Influence of polymer stabilization on the dielectric relaxations of an antiferroelectric liquid crystal

Abstract

Dielectric spectroscopic investigations are reported on an antiferroelectric liquid crystal in its pristine form and when stabilized by a polymer network without and with application of a DC bias field. While the in-phase mode associated with the helix distortion is hardly affected, the high frequency mode connected with the antiphase fluctuations shows a large decrease with increasing concentration of the photoactive monomer used to generate the polymer stabilization. With the help of X-ray data, we rule out any inter-layer nanophase segregation of the polymer strands, and argue that this strong effect is due to the large elastic interaction between the liquid crystal molecules and the polymer strands. Employing expression from a Landau theory we extract several relevant parameters, including the viscosities associated with the different relaxation modes, the elastic constant, and importantly the antiferroelectric interaction, which exhibits a substantial decrease upon polymer stabilization.

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1. Introduction

Liquid crystals stabilized by a polymer network have attracted significant attention in recent times not only due to the restricted geometry imparted by the network matrix on the liquid crystalline properties, but owing to their potential applications such as electro-optic devices (see, e.g., ref. 1 and 2). The main attraction of polymer stabilizing a liquid crystal (LC) structure is that the polymer matrix protects the director configuration of the confining LC from distortions and mechanical shock. In these systems, the LC structure is stabilized with the help of a polymer network formed by in situ photo-crosslinking of a small concentration of photoreactive monomers. The network, which can be isotropic or anisotropic depending on the conditions of cross-linking process, provides virtual surfaces inside the LC cell and thus significantly influencing the behaviour of the LC substance. A number of studies reporting such influences exist in the literature, with most concerned with the nematic or the ferroelectric smectic C^* phase. For example, in the case of polymer stabilized nematic liquid crystals, Kossyrev et al. found that the volume stabilization created by the polymer bundles improves the switch-off dynamics of the device.³ In ferroelectric LCs confined in a polymer network (PSFLC), the Goldstone dielectric strength decreases and the relaxation frequency increases, but the soft mode parameters are not affected much.⁴ In contrast to the many studies on the nematic and ferroelectric LCs, the polymer stabilized antiferroelectric (PSAFLC) phase has not received much attention, although it was shown that polymer stabilization reduces the hysteresis and flattens the electro-optic response curve, features that are useful for gray scale devices.^5,6 Specifically, the effect of the polymer network on the dielectric properties does not appear to have been reported.

The liquid crystalline state exhibits a variety of dielectric relaxation processes, classified under the categories of molecular and collective mechanisms.^7,8 The molecular processes correspond to the independent motions of individual molecules, whereas the collective processes arise from their cooperative motion. While the connection between an experimentally observed dielectric relaxation and the physical mechanism proposed has been quite straight forward in the case of a large number of liquid crystal (LC) phases, the case of the chiral antiferroelectric smectic phase (SmC^*_A) is more complicated. For example, the tilted ferroelectric SmC^* exhibits two collective modes: the Goldstone mode corresponding to the fluctuation of the azimuthal angle (phase) ϕ around the helical axis, and the soft mode associated with the fluctuation of the polar (amplitude) angle θ of the two-component tilt order parameter. In the SmA phase, wherein the molecules are upright with respect to the layer plane, these two modes become degenerate leading to the existence of the soft mode only. In contrast, in the SmC^*_A phase, owing to alternation of the tilt direction in the neighbouring smectic layers, there can be two types of azimuthal angle fluctuations,⁹ the antiphase mode and the in-phase mode. The latter is similar to the Goldstone mode in the SmC^* phase, arising due to the unwinding of the helical structure of the phase. The antiphase mode, on the other hand, represents the motion of ϕ changing in the opposite sense in the adjacent layers. Thus, the in-phase mode preserves the antiferroelectric structure, and the antiphase mode results in a deviation from the equilibrium one. Apart from these modes, the SmC^*_A phase also exhibits two molecular processes associated with the rotation of the molecules about the long and short molecular axes.^9–23 The antiphase mode has especially attracted interest from both basic science and technological points of view. It is a collective process that occurs at very high frequency, and thus has the potential to create a photonic switch with nanosecond response times.²⁴

