PHYSICAL REVIEW E 91, 062502 (2015)
Macroscopic order in a nematic liquid crystal: Perturbation by spontaneous director fluctuations Daichi Hamasuna,1 Rauzah Hashim,2 Atsuhiro Kasatani,1 Geoffrey R. Luckhurst,3 Akihiko Sugimura,1,* Bakir A. Timimi,3 and Herbert Zimmermann4 1
School of Information Systems Engineering, Osaka Sangyo University, 3-1-1 Nakagaito, Daito-Shi, Osaka 574-4853, Japan 2 School of Chemistry, University of Malaya, 50603 Kuala Lumpur, Malaysia 3 Department of Chemistry, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom 4 Department of Biophysics, Max-Planck-Institut f¨ur Medizinische Forschung, Jahnstrasse 29, D-69120 Heidelberg, Germany (Received 18 February 2014; published 4 June 2015) The dynamic alignment of the nematic director by near-orthogonal electric and magnetic fields has been investigated. The intermediate states during the relaxation process were found, with the aid of time-resolved deuterium NMR spectroscopy, to be markedly nonuniform. The macroscopic order was perturbed, although the initial and final states of the director appear to be essentially uniform. However, the initial state does have a profound influence on the uniformity of the director in the intermediate states. We have developed a fundamental model based on the effect of spontaneous director fluctuations to explain these unusual NMR observations. DOI: 10.1103/PhysRevE.91.062502
PACS number(s): 61.30.Hn, 76.60.−k
I. INTRODUCTION
Technological applications of liquid crystals, such as flatscreen display devices, depend on the anisotropic nature of these materials, commonly defined by the presence of a single director. This can be manipulated by external factors, including homogeneous electric and magnetic as well as inhomogeneous surface fields. It is therefore of prime importance to understand the alignment of the nematic director by either a magnetic and/or an electric field by careful evaluation of the hydrodynamic processes under both equilibrium and nonequilibrium conditions [1]. To complement these experiments, many theoretical models, based on continuum theory, have been developed that successfully describe the static and dynamic phenomena. Such macroscopic behavior has been investigated using deuterium nuclear magnetic resonance (2 H NMR) spectroscopy. Here the director is aligned by the magnetic field of the spectrometer, and the application of an electric field produces a nonequilibrium state, resulting in rotation of the director [1,2]. This powerful technique has proved to be especially important for the investigation of director alignment in liquid crystals [3–7]. The spontaneous fluctuations of the director alignment in a nematic phase are also well known, and as we shall see, are important in the field-induced director alignment. Their characteristic features have previously been studied with optical scattering measurements [8]. In this paper we describe the anomalous director distribution, observed and subsequently predicted, during the director reorientation in a nematic liquid crystal film subject to near-orthogonal magnetic and electric fields. Our studies are confined to low-molar-mass systems, although analogous behavior might be expected for polymeric nematics [9]. Time-resolved 2 H NMR spectroscopy has been employed to investigate the homogeneous field-induced director dynamics. This technique has the advantage of being able to explore the director distribution during the electric-field-induced rotation of the director [5]. This unique capability results from the design and construction of equipment to produce and control
*
[email protected]
1539-3755/2015/91(6)/062502(8)
the electric field in a manner precisely synchronized with the radio frequency pulses of the NMR spectrometer. Analysis of the experimental results is based on the predictions of a torque-balance equation for the homogeneous director and a simple model of the director fluctuations. The layout of this paper is as follows. In the next section we describe the time-resolved 2 H NMR experiments. The results for the dynamic behavior of the average director orientation and distribution are given in Sec. III. They are also discussed in terms of a torque-balance equation employed to calculate the time variation of the director orientation and, hence, its distribution. In addition, the factors that can influence the deviation of the director from uniform alignment are explored. Our conclusions are outlined in Sec. IV. II. EXPERIMENTAL
The nematic liquid crystal used in our experiments was 4-(1,1-d2 - pentyl)-4 -cyanobiphenyl 5CB-d2 , in which the pentyl chain is specifically deuterated at the 1 position [10,11]. The nematic-isotropic transition temperature TNI for this sample is 31.6 °C. The liquid crystal was contained in a 194.7 ± 1.0-μm-thick cell made of two transparent In2 O3 - coated glass plates. The cell construction, thickness measurement, and its filling have been described previously [4,5]. The In2 O3 surfaces of the cell were untreated and the anchoring strength was found to be small [12]; consequently, the surface interaction could not compete with the director alignment by the magnetic field of the NMR spectrometer and the electric field. In the experiments an ac electric field of 4 kHz was used. This frequency is sufficient to overcome the alignment effects of ionic conduction. The sinusoidal electric field was generated using a function generator, Wave Factory WF1943, and a high-power amplifier, model 4005, both from NF Electronic Instruments. The angle α between the electric and magnetic fields for our experiments was chosen to be close to 90 deg, that is, the initial angle between the director and the aligning electric field. To achieve this, the cell was first inserted in the NMR coil so that the glass plate surfaces were very close to being parallel to the magnetic field. Their orientation was determined from the quadrupolar splitting described in
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PHYSICAL REVIEW E 91, 062502 (2015)
200 V, and for our cell of thickness 194.7μm, this corresponds to an electric field of 1.027 MV m−1 , which is higher than the threshold field required for a magnetic field of 7.05 T. The value of EFD was fixed at 40 ms, which was sufficiently long to align the director uniformly and to reach its equilibrium orientation θ∞ at the end of this time. The value of MPT was fixed at 2 s, which is long in comparison with the magnetic relaxation time τM . To obtain an NMR spectrum with a good signal-to-noise ratio, this sequence has to be repeated many tens of thousands of times. The quality of the control of the various pulses therefore needs to be very high to ensure that the spectral lines are not broadened. The general scheme we have employed in performing the time-resolved turn-on NMR experiments is described in detail in Ref. [5]. III. RESULTS AND DISCUSSION
FIG. 1. A schematic diagram showing the pulse sequence for the turn-on dynamics NMR experiment. EFD is the time for which the electric field is on; ED is the time at which the acquisition of the free-induction decay is initiated; RD is the relaxation time following the acquisition of the transient; and MPT is the magnetic preparation time for the director alignment by the magnetic field after the electric field is switched off and before the next trigger pulse which turns it on again.
Sec. III. Then the final adjustment of the orientation of the cell was made in 0.1-deg steps with the aid of a goniometer driven by an ultrasonic motor. The details of the time lines of the electric field and the NMR pulses are shown in Fig. 1. The 2 H NMR spectra were measured via a quadrupolar echo sequence, and for this the 90-deg pulses used were of 5.25-μs duration and the receiver dead time tD was a few microseconds. In most of the experiments reported here, the value of the acquisition time tac for the free-induction decay (FID) was 20.5 ms. We now turn to the time sequence for the application of the electric field which, in combination with the constant magnetic field of 7.05 T, controls the director alignment from nonequilibrium to equilibrium states. Prior to the application of the electric field to the nematic film, the director is at equilibrium parallel to the magnetic field, since the magnetic anisotropy is positive for nematic 5CB. Application of the electric field generates a nonequilibrium state because the dielectric anisotropy is positive. The director then moves from being parallel to the magnetic field to being essentially parallel to the electric field, at which point the system has returned to equilibrium. For this turn-on experiment the spectrum was acquired after a time called the external delay (ED), while the electric field is still on. The values of ED and the relaxation delay (RD) are related to the magnetic preparation time (MPT), which is the time when the electric field is zero and the director returns to being parallel to the magnetic field with the system again at equilibrium. These times are set and controlled via the NMR spectrometer. Experiments were performed with a constant electric field, and the time for which this is applied is called the electric field duration (EFD); during this the value of α was close to 90 deg. The voltage VRMS we have used was
The spectrum expected from a pair of equivalent deuterons when the director is uniformly aligned relative to the magnetic field is a single quadrupolar doublet with the splitting given by [13] ν(θ ) = ν0 (3cos2 θ − 1)/2 ,
(1)
where θ is the angle between the magnetic field and the director, and ν0 is the quadrupolar splitting when the director is parallel to the magnetic field. On the other hand, if the director is uniformly aligned at 90 deg to the magnetic field the splitting is half of ν0 . In contrast, when the director distribution is completely random in the plane formed by the electric and magnetic fields the spectrum will be a 2D powder pattern [14]. This contains both parallel and perpendicular spectral lines, with the intensity between these features originating from intermediate orientations of the director; by these terms we mean that the quadrupolar splittings are those expected when the director is parallel or perpendicular to the spectrometer’s magnetic field. The dotted curves in Fig. 2(a) show a series of time-resolved spectra for the turn-on alignment process recorded at a shifted temperature, TNI -T =, of 16.6 °C for α = 89.7 deg and the significantly long magnetic preparation time, MPT, of 2 s, which is longer than those used in earlier experiments where α was not quite as large [5]. In Figs. 2(a) and 2(b) the outer and inner pairs of dotted lines (used as guides to the eye) show the positions of the parallel and perpendicular spectral lines, respectively. It is quite clear from Fig. 2(a) that the director distribution remains predominantly uniform; the deviation from this uniformity is largely responsible for the line broadening, as is the appearance of weak parallel and, surprisingly, intermediate spectral lines. The extent of the line broadening increases as the average director orientation with respect to the magnetic field grows, and decreases when the director tends to be orthogonal to the magnetic field. At 6 ms after the electric field was switched on the spectrum is almost identical with its initial form [cf. the spectrum for 0 V in Fig. 2(a)]. There is, however, a small change in the quadrupolar splitting and a slight asymmetric broadening of the spectral lines. By 8 ms, after the application of the electric field, a doublet appears with a smaller splitting than that of the parallel doublet. At 9 ms the splitting of this second doublet becomes still smaller, indicating that the director is moving
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FIG. 2. (Color online) (a) A series of time-resolved 2 H NMR spectra (dotted curves) for a turn-on experiment at TNI -T =16.6 °C, E = 1.027 MV m−1 , MPT = 2 s, EFD = 40 ms, tac = 41 ms, showing the nonuniform director relaxation from being parallel to the magnetic field to being almost perpendicular to it. The spectral heights are scaled so that the spectra have equal areas.(b) Expanded spectra (dotted curves) recorded in the interval 10–13 ms, in which all spectral line shapes are normalized with the largest peak height in order to show them clearly with time. Solid and dashed scales show the splittings for the major doublets and subdoublets of individual spectral line shapes, respectively. In both (a) and (b) the solid curves show the simulated spectra.
