Magneto-optic dynamics in a ferromagnetic nematic liquid crystal
We investigate dynamic magneto-optic effects in a ferromagnetic nematic liquid crystal experimentally and theoretically. Experimentally we measure the magnetization and the phase difference of the transmitted light when an external magnetic field is applied. As a model we study the coupled dynamics of the magnetization, , and the director field, , associated with the liquid crystalline orientational order. We demonstrate that the experimentally studied macroscopic dynamic behavior reveals the importance of a dynamic cross-coupling between and . The experimental data are used to extract the value of the dissipative cross-coupling coefficient. We also make concrete predictions about how reversible cross-coupling terms between the magnetization and the director could be detected experimentally by measurements of the transmitted light intensity as well as by analyzing the azimuthal angle of the magnetization and the director out of the plane spanned by the anchoring axis and the external magnetic field. We derive the eigenmodes of the coupled system and study their relaxation rates. We show that in the usual experimental set-up used for measuring the relaxation rates of the splay-bend or twist-bend eigenmodes of a nematic liquid crystal one expects for a ferromagnetic nematic liquid crystal a mixture of at least two eigenmodes.
In Ref. bropgdg () Brochard and de Gennes suggested and discussed a ferromagnetic nematic phase combining the long range nematic orientational order with long range ferromagnetic order in a fluid system. The synthesis and experimental characterization of ferronematics and ferrocholesterics, a combination of low molecular weight nematic liquid crystals (NLCs) with magnetic liquids leading to a superparamagnetic phase, started immediately rault () and continued thereafter amer (); antonio87 (); antonio2004 (); kopcansky (); ouskova (); buluy (); podoliak (). These studies made use of ferrofluids or magnetorheological fluids (colloidal suspensions of magnetic particles) rosensweig (); their experimental properties rosensweig (); odenbach2004 () have been studied extensively in modeling mario2001 (); luecke2004 (); hess2006 (); sluckin2006 (); mario2008 (); klapp2015 () using predominantly macroscopic descriptions mario2001 (); luecke2004 (); hess2006 (); mario2008 ().
On the modeling side, the macroscopic dynamics of ferronematics was given first for a relaxed magnetization jarkova2 () followed by taking into account the magnetization as a dynamic degree of freedom jarkova () as well as incorporating chirality effects leading to ferrocholesterics fink2015 (). In parallel a Landau description including nematic as well as ferromagnetic order has been presented hpmhd ().
Truly ferromagnetic NLCs have been generated alenkanature () in 2013 followed by reports of further ferromagnetic NLCs in shuaiNC (); smalyukhPNAS (), and their macroscopic static properties were characterized in detail alenkasoftmatter (). Quite recently ferromagnetic cholesteric liquid crystals have been synthesized and investigated smalyukhPRL (); smalyukhNatMat (); alenkascience (). For a review on ferromagnetic NLCs see Ref. alenkarev ().
In the present paper we describe in detail experimentally and theoretically the static and dynamic properties of ferromagnetic NLCs tilenshort (). We analyze the coupled dynamics of the magnetization and the director, initiated and controlled by an external magnetic field. We show experimentally and theoretically that dissipative dynamic coupling terms influence qualitatively the dynamics. Experimentally this is done by measuring the temporal evolution of the normalized phase difference associated with the dynamics of the director. Quantitative agreement between the experimental results and the model is reached and a dissipative cross-coupling coefficient between the magnetization and the director is accurately evaluated. It is demonstrated that this cross-coupling is crucial to account for the experimental results thus underscoring the importance of such off-diagonal effects in this first multiferroic fluid system. We also make concrete theoretical predictions of how the reversible dynamic cross-coupling terms between magnetization and director influence the macroscopic dynamics and how these effects can be detected experimentally. The experimental and theoretical dynamic results discussed in some detail in this paper for low magnetic fields in ferromagnetic NLCs demonstrate the potential for applications of these materials in displays and magneto-optic devices as well as in the field of smart fluids.
