Insights into the nanostructuring and phase behaviour of an all-aromatic prototypical nematic liquid crystal

Abstract

All-aromatic calamitic liquid crystals are an unconventional family of rigid linear mesogens that represents the closest embodiment of the idealised rod-like molecule central to liquid crystal theories. 2,6-biphenyl naphthalene (PPNPP), a prototypical all-aromatic nematogen, has recently been the focus of scientific interest for both fundamental and technological purposes. While it provides a valuable benchmark for classical theories of nematic order, its experimental study presents challenges given the high temperature of the nematic phase. Herein, molecular dynamics (MD) simulations are contrasted with X-ray diffraction (XRD) data to resolve the thermotropic phase behaviour of PPNPP and the main structural features of its liquid crystal order. The observed trend of the molecular conformation with temperature points to an unexpected and significant distortion of the molecular framework in the nematic phase, which demonstrates the substantial non-linearity of the PPNPP molecule. The simulated mesomorphic behaviour and related thermodynamic parameters are in close agreement with the experimental data. The different phases are described in terms of their molecular organisation, pair distribution function, as well as the orientational and positional order parameters. We compare the classical and extended Maier–Saupe nematic theories with the computationally (MD) and experimentally (XRD) determined orientational order parameters. The observed deviation from the theoretical models indicates substantial inadequacies of the classical theories to describe the nematic–isotropic phase transition for this class of compounds. We suggest that modifications of theory should include the effects of short-range positional order, i.e., cybotaxis.

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1. Introduction

In recent years, synthetic efforts^1–4 as well as experimental^1,2,4 and computational^2,5–8 studies have been devoted to an unconventional class of liquid crystals (LCs) composed of all-aromatic molecules that exhibit very high temperature nematic (N) phases. Unlike traditional thermotropic LCs, which typically contain a mixture of rigid and flexible components, all-aromatic LCs are made entirely of conjugated rings, exhibiting distinctive properties such as molecular rigidity, improved thermal stability and a more pronounced optical birefringence.³ The increased rigidity makes these materials suitable for use in high-performance devices where structural integrity at high-temperature or in demanding environments is critical. In addition, their highly conjugated molecular structure identifies them as possible organic semiconductors. The combination of this property with the typical features of LC mesophases (orientational order, fluidity, sensitivity to temperature and external fields, self-assembly and self-healing ability, easy processability) renders them promising materials for display and organic semiconductor technologies,⁹ photonic devices,¹⁰ and opto-electronic application.¹¹ Besides their technological interest, all-aromatic nematic LCs are attracting primary interest in LC science as their rigid and highly symmetric structure, devoid of polar groups or flexible chains, represents the closest embodiment of the idealised rod-like mesogen forming the basis of the majority of theories of the nematic to isotropic (I) phase transition.^12–17 In this regard they represent the ideal prototype or model compound to test the predictions of fundamental theories describing the nematic order.

The all-aromatic rod-like mesogen p-quinquephenyl (PPPPP) (Fig. 1) is the simplest example of a compound that closely conforms to the prototypical nematogen of the above theories¹⁸ and over the past fifteen years it has been the subject of experimental^2,4,19 and computational^2,6 investigations in an effort to quantitatively compare theory and experiment and clarify the role of the flexible chains and polar groups (absent here) in determining N behaviour. A preliminary account of the nematic order measured in d-PPPPP-d appeared in 2006¹⁸ and a further study on unlabelled PPPPP followed in 2011.¹ Those reported datasets were subsequently complemented with molecular dynamics (MD) simulations in 2014,⁶ which showed that these molecules are more flexible than expected at high temperatures, and that their nematic phase is stabilised by the polydispersity of molecular aspect ratio caused by the bending and conformational disorder of phenyl rings. More recently these studies were expanded² by presenting a comprehensive set of NMR order parameters measured on two specifically deuterium labelled PPPPP isotopomers, along with new MD simulations using an optimised force-field to explore and quantify various aspects of the N orientational order. It is worth mentioning that an additional reason for interest in an atomistic study is that PPPPP, and more generally polyphenyl derivatives, have long been prototype molecules for organic semiconductor devices,²⁰ transistors²¹ and OLED,²² along with oligothiophenes.^5,23–29 In addition, naphthalene derivatives may serve as high performance aromatic LC copolyesters with excellent mechanical features and thermal stability that can be further enhanced through molecular structure design.³⁰