further away from the parallel position. However, there is still a small part of the nematic for which the director is aligned close to the direction of the magnetic field. At 10 ms the director continues to move toward the electric field direction but with a small fraction still unmoved, close to the 0-deg orientation. Just 1 ms later the quadrupolar splitting has vanished, showing √that the director is aligned at the magic angle [θ = cos−1 (1/ 3)], although the single peak is broad. The weak parallel lines
again show that a very small fraction of the director continues to remain parallel to the magnetic field. At the same time, a weak but noticeable doublet, which we call the subdoublet, appears in the wings of the single broad peak, corresponding to an intermediate director orientation between the magic angle, √ cos−1 (1/ 3), and the electric field at α. In other words, during the dynamics investigation we effectively observe three sets of splitting patterns: the parallel and intermediate doublets, which are very weak, while the remaining strong doublet, called the major doublet, also changes with time. At 11.5 ms the parallel doublet is now barely detectable, but the major doublet and subdoublet have become more prominent. That is, the major fraction of the director is now oriented at an angle greater than the magic angle. After 12 ms the parallel lines are weak, while the fraction for the subdoublet has grown at the same rate as that for the major doublet. The quadrupolar splittings of the major doublet and subdoublet, at 12.5 ms, continue to grow, indicating the rotation of the director further from the parallel position. After 13 ms the weak, broad doublet that is the parallel lines have disappeared and the dominant spectral lines are sharper, indicating that the associated director is now aligned close to the electric field. After 20 ms the alignment of the director, essentially parallel to the electric field, is virtually complete. Before we start to understand the electric-field-induced behavior of the director dynamics, it is helpful to summarize the director behavior observed during the turn-on experiment. When the electric and magnetic fields are almost orthogonal, the director distribution starts and finishes in a uniform state, but between these two extremes the distribution is markedly nonuniform. In the basic theory, unconcerned with any spatial variation, we shall concentrate on the pure orientational distribution function for the director f (θ ), where θ is the angle between the director and the magnetic field. One reason for making this approximation is that the NMR spectrum is determined only by f (θ ) and not by f (r,θ ). Moreover, this approach is consistent with the Ockham’s razor principle; this states that the explanation of any phenomenon should make as few assumptions as possible, eliminating those that make no difference in the observable predictions of the explanatory hypothesis or theory. We proceed, therefore, with the minimum number of assumptions found to be consistent with our experimental results. When α is close to 90 deg, the angle made by the director with the aligning electric field, at the beginning of the turn-on experiment, is also close to 90 deg. At this orientation the aligning electric torque acting on the director is essentially zero and so initially the director is expected to move slowly, as observed. However, since the director distribution is not quite uniform, then there will be regions of the distribution function where the director orientation deviates from 90 deg and these are expected to move more rapidly than those close to 90 deg. The motion of the director at different rates will necessarily result in a broadening of the director distribution. But as such directors rotate to become parallel to the electric field, so the distribution function becomes more uniform. We need to evaluate to what extent this basic model for the director distribution function f (θ ) can provide a quantitative or at least semiquantitative account of the experimental, timeresolved NMR spectra. These can be simulated from f (θ )
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according to [15] I (ν) = G[ν,˜ν± (θ ),δG (θ )]f (θ ) sin θ dθ ,
PHYSICAL REVIEW E 91, 062502 (2015)
(2)
where G[ν,˜ν± (θ ),δG (θ )] denotes the normalized shape of a spectral line centered at either of the angular-dependent resonance frequencies, ν˜ + (θ ) or ν˜ − (θ ), and with a linewidth δG (θ ). Here we make a brief comment on the volume element in Eq. (2). The definition of the laboratory axis system is that the z axis is parallel to the magnetic field and the y axis is taken to be perpendicular to the EB plane. In this xyz frame the director orientation is defined by the spherical-polar angles θ and ϕ; for convenience we take ϕ to be zero when the director is in the xz plane. The angle ϕ does not enter into Eq. (2), because the frequency and linewidths depend only on the polar angle θ ; this results from the uniaxial nature of the properties about the director, that is, the nematic phase is uniaxial. The distribution function for the director f (θ,ϕ) does, in general, depend on both spherical and polar angles. However, the dependence on ϕ can be removed from the general distribution to give the reduced distribution function f (θ ) that appears in the spectral line shape I (ν). The two distribution functions are related by f (θ ) = ∫ f (θ,ϕ)dϕ. That is, the volume element in Eq. (2) is sin θ dθ . Since the deuterons exhibit residual dipolar couplings with many protons in the molecule, the spectral line shape approximates to a Gaussian, that is, 1 [ν − ν˜ ± (θ )]2 , G[ν,˜ν± (θ ),δG (θ )] = √ exp − 2δG2 (θ ) 2π δG (θ ) (3) where the linewidth δG (θ ) is defined as half the distance between the points of maximum slope of the line. Of course, the angle θ is time dependent and should be denoted by θ (t), but for the moment we shall avoid this complication in the notation. To evaluate the time dependence of the director orientation during the period EFD when the electric field is applied, we need to know the director distribution at the end of the magnetic preparation time. Given the director distribution, we can then solve the torque-balance equation to obtain the director orientations at subsequent times [3], that is, θ (t) = θ∞ + tan−1 [tan(θ0 − θ∞ )exp(−t/τON )].
(4)
Here θ0 is the initial director orientation and τON is the director relaxation time for the turn-on process, τM τON = , (5) 2 ρ + 2ρ cos 2α + 1 where μ0 γ1 , τM = χB ˜ 2
2 E ˜ε UE , ρ= = μ0 ε0 UM B χ˜
ε0 ˜ε 2 χ˜ 2 E , UM = − B . UE = − 2 2μ0
(6)
In these equations γ1 is the rotational viscosity coefficient, χ˜ is the anisotropy in the diamagnetic susceptibility, ˜ε is the anisotropy in the dielectric permittivity, μ0 is the magnetic permeability, and ε0 is the dielectric permittivity of a vacuum. The parameter ρ is the ratio of the strengths of the electric and
magnetic torques and also their energies UE and UM , when the fields are parallel to the director. The limiting value of θ (t) when t tends to infinity is given by 1 + ρ cos 2α cos 2θ∞ = . 1 + 2ρ cos 2α + ρ 2
(7)
In using these results for the director orientation, each value of the initial angle θ0 has an associated weighting factor, that is, the probability distribution function f (θ ). As a consequence, the time dependence of the director orientation can be obtained numerically by treating the different values of θ0 as originating from nematic monodomains whose orientations are mutually independent. We are, therefore, ignoring the elastic coupling, at least until the comparison between experiment and theory shows that this approximation is inadequate. The starting point for the calculations is the weighting factor for the initial director orientations, and a formula for this was originally proposed by Fan et al. [13], although for a different purpose and written in a slightly simpler form, namely, f (θ ) =
ζ 3
2[ζ − (ζ − 1)cos2 θ] 2
.
