William S.
Fall
^{ac},
Constance
Nürnberger
^{b},
Xiangbing
Zeng
*^{d},
Feng
Liu
^{a},
Stephen J.
Kearney
^{c},
Gillian A.
Gehring
*^{c},
Carsten
Tschierske
*^{b} and
Goran
Ungar
*^{ad}
^{a}State Key Laboratory for Mechanical Behaviour of Materials, School of Materials Science & Engineering, Xi'an Jiaotong University, Xian 710049, China
^{b}Department of Chemistry, Martin Luther University Halle-Wittenberg, Kurt-Mothes-Str. 2, 06120 Halle, Germany. E-mail: carsten.tschierske@chemie.uni-halle.de
^{c}Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK. E-mail: g.gehring@sheffield.ac.uk
^{d}Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, UK. E-mail: x.zeng@sheffield.ac.uk; g.ungar@sheffield.ac.uk

Received
14th December 2018
, Accepted 28th January 2019

First published on 29th January 2019

We have designed a compound that forms square liquid crystal honeycomb patterns with a cell size of only 3 nm and with zero in-plane thermal expansion. The compound is a bolaamphiphile with a π-conjugated rod-like core and two mutually poorly compatible side-chains attached on each side of the rod at its centre. The system exhibits a unique phase transition between a “single colour” tiling pattern at high temperatures, where the perfluoroalkyl and the carbosilane chains are mixed in the square cells, to a “two-colour” or “chessboard” tiling where the two chain types segregate in their respective cells. Small-angle transmission and grazing incidence X-ray studies (SAXS and GISAXS) indicate critical behaviour both below and above the transition. Both phase types are of considerable interest for sub-5 nm nanopatterning. The temperature dependence of ordering of the side chains has been investigated using Monte Carlo (MC) simulation with Kawasaki dynamics. For a 3-dimensional system with 2 degrees of freedom, universality predicts that the transition falls into the 3d Ising class; MC was therefore used to calculate observables and determine the critical exponents accessible in experiment. Theoretical values of ν, γ and, perhaps most importantly, of the order parameter β have been calculated and then compared with those determined experimentally. β found experimentally is close to the theoretical value, but ν and γ values are significantly smaller than predicted. To explain the latter, the measured susceptibility above T_{c} is compared with those from simulations of different lattice sizes. The results suggest that the discrepancies result from a reduced effective domain size, possibly due to kinetic suppression of large scale fluctuations.

## Design, System, ApplicationFor surface patterning on a sub-10 nm scale, self-assembling columnar liquid crystals with 2d long-range periodicity, are the natural choice. For application in nano-electronics, the pattern should ideally be square and have zero in-plane thermal expansivity. Moreover, it may be desirable to add a degree of complexity in the pattern, such as producing a “chessboard” pattern with alternative squares chemically different thus, e.g. combining alternating p- and n-semiconducting pathways, or electronically and ionically conducting channels, or similar. This report shows how such a system can be achieved in principle, based on an inverse columnar phase where a bistolane rod-like core provides a fixed-sized square honeycomb framework held together by H-bonding glycerol end-groups. The prismatic cells are filled alternately by a semiperfluorinated (F) and a carbosilane (Si) chain, respectively attached to either side of the linear core. The chessboard structure with 3 nm alternating square cells is achieved through an appropriate choice of the core length, side-chain area to core length ratio, and the level of incompatibility between the side-chains, as detailed in the Introduction. |

Promising and already extensively studied bolaamphiphiles consisting of a polyaromatic rod-like core, H-bonding end-groups and laterally attached flexible chains, have shown great versatility in forming a range of “inverted” columnar LC phases, i.e. honeycombs with aromatic walls (the “wax”) and the side-chains as cell fillers (the “honey”).^{21–24} These honeycomb networks prevent rotational averaging. The number of sides in the polygonal cross-section of a prismatic cell (the “tile”) is determined by the ratio between the area of the filler side-chains to the length of the rod-like core. Normally the side length of the polygon equals the length of the molecule, although in some cases double-length sides were found containing two end-linked rods.^{25,26} A range of tiling patterns have been obtained in this way, from simple triangular to highly complex nanopatterns containing up to five different coexisting tile types.^{27} Square patterns were, however, only found under special circumstances in a small temperature range.^{21,28}E.g., Recently, several members of a series of terphenyl-based bolaamphiphiles were found to exhibit non-centred (p2mm) rectangular lattices; however the cell aspect ratio was highly temperature sensitive, reaching 1:1 (square lattice) only close to isotropization temperature.^{29}

