Oddelek za fiziko

Seminar Ib - 1. letnik II. stopnja

Ferromagnetic liquid crystals

Author: Luka Pirker Mentor: Doc. Dr. Alenka Mertelj

Ljubljana, October 2014

Abstract In this seminar I will present ferromagnetic liquid crystals and their properties. Ferromagnetic liquid crystals have the characteristics of ferromagnets as of liquid crystals. Their existence was predicted over 40 years ago and, until recently, researchers were unable to experimentally observe them.

Contents 1 Introduction

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2 Liquid crystals

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3 Ferrofluids and ferromagnetism in liquids

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4 Magnetic platelets in a nematic liquid crystal

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5 Response to magnetic field

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6 Free energy density

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7 Conclusions

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Introduction

Liquid crystals have properties between those of conventional liquid and those of a solid crystal. A liquid crystal may flow like an ordinary liquid, but its molecules may arrange themselves as if they were in a crystal. In 1888, Friedrich Reinitzer, an Austrian botanical physiologist observed that cholesteryl benzoate did not melt in the same way as other compounds he was studying. He observed that it has two melting points, one at 145.5◦ C where it melts in a cloudy liquid, and the second at 178.5◦ C, where it melts again and the cloudy liquid becomes clear [1]. He also found that the new material has the ability to rotate the polarization of light. He wrote to physicist Otto Lehmann for help. Lehmann examined the cloudy fluid and reported seeing crystallites. He named the new material liquid crystals and continued studying them. The term liquid crystals describes the liquid properties of the material and their crystal-like optical properties. However, liquid crystals were not popular among scientists at the time. It took almost 80 years before scientists restarted studying them. They quickly became a topic of research in many laboratories around the world. In the years that followed many different properties of liquid crystals were discovered that theoretical models predicted. In 1970 Fran¸coise Brochard-Wyart and Pierre-Gilles de Gennes proposed that a colloidal suspension of ferromagnetic particles in nematic liquid crystals could form macroscopic ferromagnetic phases at room temperatures. Many researchers have tried to make such liquid crystals, but failed until recently.

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Liquid crystals

Liquid crystals are materials made of anisotropic molecules. There are many different types of liquid crystals. In this seminar I will write only about nematic liquid crystals that are usually composed of rod-like molecules. These molecules have an orientational order that is described by a unit vector called the director n, that represents the average molecular orientation in the nematic phase. Because there is no physical polarity along the director axis, vectors n and −n are equivalent, n ≡ −n. The orientation of the molecules can be controlled by external electric or magnetic fields or by preparing the confining surfaces so that the director at the surface has a predefined 1

direction. Nematic liquid crystals have two liquid phases. In the isotropic phase molecules do not have a defined direction, but cooling a nematic liquid crystal under a certain temperature Tiso→nem , molecules orient themselves along the director, Figure 1.

(a)

(b)

Figure 1: (a) A representation of the isotropic phase of a liquid crystal. [2] A representation of a nematic phase in a liquid crystal with rod-like molecules. [2] Introducing anisotropic particles in a nematic liquid crystal causes a distortion of the director, Figure 2. The deformation of the director field depends on the surface anchoring and on the shape of the particle. An elongated rod-like particle with surface anchoring parallel to the particle’s surface and to the particle’s long axis will orient itself in the nematic phase with its long axis parallel to n0 , where n0 is the average director orientation, Figure 2 a. If the nematic director at the particle surface is perpendicular to the surface, then the particle orients with its long axis perpendicular to the director n0 , Figure 2 b. A disk with perpendicular anchoring orients with its axis along n, Figure 2 c.

(a)

(b)

(c)

Figure 2: A schematic presentation of the distortion of the director around a anisotropic particle in a nematic liquid crystal. A rod-like particle with the surface anchoring: (a) parallel with its surface and long axis, (b) perpendicular to the surface. (c) A disk with perpendicular surface anchoring. [3]

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Ferrofluids and ferromagnetism in liquids

Ferromagnetic ordering was already experimentally observed in two liquids. In 1978 ferromagnetism was observed in superfluid 3 He, and in 1997 ferromagnetism and ferromagnetic domains 2

were observed in undercooled alloys at temperatures above 1,000 K [2]. Ferromagnetism is a phenomenon where under a certain temperature, known as the Curie temperature Tc , the material exhibits spontaneous magnetization M in the absence of an external magnetic field. Magnetization curves, that is, magnetization as a function of an external field B, show hysteresis.

