Predicting the Conditions for Homeotropic Anchoring of Liquid

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Predicting the Conditions for Homeotropic Anchoring of Liquid Crystals at a Soft Surface. 5CB on Alkylsilane Self-Assembled Monolayers Otello Maria Roscioni, Luca Muccioli, and Claudio Zannoni ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.6b16438 • Publication Date (Web): 13 Mar 2017 Downloaded from http://pubs.acs.org on March 15, 2017

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ACS Applied Materials & Interfaces

Predicting the Conditions for Homeotropic Anchoring of Liquid Crystals at a Soft Surface. 5CB on Alkylsilane Self-Assembled Monolayers Otello Maria Roscioni,∗ Luca Muccioli, and Claudio Zannoni∗ Dipartimento di Chimica Industriale “Toso Montanari” Università di Bologna, viale Risorgimento 4, IT-40136 Bologna (Italy) E-mail: [email protected]; [email protected]

Rev. February 21, 2017

Keywords: Liquid Crystals, Molecular Dynamics, Self-Assembled Monolayers, Homeotropic Alignment, Anchoring Coefficient, Surface Science, Thin Films, Organic Functional Materials

1

Abstract

We have studied, using atomistic molecular dynamics simulations, the alignment of the nematic liquid crystal 5CB on self assembled monolayers (SAMs) formed from octadecyland/or hexyl-trichlorosilanes (OTS and HTS) attached to glassy silica. We find a planar alignment on OTS at full coverage, and an intermediate situation at partial OTS coverage, due to a penetration of 5CB molecules in the monolayer, which also removes the tilt of the OTS SAM. Binary mixtures of HTS and OTS SAMs do instead induce homeotropic alignment. Comparison with the existing experimental literature is provided. ∗ To

whom correspondence should be addressed

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ACS Paragon Plus Environment


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2

Introduction

The proper alignment of liquid crystal (LC) molecules near a surface 1–6 is a key ingredient of most, if not all, LC displays (LCDs) and devices. 7–10 For example, standard twisted nematic and in-plane switching (IPS) LCDs require a planar uniform alignment of the molecules with respect to the cell surface. On the other hand, the now very popular vertical alignment LCD technology 11 requires the molecules to align essentially perpendicular to a suitably treated surface. Practical recipes have been developed to obtain the desired alignment for a given substrate and LC material. 2,12,13 However, a basic molecular understanding of the alignment is still missing and should be developed, not only for its technological applications, but also for its importance from a fundamental point of view. The difficulty of this task is witnessed also by the sensitivity of LC interfaces to external factors, like very small concentrations of analytes, that is at the core of the fascinating application of LC interfaces as sensors. 14,15 A further proof is that even light-induced trans-cis conformational changes in interfacial layers containing azobenzene derivatives are sufficient for switching from homeotropic to planar anchoring, and vice versa. 16,17 In this context, even though molecular dynamics (MD) simulations are still far from providing a simple automated tool for predicting the alignment on a surface, they are proving to be a reliable methodology to study the molecular organisation of soft organic materials near a solid surface, 18 also reproducing a number of experimental observations. 3 We have recently attacked the problem of understanding LC surface alignment and of predicting it for a given LC and substrate using atomistic molecular dynamics. 19 We have studied the ordering of 4-n-pentyl-4’-cyano biphenyl (5CB) on a hydrogen-terminated

(001) crystalline silicon surface, 5 finding a planar uniform alignment along the diagonal direction of the facet, with an enhanced orientational order with respect to the bulk of the LC film. We have found similar results for the alignment of 5CB on the (001) plane of cristobalite, a crystalline form of silicon dioxide (silica), while we have found that a planar degenerate alignment develops on silica glass, with some tilting away from the surface 2


