Magneto-optical properties of liquid-crystalline ferrofluids

Vassilios Yannopapas; Sabine H.L. Klapp; Stavros D. Peroukidis

doi:10.1364/OME.6.002681

1. Introduction

Creating suspensions of magnetic nanoparticles (MNP) in liquid crystalline (LC) matrices, i.e LC/MNP hydrid systems, is a non-trivial avenue for obtaining materials with programmable and controllable functions [1–3]. New interest on such systems was stimulated by the recent discovery of spontaneous magnetic ordering in a hybrid system of large, micron-sized magnetic plates embedded in a thermotropic low molar mass nematic LC [3]. In the present work, we focus on the opposite case of colloidal suspensions of LC rods and small MNPs, where the sizes of the species are of the same order of magnitude. An example of such systems are lyotropic suspensions of colloid pigment rods and magnetite MNPs which have been recently studied [4, 5]. Because of their chemical stability, magnetite nanostructures have become a key ingredient in technologically important applications in optoelectronics [6], magnetic storage [7] and bio-inspired devices [8]. More fundamental properties can also be probed via MNPs such as magnetic vortex formation [9].

Among the various magnetic functionalities possessed by LC/MNP hybrids, the magneto-optical (MO) response has, so far, received less attention. The MO properties of MNPs can differ markedly from those of the bulk magnetic materials [10, 11]. Furthermore, the LC/MNP hybrids offer further advances for potential MO applications in comparison to conventional ferrofluids [13]. In particular, the self-organization of MNP in anisotropic matrices, e.g. LC, provides a way of producing multi–stimuli responsive materials with controlled directional sensitivity to external fields. In the present work, we study theoretically the MO response of LC/MNP hybrids by means of rigorous electrodynamic calculations of the magneto-chiral dichroism (MCD) and the Faraday-rotation spectra. We find, in particular, that the MCD signal of a LC/MNP hybrid system may be significantly enhanced due to structural changes in the morphology of the MNP in the LC matrix upon application of an external homogeneous mag netic field. The latter can find application in monitoring the phase transitions of the LC matrices or enhancing the MCD signal of molecules attached to the MNPs.

2. Calculation methods

To explore these type of systems we have employed off-lattice canonical Monte Carlo simulations [14, 15] of a simple binary model system consisting of anisotropic Gay-Berne (GB) rodlike particles and MNPs, represented as dipolar soft repulsive spheres (DSS), i.e., spheres with an embedded central point dipole moment µ_i. Typical systems simulated here are shown in Fig. 1. A detailed analysis is presented in Refs. [14, 15]. In the systems of Fig. 1, the length (l)-to-width (σ) ratio of the rods is set to l/σ = 3 which is in the lowest accessible range for LC/MNP hybrids. The dipolar spheres have diameter σ_s = 1σ while they interact via a soft-sphere potential and standard dipole-dipole interactions. Finally, for the interactions between rods and spheres we consider a modified GB potential. For further details on the simulation methods and interaction potential see Refs. [14, 15] and references therein. The configurations (positions and orientations of the particles) obtained by MC simulations are used as input in the magneto-optic calculations that are described below.

Fig. 1 Representative simulation snapshots for mixtures with σ_s = 1σ in the absence of external magnetic field: (a) Isotropic and (b) nematic phase. In the bottom parts only the MNPs are shown for clarity. The nematic directors of the species are also indicated.

