Fluctuation modes of a twist-bend nematic liquid crystal

We report a dynamic light scattering study of the fluctuation modes in a thermotropic liquid crystalline mixture of monomer and dimer compounds that exhibits the twist-bend nematic ($\mathrm{N_{TB}}$) phase. The results reveal a spectrum of overdamped fluctuations that includes two nonhydrodynamic and one hydrodynamic mode in the $\mathrm{N_{TB}}$ phase, and a single nonhydrodynamic plus two hydrodynamic modes (the usual nematic optic axis or director fluctuations) in the higher temperature, uniaxial nematic phase. The properties of these fluctuations and the conditions for their observation are comprehensively explained by a Landau-deGennes expansion of the free energy density in terms of heliconical director and helical polarization fields that characterize the $\mathrm{N_{TB}}$ structure, with the latter serving as the primary order parameter. A"coarse-graining"approximation simplifies the theoretical analysis, and enables us to demonstrate quantitative agreement between the calculated and experimentally determined temperature dependence of the mode relaxation rates.


I. INTRODUCTION
The twist-bend nematic (N TB ) phase is a fascinating new addition to the family of orientationally-ordered, liquid crystalline states of matter. It has been described as the "fifth nematic phase" [1], complementing the uniaxial, biaxial, chiral helical (cholesteric), and blue phase nematics. Originally proposed by Meyer [2], and later elaborated on theoretically by Dozov [3], the existence of the N TB phase was suggested experimentally [4] and subsequently confirmed [1,5] in low molecular weight liquid crystals (LCs) containing achiral dimers having an oddnumbered hydrocarbon linkage between the mesogenic ends. Interest in these materials was also inspired by simulation studies [6], which predicted a nematic-nematic transition in LC dimers with odd-numbered linkages.
The N TB state possesses some remarkable properties. First, the average molecular long axis (specified by a unit vectorn called the director) simultaneously bends and twists in space. In the case of LC dimers with odd linkage, the specific tendency to bend is presumably caused by an all-trans conformation of the molecules, which results in their having a pronounced bent shape. The addition of twist allows the bend to be uniform everywhere in space. The combination of bend and twist produces an oblique helicoidal (or heliconical) winding of the director (Fig. 1), with a cone angle β (angle between then and the helicoidal axis) of magnitude 10 • . This differs from an ordinary cholesteric LC phase, where a pure twist of n results in a right-angle helicoid (β = 90 • ).
Second, the helicoidal pitch in the N TB phase is on a molecular scale -i.e., on the order of 10 nm [1,5] -com- * Electronic address: ssprunt@kent.edu pared with cholesterics, where the supramolecular pitch typically exceeds 100 nm. The much larger pitch of a cholesteric may be attributed to the relative freedom of rotations around the long molecular axes, when the latter are orthogonal to the helical axis (β = 90 • ). This configuration mitigates the chiral part of intermolecular interactions [7]. By contrast, in the N TB state (with β < 90 • ), the bend-imposed hindrance of molecular rotations results in a much shorter, nanoscale modulation, which, however, remains purely orientational in naturei.e, there is no associated variation in mass density (no Bragg peak detected by X-ray scattering [1,4,5]).
Third, and again unlike a cholesteric, the component molecules of N TB -forming LCs are typically achiral. Thus, the chiral nature of the helicoidal structure is spontaneously generated, with degenerate domains of left-and right-handed helicity.
Finally, although the N TB phase shows no evidence of a macroscopic polarization, the flexoelectric effect [8] associated with spontaneous bending ofn and the recent observation of an electroclinic effect [9] in the N TB phase suggest that a short-pitch helical polarization field is tied to the heliconical director structure (see Fig. 1). A recent theory [10] describing the transition between uniaxial and twist-bend nematic phases invokes such a polarization field as the primary order parameter.
Despite the intense experimental and theoretical efforts to explore the N TB phase, the nature of collective fluctuation modes associated with the short-pitch helicoidal structure remains an open question. It is a vital one to address, since the spectrum and dispersion of these modes are closely related to the basic structural features and to the relevant order parameter(s), and because properties of the fluctuations provide an important test of theories describing the formation of the N TB state. Although previous dynamic light scattering measurements [11] revealed a softening of the elastic constant associated with bend distortions of the director above the N-N TB transition, they did not probe fluctuation modes specifically associated with the heliconical N T B structure. Here we report, to the best of our our knowledge, the first DLS study of fluctuations within the N TB phase and their critical behavior near the transition. Our measurements reveal a pair of strongly temperature-dependent nonhydrodynamic modes plus a single hydrodynamic mode in the N TB phase, and a single nonhydrodynamic mode and pair of hydrodynamic modes (the usual director modes of a uniaxial nematic) in the higher temperature nematic phase. We demonstrate excellent agreement between the behavior of the observed modes and new theoretical predictions based on a "coarse-grained" version of a Landaude Gennes free energy for the nematic to N TB transition [10].
The coarse-graining approximation, inspired in part by earlier theoretical work on cholesterics [12] and appropriate in the limit of helical pitch much shorter than an optical wavelength, treats surfaces of constant phase in the heliconical structure as "pseudo-layers." Within this approximation, which has been previously used to explain the effect of high magnetic fields on the N TB phase [13] and to account for its flow properties [14], the normal fluctuation modes involving the director may be mapped onto those of a chiral smectic-A phase, with effective layer spacing equal to the pitch, effective director parallel to the local pitch axis, and effective elastic constants that arise from the short-pitch orientational modulation rather than from a true mass density wave.