The paper is organized as follows. In Section II we describe the experimental set-up followed in Section III by the macroscopic model. The connection between the measurements and the model is established in Section IV. In Section V we analyze the statics and in Sections VI and VII we analyze in detail the coupled macroscopic dynamics of the magnetization and the director field when switching the external magnetic field on and off, respectively. Section VIII is dedicated to a theoretical analysis of fluctuations and light scattering and in the conclusions we give a summary of the main results and a perspective.
The experimental samples have been prepared along the lines described in detail in Refs. alenkanature (); alenkasoftmatter (). In brief, the BaScFeO nanoplatelets were suspended in the liquid crystal mixture E7 (Merck, nematic – isotropic transition temperature C). The suspension was filled in liquid crystal cells with rubbed surfaces (thickness m, Instec Inc.), which induced homogeneous in-plane orientation of the NLC. The volume concentration of the magnetic platelets in the nematic low-molecular-weight liquid crystal E7 (Merck) has been estimated to be from the measurements of the magnetization magnitude alenkasoftmatter () which was A/m. E7 suspensions show long term stability, with a homogeneous response to magnetic fields and no aggregates for a period of several months. A surfactant (dodecylbenzene sulfonic acid) was used for the treatment of the nanoplatelets which favors a perpendicular orientation of the NLC molecules with respect to the nanoplatelets. Quantitative values for the Frank coefficients for E7 are available in the literature km ().
Dynamics of the director was measured by inducing director reorientation in planarly treated 20 m cells (pretilt in the range 1-3 degrees) when applying a magnetic field perpendicular to the cell plates, Fig. 1 (top). Experiments were performed on monodomain samples (see Ref. alenkarev () for a description of monodomain sample preparation) so that the director is initially at 45 degrees with respect to the crossed polarizers, Fig. 1 (bottom). Using polarizing microscopy, the monochromatic light intensity transmitted through the sample was recorded with a CMOS camera (IDS Imaging UI-3370CP, 997 fps) as a function of time on switching the magnetic field on and off. An interference filter (623.8 nm) was used to filter the light from the halogen lamp used in the microscope. The transmitted light intensity is related to the phase difference between the ordinary and the extraordinary light as will be explained below. The advantage of using polarization microscopy is that the measurements are performed in the homogeneous region of the sample without spacers or other impurities. Recording the image of the sample during the measurements also allows us to simultaneously monitor the homogeneity of the response.
With the use of a vibrating sample magnetometer alenkasoftmatter () (LakeShore 7400 Series VSM) also the equilibrium component of the magnetic moment of the sample is measured. We note that this technique is not suitable for measuring the magnetization dynamically, as several seconds per measurement are required for ambient magnetic noise averaging.
Iii Macroscopic model
Throughout the present paper we take into account the magnetization and the director field as macroscopic variables; in the following we focus on the essential ingredients of their dynamics necessary to capture the experimental results we will discuss. That is we assume isothermal conditions and discard flow effects. For a complete set of macroscopic dynamic equations for ferronematics we refer to Ref. jarkova ().
The static behavior is described by the free energy density ,
where is the magnetic constant, is the applied magnetic field, and will be assumed constant. The first term represents the coupling of the magnetization and the external magnetic field. Since , the local magnetic field is equal to , which is fixed externally, and is thus independent of the configuration. The second term describes the static coupling between the director field and the magnetization (originating from the magnetic particles). The third term describes the energy connected with the deviation of the modulus of the magnetization from . The last term is the Frank elastic energy associated with director distortions degennesbook ()
with positive elastic constants for splay (), twist (), and bend (). The saddle-splay elastic energy degennesbook () is zero in the considered geometry. While it is a good approximation to assume that , we will take into account small variations of (corresponding to large values of ).
The anchoring of the director at the plates is taken into account using a finite surface anchoring energy rapini ()
where is the anchoring strength and is the preferred direction specified by the director pretilt angle .
For the total free energy we have and the equilibrium condition requires .
where the quasi-currents have been split into reversible (, ) and irreversible, dissipative (, ) parts. The reversible (dissipative) parts have the same (opposite) behavior under time reversal as the time derivatives of the corresponding variables, i.e, Eqs. (4)-(5) are invariant under time reversal only if the dissipative quasi-currents vanish.