Here we present dielectric relaxation spectroscopy (DRS) measurements on a polymer stabilized system exhibiting a direct transition from the paraelectric SmA to the antiferroelectric SmC^*_A phase. The data demonstrate that the relaxation frequency of the antiphase mode of the bulk sample is strongly diminished upon polymer stabilization, while that of the in-phase mode is least affected, with the soft mode frequency slightly altered. We also present bias dependent measurements in the SmC^*_A phase. Employing expressions from the Landau model developed by Parry-Jones and Elston,²⁵ and its recent modification by Das et al.,²⁶ quantities such as the relevant elastic constant as well as the antiferroelectric dipolar ordering coefficient governing the relaxation parameters have been estimated. With the help of the electroclinic tilt angle measurements in the SmA phase, we have also determined the coupling coefficient for the interaction between the polymer strands and the LC molecules, and also find an estimate of the strand dimensions. We argue that the observed behaviour is due to the reduction of the antiferroelectric interaction between neighbouring layers arising due to the anchoring conditions at the LC–polymer interfaces.

2. Experimental

The antiferroelectric liquid crystal (AFLC) compound 4-(1-(trifluoromethylheptyloxy carbonyl)) phenyl 4′-octyloxybiphenyl 4-carboxylate (from Showa Shell) exhibiting the SmA–SmC^*_A transition was used as the host. The photo-polymerizable monomer used for achieving the polymer stabilization of the host is the diacrylated reactive compound RM82 (from E-Merck), which is also a liquid crystal, which helps in ensuring a homogeneous mixture of the two components. The chemical structures and transition temperatures of these two compounds are shown below.

Most of the experiments were conducted on the PSAFLC sample with a fixed concentration of X = 10 where X indicates the weight composition of RM82 in the mixture (unless stated otherwise PSAFLC sample means the X = 10 composite only). The polymerization of RM82 was facilitated by the addition of a small quantity (2%) of a photoinitiator (BME, Aldrich), and brought about by employing a low power UV lamp with a peak wavelength of 365 nm and intensity of 2 mW cm². For the dielectric measurements the samples were sandwiched between two indium tin oxide (ITO) coated glass plates having very low sheet resistance (<10 Ω sq⁻¹). Before assembling the cell, the glass plates were pre-treated with a polymer alignment layer and rubbed unidirectionally to achieve homogeneous planar alignment of the molecules. One of the relaxation processes investigated is in the MHz region. Therefore, to prevent the cell-relaxation arising due to the finite (although low) resistance of the ITO plates from affecting the sample signal, the electrode area was kept quite small (2 mm × 2 mm). The DRS measurements were carried out over a frequency range of 10² to 10⁷ Hz using an impedance analyzer (HP4194A). DC bias-dependent measurements were carried out using an LCR meter (HP4284A). Both types of measurements employed a low probing field of ∼90–170 mV μm⁻¹.

3. Results and discussion

3.1. Temperature-dependent relaxation parameters in the SmC^*_A phase

The raw DRS profiles of the frequency (f) dependence of real (ε′) and imaginary (ε′′) parts of the permittivity at a reduced temperature of T_c – 10 °C (where T_c is the SmA–SmC^*_A transition temperature) for pure AFLC and the polymer stabilized antiferroelectric LC (PSAFLC) sample are shown in Fig. 1(a) and (b), respectively. A feature common to both the samples is the presence of two relaxation processes in the SmC^*_A phase, a low frequency mode in the tens of kHz range, and another in the MHz region. In contrast, the SmA phase exhibits a single relaxation.

Fig. 1 Frequency dependence of ε′ and ε′′ in the SmC^*_A phase for the (a) pure AFLC (X = 0) and (b) the PSAFLC (X = 10) materials. The solid line through the points shows the best fit to eqn (1). The ε′′ profiles for the two resolved processes corresponding to the in-phase mode (IM) and the antiphase mode (AM) are shown separately as dashed lines.