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The electromagnetic EM modeling of the suspensions of Figs. 1 and 2 is performed via the Discrete-Dipole Approximation (DDA) technique [16–18] for MO targets (scatterers) [19, 20]. A brief description of the method is provided in the appendix. In the calculations that follow, as an exemplary case we have considered magnetite MNPs of 5 nm radius whose dielectric tensor is taken from experimental measurements of bulk magnetite. For details on the parameterization that is used see appendix, Refs. [19–22] and references therein. The DDA method is employed for LC/MPNs hybrids consisting of 400 magnetite MNPs assumed to be embedded within a LC matrix. The LC matrix is treated as a homogeneous medium with a uniaxial dielectric tensor provided by $\hat{ε} = 2.1 \hat{x} \hat{x} + 2.1 \hat{y} \hat{y} + 2.5 \hat{z} \hat{z}$ . In the isotropic case, we take $\hat{ε} = [(2 / 3) \times 2.1 + (1 / 3) \times 2.5] (\hat{x} \hat{x} + \hat{y} \hat{y} + \hat{z} \hat{z})$ . One, of course, can perform a more realistic electrodynamic calculation by taking into account the microscopic nature of the LC. In this case, the LC molecules are considered as point dipoles similarly to the MNPs. At the positions of the LC molecules (as provided by the above MC calculations) one could assign, in principle, a corresponding electric-permittivity tensor according to the orientation of the LC particle at a given position in space [23–25]. This, however, would constitute an unnecessary complexity to our problem since the MO properties of the LC/MNP hybrids are exclusively determined by: (i) the positioning of the MNPs, and (ii) the orientation of the corresponding magnetic moments. The latter determine the electric-permittivity tensor of each MNP. We note that the description of the liquid crystal as a continuous medium rather than a coarse-grained, molecular material is the standard way to treat liquid-crystalline composite materials such as metallic nanoparticles embedded in a host material [26]. Even if one assumes an inhomogeneous director field for the nematic liquid crystal, its effect on much stronger resonance phenomena, e.g., surface plasmons, is rather marginal (a very slight shift of the resonance) [23, 24]. It is, therefore, fully justified to ignore the coarse-graining of the LC matrix at the level of electrodynamic calculations and assume having a homogeneous host surrounding the MNPs. Finally, we note that the parameterization we have used in this work is representative and corresponds to realistic values [27] for systems comprising of magnetic particles in LC matrices.

Fig. 2 Representative simulation snapshots for mixtures with σ_s = 1σ that are in the I-state (when the external magnetic field is off) and in the N-state (field-on): (a) [(T^∗, ρ^∗) = (1.1,0.34)] field-off S ≃ 0.0 (b) [(T^∗, ρ^∗) = (1.1,0.34)] field-on S ≃ 0.4 (c) [(T^∗, ρ^∗) = (2.2,0.40)] field-off S ≃ 0.0 (d) [(T^∗, ρ^∗) = (2.2,0.40)] field-on S ≃ 0.6.

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The MO response of the LC/MNPs hybrids is manifested via the MCD which is defined as MCD = CD(H) − CD(H = 0), where CD is the circular dichroism defined as CD = 2(A_RCP − A_LCP)/(A_RCP + A_LCP) with A_RCP being the absorbance for right-circularly polarized light (RCP) and A_LCP for left-circularly polarized (LCP) (see appendix). Another typical MO quantity is the Faraday rotation which is substantiated via the azimuth and ellipticity rotation angles being calculated as the real and imaginary part, respectively, of the difference in the extinction coefficients between LCP and RCP incident light (see appendix).

3. Results and discussion

We begin with a brief description of the behavior of LC/MNP hybrids under or in the absence of a homogeneous magnetic field. Initially, we consider LC/MNP hybrids at various state points (T^∗, ρ^∗) in the absence of an external field: at the thermodynamic conditions considered (for details see ref. [15]), the LC matrix exhibits fully miscible isotropic (I), uniaxial nematic (N) and uniaxial smectic–B (SmB) states. Here we focus on the I– and N– states. These states are chosen as representative examples of the major effect of a magnetic field on the positional and orientational ordering of the magnetic and liquid crystalline components. In fact considerable structural changes occur, on application of an external field (see below). In dimensionless units, the temperature is T^∗ = k_BT/ε (with k_B being Boltzmann’s constant and ε is energy parameter), and the number density $ρ * = N σ_{0}^{3} / V$ (total number of particles over volume). We have obtained various morphologies of the MNP within these states. In the I-state the MNP self assembly into chains, with the dipoles in a head-to-tail configuration; in turn they self-organized into either an isotropic network of wormlike chains (strong dipolar coupling λ = µ²/k_BTσ³ = 8.2) or isotropically distributed fragmented chains (weaker coupling λ = µ²/k_BTσ³ = 4.1) within the LC matrix. Representative illustrations of these structures are given in Fig. 1(a) and Fig. 2(c), respectively. Finally, in the N– state the ferromagnetic chains spontaneously ”unwrap” and align, on average, along the director of the LC matrix (i.e. the principal axis of the uniaxial state). A representative snapshot is given in Fig. 1(b). As a result, the director of the magnetic chains coincides with the LC director. Interestingly, the ferromagnetic chains are distributed randomly ”up” and ”down” within the LC host and therefore the overall magnetization is zero. Furthermore, the chains are translationally disordered lateral to the director of the phase. We finally note that the concentration of the magnetic particles in the LC host is relatively small (less than 4.2 vol%).