An alternative approach to coarse-graining the N TB phase has recently been published [15]. Our theory is generally consistent with that work, in that both theories describe the coarse-grained N TB phase as an effective chiral smectic-A phase, with elastic constants for layer compression and layer bending. The new aspects of our approach are that it describes nonhydrodynamic as well as hydrodynamic fluctuation modes, and it relates all of the modes to microscopic fluctuations of the polarization as well as the director field. Our experimental results agree well with this theoretical approach and, perhaps more significantly, support the centrality of a helical polarization field in describing the nematic to N TB transition -an aspect which fundamentally distinguishes the N TB phase from the other known nematic LC states, including, in particular, the cholesteric phase.
The body of this paper is organized as follows: In Sec. II, we provide essential details about the experimental setup and procedures, while Sec. III describes the key experimental results. Sec. IV presents a detailed discussion of a Landau theory for a N-N TB transition and the coarse-graining approach to calculate the normal fluctuation modes associated with the twist-bend structure. The theoretical predictions are compared to the experimental results in Sec. V, and Sec. VI summarizes our findings and offers some concluding remarks. showing heliconical directorn (with cone angle β and helical pitch t0) and helical polarization field P. Right: Frame of reference used to describe spatial variations of the average director or pitch axis,t, on length scales much longer than the pitch (see Theory section). The orthogonal unit vectorsê1 andê2 form a right-handed system witht. The xyz axes are fixed in the laboratory frame.

II. EXPERIMENTAL DETAILS
DLS measurements were performed on a 30/70 wt% mixture of the monomer and dimer compounds shown in Fig. 2 [16]. This mixture has the phase sequence isotropic → (uniaxial) nematic (N) → N TB → crystal in cooling, with N to N TB transition temperature, T T B = 94.2 • C (measured with a calibrated platinum RTD in our light scattering oven). The N TB phase in this system has been characterized by a variety of techniques [5]; for our purposes, its choice afforded the possibility to obtain high quality alignment of the average director (optic axis) in either homogeneous planar or homeotropic configurations -i.e., with averagen parallel or normal to the plane of the optical substrates, respectively -using thin (5 µm) cells with appropriate surface treatments.
Our DLS measurements utilized two depolarized scattering geometries -G1 and G2, depicted in Fig. 2 -in which homodyne time correlation functions of the depolarized scattered intensity of laser light (wavelength λ = 532 nm) are collected as a function of scattering vector q and temperature T .
In geometry G1 (Fig. 2), the average director is planaraligned and oriented perpendicular to the scattering plane. We set the wavevector k i of the incident beam to an angle θ i = 0 • (measured with respect to the substrate (left) and G2 (right) described in the text, with the average director (optic axis) in the sample cell indicated by the arrow pointing out of the page for G1 (homogeneous planar alignment with averagen normal to the scattering plane) or the downward arrow for G2 (homeotropic alignment with averagê n in the plane). The orientations of polarizer and analyzer are similarly indicated. Bottom: Chemical structure of the monomer and dimer compounds utilized for the present study. The 30/70 wt% mixture exhibits a N-NTB phase transition at 94.2 • C. normal) and varied the direction of wavevector k s of the scattered light (described by scattering angle θ s relative to the substrate normal). In the nematic phase, for large θ s , this geometry probes nearly pure splay fluctuations of the director with relaxation rate Γ n 1 ∼ q 2 . In geometry G2, the average director is parallel to the substrate normal (homeotropic alignment) and lies in the scattering plane; in this case, depolarized DLS in the nematic phase probes a combination of overdamped twist and bend fluctuations ofn -the hydrodynamic twistbend director mode, with relaxation rate Γ n 2 ∼ q 2 . The incident wavevector k i was fixed at θ i = 15 • or 35 • , while the direction of k s was varied between θ s = −10 • and 50 • , with respect to averagen. When θ s = 0 • , k s lies along n , and the scattering from director fluctuations is nominally extinguished ("dark director" geometry). This choice of θ s provides an opportunity to detect fluctuation modes that do not originate fromn and contribute to the dielectric tensor in their own right. Fig. 3 shows polarizing microscope images of a homeotropic sample of the mixture during the uniaxial nematic to twist-bend (N-N TB ) transition, with the lower left part of each picture corresponding to the N and the upper right part to the N TB phase. Fig. 3(a) confirms the high quality of the homeotropic alignment of the average director and its persistence across the N-N TB transition. In the N TB phase, the average value ofn is the pitch axist of the heliconical structure, which is oriented perpendicular to the substrates (image plane in the figure). Under an applied AC voltage (5 V @ 10 KHz), a second order Freedericsz transition (reorientation of the average director in the center of the sample) is observed in the N region, while the N TB region is unchanged, Figure 3(b). In the N TB region, the reorientation occurs at higher voltage (7 V @ 10 KHz, Fig. 3(c)), and in the form of propagating focal-conic domains (FCDs), such as is usually observed in smectic liquid crystals [17,18]. The "pseudo-layered" nature of the heliconical structure [13] is reflected in the gradual relaxation of the FCDs to homeotropic alignment after removal of the field. As Fig.  3(d) indicates, the slow relaxation rate and presence of FCDs are quite distinct from the behavior observed in the nematic phase. Fig. 4 displays representative normalized DLS correlation functions recorded in the nematic and twist-bend phases of 5 µm thick samples of the LC mixture for geometries G1 and G2. In the "splay" geometry (G1), a single overdamped fluctuation mode is detected in both N and N TB phases. By scanning θ s , we determined Γ n 1 ∼ q 2 with Γ n 1 /q 2 in the range 10 −11 − 10 −10 s −1 m 2 . Thus, splay fluctuations of the optic axis are hydrodynamic on both sides of the transition.