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˚ 6 which increases as the RMS roughness of the surface increases from ∼ 1.5 to ∼ 3.2 A. Furthermore, at least in the studied cases, the crystallinity of the surface appears to produce a planar and uniform alignment of LCs and to enhance surface orientational order. This planar alignment was observed also on glass, even though with random (degenerate) orientation in the surface plane and, contrary to the previous case, an orientational order at the surface lower than that inside the bulk. While it is perhaps understandable that solid crystalline substrates are inducing order, the effects of a vacuum interface are at first surprising. Air (or vacuum) appears to be a strong aligner: homeotropic alignment of a LC at a free interface is observed experimentally and is even employed as a means of obtaining aligned films for smectics starting from freely suspended films. 13 In all our simulations of cyanobiphenyl thin films, we observed strong homeotropic alignment of the LC at the interface with vacuum, both in the case of the other surface being solid 5,6 or free as well. 20 This orientation can be understood in energetic terms as it reduces the loss of side-side interactions between the LC molecules at the free surface. A further, entropic, effect can arise for molecules such as n-alkyl-cyanobiphenyls by exposing their terminal chains to vacuum, thus gaining free energy through the augmented degrees of freedom. This last contribution may not be essential, as homeotropic alignment at the vacuum interface is also obtained for many organic crystals such as pentacene 21 and T6, 22 which do not have terminal chains. What, to the best of our knowledge, has not yet been observed using computer simulations, is a clear case of homeotropic (ı.e. vertical) alignment at a solid support surface. Some early MD studies were performed for 5CB on amorphous polyethylene surfaces 23 and polyvinyl alcohol (PVA) 24 or for 8CB on polyimide 25 but, understandably for those days, with a very small number N of 5CB molecules (from N=10 to N ≤ 32) as well as for very short time windows and with a non specifically validated force field. 23 Experimentally, homeotropic alignment is often realised by treating the support surface with coupling agents that chemically bind to it. For instance, lecithins have been traditionally

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used on glass surfaces, 2,12 exploiting the fact that the polar moiety attaches to the surface, while the fatty chains play the role of aligning agents. The importance of molecular chains and their orientation with respect to the surface is indicated also by the above mentioned case of photoresponsive alignment layers, 16 where a light-induced photoisomerisation, e.g. of chains grafted through a pivot azobenzene moiety, changes the orientation of the chains and in turn the alignment of a LC layer from planar to homeotropic. Other treatments based on surface coating with polymers 26 are also available, but their use seems to be essentially based on a trial and error procedure, because of various and possibly competing factors such as the alignment of the polymer chains, the presence of nano or microgrooves and of course the chemical nature of the polymer itself. 24,26–30 Amongst the simplest and better defined coupling agents, self-assembled monolayers are particularly interesting and well studied experimentally. 31–33 In a seminal series of papers, Abbott and co-workers reported that homeotropic anchoring of 5CB 34–36 and MBBA 37 liquid crystals is achieved on SAMs formed on gold surfaces by two-component mixtures of either short and long semi-fluorinated thiols or short and long alkanethiols. Analogous findings were reported by Fazio and co-workers for MBBA on monolayers of mixed fatty acids with different lengths of the alkyl chain. 38,39 Ruths et al. confirmed these results for 5CB, and extended them to the smectic phase of 8CB. 40 It was found that a planar anchoring of 5CB is instead obtained on single-component SAMs formed from either alkanethiols 34,35,37 or semi-fluorinated thiols, 36 which differ by the tilt angle that their chains have with respect to the normal at a surface. 41 A similarly planar alignment was obtained for 8CB on a gold substrate coated with a short SAM bearing a polar terminal carboxyl group. 42 The reported difference in the anchoring of 5CB on SAMs formed by one or two components on gold is somehow in contrast with the homeotropic anchoring of 5CB supported on a SAM of octadecyltrichlorosilane on fused silica 43 and glass. 44,45 Peek et al. suggested that this result reflects the different polarizability of silica and gold substrates,

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and hence a longer range gold-liquid crystal interaction, rather than the expected differences in the tilts of chains within SAMs. 46 For silica, Schwartz and co-workers showed that the range of interaction with the surface is of the order of 1 nm, as the random planar alignment typical of this substrate is obtained only for very short SAMs. 44,45 The contrasting observations between the anchoring of 5CB on seemingly identical SAMs hint at the delicate balance of forces acting between the 5CB and a soft interface. Indeed, the preparation of well-designed surfaces is key for the controlled alignment of soft materials at the atomic scale, as recently demonstrated on chemically patterned surfaces 47 and on the periodic grooves of black phosphorous surfaces. 48 In this paper, we investigate by means of atomistic MD simulations four realistic models of a thin film of the nematic 5CB LC supported on amorphous silica surfaces functionalised with single- or two-component mixtures of alkyl-silanes. The use of computer simulations allows us to explore in detail the morphology of the 5CB/SAM interface at a precise level of surface coverage and to establish the effect that each component has on the other one. Two samples contain a SAM of pure octadecyltrichlorosilane (OTS): one surface has a closed-packed structure and is virtually defect-free, while the other has a lower density of OTS which decreases the positional and orientational order within the SAM. The two remaining samples contain a mixture of two alkyl-silanes with opposite majorities of long (OTS) and short (HTS) chains. A detailed analysis of both the 5CB film and the SAM support is presented and discussed; the morphology of each component is characterized in terms of the relevant physical observables.