Fig. 1 PPPPP (left) and PPNPP (right): (a) molecular structure, (b) phase map measured by differential scanning calorimetry on heating: PPPPP values from ref. 18, PPNPP values from DSC data shown in Fig. S1 (ESI†); (c) DFT optimised structure determined at the B3LYP/6-311++G(d,p) level and (d) evolution of the molecular organisation of the mesophases.

Motivated by this multiplicity of fundamental and technological interests, and in an attempt to clarify the relationship between molecular shape, mesophase properties, and theoretical predictions, further investigations have been performed on the strongly-correlated, all-aromatic mesogen 2,6-biphenyl naphthalene (PPNPP).⁴ This molecule differs from PPPPP in that the central phenyl ring is substituted by a 2,6-naphthalene moiety resulting in an offset “kink” in the shape compared to PPPPP, a slightly larger aspect ratio, and a potentially different degree of flexibility due to the naphthalene core. These structural features result in a more extended N range, although shifted at higher temperatures compared to PPPPP, which makes experimental investigation even more challenging. In ref. 4, X-ray diffraction (XRD) revealed a surprisingly strong tendency towards molecular layering in the N phase, indicative of ‘‘normal cybotaxis’’, i.e., smectic A (SmA)-like stratification within clusters of mesogens,^31,32 which grows as the temperature decreases; the final result is the onset of an enantiotropic SmA phase. On the other hand, measured values of the orientational order parameter 〈P₂〉 were found to be slightly larger than those predicted by the Maier–Saupe (MS) theory over the entire N range, except for a narrow region just below the clearing point where they displayed a significant drop below the theoretical prediction. However, the challenging thermal conditions in which this experiment was carried out in terms of temperature stability control, risk of sample degradation/decomposition, and maximum achievable magnetic field, precluded an unequivocal confirmation of the departure of the observed trend from the MS theoretical prediction.

To acquire further insights into the nanostructuring and phase behaviour of PPNPP, we have performed a full atomistic MD study to resolve the nanoscopic molecular organisation over the entire mesophasic range. Here we present the results of these simulations contrasted with the previously collected XRD data. The different phases are described in terms of their molecular arrangement, orientational and positional order parameters, and mesogen conformation. Attention is paid to the short-range positional order which, although not considered in any of the classical theoretical models of the N–I transition,^12–17 is expected to play a key role in determining the peculiar cybotactic molecular clustering observed in the N phase.⁴

2. Methods

2.1 Computational details

MD simulations were performed using the NAMD 2.14 software code with the AMBER/GAFF parameters at constant atmospheric pressure and temperature using Berendsen barostat and thermostat.^33,34 The PPNPP system comprised of N = 350 molecules contained in a cubic box with periodic boundary conditions and box sides of about 60 Å. The particle mesh Ewald method was used to compute the Coulomb interactions with periodic boundary conditions, a grid spacing of 1.5 Å and a cut-off of 12 Å for the calculation in the direct space, as well as for truncating Lennard-Jones interactions. Atomic charges required for the forcefield were obtained with density functional theory (DFT) calculations using Gaussian16 at the B3LYP//6-311++G(d,p) level, with the ESP method.^35,36 The topology file for the NAMD input is detailed in the ESI.†