Beside their application potential, LCs are of considerable scientific interest, providing soft matter and low-dimensional equivalents to phenomena such as phase transitions, extensively studied in more traditional areas of condensed matter physics. Well known examples include the nematic to smectic-A transition^{30} and the twist-grain-boundary (TGB) smectic phase,^{31} both having their analogue in superconductivity, as well as the hexatic smectics being easily accessible examples of 2D melting phenomena.^{32} Ferro-, ferri- and antiferroelectric chiral smectics^{33} and the currently topical twist-bend nematic phase^{34,35} are further examples of new states of condensed matter. In contrast to numerous studies of phase transitions in nematics and smectics, transitions in thermotropic LCs with two-dimensional order have been investigated in only a few cases experimentally^{36–38} or by theory and simulation.^{27,39–44}

In the LC square patterns previously reported all squares were uniform. In order to increase functionality, the combinations of different squares, filled with different materials would be required. For this purpose a change in molecular design was required, whereby the single lateral alkyl chain of the above-mentioned terphenyls is replaced by two different and incompatible chains at the two opposite sides of the rod-like unit. The incompatibility between perfluorinated alkyl chains and hydrocarbon chains^{45} is a successfully employed approach to obtaining multicolour tiling patterns.^{24,25} However, in order to achieve adequate incompatibility the number of CH_{2} and CF_{2} units must be sufficiently large for the demixing energy to overcome the entropy penalty. However, these longer chains require more space in the prismatic cells. In order to retain square cells, the terphenyl cores forming the honeycomb were extended by the incorporation of additional ethynyl units between the benzene rings. Furthermore, in order to lower the crystal melting point, a highly branched carbosilane unit was used instead of a linear alkyl chain. The introduction of silicon as branching points into a hydrocarbon chain allows a synthesis in fewer steps than would be required for similarly branched hydrocarbons.

Here we report the self-assembly of a new LC system, formed by a bolapolyphile with a 1,4-bis(phenylethynyl) benzene (“bistolane”) core^{45} with two disparate chains attached laterally to the 2 and 5 positions (compound **1**, Scheme 1).

Scheme 1 Compound 1 and its transition temperatures (Cr = crystal, Col_{sq}^{2} = two-colour square columnar phase, Col_{sq}^{1} = single-colour square columnar phase, Iso = isotropic liquid). |

As will be shown in the following sections, this compound is found to form a square honeycomb with exceptional dimensional stability. Moreover it displays a disorder–order transition from a uniform square tiling to a two-colour chessboard-type tiling where the two side-chains segregate in separate square cells. The transition is thought to be a close real life example of the well-known and widely studied Ising model of magnetic spins.^{46} Monte Carlo simulations show the critical behaviour (e.g. development of short range fluctuations and long range order around the phase transition temperature), observed by X-ray diffraction, to closely match the 3D Ising model on both sides of the transition, provided the finite domain size is taken into account.

Powder small-angle X-ray scattering (SAXS) curves of compound **1** recorded during continuous heating at 0.5 K min^{−1} are shown in Fig. 2a. The bold trace at T_{c} = 88 °C marks the phase transition between two square columnar phases. In both phases reflections indexed as (10), (11), (20) etc. are seen, their q^{2} ratio being 1:2:4:5…. This defines the plane group as p4mm in both cases. Most notable is the decay and eventual disappearance of the strong (10) reflection of the low-T phase at T_{c}, along with the disappearance of (21) and (30) peaks. On heating above T_{c} the intensity of the remaining Bragg peaks stays almost unchanged: the (11), (20) and (22) diffraction peaks transit smoothly into the (10), (11) and (20) peaks of the high-T square phase. Remarkably also, the remaining peaks do not shift in the entire range from room temperature to the isotropization temperature T_{i}.