III. RESULTS
The spectrum and behavior of modes detected in geometry G2 are more interesting. In the nematic phase (above T T B ), two overdamped modes are observed in the range of θ s studied: the expected hydrodynamic twistbend director mode with relaxation rate Γ n 2 ∼ q 2 (see measured q 2 dependence in Fig. 5) of order ∼ 10 3 s −1 and Γ n 2 /q 2 10 −11 −10 −10 s −1 m 2 , and a faster, nonhydrodynamic mode (Fig. 5) with Γ p 2 10 5 s −1 and independent of q. (The meaning of superscript p will be clarified in the next section.) The relaxation rates of both modes were extracted from fits of the correlation data to double exponential decays.
The presence of the fast mode in the DLS correlation function is most evident in the "dark director" geometry where θ s = 0 • (see data labeled (c) in top panel of Fig. 4), although it contributes weakly for θ s = 0 • . However, even in the "dark" geometry where fluctuations inn do not contribute to the DLS to first order, we still observe the decay of the slow director mode with a significant spread in its relaxation rate. (The fit in this case used a stretched exponential, with one additional fitting parameter.) Alignment mosaicity and a consequent broadening of the scattered wavevector k s relative ton could produce a "leakage" of the slow director mode, but that does not account for the fact that no significant spread in Γ n 2 ∼ q 2 is observed for θ s off the "dark" condition. An alternative scenario based on an intrinsic coupling between the fast and slow fluctuations is argued in the Discussion section below.
In the N TB phase, the relaxation rates and q- dependence of the modes observed in geometry G2 change significantly. The twist-bend director mode, which dominates the scattering for θ s = 0, develops a large energy gap; its relaxation rate increases markedly below the transition (T = T T B ) to values in the 10 5 − 10 6 s −1 range, and, as evidenced in Fig. 5, becomes qindependent. Thus, below T T B , the twist-bend mode crosses over from a hydrodynamic to nonhydrodynamic mode. As we shall demonstrate in the Discussion section, the magnitude of the gap is consistent with a modulation ofn, whose period agrees with the FFTEM results [5] for the nanoscale periodic structure of the N TB phase. Since the effective director (or optic axis) is the pitch axist, for clarity we label its relaxation rate as Γ t 2 (replacing Γ n 2 ). Correlation data taken in the "dark director" geometry (G2 with θ s = 0 • ) in the N TB phase reveal a second, even faster nonhydrodynamic mode with a relaxation rate of 10 6 − 10 7 s −1 (see data labeled (c) in the bottom panel of Fig. 4), ∼ 10 times higher than the values of Γ p 2 for the fast mode in the nematic phase detected in the same geometry. Additionally, and again as in the nematic, a slow process -with relaxation rate comparable to that of a hydrodynamic director mode -also contributes to the correlation function.
In both phases, the total scattering intensity in the "dark" geometry, θ s = 0 • , is ∼ 10 times weaker than the intensity for neighboring angles θ s = ±10 • , where the twist-bend director mode couples to the dielectric tensor and dominates the scattering. Fig. 6 shows the temperature dependence of the relaxation rates for the two nonhydrodynamic modes (Γ t 2 and Γ p 2 ) in the N TB phase, and for the nonhydrodynamic mode (Γ p 2 ) and hydrodynamic director mode (Γ n 2 ) in the nematic phase (see figure inset). These results were obtained from analysis of correlation data taken at fixed θ i , θ s in geometry G2. The nonhydrodynamic modes FIG. 5: (Color online) Dependence of the relaxation rates of the fluctuation modes detected in geometry G2 on the magnitude of the scattering vector q. Circles and squares correspond to relaxation rates Γ n 2 and Γ p 2 of the hydrodynamic director and nonhydrodynamic polarization modes detected in scattering geometry G2 in the middle of the nematic phase (T − TT B = 25 • C). The slope of the line through the data on the log-log plot for Γ n 2 is 2, indicating Γ n 2 ∼ q 2 . Diamonds and triangles correspond to relaxation rate Γ t 2 of the nonhydrodynamic pitch axis fluctuations at temperatures T − TT B = −0.85 • C and −8.0 • C, respectively, in the NTB phase. These data are limited to higher q (or θs) due to a large component of background scattering at lower q, whose effect is exacerbated because of the low scattering intensity from fluctuations in the NTB phase in the G2 geometry.
clearly slow down significantly on approach to T T B from both sides of the transition, although on the low temperature side the present data are limited to temperatures futher than 1 • C-2 • C from the transition.
Finally, the temperature dependence of the inverse total scattered intensity (I −1 2 ), recorded in geometry G2, is plotted in Fig. 7. These data were taken at fixed θ i = 15 • , θ s = 40 • , where the dominant signal in the N TB phase comes from the nonhydrodynamic mode corresponding to Γ t 2 , and in the nematic phase from the hydrodynamic twst-bend director mode corresponding to Γ n 2 . As T → T T B from below, the decrease in I −1 2 mirrors the decrease in Γ t 2 (Fig. 6).