A series of simulations were carried out utilising a cooling down sequence from the disordered I phase configuration at 800 K, observing the spontaneous onset of order with a gradual decrease of the temperature to 650 K. The equilibration simulations were run for 40 ns and the production runs ranged from 35–40 ns to fully sample the varying mesophases from I through to the crystalline (Cr) phase. Additional simulations were performed at 670, 690 and 730 K for 20 ns to ascertain the order parameters in a larger system of N = 1000 molecules to assess the validity of our results. Statistical structural descriptors, such as the radial distribution function (RDF), the order parameters, and the orientational potential were calculated with computational algorithms³⁷ using in-house developed analysis codes. In particular, the orientational order parameter was calculated from the largest eigenvalue of the ordering matrix, except above the N–I transition temperature, where it was obtained from the intermediate eigenvalue. As explained in ref. 37 in the I phase, the averaged intermediate eigenvalue converges to the expected zero value faster than the largest eigenvalue.

2.2 Experimental details and analysis of XRD data

The experimental details of PPNPP synthesis have been described elsewhere.⁴ The mesophase sequence provided by DSC (2 heating–cooling cycles at 20 K min⁻¹, using a TA Instruments DSC2500 equipped with liquid nitrogen chiller) is shown in Fig. 1 with the corresponding transition temperatures. DSC data collected in a full heating–cooling cycle are shown in Fig. S1 (ESI†). XRD measurements were performed at the former BM26B DUBBLE beamline of the European Synchrotron Radiation Facility (ESRF), Grenoble, France. XRD data provide important structural information on molecular ordering as a function of the temperature. In particular, the azimuthal broadening of the wide-angle reflection can be used to evaluate the orientational order parameter 〈P₂〉. While this type of analysis was reported in our previous work,⁴ it was subsequently determined that the model employed to calculate 〈P₂〉 was flawed.^38,39 The problem derived from an error in the formula originally proposed by Leadbetter and Norris to link the azimuthal intensity profile of the wide-angle reflection with the molecular orientational distribution function.⁴⁰ Consequently, here we conducted a thorough re-examination of the data and recalculated the values of 〈P₂〉 using a rectified version of the model. To obtain the azimuthal intensity profile I(γ), with γ being the azimuthal angle on the detector measured with respect to the equatorial axis (horizontal in our case), the background subtracted XRD patterns were radially integrated over the experimentally accessible q range. The molecular orientational distribution function f(θ) was modelled as anticipated by the MS theory, f(θ) = Z⁻¹ exp(m cos² θ), where θ is the angle between the molecular long axis and the molecular director, m is a temperature-dependent parameter determining the degree of orientational order, and Z the partition function assuring the distribution is normalised. Within this approximation, it can be demonstrated that the azimuthal intensity profile I(γ) is related to the MS parameter m through the equation:where I_b is the background intensity, k is an intensity scale factor, D is the Dawson's integral, and I₀ is the modified Bessel function of the first kind.⁴¹ This expression corrects a previous one proposed by Davidson et al.⁴² and based on the erroneous formula of Leadbetter and Norris. By fitting the experimentally measured I(γ) with eqn (1), the m parameter can be determined and, consequently, the orientational order parameters and . The results of the fitting process are shown in Fig. S2 (ESI†).

3. Results and discussion

3.1 Mass density and molecular conformation

We first assessed the simulated mass density of PPNPP obtained as a function of temperature cooling down from the I phase (Fig. S3, ESI†). The trend shows excellent agreement with the experimentally determined phase transitions of the system, with simulated transition temperatures identified from changes of the slope at ∼750 (I–N), 690 (N–SmA) and 660 K (SmA–Cr).

To resolve the nanoscale structuring of PPNPP, we first assessed its molecular shape. While all-aromatic mesogens are significantly stiffer than conventional calamitic LCs, they still exhibit a certain degree of flexibility, which can become significant at the very high temperatures of their N phase. It has been suggested that molecular bending plays a role in determining the mesomorphic properties of PPPPP, such as its extensive N range⁶ and a temperature behaviour of 〈P₂〉 that deviates from the predictions of the MS theory.²Fig. 2a shows the probability distribution of the cosine of the bending angle β (the angle between the para axes of the two terminal biphenyl rings at the opposite ends of the molecule) at selected temperatures across the different phases. As expected, the propensity for bending increases (i.e., β deviates further from 180 deg) with the increasing temperature: approximating the cosine of the average bending angle β with the average of cos β, so that 〈β〉 = cos⁻¹ (〈cos β〉), we obtained values of 〈β〉 passing from 168.2 deg in the crystal (650 K) to 158.8 deg in the I phase (770 K).

