Magnetic nematic and cholesteric liquid crystals

Transcription

1 University of Ljubljana Faculty of Mathematics and Physics Seminar 1 Magnetic nematic and cholesteric liquid crystals Author: Peter Medle Rupnik Supervisor: doc. dr. Alenka Mertelj Ljubljana, April 2017 Abstract Suspensions of magnetic nanoplatelets in liquid crystals exhibit spontaneous magnetic ordering of the nanoplatelets mediated by long range orientational ordering of liquid crystals. The structure of material placed in chosen connement can be manipulated by applying electric and weak magnetic elds. Here three dierent cases of suspension placed in a layer are presented, each of which is distinguished by unique phenomena. 1

2 Contents 1 Introduction 2 2 Theoretical overview Free energy Sample preparation 6 4 Ferromagnetic NLC layer 6 5 Ferromagnetic CLC layer Magnetically monodomain layer Magnetically disordered layer Conclusion 10 References 10 1 Introduction Liquid crystals (LCs) are materials that exhibit orientational ordering of the constituents but only partial or no positional ordering. The average local orientation of constituents is described by unit vector n, called the director. The two directions, n and n, are indistinguishable.[5] Nematic liquid crystals (NLCs) exhibit no positional ordering and promote uniform director eld conguration. Any deformation in the director eld can be described by nematic elasticity and contributes to the increase of the free energy (Figure 1A,B,C). Cholesteric LCs (CLCs) are the chiral version of the NLCs. Unlike in the case of the NLC, energetically favourable ground state of a CLC is such that the preferred director orientation n continuously changes along the direction perpendicular to n (twist deformation), thus forming a characteristic helical structure (Figure 1B,D). Pitch P of a CLC is dened as length at which the director turns for angle 2π (Figure 1D). On scales much smaller than the length of the pitch the CLC behaves the same as NLC. Pure LCs couple with electric and magnetic elds. The magnitudes of external B required to aect the structure of LC are however very large, especially in cases of thin layers. Additionally, even when external B is applied, the relation n n still holds. The idea of a liquid crystal, that would exhibit polar magnetic ordering, and would respond well to small magnetic elds, was thought o long ago. Broschard and de Gennes proposed to obtain such a material by adding magnetic nanoparticles in the LC media.[3] The material was experimentally realised much later by using magnetic nanoplatelets, with surfaces treated to induce perpendicular alignment of the LC.[8, 9, 7, 10] Such particles in nematic environment cause deformations in the director eld (Figure 1D), that prevent particles to come too close one to another, which would lead to aggregation due to the magnetic interaction of the platelets. The suspension is stable and exhibits polar magnetic ordering, which is described by the density of magnetic moments of the nanoplatelets, i.e. the magnetisation M. On the macroscopic scale M and n are coupled, so that it is energetically favourable for the order parameters to be parallel. Consequently the suspension exhibits strong magneto-optic eect (director eld conguration 2

3 can be manipulated using small external B) and converse magneto-electric eect (magnetisation eld conguration can be manipulated with external E). Similarly the suspension can be prepared in the CLC. Figure 1: Characteristic LC deformations (A) splay, (B) twist, (C) bend. (D) Helical conguration of director eld characteristic for bulk CLC. (E) CLC placed in wedge homeotropic LC cell. (F) Idealized 2D schematic showing deformations around the magnetic nanoplatelets. On the microscopic level the magnetic CLC suspension is very similar to the NLC (the pitch is much larger than the scale of deformations around the platelets), on the macroscopic distances however the behaviour is very dierent. Both the usual and ferromagnetic CLCs form a rich variety of topological structures when conned in layers or droplets.[12, 13, 14, 11, 6, 2, 1] In connement anchoring usually promotes uniform conguration, which competes with the helical ground state of a CLC. Here the samples are examined in layer connement with boundary conditions promoting aligned director orientation perpendicularly to the layer (homeotropic anchoring). By placing the material in wedge homeotropic LC cell three regions can be distincted depending on the connement ratio d /P; here d is the cell thickness. CLC director eld conguration is homeotropic (for d /P < 1), composed of wound structures called cholesteric ngers (CFs) separated by uniform regions (for 1 < d /P < 1.5) or completely wound (for d /P > 1.5) (Figure 1E). Here the experiments were performed in the latter regime ( d /P = 1.74). The CLC suspension is lled in LC cells under dierent conditions, the sample is then unwound with external E, and the stability of unwound structure is probed by the combination 3