IV. THEORY
A successful model for the experimental fluctuation spectrum must account for: (1) the crossover from two hydrodynamic and one nonhydrodynamic mode in the nematic to one hydrodynamic and two nonhydrodynamic modes in the N TB phase; (2) the identity of the faster (nonhydrodynamic) mode detected in each phase; (3) the coupling of this fast process to slower director modes (evidenced in the data from the "dark director" geometry); and (4) the temperature dependence of the relaxation rates of the nonhydrodynamic modes. To this end, we require a model free energy density for the nematic to twist-bend transition that contains relevant hydrodynamic and nonhydrodynamic fields, and the appropriate coupling between them. Shamid et al [10] have recently analyzed the equilibrium behavior of such a model. The essential ingredient of their theory is a vector order parameter representing a polarization field P that originates, e.g., from the transverse dipole moment associated with the bent conformation of the dimer molecules that promotes the formation of the N TB phase. It is convenient to use a dimensionless form for the order parameter, p = P/P sat , where P sat corresponds to the saturated polarization at low temper- ature.
The free energy density expanded in terms of the fieldŝ n and p reads Here, K 1 , K 2 , and K 3 are the Frank elastic constants for splay, twist, and bend distortions of the directorn. The coefficient µ = µ 0 (T − T 0 ) is the temperature-dependent Landau coefficient for the polarization p (µ 0 being a constant), while ν > 0 is a higher-order, temperatureindependent Landau coefficient. The elastic constant κ penalizes spatial distortions in p, and the coefficient Λ couples p with bend distortions. The last term (not included in Ref. [10]), with η > 0, favors polarization perpendicular to the nematic director and is consistent with bend flexoelectricity. Because p is defined to be dimensionless, the Landau coefficients µ and ν carry the same units, and κ has the same units as the Frank constants.
In the N TB phase, the director field has the heliconical modulation n =ẑ cos β +x sin β cos(q 0 z) +ŷ sin β sin(q 0 z), (2) with pitch wavenumber q 0 and cone angle β. (Note that sin β was called a in Ref. [10].) Likewise, the polarization field has the helical modulation with magnitude p 0 , perpendicular ton and to the pitch axisẑ, as shown in Fig. 1 (left side). In the nematic phase, β and p 0 are both zero while q 0 is undefined; in the N TB phase, these quantities all become non-zero.
To find the ground state, we must insert Eqs. (2) and (3) into Eq. (1) for F N T B and then minimize with respect to q 0 , β, and p 0 . For this calculation, we repeat the work of Ref. [10] and generalize it to the case of weak polar elastic constant κ, which will turn out to be physically relevant. First, minimization with respect to q 0 gives and minimization with respect to β gives Equation (5) can be compared with the experiment of Ref. [5], which shows the cone angle β ∼ < 10 • within the temperature range covered by our DLS data. This result implies that p 0 (κ/K 2 ) 1/2 ∼ < 0.03. Because p 0 is a scaled polarization, which grows to order 1 at low temperature, we estimate that (κ/K 2 ) 1/2 0.03, which shows that the polarization elasticity is small compared with the Frank director elasticity.
Substituting Eqs. (4) and (5) into the free energy density and expanding for small p 0 and κ gives From this form of the effective free energy density, we can see that there is a second-order transition from the nematic to the N TB phase at the temperature This transition is unusual because the relative magnitudes of the cubic and quartic terms in Eq. (6) depends on the relative smallness of p 0 and κ. Close to the transition, where p 0 (Λ 2 κ 1/2 K 1/2 2 )/(K 2 3 ν), the cubic term dominates over the quartic term. By minimizing the effective free energy, we see that p 0 depends on temperature as This result is consistent with the scaling reported in Ref. [10], with a slight correction in the numerical co-efficient. By contrast, farther from the transition, where p 0 (Λ 2 κ 1/2 K 1/2 2 )/(K 2 3 ν), the quartic term dominates over the cubic term, and the prediction for p 0 becomes From the general form for p 0 , the crossover between these two regimes occurs at We will see below that the crossover point is extremely close to the transition, so that all of the experimental data are taken in the regime governed by Eq. (9) rather than Eq. (8).
As an aside, this theory can easily be modified to describe a first-order transition between the nematic and N TB phases, by changing the fourth-order coefficient ν to a negative value and adding a sixth-order term to F N T B in Eq. (1). We have not done so here, because the DLS data give no indication of a first-order transition. However, such a modification might be useful for analyzing the nematic-N TB transition in other systems. Now that we have determined the ground state, we will consider fluctuations about the ground state in the nematic and N TB phases.

A. Nematic phase
In the nematic phase, we must consider fluctuations in the director field about the ground staten =ẑ, and fluctuations in the polarization about the ground state p = 0. At lowest order, these fluctuations can be described by δn(r) = (n x , n y , 0) and δp(r) = (p x , p y , p z ). We insert these expressions into the free energy F N T B (Eq. (1)), and expand to quadratic order in the fluctuating components. We then Fourier transform from position r to wavevector q, and express the free energy as a quadratic form in n x (q), n y (q), p x (q), p y (q), and p z (q), By diagonalizing this quadratic form, we obtain five normal modes: (1) One hydrodynamic mode is primarily splay-bend director fluctuations, combined with some polarization fluctuations. Its relaxation rate is the ratio of the free energy eigenvalue to the relevant viscosity coefficient γ n , which gives in the limit of long wavelength (small q). Here, is the renormalized bend elastic constant [10], which shows the effect of coupling the director to the polarization. This effect accounts for the softening of bend fluctuations observed in earlier DLS studies of the director modes when T → T T B from the nematic side [19]. Specifically, Eqs. (13) and (7) imply K eff 3 = 0 at T = T T B .