3

Methods

Computer-generated models of four SAM samples were studied by means of classical MD simulations. All the calculations were carried out with the program NAMD 49 using a constant number of molecules N, volume V and temperature T (NVT ensemble) at 300

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K and 320 K, corresponding to the nematic and isotropic phases of 5CB, respectively. The temperature was controlled with a Langevin thermostat 50 with damping coefficient of 5 ps−1 . A multiple time-stepping integration scheme was used, with a time step of 1 fs for the bonded forces, 2 fs for the non-bonded forces, and 4 fs for electrostatic interactions. A ˚ was employed to truncate the short-range non-bonded interactions, cut-off radius of 12 A while long-range electrostatic interactions were evaluated through the 3D Particle Mesh ˚ for all samples. TakEwald method. 51 The simulation box measured 89.8 × 74.9 × 700.0 A ing into account the thickness of the simulated slabs, this leaves a vacuum gap of ∼ 350 ˚ which is more than enough to decouple the interactions between the top and bottom A, surfaces of the slab as required by the 3D periodic boundary conditions employed. 52 Each ˚ coated with sample is composed of an amorphous silica slab with thickness of ∼ 66.4 A, a SAM composed of ∼ 300 molecules of alkyl-silanes and, on top of the SAM, a thin film of ∼ 4100 molecules of 5CB. The SAMs investigated are composed of a mixture of octadecyltrichlorosilane (OTS) and hexyltrichlorosilane (HTS) in various proportions, as shown in Figures 1 and 2. Our simulation methodology for preparing the SAM on silica, described in detail in, 41 is inspired by the experimental procedure of Wang et al. 53 that demonstrated the formation of uniform and not cross-linked monolayers of OTS under strict anhydrous conditions. It is worth noticing that this procedure was not used in older experimental literature, with the formation of SAMs with less controlled uniformity as a possible source for the variety of finding reported. More specifically, the details of the four samples studied in this work are as follows. In sample no. 1, the SAM is composed only of OTS with a density of 4.5 molecules/nm2 , arranged in a hexagonal close-packed lattice. Similarly, the SAM in sample no. 2 is composed of OTS with a density of 4.2 molecules/nm2 . The lower density of OTS molecules leads to formation of vacancies on the resulting SAM. The SAM in sample no. 3 is composed of a mixture of 60% OTS and 40% of HTS, while the SAM in sample no. 4 is composed of a mixture of 40% OTS and 60% of HTS. The total density of SAM in samples 3

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OTS (mol/nm2 ) HTS (mol/nm2 )

sample 1

sample 2

sample 3

sample 4

4.5 0.0

4.2 0.0

2.7 1.8

1.8 2.7

Figure 1: Lateral view of 5CB films supported on various SiO2 /SAM surfaces at T = 320 K, where 5CB is in its isotropic phase. Atoms are represented as spheres and coloured according to the element: red/oxygen, yellow/silicon, grey/carbon, white/hydrogen; 5CB molecules are colour-coded according to their orientation with respect to the surface: blue/perpendicular, red/parallel. and 4 is of 4.5 molecules/nm2 , respectively. The MD simulations were carried out in the following way. First, the slab of amorphous silica was prepared as described in reference. 6 The inter-atomic interactions between silicon, oxygen and hydrogen atoms were modelled with the force field by Cygan et al. 54 Lennard-Jones parameters and bonding interactions used to describe OTS and HTS molecules were taken from the AMBER force field, 55 while torsional parameters and atomic ESP charges were derived from quantum-chemical calculations carried out with the program Gaussian 09 56 at the wB97XD/aug-cc-pVTZ level of theory. 57 The 5CB molecules were described at the united-atoms level of detail, where CH, CH2 and CH3 7