The above indexing of the diffraction pattern was also confirmed by grazing incidence SAXS (GISAXS) experiments on a surface-oriented thin film of **1** on silicon substrate, as shown in Fig. 3. Although most of the sample is homeotropic, giving diffraction peaks on the horizon line, off-equatorial reflections are also seen from areas where the columns lie parallel to the surface, rotationally averaged in the film plane. The most notable feature in Fig. 3 is the existence of strong diffuse scattering above T_{c} in place of the (10) Bragg reflections of the low-T phase – the relationship is highlighted by the light blue lines in Fig. 3. This indicates the persistence of short-range critical electron density fluctuations above T_{c}, which will be analysed quantitatively further below.

The lattice parameters a_{1} and a_{2} of the high- and low-T phases are 2.96 nm and 4.19 nm, respectively. As the ratio is exactly , the area of the unit cell of the low-T phase is twice that of the high-T phase. Electron density maps, calculated on the basis of relative diffraction intensities and the selected structure factor phases given in Table S1,† are shown in Fig. 4a and c.

Considering the molecular model, we note that a_{1} = 2.96 nm is very close to the 2.80–3.05 nm length of the molecule, as measured between the ends of the glycerol groups, depending on the glycerol conformation. This, combined with the fact that the high birefringence axis (i.e. the long axis of the π-conjugated molecular core) is perpendicular to the columns (see POM images with a λ retarder plate in Fig. 1b and d and S1c†), strongly suggests the model in Fig. 4b for the high-T phase. Such a model is consistent with the general self-assembly mode of bolapolyphile LC honeycombs, except that a square honeycomb is rather rare. In this structure the cell walls consist of molecular cores lying normal to column axis with each of the two side chains located in the two neighbouring cells. In the high-T phase the cells are filled on average with a 1:1 mixture of F and Si chains. The model in Fig. 4b has been subjected to 30 cycles of molecular dynamics annealing between room T and 700 K lasting 30 ps in total. It shows a reasonable packing density, indicating that the model is feasible. It is also consistent with the electron density map, where the light blue squares reflect the fact that the average electron density of the combined F and Si chain (572 e nm^{−3}) is somewhat higher than that of the bistolane/glycerol core (518 e nm^{−3}) – for details see Table S2.†

The doubling of the unit cell below T_{c}, the high intensity of the (10) reflection of the low-T lattice, and the reconstructed electron density map in Fig. 4c all suggest that a disorder–order transition takes place at T_{c}, with the F and Si chains preferentially occupying alternative cells, creating a chessboard tiling as in Fig. 4c and d. In a structure with perfectly segregated Si and F chains, we calculate that the contrast between Si and F cells would be 714–430 = 284 e nm^{−3}, which is >5 times higher than the contrast between the mixed cell interior and the bistolane/glycerol wall. Hence the very much stronger (10) Bragg reflection of the chessboard phase compared to that of its (11) reflection or of the (10) reflection of the high-T phase (note that the intensity scales roughly as square of the electron density contrast).

Fig. 5 (i) Allowed configurations of the square tiles and their associated energies in terms of pair interactions S, F and M (see Table 1) (ii) in plane interactions: the configurations 1 and 2 of a chain in the fully ordered state that is flipped between squares a and b. Square a contains n_{a} Si chains and 3 − n_{a} F chains, similarly for b. (iii) Out of plane interactions: the configurations 1 and 2 of a chain in the fully ordered state that is flipped between columnar interaction regions c and d. Region c contains n_{c} Si chains and 2 – n_{c} F chains, similarly for d. |

Since both the dimensionality and degrees of freedom are the same, by universality, it is sensible to conclude that the model should have the critical exponents of the 3d Ising universality class. A simple lattice model has been devised and MC simulation is used to test this conjecture. The theoretical values for the critical exponents are determined and compared with experimental values in the sections that follow. It is interesting to note that due to the introduction of pairwise interactions, we conclude that there is only one significant energy in this problem; that is the energy difference between like attractive and unlike mutually repulsive chains. The ordered states are therefore found to be exactly symmetric. Any one molecule contains one chain of each type hence flipping a molecule changes the energy state of both neighbouring tiles; this is analogous to the 3d Ising model of magnetic spins under Kawasaki spin flip dynamics.

