bar, 5 mm Full size image

As molecular auxetics are expected to have zero porosity down to the molecular length scale it is important to assess the physical structure of our LCE across length scales. By virtue of the LCE being optically transparent (Fig. 1c) we can rule out the presence of pores from millimetre length scales, which would be individually visible, down to approximately the wavelength of visible light, which would strongly scatter light causing the material to appear milky-white33.

The high contrast between the two polarising microscopy images shown in Fig. 2d confirms the high quality monodomain alignment and anisotropy of the LCEs produced. Moreover, the uniformity and black appearance of the sample when the polarisers are aligned with the LC director is further indication of the material homogeneity and lack of any light-scattering geometry down to sub-micron length scales.

Fig. 2 Micro- and nano-scopic structure of the LCE. a, b Cryo-SEM micrographs (scale bars, 2.5 µm and 250 nm, respectively) and c–e AFM height maps (scale bars, 200, 100 and 40 nm, respectively) show the structures present from molecular to microscopic length scales. Vertical colour bars accompanying AFM height maps represent height in nm. Axes used in a–e correspond to the coordinate set from Fig. 1b. f Profile of dashed line drawn across e Full size image

The potential material porosity can also be directly assessed by using scanning electron microscopy (SEM) and atomic force microscopy (AFM) to study cross sections of the LCE films exposed via freeze-fracturing.

Figure 2a, b show SEM images of yz cross sections according to the coordinate system shown in Fig. 1b. The homogeneous and largely featureless structure shown in Fig. 2a confirms our optical assessment of zero porosity down to sub-micron length scales. Although the micrograph with 10x greater magnification (Fig. 2b) reveals a more visible texture, nothing seen is indicative of pores and so it is evident that no porous geometry exists down to at least ~10 nm. In both Fig. 2a, b the only features visible are artefacts of the freeze-fracturing process, which in both cases confirms correct focusing of the exposed cross sections.

AFM, which is more sensitive to topographical features than SEM, allows us to probe the material structure down to the nanometre length scale. From the yz cross section scans of Fig. 2c–e, two features are present: an extremely fine structure on the scale of a few nanometres; and a larger slow undulation on the scale of ~10 nm and with an amplitude of ~2 nm (comparable to the length of a liquid crystal molecule). The textures visible in Fig. 2c, d are consistent with those seen via SEM in Fig. 2b and thus confirm the LCE has zero porosity down to ~10 nm. From Fig. 2e, f we can see that the variation in the surface profile has a typical amplitude of 1 nm. As the AFM tip used had a radius of curvature of 5 nm we can further conclude that no porosity exists down to ~5 nm. From Figs. 1c, d, 2 we have assessed that there is zero detectable porosity down to length scales approaching the typical length of liquid crystalline moieties and of LCE polymer chain radii of gyration34,35,36,37,38. Consequently, the physical behaviours of the LCE we report here are inherent, bulk properties of the material itself. The question of whether porosity emerges on straining the sample is considered further below.

Mechanical tests and auxetic behaviour

We studied the mechanical behaviour of the LCE films using a bespoke miniature tensile rig, which simultaneously also allows observation of the polarising microscopy texture, thus providing an insight into the LCOP30. The films were extended along the x-axis (as illustrated in Fig. 1b) in steps at various extension speeds, based on the percentage increase in sample length per minute relative to the unstrained sample length, L 0 , and at various temperatures as summarised in Table 1. For each mechanical test (each performed on a different sample) the strains \({\it{\epsilon }}_x\) and either \({\it{\epsilon }}_y\) or \({\it{\epsilon }}_z\) were recorded from photographs of the sample. In all but one test the xy plane was observed and the strains \({\it{\epsilon }}_x\) and \({\it{\epsilon }}_y\) measured; from these data \({\it{\epsilon }}_z\) could be inferred using the constant volume condition (Eq. 2, methods). From the pairs of strains, \({\it{\epsilon }}_x\) and \({\it{\epsilon }}_z\) or \({\it{\epsilon }}_x\) and \({\it{\epsilon }}_y\), the instantaneous PRs, v xz or v xy , respectively, were calculated according to the method described by Alderson et al.39 and Smith et al.40 (see equations 3–5, methods).