To extract the relaxation frequency, the ε′′ vs. f profiles data were fit to the following expression

where ε*(f) is the complex permittivity at a frequency f. The first term on the right hand side is the high frequency permittivity which includes the dielectric strengths of all the high frequency modes other than the ones under consideration, the second term is the standard Havriliak–Negami (HN) equation,²⁷ with f_R as the relaxation frequency and Δε as the dielectric strength, α and β as the symmetric and asymmetric distribution of the profile shape for the mode. In the SmA phase only one HN term was used, whereas in the SmC^*_A phase, two such terms were needed. In eqn (1), the 3^rd term is included to account for the DC conductivity (σ_o) contribution to the imaginary part of the capacitance, and the last term is used to explain the cell relaxation time arising from the finite sheet resistance of the ITO-coated glass plates used.

Fig. 2(a) and (b) display the extracted profiles of the high frequency mode in the SmC^*_A phase and the only mode in SmA at a few representative temperatures for the pure LC and the PSAFLC composite, respectively. In the SmA phase the assignment of the mode, arising from the softening of the tilt fluctuations, is unambiguous arising from the softening of the tilt fluctuations and therefore we refer to it as the soft mode. The identification of the modes in the SmC^*_A phase requires additional inputs. Even on a qualitative level, it is seen that on cooling from the SmA phase the peak frequency of the profiles (a good measure of the relaxation frequency) initially increases with decreasing temperature, but reverses its trend and then has a relatively weaker dependence on temperature, features ruling it out to be labelled as the soft mode. Noting that (i) the relaxation of the higher frequency mode occurs in the MHz range, (ii) the relaxation is seen over the broad temperature range of the SmC^*_A phase, and (iii) the fact that the mode connected with the rotation around the molecular long axis occurs in the GHz range, we can unequivocally assign the observed relaxation to be due to the antiphase process (AM). (It is possible that in the vicinity of the transition, on the SmC^*_A side, there is a substantial overlap of the AM and the soft modes, which is difficult to be resolved.) The assignment of the lower frequency mode seen in the SmC^*_A phase under the planar orientation conditions employed, has seen much debate in the literature.^{9,25,26,28–30} The cause for this is that the molecular mode connected with the fluctuations about the short axis of the molecules, as well as the in-phase mode can have relaxation frequencies at comparable ranges. However, more recently (see, e.g., ref. 14 and references therein) it has been ascertained that this mode is the in-phase mode, as proposed by Buivydas et al.⁹

Fig. 2 Dielectric absorption profiles extracted by subtracting the contribution due to conductivity and ITO layer (terms 3 and 4 in eqn (1)) at a few temperatures in the SmC^*_A (high frequency mode) and SmA (soft mode) phases of the (a) pure LC and (b) PSAFLC samples, respectively. The temperatures are given with respect to the SmA–SmC^*_A transition temperature, with positive values indicating SmC^*_A and negative values indicating SmA phase. The peak position and magnitude have a monotonic behaviour with temperature in the SmA phase, a feature that is not true in the SmC^*_A phase.

Fig. 3 shows the thermal variation of f_AM, the relaxation frequency of the antiphase mode, for the pure (X = 0) and the PSAFLC sample. For the two materials, while the thermal behaviours are quite similar with an increase in the vicinity of T_c (the SmA–SmC^*_A transition temperature), and a decrease after showing a maximum within the SmC^*_A phase, the f_R values are strongly influenced by polymer stabilization. The inset of Fig. 3 depicts the value of f_AM at a relative temperature of T_c −10 °C for the pure LC and PSAFLC samples of four different concentrations, exhibiting nearly two fold reduction for the PSAFLC samples. The in-phase mode relaxation frequency, f_IM, on the other hand, exhibits monotonic temperature dependence for the pure LC as well as the composite (see Fig. 4). Further, the magnitude of f_IM for even the X = 10 composite is hardly different from that for the pure LC.