The quasi-currents are expressed as linear combinations of conjugate quantities (thermodynamic forces); they take the form
with and where the transverse Kronecker delta projects onto the plane perpendicular to the director owing to the constraint .
In Ref. tilenshort () we focused on the dissipative quasi-currents as they had a direct relevance for the explanation of the experimental results discussed there. In the present paper we also include the reversible quasi-currents, which give rise to transient excursions of and out of the switching plane.
The dissipative quasi-currents take the form jarkova ()
Throughout the present paper we will discard the biaxiality of the material which arises for .
The reversible quasi-currents are obtained by requiring that the entropy production is zero jarkova ():
where jarkova2 ()
For solving the system Eqs. (4)-(5) a simple numerical method was used. We first discretized space into slices of width , where is the number of discretization points. Empirically it was found that using is already sufficient. After discretizing space one obtains ordinary differential equations. Due to its simplicity, we use the Euler method. One step of the Euler method for the i-th component of the director field at is
where is the time step. An analogous equation holds for the magnetization field and the equations are solved simultaneously. Since the numerical scheme for the director field is not norm preserving, we normalize the director field after each time step: .
In the discrete version, the two surface points are best treated by satisfying the same dynamic equations Eqs. (4)-(5) as the internal points, with the addition of the surface anchoring energy Eq. (3) expressed as a volume density. The divergence part of the force Eq. (7) is then replaced by its surface flux (the volume density thereof, again):
where is the surface normal pointing down (up) at the bottom (top) plate.
Iv Connection between measurements and the model
In equilibrium the magnetic-field-distorted director and magnetization fields are lying in the plane, and . In the absence of the magnetic field, the director is tilted from the axis by the pretilt , Eq. (3). The coordinate system used here is shown in Fig. 1. As explained earlier, the average component of the magnetization, , is measured by the vibrating sample magnetometer. In modeling, it is obtained by averaging the component of the magnetization field,
To derive the expression for the phase difference we start with an electric field , which is linearly polarized after the light passes through the polarizer,
where is the electric field amplitude, the initial polarization, the wave vector and the frequency of the incident light. In our case the wave vector points in the direction,, with being the wave number. The polarization of the light therefore lies in the plane and is described by the two-component complex vector . As the light passes through the sample also the components of this (Jones) polarization vector change and we analyze these changes using the Jones matrix formalism (assuming perfectly polarized light) hechtbook ().
The incident light first goes through the polarizer oriented at with respect to the axis, Fig. 1, and is linearly polarized with the initial Jones vector being . The optical axis is parallel to the director and generally varies through the cell. For any ray direction we can decompose the polarization into a polarization perpendicular to the optical axis (ordinary ray) and a polarization which is partly in the direction of the optical axis (extraordinary ray). The ordinary ray experiences an ordinary refractive index and the extraordinary ray experiences a refractive index ,
where is the extraordinary refractive index.
To calculate the intensity of the transmitted light, one first divides the liquid crystal cell into thin slices of width and describes the effect of each slice on the polarization by the phase matrix
In the limit we can express the transmission matrix of the liquid crystal cell as
where we have introduced the phase difference
In general, as we will see, the director can have also a nonzero component in the direction. In this case the simple expression for the transmission matrix Eq. (22) does not hold anymore and must be generalized.
We start the derivation of the general transmission matrix by assuming a general orientation of the director,
The azimuthal angle of the director can vary through the cell and the transformation matrix at point is
where R is the rotation matrix
Our goal is to find the transfer matrix for the whole cell,
where the arrow denotes the ordered product starting from at the right side. We first notice that
where is the identity matrix. Consequently we can write as an exponential,
where A is defined by
We can now rewrite Eq. (27) as
where we used
The exponential of the matrix from Eq. (31) reads
We then let the light pass through an analyzer at an angle ,
which gives for the final Jones vector ()
This yields the measured normalized intensity
Next we evaluate the relation between the phase difference and the measured intensity. Let be the Jones vector after the liquid crystal cell,
where and are real and . Generally . After an analyzer with we have a Jones vector
and the intensity is related to the phase difference as
Only if the director is restricted to the plane, and we have
such that the relation between the intensity and the phase difference is
where and the sign is determined by demanding that is sufficiently smooth. Generally however, the quantity obtained from the measured intensity by Eq. (42) is not the phase difference. It is the phase difference only when the director field is in the plane. For the analysis of the dynamics not confined to the plane, Sec. VI.3, we will therefore use the normalized intensity Eq. (37).