4 of E and B at dierent magnitudes and applied in dierent directions. The physical properties of a magnetic CLC layer signicantly depend on the external conditions (E and B) applied, while the suspension is placed in LC cell. If there is no external eld in the beginning, there is no initial magnetic ordering. If the sample is led in presence of external E and B, magnetically monodomain sample is obtained. In this seminar comparison of ferromagnetic LCs is given, pointing out how material behaves depending on dierent extrinsic and intrinsic geometries. The observed phenomena are interpreted by numerical modelling and stability analysis. The experimental ndings are presented after short theoretical review. 2 Theoretical overview 2.1 Free energy Important theoretical concept, required for macroscopic description of conned LC samples, is that of free energy F.[5, 10] Stable state of the observed system can be calculated by minimisation of free energy, the time evolution of unstable system can be determined using Ericksen-Leslie equations. In this seminar temperature dependence, ow and singularities in the director eld are not considered and the free energy functional can be written in the following form: ˆ ˆ F = (f M + f H + f d + f D + f Mn ) dv + f a ds T S. (1) Here f M + f H is magnetic free energy density, f d is distortion energy density, f a is contribution from anchoring of the liquid crystal on the substrates of the layer, f M,n is coupling energy density of director n with magnetisation M and f D describes coupling with external electric eld D. S is entropy of the system and T is temperature. Below all the terms are further explained. The rst term f M depends only on magnitude of magnetisation and describes ferromagnetic transition: f M = α 2 M 2 + β 4 M 4, (2) here α and β are Landau expansion coecients. In cases of constant magnetisation this term can be omitted. Distortion energy density describes continuous deformations of the director eld. Frank distortion energy density reads: f d = 1 2 K 11 ( n) K 22 (n n q 0 ) K 33 (n n) K 24 (n ( n) + n n). (3) Here K 11, K 22, K 33 and K 24 are Frank elastic constants, corresponding to splay, twist, bend and saddle-splay deformations accordingly, q 0 = 2π /P describes the intrinsic periodicity of the CLC. For the nematic liquid crystal q 0 is zero. The last term can sometimes be omitted, because it can be transformed to surface integral via Gauss theorem. In case of a layer the surface integration in equation 1 is done only over the substrate surfaces. Here it is important to include this term, especially in cases when periodic boundary conditions are introduced in numerical modelling and the length of the period is not much larger than the cell thickness. 4

5 Energy density form coupling of n and M is written with lowest symmetry allowed term in polynomial expansion: f Mn = 1 2 γµ 0 (n M) 2, (4) γ being the coupling constant. This coupling is strong and n and M are expected to have similar orientations. Free energy density from coupling with external electric eld reeds: f D = 1 D E, (5) 2 Here the electric displacement eld is D = ε 0 εe, ε 0 being vacuum permittivity and ε being linear dielectric tensor ε = ε I ε (n n). ε = ε ε is dielectric anisotropy and ε and ε are dielectric constants perpendicular and parallel to the director respectively. Free energy density from coupling with external magnetic eld reeds: f H = µ 0 H M. (6) Here µ 0 is vacuum permeability and M = M induced + M permanent is magnetisation. In the case of magnetically ordered samples the rst term describes direct coupling between n and B: ( M induced,n = µ 1 0 χ (B (B n) n) + χ (B n) n ) = µ 1 0 χ B + µ 1 0 χ (B n) n, (7) and the second term corresponds to the density of magnetic moments of the suspended nanoplatelets. In the case of magnetically disordered samples there is no macroscopic permanent magnetisation and M induced has additional term describing local ordering of magnetic nanoplatelets: M induced = M induced,n + p m N n p m (N N ), (8) where p m denote local average platelet dipole and N is number density of aligned magnetic nanoplatelets; here the arrows distinguish the two possible orientations along n and the the sign of the dierence N N determines the orientation of M. Here we approximated that the coupling constant γ is very large and n and M point in the same direction. Also in all cases the direct coupling between n and B and direct magnetic interaction between the platelets are negligible and therefore the ordering of the platelets is LC mediated. Anchoring energy reeds: f a = 1 2 W (n n s) 2, (9) n s being the preferred director orientation on the substrates (perpendicular to the surface plane for the homeotropic anchoring) and W is anchoring strength. The last entropic contribution is due to the inhomogeneity in the density of the nanoplatelets. In some cases no inhomogeneity was observed, and here this term can be omitted. In other cases the inhomogeneities in the density of the nanoplatelets are experientially evident and thus the term should be considered. The exact interaction of platelets however is not yet fully understood and it is dicult to make appropriate approximations. This term is to be considered in future in order to explain some of the following experimental ndings. 5