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OTS (mol/nm2 ) HTS (mol/nm2 )

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sample 1

sample 2

sample 3

sample 4

4.5 0.0

4.2 0.0

2.7 1.8

1.8 2.7

Figure 2: Lateral view of 5CB films supported on various SiO2 /SAM surfaces at T = 300 K, where 5CB is in the nematic phase. Atoms are represented as spheres and coloured according to the element: red/oxygen, yellow/silicon, grey/carbon, white/hydrogen; 5CB molecules are colour-coded according to their orientation with respect to the surface: blue/perpendicular, red/parallel groups are represented as single spherical interaction sites, using the parameters in the already mentioned force field for cyanobiphenyls, which was showed to reproduce the nematic-isotropic transition temperature, density and order parameters of 5CB quite successfully. 58 The SAMs were grafted on amorphous silica surfaces with the help of a computer code developed in our group, 41,57 while an isotropic film of 5CB was taken from previous simulations and placed on top. 6 MD trajectories were computed for each sample in the NVT ensemble. At first, the systems were equilibrated at 320 K for about 60 ns, followed by 60 ns of production time at 320 K. The systems were then cooled to 300 K at a rate

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of 0.5 K/ns and equilibrated for up to 300 ns, until formation of the nematic phase was observed. A trajectory lasting 120 ns was then accumulated at 300 K for the production of results.

4

Results and Discussion

We describe the molecular organization of the nematic across the film with various distributions and observables, along the lines introduced in our previous work. 5,6 We start defining the molecular orientation of each 5CB molecule in terms of the angle β between the molecular axis u, ˆ chosen as the principal axis of inertia and pointing towards the cyano group, and the normal to the surface, zˆ . Thus, a 5CB molecule with the CN bond pointing towards, away from, or parallel to the silica surface will have a cos β = −1, 1 or 0, respectively. We have computed the one-particle probability distribution function P(z, cos β), where z is the distance between the centre of mass of the molecule and the naked silica surface, as

P(z, cos β) = hδ(z − zi )δ(cos β − uˆ i · zˆ )i

(1)

where the angular brackets indicate an average over each configuration and over time. Equation (1) is normalized to 1 when integrating over z and cos β. The distribution functions P(z, cos β) computed for the isotropic (T = 320 K; top) and nematic (T = 300 K, bottom) phases are shown in Figure 3. We notice, on the vacuum side, the familiar homeotropic alignment, weak but not negligible in the isotropic phase and stronger in the nematic. The homeotropic alignment at the vacuum side is also apparent from the snapshots of samples, in Figures 1 and 2. For this film thickness, the free surface behaviour is essentially unaffected in all cases by the changes in density and composition of the SAM. At the silica-SAM surface we ˚ occupied by the bound see in general an absence of 5CB molecules for the first ∼ 25 A, 9



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alkylsilane molecules. The 5CB exclusion is of course more complete for the densest SAM in sample 1, while the presence of some 5CB molecules is observed inside the SAM in sample 2. In this case, the LC fills the empty space between OTS molecules in the SAM, which has a lower packing density compared to sample 1. A significant penetration of 5CB molecules is also seen in samples 3 and 4, just above the sublayer formed by the SAM with the short alkyl chain. The 5CB molecules in this region interact with the long chains of OTS molecules, assuming a perpendicular alignment with respect to the silica support. More detailed and quantitative density plots are shown in Figure 4. At the SAM interface, the density of the 5CB film is rather insensitive to the temperature and the main differences between the interfaces can be attributed solely to the morphology of the SAM. The penetration of 5CB into the SAM cannot even be explained in terms of the melting of the SAM, which is expected to remain ordered at the simulated temperatures. 59 The density of 5CB at 320 K is slightly smaller than at 300 K. The effect of the increasing temperature on the 5CB film is thus a thermal expansion, shifting to higher values the right outermost peak of the density profiles. The shift is only seen at the vacuum interface, since the supporting surface of SiO2 is fixed in space. Turning to our main issue, the aligning effect of the SAMs, we see that the defectfree OTS monolayer (sample 1) causes a planar alignment of the 5CB overlayer, making the film a hybrid boundary conditions one. This situation is similar to what has been observed for 5CB on alkanethiols on gold, 31,36 but different from the perpendicular alignment reported for 5CB on OTS/silica. 32,33,60 The SAM in the second sample is still composed of pure OTS, but with a lower packing density allowing, as already mentioned, for vacancies on the SAM to be filled by 5CB molecules. Accordingly, P(z, cos β) shows a change from the previous case. The 5CB molecules inside the SAM have a strict homeotropic alignment, while above the SAM interface a predominantly planar alignment is found and maintained up to the middle of the sample. It is very interesting to