In the case when the dynamics is in the plane, to compare the numerical results with the experiments and also to compare the dynamics of the director with the dynamics of the magnetization, it is convenient to introduce the normalized phase difference
where is the phase difference at zero magnetic field. The normalized phase difference is zero at and is always smaller or equal to 1. It can also assume negative values as we will see.
In this Section we present experimental and numerical results of statics and derive analytic formulae for the equilibrium configurations in the low and large external magnetic field limits.
In Fig. 2 we compare the numerical results of the equilibrium normalized phase difference to the experimental data. Below we will show in Eqs. (V.1)-(53) that the equilibrium normalized phase difference is quadratically dependent on the applied magnetic field at small magnetic fields. The normalized phase difference saturates quickly above = 10 mT at a value which is less than 1, which means there is a limit to how much the director field deforms. We also observe that the dependence of the equilibrium normalized phase difference is not symmetric with respect to the axis, which is seen in experiments as well. The reason for this is the nonzero pretilt at both glass plates.
¿From the fits to the model we extract values for the anchoring strength , the pretilt angle , the Frank elastic constant in the one constant approximation, and the static coupling coefficient :
The extracted parameters Eqs. (44)-(47) correspond to the (local) minimum of the sum of squares of residuals between the numerical and experimental values of the normalized phase difference. This minimum was sought in sensible parameter ranges (for example the Frank elastic constant was sought in the range between 5 pN and 25 pN). There are several indications that this minimum is at least very close to the global one. First, the extracted value of the Frank elastic constant is close to the value of in the pure E7 NLC. Secondly, the extracted pretilt is within the range specified by the cell provider. Moreover the value of the static coupling is similar to that estimated for the ferromagnetic NLC based on 5CB alenkasoftmatter ().
The limiting behaviors of the normalized phase difference and the normalized component of the magnetization as the magnetic field goes to zero or infinity can be calculated analytically. In all cases the boundary condition is
where is the component of the surface normal pointing upwards at and downwards at .
v.1 Low magnetic fields
The free energy density in lowest order in deviations of magnetization and director field from the equilibrium is
The equilibrium solutions for the angles are
After inserting the solutions Eqs. (50) in equations for the normalized phase difference and magnetization, one gets
where is the so called anchoring extrapolation length and . In the limit of infinite anchoring the normalized phase difference reads
One can also observe that the location of the minimum of the normalized phase difference is shifted to a value determined by the pretilt:
¿From the behavior of the normalized phase difference at low fields Eqs. (V.1)-(53) one cannot determine the value of the static coupling . It can on the other hand be determined from the low-field behavior of the magnetization. In Fig. 2 we see that the behavior is linear for low magnetic fields as can be shown analytically:
v.2 Large magnetic fields
In the large magnetic field limit we assume that both the polar angle of the director and the magnetization are either close to 0 if the applied magnetic field is positive () or close to if the applied magnetic field is negative (). The corresponding solutions will be denoted as , , , , , and .
The free energy in the case of a positive magnetic field is
The equilibrium solutions for the angles and are
with (which is proportional to the inverse “magnetization coherence length” of the director).