(8)
This distribution is normalized in the range between −90 and +90 deg, and the parameter ζ controls the uniformity of the director distribution prior to the application of the electric field; when ζ is unity the director is randomly distributed and as ζ → ∞ so the distribution tends to the Dirac δ function. It is informative to examine the detailed line shapes recorded in the time interval 10–13 ms shown in Fig. 2(b). In these expanded spectra, in which all spectral line shapes are normalized with the height of the tallest peak in order to show the spectral line shape clearly with time, there are certainly three sets of quadrupolar splittings for a given time. The splittings for the subdoublet and the major doublet apparent in Fig. 2(b) are indicated by the dashed and solid scales, respectively. These splitting frequencies give the director orientations as a function of time [see Eq. (1)]. Figure 3 shows the time dependencies of the director orientation θ for the major doublet (closed circles) and the subdoublet (open circles). At the start of the turn-on experiment there are only results for the major doublet; however, just before the end we can see a noticeable difference in the splittings, with that for the subdoublet being larger. It remains to be seen what the origin of this difference is. However, it should be noted that the spectra when the director orientation θ is close to 0 or to 90 deg are particularly insensitive to the minor variations in the director distribution. To aid in understanding the difference in the time dependence of the director orientation found for the major doublet and the subdoublet, the theoretical description of director dynamics proves to be important. For a monodomain nematic confined between untreated glass surfaces, subject to both electric and magnetic fields as used in our experiments, it seems apparent that the surface-anchoring effect and the elastic deformation may be ignored. This simplifies the LeslieEricksen based theory (see, for example, Ref. [16]) for the time dependence of the orientation of the nematic director subject to both magnetic and electric fields. In fact, the time dependence of the director orientation with respect to the magnetic field is
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of the subdirector orientation is the thermal fluctuation of the director orientation. Here, we shall treat a special case in which the director fluctuates with a small amplitude θF on the u axis, which is along each director orientation. Since a Fourier mode is sufficiently slow with respect to the time scale of the NMR experiment, we consider a time-independent function for the director fluctuation. A simplified modulation with a wavelength λ of an individual Fourier component qu is described by θ = θF sin qu u
(qu = 2π/λ) .
(9)
The orientational director distribution function fF (θ ) due to the fluctuations, independent of the position along the u axis, is given by [6] fF (θ )dθ = C0 du , FIG. 3. The time dependence of the orientation angle θ for the major doublet (closed circles) and subdoublet (open circles). Solid and dotted curves show the best fits to the experimental values of the major doublets and subdoublets, respectively.
given by Eq. (4). We can see that there are three parameters involved: the relaxation time τON , together with the initial and final director orientations. From Eqs. (5) and (7) it is apparent that τON and θ∞ are determined by the macroscopic properties of the nematic phase, and these are expected to be the same for the nematic regions associated with the major doublet and subdoublet. This then leaves the initial director orientation as the factor responsible for the difference in the director dynamics. To test this notion we need to fit the theory, that is, Eq. (4), to the experimental data in Fig. 3, and for this we require values for τON and θ∞ . To do this we use the experimental values of the parameters ρ = 2.20, α = 89.7 deg, and γ1 = 122 mPa, as well as Eqs. (4)–(7); these give θ∞ as 89.4 deg and τON as 2.18 ms. Armed with these values, we have fitted Eq. (4) to the data in Fig. 3 by varying the values of the initial director orientation. For the major doublet we find that the initial orientation is 0 deg as expected, but for the subdoublet the optimum value of the initial director orientation is 0.65 deg. This small value would certainly not be directly discernible in the initial spectra (cf. Fig. 2). The predicted values for the time dependence of the director orientation for the major doublet and subdoublet are shown as the solid and dashed lines, respectively. The agreement with experiment is clearly impressive. This supports our view that there are two director distributions distributed around mainly two orientations at an initial state and during the relaxation process of the director rotation. The time evolution of these orientations results in the major director and subdirector, which follow the torque-balance equation. That is, we do not consider an effective rotational viscosity for low-molar-mass nematics as the origin of the nonuniform director distribution, which had been introduced to explain the faster director rotation than that predicted by the hydrodynamic theory for nematic polymers [16] and for normal nematics [17]. A key question resulting from this experiment and its explanation is “What is the origin of the initial orientation of the subdirector?” One possible cause for the generation
(10)
where C0 is a constant of integration. Integration of Eq. (10) leads to u+λ π/2 fF (θ )dθ = C0 du = 1 , (11) −π/2
u
and this yields C0 as 1/λ. Equations (9)–(11) give the distribution function due to the director fluctuation, irrespective of u, in the spatial range 0 u λ (−θF θ θF ) as fF (θ ) =
1
.
(12)
2π θF2 − θ 2
This function is dependent only on the amplitude of the modulation, but independent of the Fourier component, and tends to a δ function with decreasing value of θF . In Eq. (12) the fluctuation angle θF of 0.65 deg for the subdoublet is determined from the best fit, as shown in Fig. 3. On the other hand, the orientational director distribution function for the major doublet, which initially is distributed around the magnetic field (θ0 = 0 deg), is described by Eq. (8). Since the director fluctuates around any director orientations, the modulation due to the fluctuation should be convoluted with the major director orientations. It is clear from the difference in the spectral intensities for the major doublet and subdoublet as shown in Fig. 2(b) that the director densities for both doublets are different. Since an area of the spectrum is a weighted sum of each director orientation, the ratio Rt of the director density for the subdoublet to that for the major doublet is found to be 0.3. This is an average value estimated directly from the spectral line shapes for each doublet recorded at t = 11.0, 11.5, 12.0, and 12.5 ms in Fig. 2(b). For a given value of the parameter ζ , Eqs. (8) and (12) provide an initial director distribution, which gives the probability density for finding the director at a given orientation. Figure 4(a) shows the initial director distribution function calculated for the values of √ α = 89.7 deg, θF = 0.65 deg, θ0 = 0 deg, R = 0.3, and ζ = 350, where the parameter ζ in the director distribution function of Eq. (8) is selected by the best fit of the calculated spectra with those recorded experimentally. After the application of the electric field, each director with the initial distribution as shown in Fig. 4(a) starts to rotate but by keeping the probabilities for their initial orientations constant. Applying the same typical values as those used in the calculation of the director orientation [see
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FIG. 4. (Color online) (a) The initial director distribution function simulated for the √ values α = 89.7 deg, θF = 0.65 deg, θ0 = 0 deg, Rt = 0.3, and ζ = 350. (b) The simulated time evolution of the probability distribution function f (θ ), for the director orientation angle using Eqs. (4), (8), and (12), and the initial distribution (a) at a given time.