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Figure 3: Contour map of the probability distribution function P(z, cos β) for 5CB on four silica/SAM surfaces at 320 K (top panels) and 300 K (bottom panels). The small plates show a close-up of the function P(z, cos β) near the SAM/5CB interface. compare the tilt of OTS molecules in the two cases: we see from Figure 5 that, for the hexagonal close-packed SAM in sample 1, the tilt of OTS molecules (≈23 degrees) is very similar for the SAM exposed to vacuum 41 or in contact with the 5CB film. In sample 2, the OTS molecules are again tilted in much the same way than when they are exposed to vacuum, 41 even if the distribution is now wider, given the lower coverage and, consequently, the more space available to the OTS chains. However, when the 5CB film is 11


1.0 320 K 300 K

sample 3

0.0 1.0

0.0 sample 2

Density (g/cm3)

1.0

0.0 sample 1

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sample 4


1.0

0.0

0

20

40 0

100 z (Å)

200

300

Figure 4: Density of 5CB across the film, ρ(z), for different samples at T = 300 K and 320 K (right panel) and close to the silica (left panel), computed with respect to the normal at the surface, z. The grey-shaded areas indicate the SAM. present, the penetration of 5CB molecules in the SAM has the somehow surprising effect of eliminating the tilt, aligning the OTS molecules vertically. This configuration is also evident in Figures 1 and 2. Hence the mixing of OTS and 5CB molecules heals the defects in the structure of the SAM, probably also in virtue of the similar length of OTS and 5CB molecules, creating a smooth interface resembling that of sample 1. In turn, this interface favours a planar alignment of 5CB molecules in the overlying film, even though the distribution of P(z, cos β) around cos β ∼ 0 is broader than in sample 1. For samples 3 and 4, we observe instead an essentially uniform homeotropic alignment of 5CB across the film, which seems to be of better quality for the sample with the higher concentration of the short HTS chains (sample 4). In both cases, we see from Figure 4 that the density of 5CB at the interface with the SAM displays oscillations consistent with a weak tendency to form

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sample 1 30° 20° 10°

40° 15

40°

sample 2 30° 20° 10°

5CB vacuum

10 P(θ)

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5 0 0.8

1

0.8

1

Tilt Angle (cosθ)

Figure 5: Tilt angle distributions of SAM of OTS in vacuum 41 and in presence of the 5CB film for the full coverage sample 1 (left) and for the less dense sample 2 (right). a double layer of molecules in antiparallel orientation. This disposition is also apparent from the peaks at cos β = ±1 of the probability distribution function P(z, cos β), in Figure 3. The most important observable for liquid crystals is arguably the scalar order parameter h P2 i, which represents the average degree of alignment of a molecule with respect to the local preferred direction (or director) n(r) at a certain position r. h P2 i can be obtained as the largest eigenvalue, and n(r) as the corresponding eigenvector, of the so called ordering matrix Q(r). 61 Here we are interested in studying the variation of the scalar order parameter h P2 i and of the preferred direction n across the film, i.e. as a function of the distance z from the surface and we can define Q(z) as * Q(z) =

N (z,t)

∑

i =1

[3ui (t) ⊗ ui (t) − I] 2N (z, t)

+ (2) t

where ui (t) is a unit vector giving the orientation of molecule i at time t (e.g. here the principal axis of the inertia tensor corresponding to the lowest eigenvalue), I is the identity matrix and the sum runs over all the N (z, t) molecules of the virtual slab of a chosen thickness, parallel to the surface and at distance z from it. 5,6,58 In Figure 6 we plot the order parameter h P2 i and the director components along the surface normal, nz , and in the