Fig. 2 (a) Probability distribution of cos β for PPNPP at various temperatures. The bending angle β is calculated from the para axes of the two terminal biphenyl groups at the opposite ends of the PPNPP molecule (PP–N–PP), i.e., between vectors AB and CD. Two probability distributions calculated in the same way for PPPPP are shown for comparison. (b) Calculated molecular lengths and average smectic layer spacing at 660, 670 and 680 K with (inset) the average aspect ratio as a function of temperature.

In the figure, we also contrast our results in the N phase with those computed for PPPPP at similar temperatures:⁶ for PPNPP 〈β〉 is 161.7 deg at 700 K and 160.1 deg at 740 K, while for PPPPP we have 163.2 deg at 710 K and 162.4 deg at 730 K. Overall, the behaviour of the two mesogens is very similar, with PPNPP exhibiting a slightly larger flexibility than PPPPP. As a small amount of molecular flexibility is known to stabilise the N phase,⁴³ this feature could contribute to the larger temperature range of the PPNPP N phase.

To ascertain the molecular shape of PPNPP, we evaluated its size through the dimensions L_x, L_y, and L_z of the smallest rectangular box containing the molecule and having its edges aligned along the axes of the molecule's inertial tensor principal frame (Fig. 2b). We obtained a value of ∼26.7 Å for the molecular length L = L_z, slightly decreasing with the increasing temperature due to the increased bending. As shown in the figure, this value matches well with the value of ∼26.8 Å provided by simulations for the average layer spacing in the Sm phase. This clearly indicates a SmA phase with nearly vertical molecules and consequently a high degree of orientational order.

The inset of Fig. 2b displays the temperature dependence of the average aspect ratio L/B, with the breadth B defined as the average of the two transverse molecular dimensions . A decline in the average aspect ratio is observed as the temperature increases, indicating the non-rigidity of the PPNPP molecular shape. This reflects a combination of molecular bending, which causes a decrease of L_z, and torsion of the aromatic groups around the para-axis, which induces a departure from a planar conformation and results in an increase of L_x. This behaviour showcases similarities with the class of flexible mesogens including PPPPP as well as sexithiophene.^5,6

3.2 Phase sequence and transition temperatures

In Fig. 3, we display the molecular organisation of the differing phases obtained by MD simulations on cooling from the disordered (isotropic) state (770 K) to the ordered Cr phase at 650 K. Colour coding of the molecular orientation highlights the onset of long range orientational order when the sample enters the N phase from the high temperature I melt. Upon further cooling, PPNPP starts to layer into the Sm phase at 690 K; as previously discussed, the molecules are orthogonal to the layers indicating a SmA phase.

Fig. 3 (top) Representative XRD patterns of PPNPP with the sample molecular director vertically aligned through the application of a magnetic field. Taking advantage of the symmetry of the patterns, only their left half is entirely shown. (bottom) MD simulation snapshots of the differing phases obtained on cooling down from the I phase. The colour coding indicates orientations with respect to the molecular director as from the palette.