As an external perturbation we have applied a homogeneous magnetic field (H^∗ = µH/k_BT) on the I- and N- states of the LC/MNP hybrids. The field is coupled to the permanent dipole µ_i of particle i through the potential $U_{i} = - {\hat{μ}}_{i} \cdot H$ . When the field is off in the I–state of high [Fig. 2(a)] and low [Fig. 2(c)] dipolar coupling, the total magnetization is zero and the nematic order parameter S is essentially zero. The nematic order parameter S is calculated from the ordering matrix of the rods [15]. Under a small field of H^*= 10, the nematic order parameter S jumps from nearly zero to 0.4 and the system becomes polar (macroscopic magnetization). The isotropic network [field-off see Fig. 2(a)] is replaced by nearly straight ferromagnetic chains that are oriented parallel to the direction of the magnetic field [see Fig. 2(b)]. Similar behavior is obtained for the systems of Fig. 2(c) and (d). As a general rule, the soft coupling between the magnetic particles and LC host generates a sterically-induced ordering, which can either be spontaneous (magnetic chains ordering caused by LC matrix) or forced by an external field (transformation of an isotropic state of the LC matrix caused by magnetic chains).

The magnetic nature of the LC/MNP hybrids means that these systems are expected to exhibit a nontrivial MO response. In Figs. 3(a), 3(b) we show the MCD signal for the systems of Fig. 2(b), 2(d). This is compared against the MCD signal obtained in the hypothetical case where the system of Fig. 2(a) does not undergo structural change to the system of Fig. 2(b) (N-state) under the presence of the magnetic field but remains in the I-state. Essentially, this hypothetical case corresponds to the pure MO effect stemming from the magnetite MNPs alone without taking into account the structural changes. Evidently, the actual MCD signal which incorporates the phase transition from the I- to the N-state [black curve of Fig. 3(a)] is on the average 50% higher than that of the hypothetical case (red line) without the structural changes (see also the corresponding insets). In the absence of structural changes to the LC/MNP hybrid, the MCD signal comes from the ferromagnetic contribution of the magnetite MNPs which is attributed to various electronic transitions taking place within the magnetite MNPs [28]. The external magnetic field promotes linear alignment of the chains along the field that in turn induce nematic ordering to LC matrix. The MCD signal incorporates this remarkable structural change in addition to the transitions within the magnetite MNPs leading to the enhancement of the MCD observed in Fig. 3(a). A more dramatic enhancement in the MCD signal is observed in Fig. 3(b) which corresponds to a transformation from an I-state [of smaller fragmented chains of Fig. 2(c)] to a N-state of Fig. 2(d). Namely, the actual MCD spectrum (black line) is on the average 4 times larger than the hypothetical MCD signal (red curve) if the structural change to N-state [Fig. 2(d)] does not occur and the system remains in the I-state of Fig. 2(c). The difference in the enhancement of the MCD signal between Figs. 3(a) and 3(b) may be attributed to the fact that in I-state of Fig. 2(a) there exists already an isotropic network of MNP long chains which promotes the magnetic interaction between the magnetite MNPs and shows a larger MCD signal. On the other hand, in the I-state of Fig. 2(c) (shorter) fragmented chains are formed. The presence of the MNP longer chains in the I-state of Fig. 2(a) also explains the much higher hypothetical MCD signal (about two orders of magnitude) compared to the corresponding hypothetical signal calculated for the isotropic phase of Fig. 2(c) [compare the red lines in Figs. 3(a) and 3(b)]. In the same fashion, the actual MCD signal (the one that can be measured experimentally) which encompasses the I-state to N-state transformation [black curve of Fig. 3(a)] is on the average 5 times larger than the MCD of the I-state to N-state transformation [black curve of Fig. 3(b)]. This is due to the fact the magnetic interactions in the N-state [Fig. 2(b)] are much stronger as the MNP chains are very ”tight” in comparison to the N-state of Fig. 2(d).