Table 1 Testing parameters and PR at the maximum extension Full size table

Figure 3 demonstrates how the deformations of the LCE perpendicular to the director are volume conserving and do not induce the formation of any porous microstructure. The two \({\it{\epsilon }}_z - {\it{\epsilon }}_x\) curves shown in Fig. 3a were calculated from separate tests (both under test conditions I) on separate samples where the strains \({\it{\epsilon }}_z\) were measured from direct observation of the xz plane as well as being inferred through application of the constant volume condition to the measured strains \({\it{\epsilon }}_x\) and \({\it{\epsilon }}_y\). The consistency between the direct and inferred measurements of \({\it{\epsilon }}_z\) demonstrates the validity of the constant volume condition for the LCE. As such, we can conclude that the LCE density is constant with strain—a result to be expected for non-porous elastomers that are typified by possessing bulk moduli, which are several orders of magnitude greater than their shear moduli29,41,42,43. Moreover, the volume conserving nature of the material rules out the formation of wrinkles across the width of the films.

Fig. 3 Material volume conservation and microstructure with strain. For samples tested under condition I: a z-axis strains, ε z in response to imposed x-axis strains, ε x , measured via direct observation of the LCE xz plane (green crosses) and through application of the constant volume condition to strain measurements of the xy plane (orange boxes). Errors are measurement errors (n = 1). b Photographs of the xy plane of the LCE showing consistent transparent and non-scattering, or stress-whitened, appearance. Scale bar, 5 mm. c, d Cryo-SEM images of a sample strained to the auxetic regime and freeze fractured to expose a yz cross section. Scale bars, 2.5 µm and 250 nm, respectively Full size image

Figure 3b shows photographs of a sample illuminated via transmitted light at various strain steps of a deformation. As the transparency and apparent brightness of the sample remains constant, we can deduce that no light-scattering (≲400nm) textures (such as cavitation-induced pores or strain-induced crystals) are forming within the material for the range of strains tested32,44. Fig. 3c, d shows a cryo-SEM freeze fracture of the yz plane of a LCE sample deformed into the auxetic region (cryogenic freezing prevented relaxation of the film following the fracture—see Methods). As one might expect for a highly strained sample, the apparent texture of the exposed surface differs compared to the surface seen in Fig. 2a, b for the unstrained sample. In agreement with the conclusions drawn from Fig. 3b, c also shows no evidence of cavitation or strain-induce-crystallisation features on sub-micron length scales. Figure 3d further confirms that in the auxetic region, the LCE remains non-porous as the only features visible are ~20 nm features protruding out of, and into the exposed surface. These conclusions drawn from Fig. 3b–d are in agreement with Fig. 3a as if any strain-induced pores or crystals were to develop within the sample then material would not deform at constant volume as is clearly shown in Fig. 3a45.

Figure 4a shows for conditions I–IV strains, \({\it{\epsilon }}_z\), calculated via the constant volume assumption using strains measured in the xy plane (see Methods). Like the curves in Fig. 3a, in each case the strain \({\it{\epsilon }}_z\) approaches a minimum before increasing (Table 1 and Fig. 3a). Auxetic behaviour is demonstrated in the regions of further increasing \({\it{\epsilon }}_z\) where the sample is growing thicker in the direction transverse to the increasing applied strain, \({\it{\epsilon }}_x\). Figure 4b shows for each test the instantaneous PR, v xz , against nominal strain \({\it{\epsilon }}_x\). The curves confirm the emergence of auxetic behaviour above a critical strain, \({\it{\epsilon }}_{\mathrm{c}}\), above which v xz becomes negative. The value of \({\it{\epsilon }}_{\mathrm{c}}\) for each test shows a dependence on the sample temperature and on the speed at which it is extended (Table 1), most likely a result of the different conditions allowing different levels of stress relaxation between successive extensions. However, if the behaviour of v xz for each case is considered with respect to \({\it{\epsilon }}_{\mathrm{c}}\), then we can see that the magnitude of the auxetic response is in fact infact largely identical and hence independent of extension speed and temperature (Fig. 4b). In each case shown in Fig. 4b, c, v xz for the most part monotonically decreases with strain. A minimum value of \(

u _{xz} = - 0.74 \pm 0.03\) was recorded in test IV which was strained by the largest factor (~1.7) above \({\it{\epsilon }}_{\mathrm{c}}\).