6 3 Sample preparation CLC has been prepared by mixing liquid crystal E7 and chiral dopant S811 (0.9 %wt).[4] Pitches were measured by Grandjean-Cano method. Suspensions have been prepared by quenching the suspension of scandium-doped barium hexaferrite (BaHF) single crystal nanoplatelets in either CLC or NLC from isotropic to cholesteric phase. Pitches of the CLC suspensions remained the same within the error range. The thickness of the platelets is about 5 nm, the distribution of the platelet diameter is approximately log- normal, with mean of 70 nm and standard deviation of 38 nm.[7] The nanoplatelets are magnetically monodomain with magnetic moment perpendicular to the plane of the platelets. Dodecylbenzenesulphonic acid was used as a surfactant. The suspensions were centrifuged. Aggregate free part was lled in the homeotropic LC cells (Instec, Inc., 18.3 µm) with ITO coated surfaces either without external elds or in external E (U = 10 V) and B (B = 16 mt), both applied perpendicularly to the layer. 4 Ferromagnetic NLC layer In the NLC case homogeneous alignment of the director eld (either planar or homeotropic) is achieved, when the material is placed in layer connement. Here no external eld was applied during the lling. The structure of magnetisation eld can be probed by external inplane magnetic eld and the magneto-optic response can be observed by polarising microscopy (POM). The rst exposure is especially interesting. Figure 2A shows responses of the suspension under dierent conditions. The initial response is in all cases inhomogeneous, which indicates that the sample is polydomain with characteristic size of the magnetic domains being around 2 µm. This behaviour exhibits the ferromagnetic nature of the suspension. In planar connement the domains, where magnetisation points in the same direction, join in larger domains and eventually monodomain sample is obtained. The transition from polydomain to monodomain sample is accompanied with two domain walls located near the surfaces of the layer (Figure 1B).[8] In homeotropic connement it is dicult to obtain monodomain sample from the polydomain one. There are two reasons for this; rst the demagnetisation eld is larger and secondly anchoring condition suppresses the propagation of domain walls near the substrates. 5 Ferromagnetic CLC layer Response of CLCs to homogeneous external magnetic eld diers from the NLC case. This is because the CLC favours twisted structures, while homogeneous eld forces the director eld to align. Although magnetic ordering can be obtained, in certain cases growth of magnetic domains in external B is suppressed, and after the eld is switched o, the system returns to state with no permanent magnetisation. Here two dierent systems are described. In both cases voltage was applied between the glass platelets, so that from the beginning the sample was unwound and homeotropically aligned in the ITO area. In the rst case additional magnetic eld was applied (also perpendicularly to the substrate planes), and in the second case no eld was applied. We probed both samples with in-plane B. 6

7 Figure 2: Initial (rst exposure) magneto-optic response of ferromagnetic NLC in (A) planar and (B) homeotropic connement. (C) Transition from polydomain to monodomain sample. (D) Switch of magnetization orientation in monodomain sample. The rst schematic represents the initial state and the other three show the sample congurations corresponding to the three regions on the POM image below: bright, less bright and dark respectively. 5.1 Magnetically monodomain layer If the sample was lled in homeotropic LC cell with external E and B, the magneto-optic response was homogeneous from the beginning. The sample was magnetically monodomain. The external elds were switched o and ngerprint texture appeared typical for CLC (Figure 3A). The sample was left in the ngerprint conguration in the absence of the external elds overnight and then it was unwound again by applying voltage. The sample still exhibited the magneto-optic response, however it was inhomogeneous. Features of the response clearly resembled the ngerprint pattern of the wound structure in which the sample had remained overnight. This indicates that in the wound conguration the density of magnetic nanoplatelets became inhomogeneous. This distinct inhomogeneities in magneto-optic response faded out after the sample had remained unwound in external electric eld for some short period of time (typically less than half an hour). This means that in this case the starting magnetically monodomain structure is preferred. The density of the nanoplatelets was increased in areas where director eld was less twisted. This aligned conguration of platelets is promoted either by the direct magnetic interaction of magnetic nanoplatelets or more likely by LC deformations around the nanoplatelets. As the sample was unwound, the concentration homogenised due to the entropic forces. The initial structure of the director eld, that corresponds to homogeneous magneto-optic response is shown on gure (Figure 3C). The structure is called transnationally invariant conguration (TIC), and is a wound state typical for ferromagnetic CLC in homeotropic LC cell. In magnetic case this conguration is polar (ptic). By reducing external elds these 7