The normalized component of the magnetization for large fields is
and the normalized phase difference is
It follows from symmetry that , , , and
Since the magnetization is not anchored at the boundary, in Eq. (60) it was sufficient to consider terms not higher than . On the other hand, due to the anchoring of the director field, in Eq. (V.2) we expanded the phase difference to the order . It should be noted, that the approximation for the phase difference is better if the anchoring is low, i.e., or .
where and are functions of static parameters for positive magnetic fields and . The behavior of the magnetization , Fig. 2, may at a first glance look like the Langevin function, often observed in magnetic systems. Eq. (63) tells us that this is not the case, since the Langevin function saturates with the first power in magnetic field, whereas here the saturation Eq. (63) is of second order in .
v.3 Comparison of analytic approximations with numerics
A comparison of analytic and numeric results for the director polar angle is made in Figs. 3 and 4 for small and large magnetic fields, respectively. We find a good agreement for small magnetic fields up to 0.7 mT and for large magnetic fields above 4 mT. It should be emphasized that the values of the magnetic fields at which the approximations become valid depend on the values of the static parameters. We use the values Eqs. (44)-(47) extracted from the fits to the macroscopic model.
In Fig. 5 we compare analytic and numeric results for the component of the magnetization and the normalized phase difference. Again we find a good agreement between the results at similar ranges of the magnetic field. From the insets of Fig. 5 one can conclude that for our system a magnetic field as small as 1 mT can be considered as large already. The notable discrepancy of the numeric and analytic normalized phase difference at large magnetic fields is due to the fact that one has expanded the expression for the phase difference, Eq. (23), up to the order . Since does not saturate to zero, this means that the constant term of Eq. (V.2) is slightly different from the actual value determined numerically.
The agreement between experimental data and the model for two key static properties underscores that we have solid ground for the analysis of the dynamic results which now follows.
Vi Switch-on dynamics
In this Section we present the experimental and theoretical results of the dynamics that takes place when the magnetic field is switched on.
In Fig. 6 we plot the comparison of experimental and theoretical data for the dynamics of the normalized phase difference (top) as well as the theoretical results for the normalized component of the magnetization (bottom) for two values of the applied magnetic field. As an inset we show that for small times the magnetization grows linearly, which is also obtained analytically in Sec. VI.1. As expected the rise time for the magnetization is reduced as the applied magnetic field is increased. The inset for the top graph shows that the initial phase difference is quadratic in time, which is again obtained also analytically, Sec. VI.1.
The fits for the comparison of the experimental and theoretical normalized phase difference are performed by varying the dynamic parameters taking into account the fundamental restrictions tilenshort () on their values, at fixed values of the static parameters Eqs. (44)-(47). The model captures the dynamics very well for all times from the onset to the saturation. The extracted values of the dynamic parameters are
The dissipative cross-coupling coefficient is within the allowed interval determined by the restriction tilenshort ()
The remaining two dynamic parameters do not affect the dynamics significantly and are set to and .
To extract from the time evolution of the normalized phase difference, Fig. 6 (top), a switching time as a measure of an overall relaxation rate of the dynamics, we use a squared sigmoidal model function
Remarkably, the relaxation rate, , shows a linear dependence on , Fig. 7. We were first interested in the effect of the dissipative cross coupling on . We find that a reasonably strong dynamic cross coupling is needed in order to obtain the observed linear magnetic field dependence of the relaxation rate. In the absence of this dynamic cross coupling, Fig. 7, the relaxation rate levels off already at low fields as expected since the transient angle between and gets larger, and starts to decrease for even higher magnetic fields.
The best match of the relaxation rates extracted from the experimental data and the model, Fig. 7, allows for a robust evaluation of the dissipative cross-coupling between the magnetization and the director:
vi.1 Initial dynamics
We investigate the initial dynamics of the normalized phase difference and magnetization upon application of the magnetic field. Up to linear order we also take into account the pretilt. Initially, and are parallel to . Keeping the modulus of the magnetization exactly fixed, the initial thermodynamic forces Eqs. (6) and (7) are
where is the projection of perpendicular to . With that, the initial quasi-currents are
At finite and zero it follows from Eq. (71) that the component of the director field responds linearly in time as well as linearly in the magnetic field for small times:
As a contrast, if is zero, the director responds through the nonzero molecular field due to the static coupling ,
where is the initial response of the component of the magnetization, Eq. (72). The component of the director field thus responds quadratically in time rather than linearly,
For small times we can express the refractive index Eq. (20) as
The coefficients and from Eq. (34) are then
and the normalized intensity of the transmitted light for small times is
In the lowest order of , for the phase difference one gets a linear term that is also linear in pretilt and a quadratic term which does not vanish if the pretilt is zero:
Eq. (79) will be used to extract the dissipative cross coupling coefficient and the pretilt from the experimental data. Furthermore, from Eq. (79) one can see that in the case of positive (negative) pretilt the normalized phase difference has a minimum at negative (positive) magnetic fields. By measuring the time of this minimum, Fig. 8,
one can calculate the ratio of the pretilt and the dissipative cross coupling.