Fig. 4 Measured strains and instantaneous Poisson’s ratio behaviour. a For each of the various samples deformed under conditions I–IV the strain ε z initially decreases. Beyond the minimum the behaviour becomes auxetic. b Strain-dependent Poisson’s ratios extracted from a shows that ν xy in general monotonically decreases and becomes negative above a critical strain, ε c , specific to each test condition. c The instantaneous Poisson ratios plotted relative ε c showing that the behaviour of ν xz in all cases are largely identical. d The measured principle strains in the xy plane and e the calculated instantaneous PRs, ν xy , for all test conditions I–IV. In each case ν xy begins close to zero and increases with imposed strain, ε x Full size image

Figure 4d and e show corresponding plots of the \({\it{\epsilon }}_x\)-\({\it{\epsilon }}_y\) strains measured and instantaneous PRs v xz calculated for tests I–IV. Comparing these graphs with Fig. 4a, b, it is clear that the deformation behaviour of the LCE in the xy and xz planes differs significantly—i.e., the deformation is highly anisotropic. In contrast to v xz , in each case v xy is initially near-zero and increases with strain. The anisotropy between v xy and v xz for our material should be expected for two reasons. Firstly, monodomain LCEs are well known to have inherent mechanical anisotropy28,29,46,47. Secondly, simultaneous volume conservation and Poisson’s ratio isotropy is only possible in the specific case of \(

u _{xy} =

u _{xz} = 0.5\), for which there is no auxetic behaviour48. These conditions are formalised mathematically by the condition (for volume conserving materials) \(

u _{xy} +

u _{xz} = 1\), which is derived in the methods.

Negative liquid crystal order parameter (LCOP)

Figure 5 shows polarising microscopy images of the LCE deformed under test conditions I. In the unstrained state the sample has an optical retardance \(\Gamma = {\mathrm{\Delta }}n \times d \approx 8000\;{\mathrm{nm}}\) (~14th order). The progression to birefringence colours of increasing saturation indicates a decrease in retardance (to ~1st–3rd order between \({\it{\epsilon }}_x = 0.87\)and \({\it{\epsilon }}_x = 1.06\)) that cannot be accounted for by the (initially) reducing sample thickness (Fig. 4a)30. Such behaviour is associated with a reducing LCOP within the xy plane. The black appearance at \(\left( {{\it{\epsilon }}_x = 1.14} \right)\) exists for all sample rotations with respect to the crossed polarisers (Supplementary Fig. 2) meaning there is zero birefringence and hence zero LC ordering within the xy plane—i.e., the LCE is isotropic within the xy plane. We have previously deduced that such an observation corresponds to a state of negative LCOP30. Below we confirm this behaviour by applying the theory of LCEs pioneered by Warner and Terentjev (W&T)29. Beyond \({\it{\epsilon }}_x\sim 1.20\), birefringence colours re-emerge and hence the in-plane LCOP increases, but with the director now parallel to the stress axis30.

Fig. 5 Negative LCOP order from polarising microscopy. Polarising microscopy textures at each strain step of test I. The birefringence colours indicate the retardance initially decreases, becoming zero at ε x = 1.14, before increasing again. Scale bar, 5 mm Full size image

The results presented here can be tested against the theory of W&T, which speculated about the possibility of auxetic behaviour in aligned LCEs. The theoretical suggestion has, however, never previously been observed in experiments or simulations. Under the Gaussian theory of elasticity, the polymer chain for an isotropic elastomer can be modelled as a random walk with effective steps of length l. On average, a polymer chain will, therefore, adopt a spherical conformation. However, in a nematic LCE the anisotropic ordering of the mesogenic groups is templated onto the polymer chain conformation resulting in an anisotropic and (most generally) biaxially ellipsoidal shape (Supplementary Fig. 3)29. The effective step of the polymer chain random walk is, therefore, anisotropic and is described by the effective step length tensor, \(\underline{\underline {\boldsymbol{l}}}\), which equals Diag(l 1 , l 2, l 3 ) in the principal frame29. The relationship between the effective step lengths and the polymer conformation is shown in Supplementary Fig. 3. Crucially, the coupling between the side-chain LC moieties and polymer backbone means the LCOP tensor and \(\underline{\underline {\boldsymbol{l}}}\) have a common symmetry29. As typically the LCOP has uniaxial symmetry, the effective step length tensor is given by l 1 =l ǁ and \(l_2 = l_3 = l_ \bot\) where the unique axis is aligned parallel with the nematic liquid crystal director. The ratio \(r = l_\parallel /l_ \bot\) is known as the step length anisotropy and characterises the anisotropy of the polymer backbone for uniaxial systems.