Eq. (4)], the probability distribution of the director orientation angle at a given time is calculated as shown in Fig. 4(b). The aligning electric torque acting on the director distributed around the magnetic field is small and so initially the director moves slowly. However, in regions where the director deviates from the magnetic field, the former moves more rapidly than those close to θ = 0 deg. As shown in Fig. 4(b), the motion of the director at different rates will broaden the director distribution with evolution of time to give an asymmetric distribution with a shoulder on the major peaks at t = 6−9 ms, and two peaks are clearly apparent at t = 10−13 ms. However, as the directors approach the electric field at around t = 18 ms, the distribution function becomes more uniform. From the time variation of the director distribution function shown in Fig. 4(b) and Eq. (2) the NMR spectra have been simulated, and these are shown by the solid curves in Figs. 2(a) and 2(b). The results of these simulations are found to be in good
semiquantitative agreement with experiment. Here we also make a brief comment on the volume element in Eq. (4). In principle we still need the two spherical-polar angles to define the director orientation. However, the physics of the system is such that the torques on the director from the electric and magnetic fields keep it in the EB that is the xz plane in the same coordinates as used for the simulation of the NMR spectra. This requires both the dielectric and diamagnetic anisotropies to be positive, as they are for our system. In other words, the azimuthal angle must remain zero during the field-induced alignment of the director. Since the director is confined to a plane containing the z axis, the volume element is simply dθ and so the volume element in the integral for the spectral line shape would not be sin θ dθ but dθ . This approach necessarily ignores director fluctuations, which in both situations would change the volume element. In our primitive model, however, the fluctuations have not been introduced into the torque-balance equations; indeed, to do this it would be necessary to argue that they could not be quenched by the E and B fields. As we have discussed before, it is expected that the broadening of the spectral line shape observed during the director relaxation process has its origins in the initial director distribution. In order to explore the origin of the subdoublets further, similar measurements to those performed at the shifted temperature of 16.6 °C in the turn-on experiments but with a new cell were carried out at the shifted temperatures of 15.3 °C, 19.5 °C, and 25.0 °C in the nematic phase. The N-I transition temperature of the 5CB-d2 sample was 34.5 °C, which differs slightly from that used in the previous experiments, namely 31.6 °C, and the film thickness was 202.0 ± 1.0 μm. The amplitude of the director modulation θF and the parameter ζ obtained from the best fit of the simulated 2 H NMR spectra to those measured experimentally are summarized in Table I. These values show the temperature dependences of θF and ζ , namely, that the director modulation amplitude decreases and ζ increases with decreasing temperature. Exploring further the thermal activation of both parameters, Fig. 5 gives the Arrhenius plots for θF (closed circles) and ζ (open circles). In this figure, the linear fits for both parameters clearly indicate the Arrhenius temperature dependences of θF and ζ , i.e., θF = θF0 exp(−Ea /RT ) and ζ = ζ0 exp(Ea /RT ), where Ea is the activation energy, R the gas constant, T the absolute temperature, and the pre-exponential factors, θF0 and ζ0 , which are both independent of temperature. The slopes of the two straight lines yield the activation energies of Ea = 39 kJ/mol for θF and Ea = 40 kJ/mol for ζ . Both values are essentially the same to within experimental error. This suggests that the
TABLE I. Temperature variation of the director fluctuation amplitude θF and the parameter determining the uniformity of the initial director distribution ζ . TNI − T (◦ C) 15.3 16.6 19.5 25.0
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θF (deg) 0.70 ± 0.15 0.65 ± 0.13 0.55 ± 0.10 0.40 ± 0.05
√
ζ
340 ± 5 350 ± 5 400 ± 5 450 ± 5
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FIG. 5. The Arrhenius plots for the amplitude of the director fluctuation θF and the parameter ζ . The slopes for the lines give the thermal activation energies Ea = 39 kJ/mol for θF and Ea = 40 kJ/mol for ζ .
spontaneous modulation of the director orientation and the initial director distribution are driven by the same thermally activated process. That is, during the director relaxation process the subdoublets are created by the modulation of the director orientation. Finally we discuss the reason why a remarkable nonuniform director distribution is observed for the system with nearorthogonal fields, but not for the system with α < 89 deg. In other words, why does the system with orthogonal fields enhance the modulation effect of the director fluctuation on its initial distribution? As described previously, the aligning electric torque acting on the director distributed around the magnetic field is small and so initially the director moves slowly. However, in regions where the director deviates slightly from being parallel to the magnetic field, it moves more rapidly than those close to θ0 = 0 deg [see Eq.(4)]. As we have found and shown in Table I, the director modulation angle is smaller than 0.7 deg in the nematic range measured by our experiments. It is clear from Eq. (5), however, that when the angle α between the magnetic and electric fields approaches 90 deg, the director relaxation time τON for the turn-on process increases. That is, the director moves more slowly and this amplifies the difference in the rates of the director motions distributing around θ0 = 0 deg and θ0 = θF . On the other hand, for the system with α < 89 deg, the difference in the rates of those director rotations becomes smaller and the director distribution results in a smaller linewidth. In addition to these directors’ motions, when the density of the director deviates from being parallel to the magnetic field increases, the director motions at different rates will enhance the broadening of the director distribution progressively with time. As a result, the broadening of the spectral line shape observed during the director relaxation process can be seen to originate from the initial director distribution, with its small peak intensities on either side of the major director distribution around the magnetic field. As expected, these peaks in the initial director distribution result from the director fluctuations. In addition, the modulation of the director fluctuation is a time-averaged quantity on the temporal scale of the time-resolved NMR
measurement. What is intriguing is that by applying this method with the system of orthogonal fields we are able to study directly the modulation of the director fluctuation. Our time-resolved NMR experiments have shown another surprising result, namely, the nonuniformity in the director distribution grows as the time taken to align the nematic with the magnetic field, prior to the application of the electric field, is reduced [1]. In other words, the longer the MPT, the more uniform the director distribution will be at the start of the turn-on experiment. The time-resolved NMR spectra observed in such experiments can be accounted for with the simple assumption that at the end of the magnetic preparation time the director is still not uniformly aligned, even though this time is several orders of magnitude larger than the magnetic relaxation time τM . For example, the MPT employed for the spectra recorded at TNI -T of for α = 89.7 deg [see Fig. 2(a)] was 2 s, and this large value was required to obtain the initial director distribution function necessary for reproducible results (see Fig. 6 of Ref. [1]). It would seem that this relaxation time is remarkably long in comparison with that needed to align the director distribution by the magnetic field, predicted by the torque-balance equation. Although this subject is beyond the scope of this paper, our result suggests that in addition to the time required for the director to relax to its equilibrium orientation by the external field, there is an another process with a significantly longer relaxation time needed to remove defects resulting from the thermally driven fluctuations of the director.
IV. CONCLUSION
In summary, our time-resolved NMR experiments have revealed a surprising result that the alignment of the director by essentially orthogonal fields can pass through a series of nonuniform states. This is especially true when the angle between the two fields approaches 90 deg. The spectra observed in such experiments can be accounted for with the simple assumption that at the end of the magnetic preparation time the director is still not quite uniformly aligned. As a consequence, when the two fields are almost orthogonal the different regions associated with the initial director distribution move at significantly different rates, thus enhancing the nonuniformity in the director distribution. Those regions are stimulated by the thermal director fluctuation. The study of anomalous director dynamics due to the director fluctuation gives further insight into the complex motion of the director, which may be useful in many liquid crystal applications where the director’s motion plays an important role but is usually assumed to be simple.
ACKNOWLEDGMENTS
This work was supported by a Grant-in-Aid for Scientific Research (B-24360127) of the Japan Society for the Promotion of Science (JSPS). Rauzah Hashim would like to thank the High Impact Research MoE Grant UM.C/625/1/HIR/MoE/5 from the Ministry of Education Malaysia, which funded travel during the development of the manuscript.
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[1] A. Sugimura and G. R. Luckhurst, Liquid-crystalline samples: Static and dynamic measurements of director alignment, eMagRes 2, 307 (2013). [2] A. Sugimura and G. R. Luckhurst, Deuterium NMR studies of static and dynamic director alignment for low molar mass nematics, in Nuclear Magnetic Resonance Spectroscopy of Liquid Crystals, edited by R. Y. Dong (World Scientific & Imperial College Press, Singapore, 2009), Chap. 10. [3] C. J. Dunn, G. R. Luckhurst, T. Miyamoto, H. Naito, A. Sugimura, and B. A. Timimi, A deuterium nucler magnetic resonance investigation of field induced director dynamics in a nematic slab subject to magnetic and pulsed electric fields, Mol. Cryst. Liq. Cryst. 347, 167 (2000). [4] G. R. Luckhurst, T. Miyamoto, A. Sugimura, T. Takashiro, and B. A. Timimi, The surface-induced static director distribution in thin nematic liquid crystal films: A deuterium nuclear magnetic resonance spectroscopy study, J. Chem. Phys. 114, 10493 (2001). [5] G. R. Luckhurst, T. Miyamoto, A. Sugimura, and B. A. Timimi, Director reorientation processes in a monodomain thin nematic liquid crystal film: A deuterium NMR spectroscopy study, J. Chem. Phys. 117, 5899 (2002). [6] D. Hamasuna, G. R. Luckhurst, A. Sugimura, B. A. Timimi, and H. Zimmermann, Director alignment by crossed electric and magnetic fields: A deuterium NMR study, Phys. Rev. E 84, 011705 (2011). [7] A. V´eron, A. Sugimura, G. R. Luckhurst, and A. F. Martins, Properties of the static NMR response of a confined thin nematic film of 5CB-d2 under crossed electric and magnetic fields: Theory and experiments, Phys. Rev. E 86, 051708 (2012).