Fig. 3 (a) Red line: magnetochiral dichroism (MCD) signal for the hypothetical case where the configuration of Fig. 2(a) is not modified under the action of the magnetic field and the MNPs remain at the same positions (see inset). Black lines: MCD signal for the actual case where the configuration of Fig. 2(a) undergoes structural transformation to the configuration of Fig. 2(b) under the action of the magnetic field (see inset). (b) The same as (a) but for the configurations of Fig. 2(c) and 2(d), respectively. The magnetite MNPs have 5 nm radius.

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In Figs. 4(a), 4(b) we present the spectra of the (real and imaginary part of) azimuth and ellipticity rotation angles (which quantify the Faraday effect) for the systems of Figs. 2(b) and 2(d), respectively. We observe that both rotation angles assume values within, more or less, the same range of values. The arrangement of MNPs in the N-states does not affect much the Faraday effect since MNP linear chains are present in both cases, apart from the reversed sign in the angles which, however, depends on the spatial chirality of the particular configurations of the nematic samples of Figs. 2(b) and 2(d), respectively. In both cases, given the dimensions of a calculation cube (about 50 nm), the Faraday rotation is of the order of 10⁻² milli-degrees per nanometer.

Fig. 4 Azimuth and ellipticity angles corresponding to Faraday rotation for (a) the system of Fig. 2(b) and 2(b) the system of Fig. 2(d) (also shown in the insets).

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4. Conclusions

In conclusion, we have shown numerically that liquid-crystalline/magnetite-nanoparticle hybrids show significant MCD upon application of an external magnetic field due to significant structural change the hybrids undergo from an isotropic to a nematic state. In these states a strong Faraday rotation is also observed due to the formation of long ferromagnetic chains (chains of magnetite nanoparticles) within the hybrids which favour the magnetic interactions among the nanoparticles. The enhanced MCD signal can be exploited experimentally to monitor liquid-crystalline transitions as well as to amplify the MCD signal of magnetic nanoparticles in order to keep track of the internal electron transitions in the nanoparticles.

Appendix

Our aim is to study the optical response of a collection of magnetic nanospheres within an anisotropic host based on the discrete dipole approximation (DDA) [16–18]. We consider a finite collection of point dipoles (i = 1, ···, N) each of which is located at the position r_i and corresponds to a dipole moment P_i and a (position-dependent) polarizability tensor ${\tilde{α}}_{i}$ . The above quantities are connected by

P_{i} = {\tilde{α}}_{i} E_{i}

where E_i is the electric field at i-th dipole,

E_{i} = E_{i n c, i} + \sum_{i \neq i} A_{i j} \cdot P_{j}

which is the sum of the directly incident field E_inc,i as well as the field scattered by all the other dipoles j ≠ i and it is incident on the i-th dipole [second term of Eq. (2)]. The interaction matrix A_ij is given from