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˚ For plane parallel to the surface, nx and ny , computed in discrete bins with width 17 A. sample 1, the order at the SAM interface appears to be similar or slightly lower than the one inside the sample, and close to that in the isotropic phase. This confirms our previous observations, 5,6 that the order at the interface is basically dictated by the surface and, as found experimentally for various systems, 3 that a planar degenerate anchoring tends to reduce the ordering. Looking at the director components in Figure 6, we find that the dominant component at the SAM interface is aligned along a single, but arbitrary, direction, here x. The preferred orientation at the vacuum interface is instead along z. Thus the degeneracy between the z and y components at the SAM interface has to turn into a degeneracy between the x and y components at the vacuum interface. The variation of nz above the SAM interface suggests that the energy cost of the degeneracy removal is small, ˚ allowing it to takes place before the loss of the preferred planar orientation, at z ∼ 200 A. A somewhat different situation appears for sample 2, if we consider the 5CB molecules that have penetrated the monolayer (see the density plot in Figure 4). The director components show that, starting from the silica support, the 5CB diffused inside the SAM is aligned homeotropically and then goes through an abrupt transition to a planar anchoring at the interface with the SAM. From this point onwards, the dominant component of the molecular director lies along the y direction, leaving a degeneracy between the x and z components, analogous to what observed for sample 1. A smooth transition between tilted (quasi planar) and homeotropic arrangements is observed across the film, before the latter orientation wins over thanks to the free surface effect. For this case too, the h P2 i at the borders is the same at the isotropic and nematic temperatures. The mixed SAMs definitely provide a homeotropic alignment for molecules dispersed in the SAM layer and this alignment, differently from what happens for sample 2, propagates well into the samples. We notice, from the mass density distributions in Figure 4, a modulation of the profile of the SAM/5CB interface which is due to the different chain lengths of the SAM. Sample 4, in which the SAM interface is composed mainly of

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0

100

200

sample 4

ny

nz

sample 3

nx

sample 2

Director Components

300

1 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0

sample 1

sample 4 sample 3

300 K 320 K

sample 2

1 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0

sample 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Scalar Order Parameter 〈P2〉

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0

z (Å)

100

200

300

z (Å)

Figure 6: (Left) Scalar order parameter h P2 i of different 5CB films at T = 300 K (black) and T = 320 K (green) and (Right) director components nx (red) and ny (green), on the plane parallel to the surface, and nz (blue), along the surface normal, each computed for a slice at distance z and T = 300 K. short chains of HTS with fewer protruding chains of OTS, seems to induce the strongest homeotropic alignment, both in terms of scalar ordering and of the director orientation shown by the nz component. We observe that in all cases the order is higher at the free surface. Again, this is in line with the ordering effect at the free interface that we have observed also for 8CB free standing films, where we found that the external layers are the most robust and ordered and the last to melt upon heating. 20 Complementary information of the molecular organisation across the film can be obtained by expanding P(z, cos β) in Equation 1 in terms of Legendre polynomials: 5 ρ(z) 1 3 P(z, cos β) = + h P (cos β)iz P1 (cos β) + h P2 (cos β)iz P2 (cos β) + . . . hρi 2 2 1 2

15


(3)


1.0 sample 4

0.5 0.0 −0.5 −1.0

sample 3

0.5 0.0 −0.5 −1.0

sample 2

0.5 0.0 −0.5 −1.0 〈P1(z)〉

0.5

sample 1

Order Parameters 〈P1(z)〉 and 〈P2(z)〉

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0.0 −0.5 −1.0

〈P2(z)〉 0

100

200

300

z (Å)