[8] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University Press, Oxford, 1974), p. 101. [9] A. F. Martins, P. Esnault, and F. Volino, Measurement of the Viscoelastic Coefficients of Main-Chain Nematic Polymers by an NMR Technique, Phys. Rev. Lett. 57, 1745 (1986). [10] G. W. Gray and A. Mosley, The Raman Spectra of 4-Cyano4’-pentylbiphenyl and 4-Cyano-4’-pentyl-d11 -biphenyl, Mol. Cryst. Liq. Cryst. 35, 71 (1976). [11] H. Zimmermann, Specifically deuteriated intermediates for the synthesis of liquid crystals and liquid-crystalline polymers, Liq. Cryst. 4, 591 (1989). [12] A. Sugimura, T. Miyamoto, M. Tsuji, and M. Kuze, Determination of the unified-anchoring energy of a nematic liquid crystal on a polymer substrate, Appl. Phys. Lett. 72, 329 (1998). [13] S. M. Fan, G. R. Luckhurst, and S. J. Picken, A deuterium NMR investigation of local orientation and director kinetics in aramid solutions, J. Chem. Phys. 101, 3255 (1994). [14] G. R. Luckhurst, B. A. Timimi, T. Miyamoto, and A. Sugimura, The alignment of the smectic A phase of 4-octyl4’-cyanobiphenyl induced by an electric field. A time-resolved deuterium NMR study, Mol. Cryst. Liq. Cryst. 402, 103 (2003). [15] D. Hamasuna, G. R. Luckhurst, A. Sugimura, B. A. Timimi, K. Usami, and H. Zimmermann, Deuterium NMR spectra of a monodomain nematic: Angular dependence of the linewidths, Thin Solid Films 517, 1394 (2008). [16] See, for example, I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals (Taylor & Francis, London, 2004), p. 133. [17] A. Sugimura and A. V. Zakharov, Field-induced periodic distortions in a nematic liquid crystal: Deuterium NMR study and theoretical analysis, Phys. Rev. E 84, 021703 (2011).
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Macroscopic order in a nematic liquid crystal: Perturbation by spontaneous director fluctuations Daichi Hamasuna,1 Rauzah Hashim,2 Atsuhiro Kasatani,1 Geoffrey R. Luckhurst,3 Akihiko Sugimura,1,* Bakir A. Timimi,3 and Herbert Zimmermann4 1
School of Information Systems Engineering, Osaka Sangyo University, 3-1-1 Nakagaito, Daito-Shi, Osaka 574-4853, Japan 2 School of Chemistry, University of Malaya, 50603 Kuala Lumpur, Malaysia 3 Department of Chemistry, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom 4 Department of Biophysics, Max-Planck-Institut f¨ur Medizinische Forschung, Jahnstrasse 29, D-69120 Heidelberg, Germany (Received 18 February 2014; published 4 June 2015) The dynamic alignment of the nematic director by near-orthogonal electric and magnetic fields has been investigated. The intermediate states during the relaxation process were found, with the aid of time-resolved deuterium NMR spectroscopy, to be markedly nonuniform. The macroscopic order was perturbed, although the initial and final states of the director appear to be essentially uniform. However, the initial state does have a profound influence on the uniformity of the director in the intermediate states. We have developed a fundamental model based on the effect of spontaneous director fluctuations to explain these unusual NMR observations. DOI: 10.1103/PhysRevE.91.062502
PACS number(s): 61.30.Hn, 76.60.−k
I. INTRODUCTION
Technological applications of liquid crystals, such as flatscreen display devices, depend on the anisotropic nature of these materials, commonly defined by the presence of a single director. This can be manipulated by external factors, including homogeneous electric and magnetic as well as inhomogeneous surface fields. It is therefore of prime importance to understand the alignment of the nematic director by either a magnetic and/or an electric field by careful evaluation of the hydrodynamic processes under both equilibrium and nonequilibrium conditions [1]. To complement these experiments, many theoretical models, based on continuum theory, have been developed that successfully describe the static and dynamic phenomena. Such macroscopic behavior has been investigated using deuterium nuclear magnetic resonance (2 H NMR) spectroscopy. Here the director is aligned by the magnetic field of the spectrometer, and the application of an electric field produces a nonequilibrium state, resulting in rotation of the director [1,2]. This powerful technique has proved to be especially important for the investigation of director alignment in liquid crystals [3–7]. The spontaneous fluctuations of the director alignment in a nematic phase are also well known, and as we shall see, are important in the field-induced director alignment. Their characteristic features have previously been studied with optical scattering measurements [8]. In this paper we describe the anomalous director distribution, observed and subsequently predicted, during the director reorientation in a nematic liquid crystal film subject to near-orthogonal magnetic and electric fields. Our studies are confined to low-molar-mass systems, although analogous behavior might be expected for polymeric nematics [9]. Time-resolved 2 H NMR spectroscopy has been employed to investigate the homogeneous field-induced director dynamics. This technique has the advantage of being able to explore the director distribution during the electric-field-induced rotation of the director [5]. This unique capability results from the design and construction of equipment to produce and control
*
[email protected]
1539-3755/2015/91(6)/062502(8)
the electric field in a manner precisely synchronized with the radio frequency pulses of the NMR spectrometer. Analysis of the experimental results is based on the predictions of a torque-balance equation for the homogeneous director and a simple model of the director fluctuations. The layout of this paper is as follows. In the next section we describe the time-resolved 2 H NMR experiments. The results for the dynamic behavior of the average director orientation and distribution are given in Sec. III. They are also discussed in terms of a torque-balance equation employed to calculate the time variation of the director orientation and, hence, its distribution. In addition, the factors that can influence the deviation of the director from uniform alignment are explored. Our conclusions are outlined in Sec. IV. II. EXPERIMENTAL
The nematic liquid crystal used in our experiments was 4-(1,1-d2 - pentyl)-4 -cyanobiphenyl 5CB-d2 , in which the pentyl chain is specifically deuterated at the 1 position [10,11]. The nematic-isotropic transition temperature TNI for this sample is 31.6 °C. The liquid crystal was contained in a 194.7 ± 1.0-μm-thick cell made of two transparent In2 O3 - coated glass plates. The cell construction, thickness measurement, and its filling have been described previously [4,5]. The In2 O3 surfaces of the cell were untreated and the anchoring strength was found to be small [12]; consequently, the surface interaction could not compete with the director alignment by the magnetic field of the NMR spectrometer and the electric field. In the experiments an ac electric field of 4 kHz was used. This frequency is sufficient to overcome the alignment effects of ionic conduction. The sinusoidal electric field was generated using a function generator, Wave Factory WF1943, and a high-power amplifier, model 4005, both from NF Electronic Instruments. The angle α between the electric and magnetic fields for our experiments was chosen to be close to 90 deg, that is, the initial angle between the director and the aligning electric field. To achieve this, the cell was first inserted in the NMR coil so that the glass plate surfaces were very close to being parallel to the magnetic field. Their orientation was determined from the quadrupolar splitting described in
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200 V, and for our cell of thickness 194.7μm, this corresponds to an electric field of 1.027 MV m−1 , which is higher than the threshold field required for a magnetic field of 7.05 T. The value of EFD was fixed at 40 ms, which was sufficiently long to align the director uniformly and to reach its equilibrium orientation θ∞ at the end of this time. The value of MPT was fixed at 2 s, which is long in comparison with the magnetic relaxation time τM . To obtain an NMR spectrum with a good signal-to-noise ratio, this sequence has to be repeated many tens of thousands of times. The quality of the control of the various pulses therefore needs to be very high to ensure that the spectral lines are not broadened. The general scheme we have employed in performing the time-resolved turn-on NMR experiments is described in detail in Ref. [5]. III. RESULTS AND DISCUSSION
FIG. 1. A schematic diagram showing the pulse sequence for the turn-on dynamics NMR experiment. EFD is the time for which the electric field is on; ED is the time at which the acquisition of the free-induction decay is initiated; RD is the relaxation time following the acquisition of the transient; and MPT is the magnetic preparation time for the director alignment by the magnetic field after the electric field is switched off and before the next trigger pulse which turns it on again.