Fig. 3 Thermal variation of the relaxation frequency of the antiphase mode (f_AM) in the SmC^*_A phase (T − T_c < 0), and soft mode (f_s) in the SmA phase (T − T_c > 0), for the pure LC (X = 0) and the PSAFLC (X = 10) materials. (Here T_c is the SmA–SmC^*_A transition temperature.) The inset displays the concentration dependence of f_AM at a fixed reduced temperature of T_c −10 °C. The line through the data is merely a guide to the eye.

Fig. 4 Relaxation frequency of the in-phase mode (f_IM) versus reduced temperature for the pure AFLC (X = 0) and the PSAFLC (X = 10) materials, exhibiting practically no difference in the values over the entire temperature range of measurement.

3.2. Bias-dependent relaxation parameters in the SmC^*_A phase

Anticipating the comparison of the experimental findings with the predictions of the theoretical model, to be discussed below, we determined the influence of the DC bias field on the dielectric parameters. Fig. 5 shows the bias field dependence of ε′ measured at a frequency of 1 kHz at a reduced temperature of T_c −10 °C for the pure LC and the PSAFLC composite. The ground state value remaining constant up to a certain threshold field and increasing beyond that to reach a maximum (unwound antiferroelectric state) and then achieve a much lower limiting value at higher bias values, are features expected for the bias-dependent permittivity in the SmC^*_A phase.²⁵ For neither of the samples the maximum bias available (40 V) was sufficient to achieve the final ferroelectric state, albeit much thinner (3 μm) cells were used than in the DRS measurements (6 μm) described above. Although the gross features are the same for the two materials, there are essential differences. (i) The ground state (zero bias) value ε_0V, and ε_max, the maximum value, are comparable for the pure LC and the composite. (ii) Within the errors, the threshold field remains unchanged. (iii) For the composite the overall profile appears different with a much gradual rise in ε′ above the threshold field. The DRS profiles at a few representative DC bias (V_DC) values for the pure LC and PSAFLC composites are presented in Fig. 6(a) and (b). The bias field increases the strength (Δε) of both the in-phase and anti-phase modes, although to different extents; the changes are also dependent on the absence/presence of the network. Upon application of ∼3 V μm⁻¹ bias field Δε_IM gets doubled for the pure LC sample, but increases by ∼27% for the composite. The strength of the antiphase mode Δε_AM has a similar feature for the two materials, except that the changes are much smaller: 27% and 12% for the pure LC and composite, respectively. Concomitantly, the relaxation frequency of the in-phase mode has the opposite behaviour, changing by the same magnitude as the strength, but decreasing upon application of bias field. However, the relaxation frequency of AM is not affected by the application of DC bias. We now analyze these data using a Landau–Ginzburg model which describes the collective dielectric modes of the antiferroelectric liquid crystal as a function of frequency and an applied DC bias field.²⁵ We begin by giving a summary of the predictions of the model employed.

Fig. 5 The DC bias field dependence of ε′ in the SmC^*_A phase, at a fixed reduced temperature of T_c – 10 °C for the pure LC and X = 10 composites.

Fig. 6 DRS profiles for the pure LC (a) and the X = 10 composite (b) at DC bias fields indicated by numbers (in V μm⁻¹) against each data set. The bias field induced an increase in the dielectric strength of both the in-phase and anti-phase modes is more for the pure LC than the PSAFLC material. The solid line through the points shows the best fit to eqn (1).