Figure 7: Order parameters h P1 (z)i (red) and h P2 (z)i (black) of different 5CB films at T = 300 K across the film, computed with respect to the normal at the surface, z. where ρ(z) is the density per unit of length, cos β ≡ u · z, h P1 (cos β)iz ≡ h P1 (z)i is the polar order parameter, h P2 (cos β)iz ≡ h P2 (z)i is the quadrupolar order parameter and so on. Each term is calculated at a distance z from the solid surface. The results for h P1 (z)i and h P2 (z)i, computed in the nematic phase, are given in Figure 7. For all the samples,

h P1 (z)i has a broad negative peak at the interface with vacuum, corresponding to the alkyl tails of 5CB molecules oriented toward the exterior of the film. 62 This peak is followed by small oscillations, corresponding to the formation of polar layers of molecules oriented anti-parallel to each other. 63 An analogous but weaker polar ordering is also observed at the interface with the SAM for samples 2, 3 and 4. By taking h P2 (z)i into account, we see that a planar ordering, corresponding to h P2 (z)i < 0, is established in samples 1 and 2 just above the SAM interface. For sample 2, we see that the 5CB molecules mixed with the OTS inside the SAM form a disconnected domain, with an abrupt transition at the SAM

16


0.1

2

0.05

1

0

0

0.15

3

0.1

2

0.05

1

0

0

0.15

3

0.1

2

0.05

1

0

0

0.15

3

0.1

2

0.05

1

0

0

100

200

sample 4

3

sample 3

4

sample 2

0.2 0.15

sample 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 Rapini−like Anchoring Coefficient −wA 2 (J/m )

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0 300

z (Å)

Figure 8: Rapini-like anchoring coefficient w2A for 5CB as a function of the distance from the SAM interface at 300 K (solid blue circles). The coefficients are calculated for sections ˚ The surface density of molecules in each layer of the sample with a thickness of 12 A. 2 (dashed line, molecules/nm ) is reported on the right-hand axis. interface which is mirrored by the negative value of the scalar order parameter h P2 i, in Figure 6. A similar transition region is present at the interface with the SAM in sample 3. Nevertheless in this case, the homeotropic alignment is maintained throughout the interface and the local value the scalar order parameter h P2 i is positive and higher than in sample 2. For sample 4, a clear homeotropic alignment is maintained from the SAM to the vacuum interfaces, which is confirmed by positive h P2 (z)i values across the whole film. We now try to quantify the anchoring energy for the various samples. In our previous work on 5CB on silicon 5 and silica 6 solid surfaces, we have obtained an orientational anchoring energy as a function of distance from the surface by a Boltzmann inversion of the positional–orientational distribution function, obtained from the MD simulations

17



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(Eq. 1) as

Z zi +∆z/2

  N W (zi , cos β) = −k B T i ln  Z A

zi −∆z/2 Z 1 lz 0

dz

−1

dz P(z, cos β)

d cos β P(z, cos β)

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   

(4)

where the subscript i indicates that zi is a discrete variable, since the probability is stored as a histogram calculated from the simulation trajectory, and lz is the length of the simulation box. The conversion factor Ni /A takes into account the number of molecules per ˚ giving a density of ≈ 3.5 unit of area, here computed in bins with a thickness of 12 A, molecules of 5CB/nm2 . The effective potential W (z, cos β) can be fitted to the classical Rapini 31 form: 1 WR (z, cos β) = w0A (z) − w2A (z) sin2 ( β − βdeq (z)) 2

(5)

We fitted equation (5) using the non-linear least-squares Marquardt-Levenberg algorithm. 64 For each bin, the easy axis with orientation βdeq (z) was set to the local director n of 5CB molecules. Since in the nematic phase the director n ≡ −n, the functions P(zi , cos β) and W (z, cos β) are symmetrical around cos β = 0. We took this into account in the fitting procedure; for a more detailed explanation, we refer to the supplementary material. The resulting Rapini-like anchoring coefficients w2A are shown in Figure 8. The plots show a value of about 0.05 J/m2 in the centre of each film, which is the reference value for the nematic phase of 5CB. At the interface with vacuum, the anchoring energy is about 0.15 J/m2 , in agreement with previous determinations (see Supporting Information). At the interface with the SAM, the anchoring coefficients have a value similar to that of the bulk. This is not surprising, given the somewhat similar chemical nature of the SAM and 5CB phases. The larger errors bars are due to the smaller number of molecules present at the SAM/5CB interface. For sample 1, the anchoring coefficient is slightly higher than the bulk and originates from the peak of the probability distribution function P(z, cos β) just above the SAM. In sample 2, we found a marked decrease of the anchoring coefficient 18


Page 19 of 30

P(cosθ)

Density (g/cm3)

1 0.5 0 −10

0

10 z (Å)