If , the time of the minimum decreases more slowly with increasing magnetic field:
The normalized phase difference evaluated at is of second order in the pretilt:
The minimum value Eq. (82) is independent of the applied magnetic field. This can be explained by the fact that the director field goes through an intermediate state which is approximately aligned with the glass plates of the cell.
We note that if both the dissipative cross coupling coefficient and the pretilt are zero, the normalized phase difference initially grows as .
Assuming a negative pretilt, Eq. (79) predicts a minimum for positive magnetic fields, which is also seen in experiments, Fig. 6 (top). In Fig. 8 we show experimental inverse times of the minima. The large error at high magnetic fields is due to the time resolution limitations (1 ms). ¿From the linear behavior predicted by Eq. (80) we can extract the ratio between the dissipative cross-coupling and the pretilt. Independently we can extract the pretilt by measuring the values of the minima, Fig 9.
Fitting Eq. (79) to the initial time evolution of measured normalized phase differences (like those presented in Fig. 6) for several values of the magnetic field , we determine the parameters and shown in Fig. 10 and Fig. 11, respectively. Therefrom we extract the value of the dissipative cross-coupling parameter between director and magnetization,
and from the parameter of Eq. (79) we extract the pretilt,
The normalized component of the magnetization Eq. (18) is linear in :
which is in accord with Fig. 6 (bottom). ¿From the initial behavior one can therefore directly determine the dissipative coefficient .
Let us define the initial rate of the director reorientation as the time derivative of the director component at ,
For a nonzero dissipative cross-coupling coefficient the initial rate, Eq. (73), is
However if , the initial rate of the director reorientation is proportional to the component of the magnetization, Eq. (74),
The relaxation rates Eqs. (87) and (88) describe two different mechanisms of the director reorientation. The former is associated with the dynamic coupling of the director and the magnetization, whereas the latter is governed by the static coupling of the director and the magnetization. Here a deviation of the magnetization from the director is needed to exert a torque on the director.
vi.2 Dissipative cross-coupling
We have demonstrated that the dissipative cross-coupling of the director and the magnetization, i.e., the terms of Eqs. (8) and (9), affects the dynamics decisively and is crucial to explain the experimental results. It is described by the parameters and of Eq. (10). Here we check the sensitivity of the dynamics to the values of these two parameters. Varying while keeping , Fig. 12, we see that the influence of is rather small and is not substantial. Moreover, the initial dynamics is not affected, Fig. 12 (inset).
On the other hand, increasing strongly reduces the rise time of the normalized phase difference, Fig. 13, and also strongly affects the initial behavior (inset). For large values of one also observes an overshoot in the normalized phase difference.
By inspecting Eq. (10) one sees that the influence of is largest when , and . On the other hand, the influence of is largest when . Since and are initially parallel and moreover the transient angle between them never gets large due to the strong static coupling compared to the magnetic fields applied, it is understandable that affects the dynamics more than .
vi.3 Reversible cross-coupling
The reversible cross-coupling of the director and the magnetization, described by the terms of Eqs. (12) and (13), has not been considered up to this point. We focus on the reversible cross coupling coefficient and put both reversible tensors and of Eqs. (III) and (III) to zero.
If the reversible currents are included, both variables wander out of the plane dynamically, which will be described by the azimuthal angles and of the magnetization and the director, respectively, defined by , . The dynamic behavior of both azimuthal angles is shown in Fig. 14.
Contrary to the polar angles we find that the response of the azimuthal angle of the director is faster than that of the magnetization. ¿From Fig. 14 we read off that the maximum azimuthal angles increase with , being higher for the magnetization than for the director.