Each LC molecule is comprised of two different side chains one silicone (S) and one fluorine (F) which may pairwise interact. Like interactions are more strongly attractive than unlike interactions which results in an order–disorder transition. Within each supramolecular square the side chains can interact equally through 3 different pair interactions; these are S: Si–Si, F: F–F and M: Si–F. Crucially, chain–chain interactions within a square carry equal weight and the positioning inside the supramolecular square is neglected. The total possible number of interactions which exist in any given square describes the squares energy, that is for n Si chains and (4 − n) F chains the total energy can be expressed as the number of distinct interactions; Si–Si: n(n − 1)/2, F–F: ((4 − n)(3 − n))/2 and Si–F: n(4 − n).

The 5 possible configuration energies for an arbitrary square containing n Si chains and (4 − n) F chains and their multiplicities are given in Table 1. The multiplicities add to 2^{4} = 16 with the largest contribution coming from configuration with an equal number of Si and F chains; indicating the importance of entropy in driving this order–disorder transition.

No. Si | No. F | Energy | Multiplicity |
---|---|---|---|

4 | 0 | 6S | 1 |

3 | 1 | 3S + 3M | 4 |

2 | 2 | S + F + 4M | 6 |

1 | 3 | 3F + 3M | 4 |

0 | 4 | 6F | 1 |

In Fig. 5(i) the configurations before and after a molecule has been flipped on the border between two squares is shown; this is an example of Kawasaki dynamics where the flipping of a single molecule directly changes the energy of both neighbouring squares involved. Interactions between neighbouring molecules out of plane are also included, depicted in Fig. 5(ii); these are taken to be half those in plane. The initial and final configuration energy of the 2 states, U_{1} and U_{2} respectively, can be expressed in terms of the pair interactions and number of Si chains in and out of plane

(1) |

(2) |

(3) |

The constant U_{0} = [S + F − 2M] is therefore the only significant energy involved in the model since any chain flip affects the chain number in both the neighbouring squares. In our case U_{0} is always negative because like chains attract more strongly; a chessboard ordering can occur only for negative U_{0}.

The 3d lattice is broken down into columns of L 2d chessboards each containing two identical sublattices A and B, this is depicted in Fig. 6. A fully ordered 2-colour tiling, (Fig. 6a) consists of alternating columns of supramolecular squares comprising of 4 Si or 4F chains such that the number of Si or F chains in either sublattice is maximised, see Fig. 6b. For a single colour tiling the columns comprise of a mixture of Si or F chains where, on average, the number of Si or F chains in any cell are equal, Fig. 6c and d. The order parameter, σ_{i}, for one sample configuration is defined as

(4) |

(5) |

A plot of χ is as a function of temperature is shown in Fig. 7b for 5 different lattice sizes. An initial thermalisation of 10^{4} MC steps is performed and 10^{5} measurements were taken at each temperature. The ensemble averages for the order parameter 〈σ〉 and susceptibility χ are calculated in order to construct the phase diagram. According to universality, the transition should fall into the 3d Ising universality class. By considering the scaling functions for the order parameter, and susceptibility we can confirm this by fitting to the Ising exponents, the scaling functions are given by^{48}

(6) |

σ = [(T_{c} − T)/T_{c}]^{β} | (7) |

We make the assumption that the sublattice magnetization σ is proportional to the amplitude of diffraction, i.e. that the flip of direction of side groups at a side of a square cell (equivalent to a flip of spin), results in a constant increase (or decrease) to the overall structure factor F_{(10)} of the (10) diffraction peak of the two coloured square phase. As the diffraction intensity I_{(10)} is proportional to |F_{(10)}|^{2}, it varies as

I_{(10)} ∝ (T_{c} − T)^{2β} | (8) |

The critical exponent β can be found by fitting the diffraction intensity I_{(10)} as a function of temperature to the above equation.

The best-fit to the experimentally determined intensities suggests that 2β = 0.766 and β = 0.383 (Fig. 2a and 9a). This is slightly larger than the value of β = 0.326 expected for a 3d-Ising model and also derived by our Monte Carlo simulation. For comparison, the theoretically predicted curve from the 3d Ising model is shown also in Fig. 9a (broken curve). It is evident that the observed experimental data follows the theoretical curve closely, even though the experimentally observed transition is less sharp than predicted by theory. This slight discrepancy could be attributed possibly to the system not residing in thermal equilibrium, despite the relatively slow heating rate used (0.5 °C min^{−1}).