3.3. Theoretical background

Employment of Landau analysis to theoretically predict the dielectric spectrum of the antiferroelectric liquid crystal has been carried out by a few groups.^23,25,26 Here we use the predictions made²⁵ by Parry-Jones and Elston (PJE). Their model is based on continuous order parameters ϕ_a and ϕ_b, defined as ϕ_a = (ϕ_e + ϕ_o)/2 and ϕ_b = (ϕ_e − ϕ_o)/2, with ϕ_e and ϕ_o being the azimuthal angles of the director in the even and odd layers, respectively. Variations of ϕ_a and ϕ_b are associated with the in-phase and anti-phase fluctuations. Specifically, ϕ_b = 0 points to a ferroelectric ordering, and ϕ_b = π/2 to an antiferroelectric one. The authors employed the following free energy expressionwhere E is the applied electric field, P_s is the spontaneous polarization, K is an elastic constant describing the elasticity associated with the helical structure, x is the distance along the helix with a pitch p and Γ is a dipolar ordering coefficient that, when positive, favours antiferroelectric ordering between adjacent layers. Das et al.²⁶ have recently used a slightly modified version of eqn (2) by considering an additional term for the coupling between E and the interlayer interaction for a pair of smectic layers with interaction strength of V_o. Ignoring the contribution due to V_o reduces the expressions derived by Das et al.²⁶ to those obtained in the PJE model. While deriving the expression for the different dielectric modes, the PJE model considers a general case for the applied field E taking it to be a sum of a DC bias field E_b and an oscillatory probing field E₀ exp iωt so that E = E_b + E₀ exp iωt (where ω = 2πf). The derived expressions for the dielectric strength Δε_R and the relaxation frequency f_R (R = IM, AM) are:

In-phase mode.

Anti-phase mode. here η_a and η_b are the viscosities of the modes. Apart from the quantities P_s and p which can be experimentally determined, the above expressions involve parameters such as K, Γ, η_a and η_b. Interestingly the PJE model also has additional predictions for the bias field dependence of the low frequency dielectric constant, involving specifically the parameter Γ. According to the model, the difference ε_0V − ε_max is related to the antiferroelectric interaction strength Γ as,

3.4. Comparison of the experimental data with the model predictions

We start by determining the antiferroelectric interaction strength Γ with the help of eqn (7). For this purpose we measured the spontaneous polarization P_s using the triangular wave method.³¹ At T_c −10 °C, the P_s values are 121 nC cm⁻² and 84 nC cm⁻² for the pure LC and X = 10 composite, yielding Γ values of 19 kJ m⁻³ and 9.7 kJ m⁻³, respectively; these values are comparable to those obtained for another antiferroelectric LC.²⁵ The present results indicate a significant reduction of antiferroelectric interaction in the composite.

Let us recall from the results presented above the fact that both the strength and the frequency of the in-phase mode are altered by the bias whereas only the strength of the AM is changed. From eqn (3)–(6) of the PJE model, it can be concluded that for such a behaviour the helix pitch value must be the parameter mainly responsible. This can be understood especially for the in-phase mode, since it is associated with the helix distortion, and application of bias does distort the helix. Thus, taking the ratio of the f_R values (for the in-phase mode) without and with bias, it appears that the DC bias changes the pitch by ∼47% for the pure LC, and by ∼16% for the composite.

To extract the parameters K, η_a and η_b using eqn (3)–(6), we take the value of the undisturbed pitch to be 1 μm, as reported in ref. 32 and assume it to be the same for both pure LC and the composite. From eqn (4) it is noticed that Δε_IM for the in-phase mode is zero in the absence of the bias field. However, in the experiments we observe that the in-phase relaxation is present even in the absence of bias field. Therefore, in eqn (4) we substitute, E_b by E_b + E_o. Thus according to eqn (4), to a first approximation, Δε_IM varies linearly as a function of E_b² with the slope being equal to , from which K can be calculated. In the model by Das et al.,²⁶ the inclusion of the smectic layer interaction represented by the parameter V_o, adds a term which is linear in E_b. Attempts to fit Δε_IM data for the combination of terms linear and quadratic in E_b, yielded very similar quality of the fitting. However, the coefficient of the linear term, which was an order of magnitude smaller than the quadratic term, has a very large error; these features were true for the pure LC as well as the composite. Therefore we assume that for the present system the smectic layer interaction can be safely neglected.