Sec. III. Then the final adjustment of the orientation of the cell was made in 0.1-deg steps with the aid of a goniometer driven by an ultrasonic motor. The details of the time lines of the electric field and the NMR pulses are shown in Fig. 1. The 2 H NMR spectra were measured via a quadrupolar echo sequence, and for this the 90-deg pulses used were of 5.25-μs duration and the receiver dead time tD was a few microseconds. In most of the experiments reported here, the value of the acquisition time tac for the free-induction decay (FID) was 20.5 ms. We now turn to the time sequence for the application of the electric field which, in combination with the constant magnetic field of 7.05 T, controls the director alignment from nonequilibrium to equilibrium states. Prior to the application of the electric field to the nematic film, the director is at equilibrium parallel to the magnetic field, since the magnetic anisotropy is positive for nematic 5CB. Application of the electric field generates a nonequilibrium state because the dielectric anisotropy is positive. The director then moves from being parallel to the magnetic field to being essentially parallel to the electric field, at which point the system has returned to equilibrium. For this turn-on experiment the spectrum was acquired after a time called the external delay (ED), while the electric field is still on. The values of ED and the relaxation delay (RD) are related to the magnetic preparation time (MPT), which is the time when the electric field is zero and the director returns to being parallel to the magnetic field with the system again at equilibrium. These times are set and controlled via the NMR spectrometer. Experiments were performed with a constant electric field, and the time for which this is applied is called the electric field duration (EFD); during this the value of α was close to 90 deg. The voltage VRMS we have used was
The spectrum expected from a pair of equivalent deuterons when the director is uniformly aligned relative to the magnetic field is a single quadrupolar doublet with the splitting given by [13] ν(θ ) = ν0 (3cos2 θ − 1)/2 ,
(1)
where θ is the angle between the magnetic field and the director, and ν0 is the quadrupolar splitting when the director is parallel to the magnetic field. On the other hand, if the director is uniformly aligned at 90 deg to the magnetic field the splitting is half of ν0 . In contrast, when the director distribution is completely random in the plane formed by the electric and magnetic fields the spectrum will be a 2D powder pattern [14]. This contains both parallel and perpendicular spectral lines, with the intensity between these features originating from intermediate orientations of the director; by these terms we mean that the quadrupolar splittings are those expected when the director is parallel or perpendicular to the spectrometer’s magnetic field. The dotted curves in Fig. 2(a) show a series of time-resolved spectra for the turn-on alignment process recorded at a shifted temperature, TNI -T =, of 16.6 °C for α = 89.7 deg and the significantly long magnetic preparation time, MPT, of 2 s, which is longer than those used in earlier experiments where α was not quite as large [5]. In Figs. 2(a) and 2(b) the outer and inner pairs of dotted lines (used as guides to the eye) show the positions of the parallel and perpendicular spectral lines, respectively. It is quite clear from Fig. 2(a) that the director distribution remains predominantly uniform; the deviation from this uniformity is largely responsible for the line broadening, as is the appearance of weak parallel and, surprisingly, intermediate spectral lines. The extent of the line broadening increases as the average director orientation with respect to the magnetic field grows, and decreases when the director tends to be orthogonal to the magnetic field. At 6 ms after the electric field was switched on the spectrum is almost identical with its initial form [cf. the spectrum for 0 V in Fig. 2(a)]. There is, however, a small change in the quadrupolar splitting and a slight asymmetric broadening of the spectral lines. By 8 ms, after the application of the electric field, a doublet appears with a smaller splitting than that of the parallel doublet. At 9 ms the splitting of this second doublet becomes still smaller, indicating that the director is moving
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FIG. 2. (Color online) (a) A series of time-resolved 2 H NMR spectra (dotted curves) for a turn-on experiment at TNI -T =16.6 °C, E = 1.027 MV m−1 , MPT = 2 s, EFD = 40 ms, tac = 41 ms, showing the nonuniform director relaxation from being parallel to the magnetic field to being almost perpendicular to it. The spectral heights are scaled so that the spectra have equal areas.(b) Expanded spectra (dotted curves) recorded in the interval 10–13 ms, in which all spectral line shapes are normalized with the largest peak height in order to show them clearly with time. Solid and dashed scales show the splittings for the major doublets and subdoublets of individual spectral line shapes, respectively. In both (a) and (b) the solid curves show the simulated spectra.
further away from the parallel position. However, there is still a small part of the nematic for which the director is aligned close to the direction of the magnetic field. At 10 ms the director continues to move toward the electric field direction but with a small fraction still unmoved, close to the 0-deg orientation. Just 1 ms later the quadrupolar splitting has vanished, showing √that the director is aligned at the magic angle [θ = cos−1 (1/ 3)], although the single peak is broad. The weak parallel lines
again show that a very small fraction of the director continues to remain parallel to the magnetic field. At the same time, a weak but noticeable doublet, which we call the subdoublet, appears in the wings of the single broad peak, corresponding to an intermediate director orientation between the magic angle, √ cos−1 (1/ 3), and the electric field at α. In other words, during the dynamics investigation we effectively observe three sets of splitting patterns: the parallel and intermediate doublets, which are very weak, while the remaining strong doublet, called the major doublet, also changes with time. At 11.5 ms the parallel doublet is now barely detectable, but the major doublet and subdoublet have become more prominent. That is, the major fraction of the director is now oriented at an angle greater than the magic angle. After 12 ms the parallel lines are weak, while the fraction for the subdoublet has grown at the same rate as that for the major doublet. The quadrupolar splittings of the major doublet and subdoublet, at 12.5 ms, continue to grow, indicating the rotation of the director further from the parallel position. After 13 ms the weak, broad doublet that is the parallel lines have disappeared and the dominant spectral lines are sharper, indicating that the associated director is now aligned close to the electric field. After 20 ms the alignment of the director, essentially parallel to the electric field, is virtually complete. Before we start to understand the electric-field-induced behavior of the director dynamics, it is helpful to summarize the director behavior observed during the turn-on experiment. When the electric and magnetic fields are almost orthogonal, the director distribution starts and finishes in a uniform state, but between these two extremes the distribution is markedly nonuniform. In the basic theory, unconcerned with any spatial variation, we shall concentrate on the pure orientational distribution function for the director f (θ ), where θ is the angle between the director and the magnetic field. One reason for making this approximation is that the NMR spectrum is determined only by f (θ ) and not by f (r,θ ). Moreover, this approach is consistent with the Ockham’s razor principle; this states that the explanation of any phenomenon should make as few assumptions as possible, eliminating those that make no difference in the observable predictions of the explanatory hypothesis or theory. We proceed, therefore, with the minimum number of assumptions found to be consistent with our experimental results. When α is close to 90 deg, the angle made by the director with the aligning electric field, at the beginning of the turn-on experiment, is also close to 90 deg. At this orientation the aligning electric torque acting on the director is essentially zero and so initially the director is expected to move slowly, as observed. However, since the director distribution is not quite uniform, then there will be regions of the distribution function where the director orientation deviates from 90 deg and these are expected to move more rapidly than those close to 90 deg. The motion of the director at different rates will necessarily result in a broadening of the director distribution. But as such directors rotate to become parallel to the electric field, so the distribution function becomes more uniform. We need to evaluate to what extent this basic model for the director distribution function f (θ ) can provide a quantitative or at least semiquantitative account of the experimental, timeresolved NMR spectra. These can be simulated from f (θ )
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according to [15] I (ν) = G[ν,˜ν± (θ ),δG (θ )]f (θ ) sin θ dθ ,
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(2)
where G[ν,˜ν± (θ ),δG (θ )] denotes the normalized shape of a spectral line centered at either of the angular-dependent resonance frequencies, ν˜ + (θ ) or ν˜ − (θ ), and with a linewidth δG (θ ). Here we make a brief comment on the volume element in Eq. (2). The definition of the laboratory axis system is that the z axis is parallel to the magnetic field and the y axis is taken to be perpendicular to the EB plane. In this xyz frame the director orientation is defined by the spherical-polar angles θ and ϕ; for convenience we take ϕ to be zero when the director is in the xz plane. The angle ϕ does not enter into Eq. (2), because the frequency and linewidths depend only on the polar angle θ ; this results from the uniaxial nature of the properties about the director, that is, the nematic phase is uniaxial. The distribution function for the director f (θ,ϕ) does, in general, depend on both spherical and polar angles. However, the dependence on ϕ can be removed from the general distribution to give the reduced distribution function f (θ ) that appears in the spectral line shape I (ν). The two distribution functions are related by f (θ ) = ∫ f (θ,ϕ)dϕ. That is, the volume element in Eq. (2) is sin θ dθ . Since the deuterons exhibit residual dipolar couplings with many protons in the molecule, the spectral line shape approximates to a Gaussian, that is, 1 [ν − ν˜ ± (θ )]2 , G[ν,˜ν± (θ ),δG (θ )] = √ exp − 2δG2 (θ ) 2π δG (θ ) (3) where the linewidth δG (θ ) is defined as half the distance between the points of maximum slope of the line. Of course, the angle θ is time dependent and should be denoted by θ (t), but for the moment we shall avoid this complication in the notation. To evaluate the time dependence of the director orientation during the period EFD when the electric field is applied, we need to know the director distribution at the end of the magnetic preparation time. Given the director distribution, we can then solve the torque-balance equation to obtain the director orientations at subsequent times [3], that is, θ (t) = θ∞ + tan−1 [tan(θ0 − θ∞ )exp(−t/τON )].