The MD predicted transition temperatures were determined by evaluating the system enthalpy as a function of the temperature using 〈H(T)〉 = 〈U(T)〉 + p〈V(T)〉. Here p is the external pressure, 〈U(T)〉 and 〈V(T)〉 are the average values of the force field energy and volume of the box at a given temperature. The enthalpy values around the I–N and the N–SmA phase transitions are shown in Fig. 4 and Fig. S4 (ESI†), respectively. The corresponding temperatures are identified by a discontinuity in the 〈H(T)〉 curves, suggesting a weakly first-order nature for both transitions, leading to T_N–I = 752.5 K and T_SmA–N = 692.5 K; both phase transition temperatures are in excellent agreement with the values obtained on heating from DSC and optical microscopy measurements, T_N–I = 753 K and T_SmA–N = 691 K, respectively. The measured transition enthalpy estimated from simulations for the I–N phase transition is ΔH_N–I = 3.31 kJ mol⁻¹, in good agreement with the values provided by DSC, 1.6 kJ mol⁻¹ on heating and 2.6 kJ mol⁻¹ on cooling. On the other hand, for the N–SmA transition our simulations provides ΔH_SmA–N = 3.94 kJ mol⁻¹, while DSC indicates a second-order transition with no measurable enthalpy change. This discrepancy is probably due to the limited size of the simulation sample which favours more ordered mesophases and makes it impossible to follow the gradual growth of smectic order fluctuations (cybotaxis) while the nematic is cooled down towards T_SmA–N. Simulations on a much larger system are in progress to clarify this important point.

Fig. 4 Simulated enthalpy H as a function of temperature T for PPNPP (purple squares) with the determined T_N–I transition temperature at 752.5 K (yellow line). The different linear trends of H(T) in the I and N temperature range is shown by the green and blue line, respectively.

The phase sequence predicted by MD simulations was experimentally confirmed by XRD measurements. Fig. 3 shows the typical XRD patterns of the different phases for a sample aligned by a vertical magnetic field. In this case, the transition temperatures were identified by changes in the patterns at ∼745 K (N–I) and at ∼670 K (SmA–N). The main features of the XRD pattern and its evolution with temperature are indicative of ‘‘normal cybotaxis’’ in the N phase (i.e. local SmA-like stratification within clusters of mesogens),^31,32 as previously discussed in ref. 4.

In the SmA and N phases, the patterns are characterised by a pair of broad wide-angle peaks symmetrically placed on the equatorial axis (the axis orthogonal to the alignment direction) and a pair of narrower low-angle peaks on the meridional axis (the vertical alignment direction). The former corresponds to a d-spacing of ∼5 Å and is related to the average transverse intermolecular distance, whereas the latter reflects the intermolecular positional correlation in the longitudinal direction. In the SmA phase the low-angle peaks are particularly intense and narrow,⁴ a signature of the long-range layering of the molecules, and correspond to a d-spacing of 25.9 Å, only slightly shorter than the layer spacing and molecular length values provided by MD simulations (Fig. 2b). Although stronger at low temperatures, the layering is evident well above the SmA–N transition, with the associated longitudinal correlation length exhibiting surprisingly large values, which exceed five molecular lengths over most of the N range.⁴ The longitudinal d-spacing in the N phase, which represents the average intermolecular distance along the director, is shown in Fig. 5: the values are in good agreement with the molecular length estimated by MD simulations but exhibit a stronger dependence with the temperature, passing from 25.8 Å just above the SmA–N transition to 28.3 Å at the clearing point. This evidence suggests that the effects of a decrease in density on longitudinal positional ordering exceed those of growing orientational disorder as the temperature increases.

Fig. 5 Experimental values of the longitudinal d-spacing obtained at selected temperatures across the SmA and N phases from XRD data.

In the Sm and N phases, the patterns also show a sequence of weaker meridional reflections spanning the intermediate and wide-angle region of the investigated q-range. This is a characteristic feature of the diffraction patterns from all-aromatic nematogens⁴ and may be ascribed to a combination of effects: on the one hand, the lack of electron-poor aliphatic tails contrasting with the electron-rich aromatic core results in a molecular electron density profile which is only weakly modulated; on the other, the lack of flexible terminal tails decreases the polydispersity of the molecular length, thus favouring a more ordered arrangement of the molecules in the longitudinal direction. Both mechanisms result in a less sinusoidal and more square-like longitudinal electron density profile, and hence in a large number of higher order harmonics in the diffraction pattern.