Figure 3: (a) Monodomain nanoparticles are randomly oriented in a ferrofluid in absence of an external magnetic field B. [2] (b) Particles orient along B. [2] Ferrofluids are not ferromagnetic, they are composed of ferromagnetic monodomain nanoparticles that act as nanomagnets. Ferrofluids are superparamagnetic liquids which become strongly magnetized in the presence of a magnetic field. In the absence of an external magnetic field, the average magnetic interaction between two nanoparticles is smaller than thermal energy, and the magnetic moments of the particles are randomly oriented within the fluid, Figure 3. So ferrofluids do not have macroscopic spontaneous magnetization in the absence of an external magnetic field and thus are not ferromagnetic. When we apply an external magnetic field, magnetic moments of the nanoparticles orient along the external magnetic field. The magnetic interaction between the particles becomes larger than kB T. Because the nanomagnets attract each other, if they are oriented in the same way, chaining of the particles occurs, which leads to an increase of viscosity of ferrofluids [4]. Ferromagnetic nanoparticles in the isotropic phase of a liquid crystal behave as any other ferrofluid. In the nematic phase, as explained before, anisotropic particles adopt a certain orientation. If they have magnetic moments pinned to their shape, the nematic phase will also orient magnetic moments. Because nematic phase is non-polar (n ≡ −n), this is not enough to induce ferromagnetic ordering of the moments, Figure 4b. For that, strong enough magnetic interaction between the particles is needed.

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Magnetic platelets in a nematic liquid crystal

It has been shown that nanometre-sized ferromagnetic platelets with perpendicular surface anchoring can be used to make a stable ferromagnetic phase in nematic liquid crystal [2]. A stable nematic suspension is the result of the balance between nematic-mediated force and the magnetic interaction between particles. When particles are put in a liquid crystal, the direction of the director around them is changed, as seen on Figure 5. Other particles are affected by the distortion of the director and the interaction between them can be repulsive or attractive. The symmetry of the deformation determines if the nematic force is dipolar or quadrupolar. In this case the nematic force is quadrupolar, and gives the strongest attraction when the line joining two particles makes 3

(a)

(b)

Figure 4: (a) Ferrofluid on a glass surface in a magnetic field that is produced by a magnet. [5] (b) Platelet nanoparticles in a nematic liquid crystal. If the magnetic interaction is not strong enough, magnetic moments (red arrows) will be randomly oriented. [2] an angle of about 50◦ with respect to n0 . A stable ferromagnetic phase in nematic liquid crystal can not be made with rod-shaped particles. The deformation of the nematic field, using rod-shaped particles, is small and the nematic interaction is too weak to prevent aggregation. Because the vectors n and −n are equivalent, the ferromagnetic phase will not appear unless the magnetic interaction between magnetic dipoles is strong enough and such that magnetic moments of the particles orient ferromagnetically, that is, in the same direction. In Figure 5 b there is a schematic representation of the magnetic field and the distortion of the director. It turns out, that the nematic quadrupolar interaction combined with sufficient magnetic interaction gives a ferromagnetic phase. Spontaneous magnetization is along n0 which can be seen with polarized microscopy imaging in an external magnetic field. A suspension of magnetic platelets in a liquid crystal in the isotropic phase was used to fill planar glass cells. The cells were prepared so that the director was parallel with the interior surface of the cell, Figure 6. To achieve an optimal sample, the cell must be quenched into the nematic phase. Quenched samples remain stable and no additional aggregation occurs even after several months or when exposed to external magnetic fields. If the cell is cooled slowly, aggregates form. The magnetization in the cell is also parallel with the glass surface.

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Response to magnetic field

Samples were examined using polarizing microscopy. The sample is put between a polarizer and an analyser, Figure 6 b. The polarizer and analyser are optical filters that pass light of a specific polarization. When they are crossed at an angle of 90◦ , no light can go through and a dark picture is obtained. When the sample is put between the filters and the external magnetic field B is 0, once again a dark image is obtained. This means that no light goes through the analyser. This is because the director is aligned within the sample, and it can not change the polarization of the transmitted light. When a magnetic field parallel to the director is applied, dark and bright domains 4

(a)

(b)

Figure 5: (a) TEM image of magnetic platelets. [2], (b) A schematic representation of the ferromagnetic liquid crystal. Blue lines represent the distortion of the director, orange the magnetic field, red horizontal lines represent magnetic platelets and red arrows the direction of magnetic moments. [2]