(4)
Here θ0 is the initial director orientation and τON is the director relaxation time for the turn-on process, τM τON = , (5) 2 ρ + 2ρ cos 2α + 1 where μ0 γ1 , τM = χB ˜ 2
2 E ˜ε UE , ρ= = μ0 ε0 UM B χ˜
ε0 ˜ε 2 χ˜ 2 E , UM = − B . UE = − 2 2μ0
(6)
In these equations γ1 is the rotational viscosity coefficient, χ˜ is the anisotropy in the diamagnetic susceptibility, ˜ε is the anisotropy in the dielectric permittivity, μ0 is the magnetic permeability, and ε0 is the dielectric permittivity of a vacuum. The parameter ρ is the ratio of the strengths of the electric and
magnetic torques and also their energies UE and UM , when the fields are parallel to the director. The limiting value of θ (t) when t tends to infinity is given by 1 + ρ cos 2α cos 2θ∞ = . 1 + 2ρ cos 2α + ρ 2
(7)
In using these results for the director orientation, each value of the initial angle θ0 has an associated weighting factor, that is, the probability distribution function f (θ ). As a consequence, the time dependence of the director orientation can be obtained numerically by treating the different values of θ0 as originating from nematic monodomains whose orientations are mutually independent. We are, therefore, ignoring the elastic coupling, at least until the comparison between experiment and theory shows that this approximation is inadequate. The starting point for the calculations is the weighting factor for the initial director orientations, and a formula for this was originally proposed by Fan et al. [13], although for a different purpose and written in a slightly simpler form, namely, f (θ ) =
ζ 3
2[ζ − (ζ − 1)cos2 θ] 2
.
(8)
This distribution is normalized in the range between −90 and +90 deg, and the parameter ζ controls the uniformity of the director distribution prior to the application of the electric field; when ζ is unity the director is randomly distributed and as ζ → ∞ so the distribution tends to the Dirac δ function. It is informative to examine the detailed line shapes recorded in the time interval 10–13 ms shown in Fig. 2(b). In these expanded spectra, in which all spectral line shapes are normalized with the height of the tallest peak in order to show the spectral line shape clearly with time, there are certainly three sets of quadrupolar splittings for a given time. The splittings for the subdoublet and the major doublet apparent in Fig. 2(b) are indicated by the dashed and solid scales, respectively. These splitting frequencies give the director orientations as a function of time [see Eq. (1)]. Figure 3 shows the time dependencies of the director orientation θ for the major doublet (closed circles) and the subdoublet (open circles). At the start of the turn-on experiment there are only results for the major doublet; however, just before the end we can see a noticeable difference in the splittings, with that for the subdoublet being larger. It remains to be seen what the origin of this difference is. However, it should be noted that the spectra when the director orientation θ is close to 0 or to 90 deg are particularly insensitive to the minor variations in the director distribution. To aid in understanding the difference in the time dependence of the director orientation found for the major doublet and the subdoublet, the theoretical description of director dynamics proves to be important. For a monodomain nematic confined between untreated glass surfaces, subject to both electric and magnetic fields as used in our experiments, it seems apparent that the surface-anchoring effect and the elastic deformation may be ignored. This simplifies the LeslieEricksen based theory (see, for example, Ref. [16]) for the time dependence of the orientation of the nematic director subject to both magnetic and electric fields. In fact, the time dependence of the director orientation with respect to the magnetic field is
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of the subdirector orientation is the thermal fluctuation of the director orientation. Here, we shall treat a special case in which the director fluctuates with a small amplitude θF on the u axis, which is along each director orientation. Since a Fourier mode is sufficiently slow with respect to the time scale of the NMR experiment, we consider a time-independent function for the director fluctuation. A simplified modulation with a wavelength λ of an individual Fourier component qu is described by θ = θF sin qu u
(qu = 2π/λ) .
(9)
The orientational director distribution function fF (θ ) due to the fluctuations, independent of the position along the u axis, is given by [6] fF (θ )dθ = C0 du , FIG. 3. The time dependence of the orientation angle θ for the major doublet (closed circles) and subdoublet (open circles). Solid and dotted curves show the best fits to the experimental values of the major doublets and subdoublets, respectively.
given by Eq. (4). We can see that there are three parameters involved: the relaxation time τON , together with the initial and final director orientations. From Eqs. (5) and (7) it is apparent that τON and θ∞ are determined by the macroscopic properties of the nematic phase, and these are expected to be the same for the nematic regions associated with the major doublet and subdoublet. This then leaves the initial director orientation as the factor responsible for the difference in the director dynamics. To test this notion we need to fit the theory, that is, Eq. (4), to the experimental data in Fig. 3, and for this we require values for τON and θ∞ . To do this we use the experimental values of the parameters ρ = 2.20, α = 89.7 deg, and γ1 = 122 mPa, as well as Eqs. (4)–(7); these give θ∞ as 89.4 deg and τON as 2.18 ms. Armed with these values, we have fitted Eq. (4) to the data in Fig. 3 by varying the values of the initial director orientation. For the major doublet we find that the initial orientation is 0 deg as expected, but for the subdoublet the optimum value of the initial director orientation is 0.65 deg. This small value would certainly not be directly discernible in the initial spectra (cf. Fig. 2). The predicted values for the time dependence of the director orientation for the major doublet and subdoublet are shown as the solid and dashed lines, respectively. The agreement with experiment is clearly impressive. This supports our view that there are two director distributions distributed around mainly two orientations at an initial state and during the relaxation process of the director rotation. The time evolution of these orientations results in the major director and subdirector, which follow the torque-balance equation. That is, we do not consider an effective rotational viscosity for low-molar-mass nematics as the origin of the nonuniform director distribution, which had been introduced to explain the faster director rotation than that predicted by the hydrodynamic theory for nematic polymers [16] and for normal nematics [17]. A key question resulting from this experiment and its explanation is “What is the origin of the initial orientation of the subdirector?” One possible cause for the generation
(10)
where C0 is a constant of integration. Integration of Eq. (10) leads to u+λ π/2 fF (θ )dθ = C0 du = 1 , (11) −π/2
u
and this yields C0 as 1/λ. Equations (9)–(11) give the distribution function due to the director fluctuation, irrespective of u, in the spatial range 0 u λ (−θF θ θF ) as fF (θ ) =
1
.
(12)
2π θF2 − θ 2
This function is dependent only on the amplitude of the modulation, but independent of the Fourier component, and tends to a δ function with decreasing value of θF . In Eq. (12) the fluctuation angle θF of 0.65 deg for the subdoublet is determined from the best fit, as shown in Fig. 3. On the other hand, the orientational director distribution function for the major doublet, which initially is distributed around the magnetic field (θ0 = 0 deg), is described by Eq. (8). Since the director fluctuates around any director orientations, the modulation due to the fluctuation should be convoluted with the major director orientations. It is clear from the difference in the spectral intensities for the major doublet and subdoublet as shown in Fig. 2(b) that the director densities for both doublets are different. Since an area of the spectrum is a weighted sum of each director orientation, the ratio Rt of the director density for the subdoublet to that for the major doublet is found to be 0.3. This is an average value estimated directly from the spectral line shapes for each doublet recorded at t = 11.0, 11.5, 12.0, and 12.5 ms in Fig. 2(b). For a given value of the parameter ζ , Eqs. (8) and (12) provide an initial director distribution, which gives the probability density for finding the director at a given orientation. Figure 4(a) shows the initial director distribution function calculated for the values of √ α = 89.7 deg, θF = 0.65 deg, θ0 = 0 deg, R = 0.3, and ζ = 350, where the parameter ζ in the director distribution function of Eq. (8) is selected by the best fit of the calculated spectra with those recorded experimentally. After the application of the electric field, each director with the initial distribution as shown in Fig. 4(a) starts to rotate but by keeping the probabilities for their initial orientations constant. Applying the same typical values as those used in the calculation of the director orientation [see
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FIG. 4. (Color online) (a) The initial director distribution function simulated for the √ values α = 89.7 deg, θF = 0.65 deg, θ0 = 0 deg, Rt = 0.3, and ζ = 350. (b) The simulated time evolution of the probability distribution function f (θ ), for the director orientation angle using Eqs. (4), (8), and (12), and the initial distribution (a) at a given time.