3.3 Positional and orientational order

To provide further insights into the local structure of the PPNPP phases, we analysed each temperature using the radial distribution function (RDF) derived from MD simulations and showcased in Fig. 6. At 650 K we observe a sequence of peaks, with the first one at 5 Å. The distinct peaks indicate a periodic ordered crystalline structure; the absence of a peak at 3.5 Å confirms that the molecule is not planar and accordingly the conventional π–π packing of aromatics is not present in the PPNPP system. The second peak at ∼10 Å displays a slight splitting which implies a local hexagonal structure in the crystalline phase.⁴⁴ The sharp peaks (∼5, 10 and 15 Å) at this low temperature confirm the ordered nature which contrasts with the Sm phase identified at 670 K; though some ordering is present at 5 Å, we observe a loss in the sharpness of the peaks at extended interatomic distances, noticeably at ∼10 and 15 Å, confirming the nearest-neighbour liquid-like order within each smectic layer. In the high temperature N (700 and 740 K) and I (770 K) phases of PPNPP, disordered structuring is evident through the absence of distinct peaks beyond the weak 5 Å peak that is responsible for the diffuse wide-angle reflection in the XRD patterns. For the N and I states, the g₀(r) value of 1 is attained at very short intermolecular distances as expected for a fully fluid phase (Fig. 6).

Fig. 6 Radial distribution functions of the different phases at 650, 670, 700, 740 and 770 K.

To fully resolve the molecular organisation in the observed mesophases, we evaluated the temperature dependent behaviour of the positional and the orientational order parameters. The positional (smectic) order parameters τ_n = 〈cos(2πnz/d)〉, with d being the smectic layer spacing, represent the Fourier expansion coefficients of the probability of finding a molecule at a position z along the normal to the layers.⁴⁵ The first three positional order parameters (τ₁, τ₂, τ₃) computed from our simulations are displayed in Fig. 7. A sharp rise in the first smectic order parameter τ₁ is evident below the N temperature range confirming the onset of smectic order.^5,6 In particular, τ₁ grows steadily from a value of ∼0.2 at the lower end of the N temperature range up to ∼0.7 at the lower end of the underlying Sm phase, with a behaviour similar to that reported for PPPPP.⁶ It is interesting to observe that the increase of τ₁ above the baseline value starts well above the SmA–N transition temperature, indicating a certain degree of positional order (cybotaxis) also in the N phase, as already suggested from XRD data.⁴ At the opposite end of the investigated temperature range, a further increase of τ_n, now exceeding 0.8 below 660 K, marks the transition to the Cr phase.

Fig. 7 Simulated smectic order parameters τ₁, τ₂ and τ₃ as a function of temperature.

Fig. 8 displays the simulated orientational order parameters 〈P₂〉 and 〈P₄〉 as a function of the reduced temperature T/T_NI, compared with the experimental values obtained from the XRD data and with the trend predicted by the MS theory. To validate our results in the Sm and N phases, we simulated a larger sample of N = 1000 molecules of PPNPP at selected temperatures. At 670, 690, and 730 K, we determined a 〈P₂〉 value of 0.89, 0.79, and 0.62 respectively, confirming good agreement with the original system (N = 350 molecules) whereby 〈P₂〉 was calculated as 0.91, 0.85, and 0.65 at the three respective temperatures. Overall, the dependence of the order parameter on the sample size, with 〈P₂〉 being larger and transitions being smoother the smaller the N value, is qualitatively similar to those reported for sexithiophene, another all-aromatic LC.⁵ Nevertheless, even when these effects are considered, the MD values for both 〈P₂〉 and 〈P₄〉 are significantly different from the MS predictions. This departure from the theory is confirmed by the experimental values, whose trend is intermediate between those of simulations and theory. While the continuous decrease of the order parameters on approaching the clearing point, contrasting with the abrupt change theoretically expected for a first order transition, could be explained by the limited sample size for MD simulations and with slight instabilities and/or inhomogeneities in the sample temperature for XRD experiments, the observed difference with the MS trend at lower temperatures seems indicative of the inability of MS theory to correctly model our system.