8 Figure 3: (A) POM images of wound ngerprint conguration (left), magneto-optic response immediately after unwinding (center) and after several minutes (right). (B) POM image of 2D periodic pattern caused by critical uctuations in the director eld and two schematics showing the shape of the instability in two regions of the layer; near the bottom and near the top substrate. Gray arrows show the direction of the wave vector of the two uctuations. (C) Schematics of ptic, ptic2 and reversed ptic corresponding to areas on the POM images above. (D) Stable periodic structure that evolved from ptic2. two structures destabilised into dierent congurations, depending on the magnitudes of B and E. Besides the inhomogeneous instabilities (nucleating cholesteric ngers), which are common in the usual CLC, homogeneous instabilities were observed as periodic 1D and 2D patterns. These periodic structures arise from uctuations in the director eld and are unique instabilities for ferromagnetic CLC. The transient structure of the 2D periodic instability is shown on Figure 3B. The situation is dierent if stable ptic is subjected to a sudden reversal of magnetic eld. In this case the starting ptic conguration also becomes unstable, however nucleation of cholesteric ngers and homogeneous instabilities are here suppressed. The relaxation of the director eld depends on the magnitude of external E. If external E is strong, the director eld relaxes trough homeotropic conguration to the reversed ptic (resulting magneto-optic eect is the same as the initial). In cases of low E, the situation is dierent. The magneto optic response at rst slightly darkens and after some time white lines start to grow around 8

9 the LC cell spacers. In regions within the lines the magneto-optic response is the same as the initial. Both states, outside and within the lines are stable (Figure 3C). In region outside the white lines the director eld near the two substrates rotates from two sides, while on the middle of the sample it remains in the initial position. This results in more twisted conguration ptic2. This new conguration also exhibits interesting homogeneous periodic destabilisation in case when the magnitude of B is slowly reduced (Figure 3D). While in the previous cases the periodic structures were transient, here this structure is stable even when B and E are turned o. In regions within the domain walls the conguration is reversed ptic (the same as in the rst case). 5.2 Magnetically disordered layer Figure 4: (A) POM images showing magneto-optic response only on areas, where CF has magnetically ordered the sample. (B) Growing of magnetic domains. (C) Proposed structure of the magnetic domains. (D) Anisotropic response of the domains in slightly tilted sample. (E) Structure of magnetic domains after the magnetic eld is switched o. (F) Vanishing of the magnetic domains in unwound sample; the sample was probed every few minutes by weak magnetic eld. 9

10 If the sample was lled in homeotropic LC cell with applied voltage but without external B, no magneto-optic response to in-plane B was observed from beginning. There was also no permanent magnetisation at the beginning. The voltage was then slowly reduced at constant in-plane B. As the cholesteric ngers appeared, voltage was again increased and the sample was again unwound. In areas where CFs previously appeared, weak magneto-optic response was observed by polarising microscopy (Figure 4A). This means that formation of CF caused local magnetic order. After some time this magneto-optic response strengthened and evolved into a stripe pattern across the entire ITO region (Figure 4B). By slightly tilting B out of the plane of the layer (few degrees) the intensity of every two successive stripes changed, one becoming brighter and the other darker. Reversing the direction of B resulted in obtaining reversed subsequence of darkened and brightened stripes (Figure 4D). The layer was analysed by rotating the crossed polarisers. In case when polariser was perpendicular to the external B almost completely dark state was obtained, which suggests that there was only little twist deformation along the direction perpendicular to the plane of the layer. The proposed structure is shown on Figure 4C, this scheme is however not the result of numerical modelling. On the schematics one may also note that in certain areas twist deformations of reversed handedness are present, which is especially energetically unfavourable for the CLC. This polydomain structure was stable in case of applied B. By turning o B the sample relaxed to polydomain conguration on Figure 4E. Although local magnetic ordering is present here, the macroscopic magnetic moment P m = MdV integrated over several periods goes towards zero. After some time this magnetic structure vanished (Figure 4F), and the sample returned to the initial state with no permanent magnetisation. The preferred state of this sample is therefore magnetically disordered. 6 Conclusion Magnetic behaviour of suspensions of magnetic nanoplatelets in LCs is very interesting and unique. The system can be to an extent compared to the solid magnetic materials. The direct classical magnetic dipole interaction of intrinsic magnetic moments of the electrons in crystals is not enough to cause spontaneous magnetic ordering. Instead the origin is quantum mechanical and the type of magnetic order is determined by exchange interaction, which depends on the electron conguration of atoms in the lattice and the crystal eld. In the suspension the direct magnetic interaction of the nanoplatelets is small, and ordering is LC mediated, here the long range ordering is due to the orientational ordering of the LCs. In both cases magnetic properties depend on the structure of the material, which is in the case of LCs easily manipulated. It is therefore not only the magnetic structure, but also the characteristics of magnetic response that can be manipulated. The CLC suspension is especially interesting since helical magnetic structure can be obtained - this is analogues to the helimagnets in solids.[15] On the other hand it would be interesting to study CLC suspension where pitch length would be set beyond the characteristic length of magnetic domains. Here formation of magnetic domains would be suppressed and microscopic picture should be examined more closely. 10

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