From the knowledge of Γ and K, eqn (3) and (5) can be used to calculate η_a and η_b. Fig. 7 depicts the Δε_IM variation as a function of the square of the DC bias field. The straight line fit to the data above the threshold field was used for the calculations mentioned above. Table 1 shows the different parameters determined for the pure LC and the X = 10 composite. Whereas the viscosity associated with the antiphase mode decreases slightly, that for the in-phase mode shows a larger increase for the network-system. The four-fold increase seen for the elastic constant K needs a comment. It may be recalled here that a factor of 5 increases in the splay elastic constant, and two-orders of magnitude enhancement in the bend elastic constant has been seen in a nematic confined in a gel matrix.³³ Increase of a similar magnitude has also been reported³⁴ in a ferroelectric liquid crystal confined in a polymer network. These results suggest that the interactions of the LC molecules with the walls of the polymer network perhaps increases the effective elastic constant. The possible reason for this could be the fact that it is more difficult to bring about bend/twist deformation of the polymer strands, than of the small LC molecules.

Fig. 7 The variation of Δε_IM as a function of the square of the DC bias field, shown on a semi logarithmic scale to emphasize the presence of a threshold field in both the X = 0 and X = 10 materials. The line through the data points represents fits to two different linear equations above and below the threshold fields. The inset shows the straight line fit to the data above the threshold used to extract the elastic constant K.

Table 1 Parameters extracted from the Landau analyses discussed in the text for the pure LC and the PSAFLC (X = 10) samples

Parameter	Pure LC	PSAFLC (X = 10)
Γ (kJ m^−3)	19 ± 1.4	9.7 ± 1.4
Ps (10^−5 C m^−2)	121 ± 4	84 ± 6
K (pN)	6.6 ± 0.3	19.3 ± 0.6
ηa (mPa s)	5.0 ± 0.2	12.6 ± 0.2
ηb (mPa s)	3.8 ± 0.2	2.7 ± 0.4
α (kJ m^−3 K^−1)	46.2 ± 2.7	8.8 ± 1.7
ηs (mPa s)	36.1 ± 2.7	9.6 ± 2.1

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3.5. Structural aspects

Now we look for possible structural reasons for the drastic reduction in f_AM when the host liquid crystal is polymer stabilized. Polymer stabilization actually involves creation of a network of polymer strands arising from the polymerization of the photoactive monomer. When the network is well established, the host LC molecules get trapped between the polymer strands. For a sufficient concentration of the polymer, such a trapping can yield confinement effects on the LC sample. In other words, the finite dimension between the enclosing polymer strands can restrict the extent of correlation of the phase structure of the LC material. In the present case, such a confinement hardly affects the relaxation frequency of the in-phase mode.

The fact that f_AM is a drastically lowered means a reduction in the antiferroelectric interaction between the neighbouring layers, as indeed is seen by the Γ value which is reduced by 50% for the PSAFLC composite. It must be recalled that the AM process mimics a situation of rotation of the molecules in the neighbouring layers in opposite azimuthal direction reflecting a scenario of reduced antiparallel correlation. Therefore a decrease in Γ and consequently in f_AM suggests that structurally there is less favour for the SmC^*_A phase. A possible structural reason for the reduction of antiferroelectric interaction could be that the polymer strands interleave between the layers diluting the inter-layer communication. Such an incursion of polymer strand between layers, referred to as nanophase segregation,³⁵ is known in polymer stabilized composites of small flexible monomers, such as HDDA, but not with rod-shaped mesogen-like monomers, such as RM82. For a further confirmation of the absence of nanophase segregation in the present system we performed X-ray measurements. The diffraction profiles in the low angle region obtained in the SmC^*_A phase (see Fig. 8) are identical for pure LC and PSAFLC materials, except for a small increase in the spacings. Particularly important is the fact that the ratio of the peak intensities of the second harmonic to that of the fundamental is essentially the same for the pure LC and the composite. The value of this ratio can be taken to indicate the nature of the mass density profile along the layering direction. The ratio would be quite high, if the mass density profile is like a square-wave, and zero for a perfect sinusoidal profile. Thus, the ratio being the same for the two materials points to the nature of the layer structure remaining unaltered for the polymer stabilized sample.

Fig. 8 XRD profiles in the low angle region of the SmC^*_A phase for the X = 0 and X = 10. The ratio of intensities of the second harmonic to the fundamental is essentially the same for the Pure LC and the composite, as seen in the inset.