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(b)

Figure 6: (a) A schematic representation of the cell filled with a suspension of magnetic platelets in a liquid crystal. [6], (b) A schematic representation of the experiment using polarizing microscopy. [2] are observed, Figure 7 b,c. This is because the sample is made out of domains that have opposite magnetization. The domains in which the magnetization points in the same direction as the external magnetic field, will appear dark. The magnetic platelets will point in the same direction as before and thus the director around them will not change. The domains, in which the magnetization 5

points in the opposite direction as the external magnetic field, will appear bright. This is because the magnetic platelets within the sample will rotate because their magnetic moments want to align with the external magnetic field, Figure 8. At the cell surface the director will not change because the surface anchoring is too strong. This will cause the director to twist which will cause a change in the polarization of the transmitted light. Some of that light will go through the analyser and we will observe it as a bright area. If the field is reversed, bright domains become dark and vice versa, Figure 7 c. If a field perpendicular to the director is applied, domain walls become visible, Figure 7 d. The alteration of light properties with a magnetic field is called a magneto-optic effect. This experiment shows that spontaneous magnetization is present along the director and that the two types of domains have opposite magnetization.

Figure 7: Polarizing microscopy images of a polydomain sample in an external magnetic field. A shows the direction of the analyser and P the direction of the polarizer. The scale bar in the first image is 40 µm. When an external magnetic field parallel with n is applied, dark and bright domains are observed. Dark domains become bright when the field is reversed and previously dark domains become bright. When the external field is perpendicular to n, the domain walls are visible. [2]

Figure 8: A scheme showing two domains with opposite magnetization. When there is no external magnetic field, no light is transmitted. When an external magnetic field parallel with n is applied, the director in one of the domains twists. This causes a change in the polarization of transmitted light, and is visible as a bright area. [2] Monodomain samples can also be created by quenching from the isotropic to the nematic phase in an external magnetic field parallel to the director. Magnetization curves of a monodomain sample are obtained using a vibrating sample magnetometer, Figure 9. 6

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(b)

Figure 9: (a) A scheme of a vibrating sample magnetometer. [7] (b) LakeShore 7400 Series vibrating-sample magnetometer used to measure the magnetization curve. [8] The sample is placed in a uniform magnetic field and then physically vibrated in a sinusoidal pattern using a piezoelectric material. Stationary pickup coils are mounted on the poles of the electromagnet. The change in magnetic flux originating from the vertical movement of the magnetized sample induces a voltage Uind in the coils. The induced voltage can be written as ∂φ , (1) ∂t Where φ is the magnetic flux. For the pickup coils with a flat surface A and n windings, one can write Uind = −

∂B , (2) ∂t Where B is the magnetic field density. For a sample with magnetization M in a homogeneous magnetic field H0 , the magnetic field density can be written as Uind = −nA

B = µ0 H0 + µ0 M.

(3)

The magnetization M is defined as the density of magnetic dipole moments in the sample. In a constant magnetic field H0 we have ∂B ∂M = µ0 . ∂t ∂t 7

(4)

From this we can see that the induced voltage is proportional to the magnetic moment of the sample. By measuring it as a function of the external magnetic field, a magnetization curve can be obtained. The magnetization curve of a monodomain sample was measured. When the external field is applied in the same direction as the magnetization of the sample, the magnetization does not change because the magnetic moments of all particles are already oriented in this direction. When the field is applied in the opposite direction, magnetization along the director starts to decrease. At the coercive external magnetic field, the magnetization changes direction and at larger magnetic fields it saturates. The absolute value of saturated magnetization is the same as at the beginning. This means that all platelets have rotated for 180◦ . When the field is again reversed the sample magnetization returns at its initial value and a magnetic hysteresis is observed. At low concentrations of the particles, the hysteresis is asymmetric because the director at the cell surface is pinned, and so magnetization cannot completely switch in the external magnetic field. For larger concentrations, the magnetization and director reverse also at the surface and a hysteresis curve is observed. The magnetization and director reversal at the surface happens by motion of surface domain walls. The surface domain walls are seen by polarizing microscopy as white lines, Figure 11.