Fig. 8 Average simulated order parameters (MD) 〈P₂〉 and 〈P₄〉, XRD experimentally determined (Exp) 〈P₂〉 and 〈P₄〉, and theoretical predictions from MS and EMS theories plotted as a function of reduced temperature T/T_NI.

A similar discrepancy has been reported for PPPPP, with both experimental (from NMR measurements) and simulated 〈P₂〉 values being larger than MS predictions.² For PPPPP, a much better agreement with the theory can be obtained by considering the extended MS theory (EMS), with the maximum scaling exponent of 3.⁴⁶ While the standard MS theory describes the tendency of the LC molecules to align parallel to each other through a molecular field orientational potential U(cos θ) = −ε〈P₂〉P₂(cos θ), with ε being a constant parameter expressing the potential strength, the EMS theory modifies the MS model by assuming a temperature dependent ε parameter (the scaling exponent of 3 used to plot 〈P₂〉 in Fig. 8 corresponds to ε ∝ T⁻²).⁴⁶ Even though the EMS theory has been developed for polymeric LCs, so that its application to low molecular weight all-aromatic compounds seems not well justified, similar dependences of ε on the temperature can be found in other more general extensions of the MS theory, e.g. as an indirect consequence of the dependence of ε on the volume.⁴⁷ To explicate the limits of the standard MS theory, we fitted the orientational energies obtained from simulations at each temperature within the N range with the MS potential U(cos θ) = U₀ − ε〈P₂〉P₂(cos θ) with U₀ and ε used as fit parameters. Although the energy data were fitted separately for each temperature, the MS potential was unable to adequately reproduce the simulation data, especially at the lowest temperatures (Fig. 9a). Following ref. 37 and 47, we found that a much better fit of the MD orientational energies could be obtained by considering the fourth-rank expression of the potential U(cos θ) = U₀ − ε[〈P₂〉P₂(cos θ) + λ〈P₄〉P₄(cos θ)], with the U₀ and ε dependent on the temperature and λ fixed (Fig. 9b). The fit provides a value of λ = 0.146 ± 0.004 of the same order of magnitude as the one reported for 4,4′-diethoxyazoxybenzene, λ = 0.116,⁴⁷ and the values of ε shown in Fig. S5a (ESI†). The latter parameter decreases with the increasing temperature approximately following a power law with exponent −1.9 ± 0.4 relatively close to the value of −2 assumed to draw the EMS curve of 〈P₂〉(T) in Fig. 8. In a similar fashion, if the value of ε is related to the molar volume V through the relation ε = ε₀V^−γ as in ref. 47, with V obtained from the computed density curve in Fig. S3 (ESI†), we get an exponent γ = 1.5 ± 0.2 (Fig. S5b, ESI†).

Fig. 9 Orientational energy obtained from simulations at selected temperatures within the N phase (dots) and corresponding best fit curves (lines). Fitting function: (a) U(cos θ) = U₀ − ε〈P₂〉P₂(cos θ) (χ_red² = 0.0045); (b) U(cos θ) = U₀ − ε[〈P₂〉P₂(cos θ) + λ〈P₄〉P₄(cos θ)], with the parameter λ forced to be independent of the temperature (χ_red² = 0.0014). In both cases, the fit parameters U₀ and ε are temperature dependent.

4. Conclusions

We have performed detailed MD simulations of the nanoscopic structuring and phase transitions for the prototypical rod-like nematic LC PPNPP. The mesophase sequence, transition temperatures and relevant thermodynamic parameters are in close agreement with the data from XRD and DSC analysis. Unlike what is expected based on the low-temperature, intrinsic rigidity of the naphthalene core, the observed trend of the molecular conformation with temperature points to a significant distortion of the molecular framework in the N phase, which indicates substantial non-rigidity (non-linearity) of the PPNPP molecule at these high temperatures.