In fact, reduction in f_AM has been reported²⁴ when a FLC is added to the host AFLC. With increasing concentration of the second component, f_AM decreases, and is argued to be due to the decrease in the elastic constant between the two adjacent layers. This case is straight-forward since the second component added stabilizes (destabilizes) the ferroelectric (antiferroelectric) phase, unlike the presently studied polymer-stabilized system, retaining the phase sequence of the host LC. To estimate the influence of the polymer network surface we consider the free energy density of the system near T_c with an additional term which regards³⁶ the polymer network to be exerting a field-like component on the local LC director. The additional part is a Hookean elastic coupling term, W_pθ² (in the small θ limit) with W_p being the elastic interaction coefficient. With a similar term Petit et al. used³⁴ a simple expression to calculate W_p, ΔT_c = T_c − T^′_c = W_p/α_L. Here T_c and T^′_c are the SmA–SmC^*_A transition temperatures for the pure AFLC and the PSAFLC samples, respectively, and α_L the usual Landau coefficient of the quadratic term (in θ) of the free energy. We obtained α_L from electroclinically induced tilt angle (δθ) measurements in the SmA, and employing the standard equation,³⁷ . Here χ is the high frequency susceptibility, ε_o, the permittivity of free space, E, the applied field, and C, the polarization (P)–θ bilinear coupling coefficient. For a fixed E value, and T − T_c (or T^′_c) = 1 °C, δθ reduces from 6.5° for AFLC to 5.1° for the PSAFLC sample suggesting perhaps that the polymer strands have a reduced favour for tilting of the molecules, a feature reflected in T^′_c being 5 °C lower (for X = 10) than T_c. From measurements of P and θ the value of C was computed to be 33 × 10⁶ V m⁻¹ and 6.7 × 10⁶ V m⁻¹ for X = 0 and X = 10 samples, respectively, these values are in the range obtained for the PSFLC case.³⁸ Using the slope of the temperature-dependent δθ, α_L was found to be a factor of 5 lower for the polymer stabilized sample (α_L = 46.2 ± 2.7 kJ m⁻³ K⁻¹ and 8.8 ± 1.7 kJ m⁻³ K⁻¹ for X = 0 and X = 10), which is at variance with that in ref. 39, reporting no difference in the α_L values for the FLC host and its polymer stabilized version. It must, however, be mentioned that the α_L values quoted in ref. 39 are for a system with only 2% of the photoactive monomer, whereas in the present system the concentration is 10%. It is thus possible that the polymer morphology changes with higher concentration of the monomer from a completely phase-separated bi-continuous structure to one which approaches a cross-linked network, the latter type of system is known to have lower α_L values.³⁴ Using the values of α_L and T_c–T^′_c, we calculated W_p to be 40.8 kJ m⁻³, a value which is a factor of 2 higher than for the PSFLC system,³⁸ even after considering a linear increase with polymer concentration, pointing to a better elastic interaction between the AFLC molecules and the polymer network in the present case.

3.6. Soft mode behaviour in the SmA phase

As mentioned already, the only collective relaxation mode in the chiral SmA phase is the soft mode originating due to the softening of the elastic constant governing the upright orientation of the director. The softening of the elastic constant results in a lowering of the associated relaxation frequency on approaching the transition. Fig. 9 shows such a behaviour for the pure LC and also the X = 10 composite. Although the absolute values are not very different between the two materials, the temperature dependence of the relaxation frequency, f_s, appears to be slightly altered by polymer stabilization. The thermal behaviour of the soft mode in the SmA phase is described by the following Landau expression³⁷Here, μ is the flexoelectric coefficient, q is the wavevector of the helix and η_s is the viscosity associated with the soft mode. By considering only the data in the vicinity of the transition (up to T_c + 3 °C) we have done the fitting to eqn (8). From the slope term of this equation the value η_s can be obtained, with the knowledge of α_L, Table 1 also lists the values of η_s thus obtained. Polymer stabilization is seen to lower the soft mode viscosity by a factor of ∼4, the cause for which is not clear to us.