(2) Another hydrodynamic mode is primarily twistbend director fluctuations, combined with some polarization fluctuations. Its relaxation rate is again with the renormalized bend elastic constant K eff 3 . (3, 4) Two nonhydrodynamic modes are mostly polarization fluctuations p x and p y , combined with some director fluctuations. In the limit of q → 0, these modes have relaxation rate (5) Another nonhydrodynamic mode is polarization p z by itself. In the limit of q → 0, it has relaxation rate Here, γ p and γ p are the mode viscosities.
Overall, we should emphasize the contrast between the nematic phase of the N TB -forming material studied here and a typical nematic phase. In the N TB -forming material, we observe a nonhydrodynamic mode with a relaxation rate that decreases with temperature, as the system approaches the transition to the N TB phase. The theory attributes this mode to polarization fluctuations, which become less energetically costly as the system develops incipient polar order. By contrast, in a typical nematic phase, no such mode can be observed in DLS experiments; presumably polarization fluctuations decay too rapidly to be detected.

B. Twist-bend phase
In the N TB phase, the analysis of normal modes is complicated because of the nonuniform, modulated director structure. However, as mentioned in the Introduction, we can simplify this calculation through a coarse-graining approximation, which averages over the director modulation to find the larger-scale properties of the phase. Such coarse graining has previously been done for the cholesteric phase [12], and it shows that the cholesteric has the same macroscopic elastic properties as a smectic phase. In this section, we generalize the coarse-graining procedure to the more complex case of the N TB phase. Indeed, it should be an even better approximation for the N TB than for the cholesteric phase, because the pitch of the N TB is so short.
The basic concept of the coarse-graining procedure is illustrated in Fig. 1. We suppose that the director field has a rapid heliconical modulation with respect to a local orthonormal reference frame (ê 1 (r),ê 2 (r),t(r)), and this orthonormal frame varies slowly in space. Furthermore, the heliconical modulation might be displaced upward or downward by a phase φ(r), which also varies slowly in space. Hence, the director field can be written aŝ n(r) =t(r) cos β +ê 1 (r) sin β cos(q 0 z + φ(r)) +ê 2 (r) sin β sin(q 0 z + φ(r)).
In this expression,t(r) is the coarse-grained director, which would be measured in any experiment that averages over the nanoscale heliconical modulation. By analogy with the director field, the polarization field has a rapid helical modulation with respect to the same local orthonormal reference frame, which can be written as p(r) =ê 1 (r)p 0 sin(q 0 z + φ(r)) −ê 2 (r)p 0 cos(q 0 z + φ(r)) + δp(r). (18) Here, δp(r) = δp xx + δp yŷ + δp zẑ is a fluctuating additional contribution to the polarization, which varies slowly in space. It is allowed because p is not restricted to be a unit vector. The contribution δp(r) is the coarsegrained polarization, which would be measured in any experiment that averages over the nanoscale helical mod-ulation.
From Eqs. (17)(18), we can see that the pseudo-layers are surfaces of constant q 0 z + φ(r) = q 0 (z − u(r)), where u(r) = −φ(r)/q 0 is the local pseudo-layer displacement. The local helical axis (or pseudo-layer normal) is given by the gradient We now consider the case of a well-aligned sample, as in a light-scattering experiment. In this case, the coarse-grained directort(r) has small fluctuations about z, while the phase φ(r) and coarse-grained polarization δp(r) have small fluctuations around 0. The full orthonormal reference frame can be written aŝ to quadratic order in t x (r) and t y (r). One might think that another variable would be needed to specify the vectorsê 1 andê 2 in the plane perpendicular tot. However, rotations in this plane can be included in the choice of the phase φ. As discussed in Ref. [12] for the cholesteric case, such rotations are analogous to gauge transformations. Hence, we make the specific choice of gauge in Eq. (20). With this choice, our orthonormal basis has small fluctuations away from (x,ŷ,ẑ).
We insert Eqs. (17)(18) for the director and polarization fields, together with Eq. (20) for the orthonormal basis, into Eq. (1) for the free energy of the N TB phase. We then make the coarse-graining approximation: We integrate over the rapid variations of cos q 0 z and sin q 0 z, assuming that the slowly varying fields are constant over the length scale of the pitch. We thus obtain an effective free energy in terms of the six coarse-grained variables φ(r), t x (r), t y (r), δp x (r), δp y (r), and δp z (r). We expand the free energy to quadratic order in these fields, and Fourier transform it from position r to wavevector q, to obtain Here, M(q) is a matrix of wavevector-dependent coefficients, which must be diagonalized to find the normal modes.
It is most convenient to understand the mode structure in the limit of q → 0. In this limit, the matrix simplifies to the block-diagonal form where From this block-diagonal form, we can extract the following six normal modes: (1) The phase φ = −u/q 0 is itself a normal mode. This mode is hydrodynamic, with zero energy (and zero relaxation rate) in the limit of q → 0. It is analogous to the layer displacement of a smectic-A phase, which costs zero energy for uniform displacement. It is also analogous to the hydrodynamic director mode in a cholesteric phase (which is called the pure twist mode in the theory of cholesteric light scattering [21]). It is visualized in terms of pseudo-layers in Fig. 8a,b. (2, 3) The coarse-grained director tilt t x and polarization δp y are coupled by the helicity of the N TB phase. Together, they form a pair of normal modes, both of which are non-hydrodynamic, with non-zero energy (and non-zero relaxation rate) in the limit of q → 0. In the limit of weak coupling, which is given by the criterion m 22 m 33 m 2 23 , Here, γ t and γ p are phenomenological viscosities associated with the normal modes. The two modes are analogous to tilt and polarization fluctuations in a chiral smectic-A phase. The tilt mode is also analogous to the non-hydrodynamic director mode in a cholesteric phase (which is called the umbrella mode in the theory of cholesteric light scattering [21]). The tilt mode is visualized in Fig. 8c; the polarization mode is not visualized.