In their theory of LCEs, W&T speculated that an auxetic response of LCEs may be observed to begin at a deformation given by \((\lambda _x,\lambda _z) = \left( {r_0^{1/3},r_0^{ - 1/6}} \right)\), where r 0 is the step length anisotropy for the unstrained LCE and the deformations λ i \(\left( { = {\it{\epsilon }}_i + 1} \right)\), are components of the deformation gradient tensor, \(\underline{\underline \lambda }\). Considering data for test I (chosen as we show corresponding polarising microscopy images in Fig. 5) from Fig. 4a, we calculate using the critical strains of \(({\it{\epsilon }}_x,{\it{\epsilon }}_z) = (1.02 \pm 0.02, - 0.30 \pm 0.02)\) values of \(r_0 = 8.2 \pm 0.3\) and \(r_0 = 8.5 \pm 0.7\), which are comfortably self-consistent.

Further independent calculations of r 0 can be made by applying W&T theory (see methods) to the strain at which we have observed zero birefringence (Fig. 5). Our ability to use this theory stems from the following deductions about the symmetry of \(\underline{\underline {\boldsymbol{l}}}\) for the LCE in the unstrained state and at the black state seen in Fig. 5. For the unstrained LCE, the LCOP has nematic (uniaxial) symmetry and the director lies along the y-axis—therefore meaning \(\underline{\underline {\boldsymbol{l}}} = \underline{\underline {\boldsymbol{l}}} ^0{\mathrm{ = Diag}}(l_ \bot ^0,l_\parallel ^0,l_ \bot ^0)\). As in the black state the LCOP is deduced to be isotropic within the xy plane, the effective step lengths l 1 and l 2 must in this state be equal—therefore meaning \(\underline{\underline {\boldsymbol{l}}} = \underline{\underline {\boldsymbol{l}}} ^\prime {\mathrm{ = Diag}}(l_ \bot ^\prime ,l_ \bot ^\prime ,l_\parallel ^\prime )\), which has principal axes parallel with those of \(\underline{\underline {\boldsymbol{l}}} ^0\). Further details of how these deductions are used to calculate values for r in the unstrained and black states are given in the methods.

From Fig. 5 we can deduce for test I that at the state of a negative LCOP \(\lambda _x = 2.15 \pm 0.05\) and \(\lambda _y = 0.67 \pm 0.05\) (measurement errors, n = 1). By Inserting these values in to Eq. 9 (methods) and solving, we find \(r_0 = 10.2 \pm 1.6\) and \(r^\prime = 0.103 \pm 0.015\) (propagation of measurement errors through Eq. 9, n = 1). This calculation of \(r^\prime < 1\) corresponds to the system adopting a uniaxial oblate polymer conformation. Previously we have determined values of r 0 = 9.3 and r 0 = 3.8 from opto-mechanical and thermal tests, respectively11. At the time, comparisons of these values to those from LCEs of similar chemistries led us to conclude that the latter value was more likely to be accurate30. However, the several independent calculations of r 0 ~9 presented here now leads us to believe that this value is actually most likely to be correct. The self-consistency of values presented here demonstrates that W&T theory describes the physical behaviour of our material well—hinting that the auxetic response of future materials could be tuned by controlling the magnitude of r 0 .

Using the calculated values of r we can also extract values for the polymer chain backbone order parameter, Q b , using the equation49

$$r = \frac{{1 + 2Q_{\mathrm{b}}}}{{1 - Q_{\mathrm{b}}}}\therefore Q_{\mathrm{b}} = \frac{{r - 1}}{{r + 2}}$$ (1)

which gives \(Q_{\mathrm{b}}^0 = 0.74 \pm 0.03\) and \(Q_{\mathrm{b}}^\prime = - 0.41 \pm 0.01\) (propagation of measurement errors through Eq. 1, n = 1) which, given the shared symmetry between the LCOP and \(\underline{\underline {\boldsymbol{l}}}\) tensors, are consistent with our present and previous deductions of the LCOP symmetry30.