While the change of order parameter below critical temperature carries information of β, two other critical exponents, γ and ν, are related to the local fluctuation-correlation of the order parameter above the critical point, i.e. <σ_{i}σ_{j}>. Similarly to what has been discussed before, such fluctuations will be reflected in the local fluctuation of structure factor (F_{i} at lattice point i and F_{i} proportional to σ_{i}) and leads to diffuse scattering, with an intensity proportional to , around q_{(10)}. The experimental diffuse scattering peaks just below and above the critical point are shown in Fig. 9b.

For a 3d Ising model, the correlation function G(r) takes the form

(9) |

(10) |

Since for the 3D Ising model critical exponent η is extremely small (0.036), the diffuse peak can be treated as Lorentzian. Therefore, we have fitted the diffuse peaks as shown Fig. 9c and d to a Lorentzian function and retrieved the peak height h_{D} in addition the full width at half maximum (FWHM) values as a function of temperature (Fig. 9, data shown by empty circles). Above the critical point the correlation length ξ is proportional to (T − T_{c})^{−ν}. The peak height h_{D} is proportional to the susceptibility χ, and (T −T_{c})^{−γ} where γ = (2 − η)ν; and the FWHM = 2/ξ is inversely proportional to the correlation length, therefore^{50}

(11) |

The best-fit and theoretical curves from fitting of peak height and FWHM of the diffuse scattering are compared in Fig. 9.

Such reduction of diffuse peak height or measured susceptibility close to the critical temperature can in fact be seen in our Monte Carlo simulation with different sample sizes (number of cells). Fig. 10a shows the simulated susceptibility for four different domain sizes (12^{3}, 24^{3}, 48^{3}, 96^{3} and 192^{3} cells respectively), and a significant reduction in the susceptibility close to the critical temperature can be clearly seen for domain sizes of 12^{3} and 24^{3} cells. In fact the trend in the 24^{3} cells is very close to that experimentally observed (Fig. 10a), suggesting a domain size of only ∼70 nm. On the other hand, the width of the (10) Bragg diffraction peak of the two-colour square phase below T_{c}, and of the (10) peak of the single-colour square phase above T_{c}, suggest a much larger domain size of ∼250 nm. We speculate that large scale fluctuations of local two-coloured patches are likely to be restricted kinetically, as they involve converting mixed cells at the boundaries of such local patches by exchanging the directions of the two side groups of molecules (Fig. 10b). While in a spin model an inversion of spin has no energy barrier, in our system the corresponding change molecular orientation always has one. Therefore, the chance of large scale fluctuations will be significantly reduced when compared to the situation when no such energy barrier is present. This would result in a smaller effective domain size for the two-colour fluctuations.

Having learned the basic nano-architectonic principles of creating 1- and 2-colour square patterns from the above results, in a follow-up study we undertook to increase the cell size. Thus a series of related bolapolyphiles were synthesised, having a longer oligo(p-phenyleneethynylene) core involving five benzene rings instead of only three in the bistolane core and suitably extended side-chains.^{51} Both, the mixed-chain and the chessboard phases were observed for compounds with appropriate chain lengths, showing that the principles of architectural design learned from the present work are applicable to a wider class of materials and are generic.

(12) |

As the observed diffraction intensity I(hk) is only related to the amplitude of the structure factor |F(hk)|, the information about the phase of F(hk), ϕ_{hk}, cannot be determined directly from experiment. However, since the plane group p4mm is centrosymmetric, the structure factor F(hk) is always real and ϕ_{hk} is either 0 or π. This makes it possible for a trial-and-error approach, where candidate electron density maps are reconstructed for all possible phase combinations, and the “correct” phase combination is then selected on the merit of the maps, helped by prior physical and chemical knowledge of the system.

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## Footnote |

† Electronic supplementary information (ESI) available: Optical micrographs, DSC traces, X-ray diffraction data, volume fractions and electron densities of constituent moieties, details of synthesis and chemical characterization, energies used in MC simulation. See DOI: 10.1039/c8me00111a |

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