A coupling between tilt and polarization (even if weak -i.e., small m 23 ) has an important physical significance. If an electric field is applied in the y-direction, it induces a polarization δp y . Because of the coupling, it must also induce a tilt t x . Hence, the N TB phase has an electroclinic effect, analogous to a chiral smectic-A phase. The sign of the electroclinic effect depends on the sign of m 23 , which is controlled by the sign of the helicity q 0 . For that reason, domains of right-and left-handed helicity must have opposite electroclinic effects. In earlier work, a weak electroclinic effect was observed experimentally and mod-eled by a different theoretical method [9]. Here, we see that it is a consequence of the coarse-grained free energy.
(4, 5) The coarse-grained director tilt t y and polarization δp x form another pair of nonhydrodynamic normal modes, which is degenerate with the previous pair.
( 6) The polarization component δp z is itself a nonhydrodynamic normal mode. Its relaxation rate is where γ p is the viscosity of this mode. If the wavevector q is small but nonzero, the five nonhydrodynamic modes are only slightly changed. To model their relaxation rates, we can still use Eqs. (24) and (25) derived above. However, the hydrodynamic mode is more significantly changed. We can consider the cases of q parallel and perpendicular to the z-direction separately: For q in the z-direction, the hydrodynamic mode still involves the phase φ by itself, not coupled with any other coarse-grained degrees of freedom. This mode is visualized in Fig. 8d. It is a z-dependent rotation of the heliconical director fieldn(r), which does not change the coarse-grained directort. Equivalently, this mode can be regarded as a z-dependent displacement u = −φ/q 0 of the pseudo-layers, leading to alternating compression and dilation of the pseudo-layer structure. In the limit of long wavelength (small q), the free energy cost of this fluctuation is 1 2 Hence, the relaxation rate is Γ u (q z ) = 1 2 γ −1 u B eff q 2 z , where γ u is the relevant viscosity.
For q in the x-direction, the hydrodynamic normal mode is a linear combination of φ, t x , and δp y , as visualized in Fig. 8e. This mode is an x-dependent rotation of then(r), or equivalently an x-dependent displacement of the pseudo-layers, leading to curvature of the pseudolayer structure. This displacement is accompanied by a tilt of the coarse-grained director in the x-direction, so that the localt remains normal to the local pseudo-layers. If q is in any other direction in the (x, y) plane, the same description applies with the corresponding rotation. The free energy cost of this fluctuation is 1 2 to lowest order in small β. Hence, the relaxation rate is Γ u (q ⊥ ) = 1 2 γ −1 u K eff q 4 ⊥ . In both cases, the effective elasticity of the N TB phase is equivalent to a smectic-A phase, with B eff and K eff playing the roles of the elastic moduli for compression and bending of the smectic layers, respectively. In that way, our coarse-graining of the N TB phase is analogous to earlier work on coarse-graining of the cholesteric phase, which also has effective smectic elasticity [12]. Hydrodynamic mode with wavevector q = 0, with uniform rotation ofn(r) and hence uniform displacement of pseudo-layers; this mode has no energy cost with respect to the ground state. (c) Nonhydrodynamic tilt mode, with the coarse-grained directort (average ofn(r)) tilted with respect to pseudo-layer normal. (d) Hydrodynamic mode with q = qẑ, with z-dependent rotation ofn(r) and z-dependent displacement of pseudo-layers (leading to compression and dilation). (e) Hydrodynamic mode with q = qx, with x-dependent rotation of n(r) and x-dependent displacement of pseudo-layers (leading to curvature), accompanied by tilt so thatt remains normal to pseudo-layers.

V. DISCUSSION
We can now compare the calculated normal modes with the light scattering experiment.

A. Nematic phase
The fluctuating part of the dielectric tensor can be expressed in terms of the normal modes using the relation ij (r) = ∆ n n i n j + ∆ p sat p i p j where (i, j) = (x, y, z), ∆ n is the dielectric anisotropy associated with the orientational ordering ofn, and ∆ p sat is the saturated value of the dielectric anisotropy associated with the p ordering.
In geometry G1, with q z = 0, the fluctuations inn and p decouple. The former yield the usual pair of hydrodynamic director modes (n 1 , n 2 ), while the latter produce a doubly degenerate nonhydrodynamic mode associated with p x , p y , plus an independent nonhydrodynamic mode associated with p z . Assuming large coefficient η in Eq. (1), we can neglect p z . Since the incident polarization in geometry G1 is alongẑ, the relevant elements of ij for depolarized scattering are xz and yz . Assuming negligible p z , these elements are dominated by the director modes, and specifically in our experiment for large θ s , by the splay fluctuations in the normal mode n 1 . Therefore, in agreement with our experimental results for geometry G1, the model with large η predicts that the DLS correlation function is described by a single exponential decay (with relaxation rate Γ n 1 ), and that the contribution from nonhydrodynamic polarization fluctuations is not observable.