Figure 10: The magnetization curve at different concentrations c of ferromagnetic platelets. [2]

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Figure 11: Sequence of images showing the complete switching of a monodomain sample. There are two white lines, one for each surface of the sample. [2]

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Free energy density

Nematic liquid crystals, e.g. 5CB, have a very small magnetic anisotropy χa =1.6 10−6 , where χa is the difference of magnetic susceptibility, parallel and perpendicular to n. This means that magnetic coupling of the liquid crystal with the external magnetic field is very weak. In a ferromagnetic liquid crystal, the orientation of n is coupled to the orientation of platelets, i.e. magnetic moments. So when an external magnetic field is applied, it reorients the platelets and consequently also the orientation of n is affected. In the Landau-de Gennes description, the free energy density f of ferromagnetic liquid crystals can be written as: f = fnem +

α 2 β 4 1 1 M + M − µ0 M · H − γµ0 (n · M)2 − χa µ0 (n · H)2 , 2 4 2 2

(5)

where fnem is the free energy density of nematic phase [4]. The following three terms, α2 M 2 + β 4 4 M − µ0 M · H, give the magnetic free energy density, with µ0 being vacuum permeability, α and β the Landau expansion coefficients describing the ferromagnetic transition, and H the external magnetic field. The term 21 γµ0 (n · M)2 describes the coupling of the nematic director with the magnetization M , where γ is a constant. The last term in the equation, 12 χa µ0 (n · H)2 , describes the coupling of the nematic director with the external magnetic field. The coupling constant γ is around 100 and is 8 orders of magnitude larger than magnetic anisotropy χa [2]. Consequently, ferromagnetic liquid crystals are very sensitive to small magnetic fields. Constant γ is obtained by measuring the threshold field Bc at which the reversal of M begins. The threshold field is written as Bc = γ

πµ0 KMs , + γµ0 Ms2 d2

Kπ 2

(6)

where Ms is spontaneous magnetization, K elastic constant and d the thickness of the sample. Measured Bc is around 1mT which gives an estimate for γ of 100 [2]. 9

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Conclusions

Over 40 years, researchers have struggled to experimentally realize ferromagnetic liquid crystals. Ferromagnetic liquid crystals have the properties of ferromagnets as well as the properties of nematic liquid crystals. Monodomain and polydomain samples can be obtained. Both respond to an external magnetic field as expected for a ferromagnet. The new material switches at very small magnetic fields which may lead to new magneto-optic devices. Mixing ferromagnetic platelets in a chiral or smectic liquid crystal could give new magnetic phenomena in complex fluids.

References [1] http://en.wikipedia.org/wiki/Liquid crystal (18.10.2014) ˇ c: Ferromagnetism in suspension of magnetic [2] A. Mertelj, D. Lisjak, M. Drofenik, M. Copiˇ platelets in liquid crystal, Nature, Vol. 504, (2013) [3] B. Senyuk, D. Glugla, I. I. Smalyukh: Rotational and translational diffusion of anisotropic gold nanoparticles in liquid crystals controlled by varying surface anchoring, Physical Review, E 88, (2013) [4] S. Odenbach: Colloidal Magnetic Fluids: Basics, Development and Application of Ferrofluids, p. 270-275, Springer, Germany, (2009) [5] http://en.wikipedia.org/wiki/Ferrofluid (24.5.2014) ˇ c: Magneto-optic and converse magnetoelectric [6] A. Mertelj, N. Osterman, D. Lisjak, M. Copiˇ effects in a ferromagnetic liquid crystal, Soft Matter, (2014) [7] http://en.wikipedia.org/wiki/Vibrating sample magnetometer (24.5.2014) [8] http://www.lakeshore.com/products/Vibrating-Sample-Magnetometer/7400-SeriesVSM/Pages/Overview.aspx (24.5.2014) [9] A. Boczkowska: Advanced Elastomers - Technology, Properties and Applications, p. 150-160, InTech, USA, (2012) [10] http://phelafel.technion.ac.il/∼hilag/tutorial/LCPhases.html (24.5.2014) [11] F. Brochard, P.G. de Gennes: Theory of magnetic suspensions in liquid crystals, Le Journal de Physique, Vol. 7, (1970) [12] W. Burgei, M.J. Pechan, H. Jager: A simple vibrating sample magnetometer for use in a materials physics course, American Journal of Physics, Vol. 71, (2003) [13] S. Ovtar, D. Lisjak, M. Drofenik: Barium hexaferrite suspension for electrophoretic deposition, Journal of Colloid and Interface Science, Vol. 337, (2009)

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