The different mesophases have been described in terms of their molecular organisation, pair distribution function and orientational and positional order parameters. As observed for PPPPP,² the differences of the orientational order parameters between theory and experiment (as summarised in Fig. 8) may point to a common issue with all classical theories of the N–I phase transition, i.e., a failure to consider short range positional order (cybotaxis) in the nematic. For PPNPP, previous XRD data⁴ show evidence of unexpectedly large cybotactic domains compared with those commonly found in bent-core nematogens,^31,32 featuring longitudinal and transversal sizes well above five molecular lengths and five molecular diameters, respectively, over most of the N range. Although cybotaxis cannot be studied in detail with our simulations because of the restricted sample size, the increase of the positional order parameter τ₁ well above the N–Sm transition temperature is strongly suggestive of the emergence of cybotactic order.

In conventional nematics short-range order is simply considered to be of the liquid-like type whereas cybotactic nematics exhibit a stronger level of short-range order associated with the Sm-like stratification of the mesogens within nanoclusters. Observing cybotaxis in all-aromatic mesogens like PPNPP is quite surprising as in principle these molecules do not possess the features conducive to molecular layering, neither in the molecular shape, as it occurs in bent-core mesogens,³² nor in their chemical constitution: the PPNPP architecture is devoid of diverse chemical components (aromatic and aliphatic molecular segments) known to promote smectic layering in conventional, low molar mass thermotropic mesogens. At this stage, we may only suggest that the interplay between the PPNPP departure of linearity at elevated temperatures and the aromatic ring stacking interactions may exacerbate near-neighbour positional correlations in the N phase favouring a Sm-like local stratification which is inherent to the cybotactic nanoscale molecular structuring.^31,32 In turn, the molecular layering could promote a stronger orientational order within cybotactic groups, which could explain why 〈P₂〉 exceeds the MS prediction over most of the N range.

However, the true nature of the short-range order underlying the stratified clustering in cybotactic nematics has yet to be fully understood. Open questions regard the coupling between cybotactic order and nematic orientational order, a static (persistent clusters) vs. dynamic (transient fluctuations) picture of cybotactic groups, the main factors driving the local molecular layering, such as the mesogen (statistical) shape and its chemical structure (linear vs. bent, rigid vs. flexible, polar vs. apolar, aromatic vs. aliphatic), the temperature dependence of cybotactic order and its relationship with the possible presence of an underlying Sm phase (cybotaxis as pretransitional effect vs. cybotaxis as a more general feature of N order). In order to answer these questions, it would be necessary to study and compare many different systems through a combination of experiments and simulations. From the experimental point of view, this requires the acquisition of high-resolution XRD data and its proper elaboration to fully characterise the short-range positional order of the cybotactic nematic. Accordingly, from the standpoint of simulations, a challenging extension of the box size is required to explore spatial regions larger than the volume of a single cybotactic cluster. System sizes of the order of 10⁴ molecules would be required to properly investigate the spatial decay of longitudinal positional correlations.⁴⁸ Finally, a comprehensive theory of the N phase which includes cybotactic order and considers its effect on the orientational order is still missing. We believe that a good starting point in this direction could be a proper extension of the modified MS theory for cybotactic nematics proposed by Droulias et al.,⁴⁹ or that of Onsager theory proposed by Xiao and Sheng⁵⁰ but in addition to the internal (i.e. within the clusters) orientational order, also the internal positional order associated to the Sm-like layering should be included. Experimental, simulation and theoretical efforts designed to address these topics are underway.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this study's findings are available from the corresponding authors upon reasonable request. Analysed data supporting this article have been included as part of the ESI.†

Acknowledgements

We acknowledge the European Synchrotron Radiation Facility (ESRF) for provision of synchrotron radiation facilities under proposal number A26-2-618 and we would like to thank G. Portale, D. Hermida Merino for assistance and support in using the BM26B DUBBLE beamline and BM28-XMaS staff (ESRF) for providing the superconducting magnet. The authors acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. The authors thank Stephanie de Matos for her graphical expertise in constructing Fig. 1 and 3.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5cp01423a

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