A_{i j} = \frac{\exp (i k r_{i j})}{r_{i j}} [k^{2} ({\hat{r}}_{i j} {\hat{r}}_{i j} - 1_{3}) + \frac{i k r_{i j} - 1}{r_{i j}^{2}} (3 {\hat{r}}_{i j} {\hat{r}}_{i j} - 1_{3})], i \neq j

where 1₃ is the 3 × 3 unit matrix, r_ij = r_i − r_j,

{\hat{r}}_{i j} = r_{i j} / | r_{i j} |

. By combining Eqs. (1) to (3) we obtain linear system of equations, i.e.,

\sum_{j = 1}^{N} A_{i j} P_{j} = E_{i n c, i}

where the diagonal elements of the interaction matrix are essentially the inverse of the polarizability tensor of each dipole, i.e.,

A_{i j} = {[{\tilde{α}}_{i}]}^{- 1} .

For a generally anisotropic sphere characterized by a dielectric tensor ${\tilde{ε}}_{s}$ and is immersed within an anisotropic host of dielectric tensor ${\tilde{ε}}_{h}$ , the polarizability tensor of the sphere is given by the Clausius-Mossoti formula for anisotropic spheres [26], i.e.,

\tilde{α} = V_{s} \frac{2 ε_{h; ⊥} + ε_{h; ∥}}{4 π} [{\tilde{ε}}_{s} - {\tilde{ε}}_{h} 1_{3}] {[{\tilde{ε}}_{s} + 2 {\tilde{ε}}_{h} 1_{3}]}^{- 1} .

For a gyrotropic material, the permittivity tensor is provided by [19,20]

{\tilde{ε}}_{s} = (\begin{matrix} ε_{x x} & - i Q m_{z} & i Q m_{y} \\ i Q m_{z} & ε_{y y} & - i Q m_{x} \\ - i Q m_{y} & i Q m_{x} & ε_{z z} \end{matrix})

where Q is magneto-optical Voigt parameter and m_x, m_y, m_z are the direction cosines for the magnetization vector. For nematic LC host the corresponding tensor is written as

{\tilde{ε}}_{h} = (\begin{matrix} ε_{⊥} & 0 & 0 \\ 0 & ε_{⊥} & 0 \\ 0 & 0 & ε_{∥} \end{matrix})

Having determined the dipole moment P_i at each point dipole, one can calculate quantities such as the scattering, extinction and absorption cross sections, i.e.,

C_{s c} = \frac{4 π k}{| E_{i n c} |^{2}} \sum_{i = 1}^{N} [\frac{2}{3} k^{3} | P_{i} |^{2} - ℑ (P_{i} \cdot E_{self, i}^{*})]

C_{e x t} = \frac{4 π k}{| E_{i n c} |^{2}} \sum_{i = 1}^{N} ℑ (P_{i} \cdot E_{i n c, i}^{*})

C_{a b s} = \frac{4 π k}{| E_{i n c} |^{2}} \sum_{i = 1}^{N} [ℑ (P_{i} \cdot E_{i}^{*}) - \frac{2}{3} k^{3} | P_{i} |^{2}]

where E_self,_i = E_i − E_inc,i. Evidently, C_ext = C_sc +C_abs.

The Faraday rotation, θ, and ellipticity, η, are provided by [20]

θ + i η = \frac{π k}{E_{0}} \sum_{i = 1}^{N} (P_{i}^{L} - P_{i}^{R}) \exp (- i k \hat{z} \cdot r_{i})

where

P_{i}^{L, R} = P_{i} \cdot {\hat{e}}_{L, R}^{*}

{\hat{e}}_{L, R} = \frac{1}{\sqrt{2}} (\hat{x} \pm i \hat{y})

The absorption chiral dichroism is defined as

C D = C_{a b s}^{R C P} - C_{a b s}^{L C P}

where RCP (LCP) stands for right (left-circularly polarized light. The magnetic chiral dichroism (MCD) is defined as

M C D = C D (H) - C D (H = 0) .

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Magneto-optical properties of liquid-crystalline ferrofluids

Abstract

1. Introduction

2. Calculation methods

3. Results and discussion

4. Conclusions

Appendix

References and links

Cited By

Figures (4)

Equations (16)

Optical Materials Express