The situation is different in geometry G2, where q = q xx + q zẑ for scattering in the x-z plane. (The choice of x-z or y-z is arbitrary.) In depolarized DLS, with the incident light polarized alongŷ, we probe fluctuations zy and xy . From the former, we expect and observe the n 2 (twist-bend) hydrodynamic mode. The latter ( xy ) couples to nonhydrodynamic polarization fluctuations transverse to the nematic ordering axis, which contribute maximally to the DLS signal in the "dark director" limit of G2, where zy → 0 and xy dominates.
Since xy is quadratic in p fluctuations (Eq. (28)), DLS probes the higher-order time correlation function C(p x , p y ) = p * x (0)p * y (0)p x (τ )p y (τ ) (τ = delay time). Based on the normal mode structure of the free energy for q z = 0 and assuming the fluctuations are Gaussian random variables with zero mean, C(p x , p y ) can be reduced to p * x (0)p x (τ ) p * y (0)p y (τ ) . The normal modes are linear combinations of n 1 , p x and of n 2 , p y , yielding for K 1 ≈ K 2 a pair of nearly degenerate hydrodynamic modes and a pair of nearly degenerate nonhydrodynamic modes.
In the limit that the energy associated with p fluctuations is much greater than that of the p −n coupling, and that the latter is much greater than the elastic energy ofn fluctuations, the correlation function is a double exponential decay, as observed in our experiment [20], with the faster decay characterized by relaxation rate Γ p 2 ∼ (constant in q) for the p fluctuations, and the slower characterized by a rate Γ n 2 ∼ q 2 representing a mixture of director modes. This mixture could explain the broadening of the slower decay indicated by our data analysis.
Outside of the "dark director" geometry, the twistbend director scattering from zy , which is linear in n 2 , prevails, and the fast decay makes only a weak contribution to the DLS correlation function -again in agreement with the experiment. The relaxation rate of the director mode (Fig. 6) decreases as T → T T B from above, but only by a factor ∼ 1.6. This modest decrease remains consistent with the expected softening of bend fluctuations, K eff 3 → 0 as T → T T B [see Eqs. (11) and (7)], since Eq. (12) indicates that the relaxation rate Γ n 2 K 2 q 2 ⊥ /γ n for the condition q 2 ⊥ q 2 z in geometry G2. Thus, in the scattering geometry used, Γ n 2 is not very sensitive to the temperature dependence of K eff 3 .

B. Twist-bend phase
The spectrum of modes is related to fluctuations of the dielectric tensor through a modified version of Eq. (28), where ∆ n n i n j is replaced by ∆ t t i t j . The hydrodynamic mode is the extension into the N TB phase of the splay-bend director mode n 1 , which is observed in the nematic phase in geometry G1. The nonhydrodynamic tilt mode is the extension of the hydrodynamic twistbend director mode n 2 , which is observed in the nematic phase in geometry G2. It acquires a large energy gap when the heliconical structure forms, analogous to the gap in n 2 that develops in a smectic-A phase due to the large energy cost of tilting the director away from the layer normal. The coarse-grained model thus accounts for both the slow hydrodynamic mode (data labeled (a) in the bottom panel of Fig. 4) and the slower of the pair of nonhydrodynamic modes (data labeled (b) in the bottom panel of Fig. 4), which are observed in experimental geometries G1 and G2, respectively.
The faster nonhydrodynamic mode in the N TB phase is detected in the "dark director" limit of geometry G2 (see correlation data labeled (c) in the bottom panel of Fig. 4). As in the nematic case, it can be associated with fluctuations of the polarization (δp x , δp y ). Because the polarization fluctuations are only observed for a scattering geometry where thet fluctuations are "dark," the coupling between tilt and polarization fluctuations must be weak.
The coarse-grained theory predicts additional terms in the expression for the energy gap of these fluctuations in the N TB phase compared with the nematic phase [see Eq. (24b) compared with (15)]. These terms imply an increase in the relaxation rate Γ p of the polarization mode at T T B , which is consistent with the experimentally observed behavior (Fig. 6). According to the model, this increase in Γ p signals a transition to a heliconical structure with β = 0 and p 0 = 0.
The slow relaxation process that mixes with the fast polarization fluctuations in the correlation function is also explained by the theory: When q z and q ⊥ are nonzero, as is generally the case in the G2 geometry, δp x and δp y mix with the slow hydrodynamic variable φ and witĥ t, and thus the correlation function contains a slow component corresponding to undulation of the pseudo-layers and splay oft.
The final nonhydrodynamic mode predicted by the theory, related to δp z , has an even higher relaxation rate, which is not detected in our experiment. This high relaxation rate implies a relatively large value for the coefficient η in the free energy of Eq. (1).
To fit the experimental data for relaxation rates as functions of temperature, we combine Eq. (15) in the nematic phase and Eqs. (23-24) in the N TB phase. For the equilibrium cone angle β and pitch wavenumber q 0 , we use the leading terms in Eqs. (4)(5) near the secondorder transition, which give sin 2 β ≈ p 0 (κ/K 2 ) 1/2 and q 0 ≈ (Λ/K 3 )p 1/2 0 (K 2 /κ) 1/4 . For the equilibrium polarization p 0 , we use the approximation of Eq. (9), derived with the assumption of small polarization elasticity κ. The predicted relaxation rates then become We can compare Eqs. (29) directly with the data in Fig. 6.