20

30°

10°

sample 1 sample 2 sample 3 sample 4

20 10 0

30

0.4

0.6 0.8 Tilt Angle (cosθ)

1

1.2 −1

P(Length) (Å )

0.6 0.4 0.2 0

50°

30

1.5

P(Thickness) (Å−1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

10

15 20 25 Thickness (Å)

30

0.9 0.6 0.3 0

5

10

15 20 Length (Å)

25

30

Figure 9: Density (top left panel), tilt angle (top right panel), thickness (bottom left panel) and length distribution (bottom right panel) distributions of SAM in the four samples. at the SAM/5CB interface, which originates from the change between the homeotropic alignment of 5CB molecules diffused in the SAM (first point) and the planar alignment of the 5CB film. In sample 3, where the homeotropic alignment of the SAM interface is maintained, the transition is less abrupt. A smooth increase of the anchoring coefficients, from the SAM to the free interface, is instead observed in sample 4. The internal structure of the SAM is characterized in terms of density and thickness, and the conformation of OTS and HTS molecules in terms of tilt angle and length of the alkyl chain, as shown in Figure 9. The density profile of the SAM is dominated by a broad ˚ due to the −O−Si(OH)2 end group of the alkyl-silane molecules. peak around z = 0 A, The density profile also reveals that the alkyl chains of OTS occupy the same region of space in all samples, regardless of their composition. This finding can be easily explained by taking the tilt angle into the picture. As already mentioned, OTS molecules in sample 19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 are tilted of about 23◦ w.r.t. the surface normal. In contrast, the tilt angle of OTS in sample 2 and of OTS plus HTS in samples 3 and 4 is ∼ 0. As a consequence, the thickness ˚ indicates the of samples 2–4 is greater than the one of sample 1. The peak at around 11.7 A thickness of the HTS sub-lattice in samples 3 and 4. Finally, the distribution of molecular ˚ due to alkyl chains in the lengths for OTS shows an intense outermost peak at z ∼ 27.0 A all-trans conformation, with a second broad peak at smaller values indicating molecules with one kink. 41 A similar distribution is also found for HTS molecules in samples 3 and ˚ 4, as revealed by the two peaks at z ∼ 11.3 and ∼ 11.9 A.

5

Conclusions

We have presented MD simulations of thin films (≈ 30 nm) of the nematic 5CB liquid crystal supported over amorphous silica surfaces functionalised with different mono-layers of long (OTS) alkyl silanes, as well as of long and short OTS:HTS mixtures. The orientation of the 5CB above the SAM and across the films has been first determined as a function of the surface coverage of the alkyl silane with the long, C18 alkyl chains, i.e. OTS. The results show that a defect-free SAM surface with hexagonal close packing induces a planar anchoring of the liquid crystal, whereas a lower packing density allows 5CB molecules to diffuse inside the SAM in a strict homeotropic orientation, which reduces to zero the tilt angle of OTS from the surface normal and fills the gaps in the structure of the SAM, yielding a regular interface which in turn induces a planar alignment on the overlying film of 5CB. We have found that mixed OTS:HTS SAMs (60:40 and 40:60 ratios) give instead a homeotropic anchoring of 5CB. The most effective aligning monolayer appears to be the one with a majority of short chains. These results suggest that homeotropic alignment is obtained with a surface density of OTS lower than 2.7 molecules/nm2 , which allows the mixing with 5CB and the formation of a uniform interface where both the 5CB and the OTS molecules are perpendicular w.r.t. the surface. The present work demonstrates that

20


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the anchoring of a liquid crystal depends not only on the composition and morphology of the substrate but, critically, on the degree of mixing with the underlying SAM. A fine control over the assembly of uniform and defect-free SAM is therefore mandatory in order to obtain the desired alignment.

6

Acknowledgements

We acknowledge the European project MINOTOR (grant no. FP7-NMP-228424) for funding this research and the CINECA Supercomputing Center for providing computer time through an ISCRA grant. OMR wishes to acknowledge R. F. De Sousa for useful discussions on continuum theory.

Supporting Information Available Detailed explanation of the calculation of Rapini-like anchoring coefficients. Anchoring coefficients of four samples studied in previous works.

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Predicting the Conditions for Homeotropic Anchoring of Liquid

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