Fig. 9 Temperature dependence of the soft mode relaxation frequency f_s, in the SmA phase for the pure LC and composite. Polymer stabilization lowers the slope with which f_s increases on moving away from T_c. The lines through the data are fits to eqn (8).

3.7. Molecular arrangement after polymer stabilization

Finally, we look at the possible director orientation when the polymer strands are present. The possibility that the polymer strands intersperse the smectic layers (Fig. 10a) is, as mentioned earlier, ruled out by the X-ray data. The RM82 monomers themselves exhibiting an LC phase have a strong tendency to stay within the layers avoiding nanophase segregation. This is especially true in the polymerized state, with the polymer strands running through the layers, giving rise to two other possibilities. The first of these requires an elastic interaction between the AFLC molecules and the polymeric ones strong enough to stabilize the alternate tilting of the AFLC molecules in the neighbouring layers even in the immediate vicinity of the polymer strands, or rather have the segment of the strand also adapt the same tilting direction as the AFLC molecules (Fig. 10b). This would result in an unaltered f_AM value, contrary to the experimental results. The last possibility considers three different scenarios for the orientation of the surface molecules (molecules in the vicinity of the polymer strand): (a) synclinic tilt (see Fig. 10c), i.e., all the surface molecules have the same tilt direction unlike the anticlinic arrangement in the “bulk”, (b) a randomization of the tilt direction along the strand axis induced by the requirement that the surface molecules want to be parallel to the local orientation direction of the strand and (c), a more generalized situation of the surface molecules assuming an SmA kind of ordering (Fig. 10d). Each of these cases would lead to the formation of a defect in the vicinity of the strand. While a straight forward differentiation amongst these three scenarios is not possible, we can perhaps state the following. The randomization of the tilt or the SmA-type of ordering will lead to a substantial reduction of the antiferroelectrically coupled regions whereas in the synclinic case in half the number of layers will be compatible with the tilt direction of the surface molecules. The effect of polymerization being quite strong on the f_AM values (Fig. 3) perhaps indicates that the synclinic situation is less favoured. Although f_IM is not influenced, finite size effects on f_AM, caused by the virtual surfaces of the polymer strands, cannot be fully ruled out. Both the elastic and finite size effects, becoming stronger with increasing polymer concentration, can be viewed in terms of an additional field causing slowing down of the antiphase dynamics. Such a field would be similar to that used in the case of twist grain boundary phases,⁴⁰ and dependent on the molecular anchoring at the polymer strands, the distance between the strands and an effective elastic constant of the LC molecules.

Fig. 10 Schematic diagram illustrating three different scenarios for the orientation of the liquid crystal molecules (ellipses) in the SmC^*_A phase in the polymerized situation. In (a) the layers undergo nanophase segregation with the polymer strand (blue line) intervening between smectic layers. In (b)–(d) the polymer strand runs through the layers, but with a difference: in (b) and (c) the molecules in the vicinity of the strand are also tilted, anticlinic (b) and synclinic (c) fashion, whereas in (d) the molecules in the neighbourhood of the strand are SmA-like (parallel to the layer normal). In the non-segregated cases (b–d) the possible surface influence of the strand is shown as a tube having the same contour as that of the strand.

In summary, with the help of dielectric spectroscopic response we find that confining an antiferroelectric liquid crystal in a polymer network results in a substantial reduction of antiphase fluctuation frequency while hardly affecting the in-phase relaxation frequency. Employing data obtained as a function of DC bias, several parameters, including the antiferroelectric coupling coefficient, the relevant elastic constant, and the viscosities have been extracted. Particularly interesting is the substantial reduction of the antiferroelectric coupling coefficient which should be responsible for the decrease in the antiphase mode frequency. The analysis of the data also shows that the quadratic coefficient of the Landau free energy decreases by a factor of 5 for the polymer stabilized system. The Hookean elastic constant representing the elastic interaction between the liquid crystalline molecules and the strands of the polymer network, is quite large and argued to play an important role for the observed behaviour. The present findings are expected to open up a new way to understand collective modes in restricted geometries.

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