In this comparison, we assume that the orientational viscosities γ t and γ p do not vary strongly with temperature. First, fitting Eq. (29a) to the data for Γ p in the nematic phase, we find µ 0 /γ p = 3600 s −1 K −1 and Λ 2 /(K 3 µ 0 ) = 30 K. The fit is shown as a solid line in Fig. 6 (bottom panel, T > T T B ).
Second, the data for Γ t in the N TB phase are consistent with the linear dependence in Eq. (29b). This consistency confirms that the experiment is in the regime where p 0 follows the the approximation of Eq. (9) rather than Eq. (8). The experimental slope corresponds to the combination of parameters Λ 2 (K 1 + K 2 )µ 0 /(2γ t K 2 3 ν) = 84000 s −1 K −1 . This fit is shown as a solid line in Fig. 6 (top panel, T < T T B ).
Third, the data for Γ p in the N TB phase can be fit to the expression in Eq. (29c), as shown by the solid line in Fig. 6 (bottom panel, T < T T B ). In this fit, we use the parameters µ 0 /γ p = 3600 s −1 K −1 and Λ 2 /(K 3 µ 0 ) = 30 K obtained from the analysis of Γ p in the nematic phase. The fit yields ηκ 1/2 (K 2 µ 0 ν) −1/2 = 1200 K 1/2 .
We now combine the last fit result with two estimates. From the argument after Eq. (5), we have (κ/K 2 ) 1/2 0.03. Furthermore, if we use Eq. (9) and take p 0 0.1 at T T B − T = 1 K, we find (µ 0 /ν) 1/2 0.1 K −1/2 . Together with the fit result, these estimates give η/µ 0 4.0 × 10 5 K. This large value indicates that the relaxation rate Γ p of longitudinal polarization fluctuations in Eqs. (16) and (25) is much larger than Γ p , and hence explains why those fluctuations are not observed in our experiment.
We may also verify two conditions on which our analysis is predicated: (1) that T T B − T > ∆T x = 9Λ 4 κK 2 /(4K 4 3 µ 0 ν) (see Eq. (10) and accompanying discussion above) over the temperature range of our data in the N TB phase, meaning Eq. (9) applies, and therefore Eq. (29b) is valid; and (2) that m 22 m 33 m 2 23 , which validates the decoupling approximation for the polarization and tilt modes, and hence the use of Eqs. (29a) and (29b) for their relaxation rates.
Finally, consider the data for the inverse scattering intensity I −1 2 in Fig. 7. These data were recorded in geometry G2 for θ i = 15 • , θ s = 40 • , where the scattering is dominated by optic axis fluctuations (i.e.,n ort). In each phase, I −1 2 is proportional to the free energy density of these fluctuations. On this basis, we can make two useful comparisons between experiment and theory: (1) Since I −1 2 ∝ γ t Γ t in the N TB phase, and since Γ t is essentially linear in T T B − T (Fig. 6), we expect and observe the same for I −1 2 (Fig. 7).
(2) In the nematic phase, the free energy density of director fluctuations is given by 1 2 K 2 q 2 ⊥ (from the Frank free energy with the experimental condition K 2 q 2 ⊥ K 3 q 2 z appropriate for geometry G2). In the N TB phase, the free energy density of coarse-grained director fluctuations is given by 1 2 (K 1 +K 2 )q 2 0 sin 2 β [from Eq. (24a) for γ t Γ t combined with the result p 0 ≈ (K 3 /Λ)q 0 sin β near the transition]. Hence, the ratio of scattering intensities in the two phases should be From Ref. [5], using relative values of the optical birefringence at T = T T B and T − T T B = −5 • C, we estimate β = 7.5 • . From the same reference, FFTEM textures show that the pitch is 2π/q 0 = 9.3 nm. In our experimental geometry, q ⊥ = 2π(sin θ i + sin θ s )/λ = 0.011 nm −1 . Combining these numbers gives By comparison, the experimental intensity ratio in Fig. 7 (between the nematic phase just above the transition and the N TB in the middle of its range, 5 • C below the transition) is approximately 60. This quantitative similarity gives additional support to the theory.

VI. CONCLUSION
Our DLS study of a twist-bend nematic liquid crystal demonstrates the presence of a pair of temperaturedependent, nonhydrodynamic fluctuation modes connected to the N TB structure. One of these modes is associated with twist-bend director fluctuations in the presence of a short-pitch heliconical modulation ofn, while the other is accounted for by fluctuations in a vector or-der parameter that corresponds to a helical polarization field coupled to the director modulation. The behavior of both modes, as well as the presence of a single hydrodynamic mode in the N TB phase (associated with splay fluctuations of the helical pitch axis), are quantitatively explained by a theoretical model based on two components: (1) a Landau-de Gennes free energy density, which is expanded in the director and polarization fields; and (2) A "coarse-graining" of this free energy that maps the heliconical structure onto a smectic-like system characterized by a "pseudo-layer" displacement field and an effective director normal to the layers. This model predicts one hydrodynamic and one non-hydrodynamic "layer"-director mode, and also reveals how the distortions of the pseudo-layers couple to fluctuations in the polarization field.
It will be interesting to test this mapping further -for example, by designing experiments to determine the magnitude of the effective elastic constant for layer compression as a function of heliconical pitch [22]. It could also be illuminating to probe the response of the polarization mode to an applied electric field. Finally, extending the Landau-deGennes theory to include a first-order N-N TB transition may prove useful for understanding experimental results on a wider range of dimers or monomer/dimer mixtures.