Eq. (4)], the probability distribution of the director orientation angle at a given time is calculated as shown in Fig. 4(b). The aligning electric torque acting on the director distributed around the magnetic field is small and so initially the director moves slowly. However, in regions where the director deviates from the magnetic field, the former moves more rapidly than those close to θ = 0 deg. As shown in Fig. 4(b), the motion of the director at different rates will broaden the director distribution with evolution of time to give an asymmetric distribution with a shoulder on the major peaks at t = 6−9 ms, and two peaks are clearly apparent at t = 10−13 ms. However, as the directors approach the electric field at around t = 18 ms, the distribution function becomes more uniform. From the time variation of the director distribution function shown in Fig. 4(b) and Eq. (2) the NMR spectra have been simulated, and these are shown by the solid curves in Figs. 2(a) and 2(b). The results of these simulations are found to be in good
semiquantitative agreement with experiment. Here we also make a brief comment on the volume element in Eq. (4). In principle we still need the two spherical-polar angles to define the director orientation. However, the physics of the system is such that the torques on the director from the electric and magnetic fields keep it in the EB that is the xz plane in the same coordinates as used for the simulation of the NMR spectra. This requires both the dielectric and diamagnetic anisotropies to be positive, as they are for our system. In other words, the azimuthal angle must remain zero during the field-induced alignment of the director. Since the director is confined to a plane containing the z axis, the volume element is simply dθ and so the volume element in the integral for the spectral line shape would not be sin θ dθ but dθ . This approach necessarily ignores director fluctuations, which in both situations would change the volume element. In our primitive model, however, the fluctuations have not been introduced into the torque-balance equations; indeed, to do this it would be necessary to argue that they could not be quenched by the E and B fields. As we have discussed before, it is expected that the broadening of the spectral line shape observed during the director relaxation process has its origins in the initial director distribution. In order to explore the origin of the subdoublets further, similar measurements to those performed at the shifted temperature of 16.6 °C in the turn-on experiments but with a new cell were carried out at the shifted temperatures of 15.3 °C, 19.5 °C, and 25.0 °C in the nematic phase. The N-I transition temperature of the 5CB-d2 sample was 34.5 °C, which differs slightly from that used in the previous experiments, namely 31.6 °C, and the film thickness was 202.0 ± 1.0 μm. The amplitude of the director modulation θF and the parameter ζ obtained from the best fit of the simulated 2 H NMR spectra to those measured experimentally are summarized in Table I. These values show the temperature dependences of θF and ζ , namely, that the director modulation amplitude decreases and ζ increases with decreasing temperature. Exploring further the thermal activation of both parameters, Fig. 5 gives the Arrhenius plots for θF (closed circles) and ζ (open circles). In this figure, the linear fits for both parameters clearly indicate the Arrhenius temperature dependences of θF and ζ , i.e., θF = θF0 exp(−Ea /RT ) and ζ = ζ0 exp(Ea /RT ), where Ea is the activation energy, R the gas constant, T the absolute temperature, and the pre-exponential factors, θF0 and ζ0 , which are both independent of temperature. The slopes of the two straight lines yield the activation energies of Ea = 39 kJ/mol for θF and Ea = 40 kJ/mol for ζ . Both values are essentially the same to within experimental error. This suggests that the
TABLE I. Temperature variation of the director fluctuation amplitude θF and the parameter determining the uniformity of the initial director distribution ζ . TNI − T (◦ C) 15.3 16.6 19.5 25.0
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θF (deg) 0.70 ± 0.15 0.65 ± 0.13 0.55 ± 0.10 0.40 ± 0.05
√
ζ
340 ± 5 350 ± 5 400 ± 5 450 ± 5
MACROSCOPIC ORDER IN A NEMATIC LIQUID . . .
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FIG. 5. The Arrhenius plots for the amplitude of the director fluctuation θF and the parameter ζ . The slopes for the lines give the thermal activation energies Ea = 39 kJ/mol for θF and Ea = 40 kJ/mol for ζ .
spontaneous modulation of the director orientation and the initial director distribution are driven by the same thermally activated process. That is, during the director relaxation process the subdoublets are created by the modulation of the director orientation. Finally we discuss the reason why a remarkable nonuniform director distribution is observed for the system with nearorthogonal fields, but not for the system with α < 89 deg. In other words, why does the system with orthogonal fields enhance the modulation effect of the director fluctuation on its initial distribution? As described previously, the aligning electric torque acting on the director distributed around the magnetic field is small and so initially the director moves slowly. However, in regions where the director deviates slightly from being parallel to the magnetic field, it moves more rapidly than those close to θ0 = 0 deg [see Eq.(4)]. As we have found and shown in Table I, the director modulation angle is smaller than 0.7 deg in the nematic range measured by our experiments. It is clear from Eq. (5), however, that when the angle α between the magnetic and electric fields approaches 90 deg, the director relaxation time τON for the turn-on process increases. That is, the director moves more slowly and this amplifies the difference in the rates of the director motions distributing around θ0 = 0 deg and θ0 = θF . On the other hand, for the system with α < 89 deg, the difference in the rates of those director rotations becomes smaller and the director distribution results in a smaller linewidth. In addition to these directors’ motions, when the density of the director deviates from being parallel to the magnetic field increases, the director motions at different rates will enhance the broadening of the director distribution progressively with time. As a result, the broadening of the spectral line shape observed during the director relaxation process can be seen to originate from the initial director distribution, with its small peak intensities on either side of the major director distribution around the magnetic field. As expected, these peaks in the initial director distribution result from the director fluctuations. In addition, the modulation of the director fluctuation is a time-averaged quantity on the temporal scale of the time-resolved NMR
measurement. What is intriguing is that by applying this method with the system of orthogonal fields we are able to study directly the modulation of the director fluctuation. Our time-resolved NMR experiments have shown another surprising result, namely, the nonuniformity in the director distribution grows as the time taken to align the nematic with the magnetic field, prior to the application of the electric field, is reduced [1]. In other words, the longer the MPT, the more uniform the director distribution will be at the start of the turn-on experiment. The time-resolved NMR spectra observed in such experiments can be accounted for with the simple assumption that at the end of the magnetic preparation time the director is still not uniformly aligned, even though this time is several orders of magnitude larger than the magnetic relaxation time τM . For example, the MPT employed for the spectra recorded at TNI -T of for α = 89.7 deg [see Fig. 2(a)] was 2 s, and this large value was required to obtain the initial director distribution function necessary for reproducible results (see Fig. 6 of Ref. [1]). It would seem that this relaxation time is remarkably long in comparison with that needed to align the director distribution by the magnetic field, predicted by the torque-balance equation. Although this subject is beyond the scope of this paper, our result suggests that in addition to the time required for the director to relax to its equilibrium orientation by the external field, there is an another process with a significantly longer relaxation time needed to remove defects resulting from the thermally driven fluctuations of the director.
IV. CONCLUSION
In summary, our time-resolved NMR experiments have revealed a surprising result that the alignment of the director by essentially orthogonal fields can pass through a series of nonuniform states. This is especially true when the angle between the two fields approaches 90 deg. The spectra observed in such experiments can be accounted for with the simple assumption that at the end of the magnetic preparation time the director is still not quite uniformly aligned. As a consequence, when the two fields are almost orthogonal the different regions associated with the initial director distribution move at significantly different rates, thus enhancing the nonuniformity in the director distribution. Those regions are stimulated by the thermal director fluctuation. The study of anomalous director dynamics due to the director fluctuation gives further insight into the complex motion of the director, which may be useful in many liquid crystal applications where the director’s motion plays an important role but is usually assumed to be simple.
ACKNOWLEDGMENTS
This work was supported by a Grant-in-Aid for Scientific Research (B-24360127) of the Japan Society for the Promotion of Science (JSPS). Rauzah Hashim would like to thank the High Impact Research MoE Grant UM.C/625/1/HIR/MoE/5 from the Ministry of Education Malaysia, which funded travel during the development of the manuscript.
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