Issue 
Eur. Phys. J. Appl. Phys.
Volume 87, Number 1, 2019



Article Number  10201  
Number of page(s)  9  
Section  Physics of Organic Materials and Devices  
DOI  https://doi.org/10.1051/epjap/2019190181  
Published online  13 September 2019 
https://doi.org/10.1051/epjap/2019190181
Regular Article
Study of the orientational ordering in the fieldinduced intermedate phase from the SmC *Fi_{2} phase in chiral smectic liquid crystals
Université Tunis ElManar, Faculté des Sciences de Tunis, Laboratoire de physique de la matière molle et de modélisation électromagnétique (LP3ME), Campus Universitaire Farhat Hached, 2092 Tunis, Tunisia
^{*} email: hassen.dhaouadi@ipeit.rnu.tn
Received:
3
June
2019
Received in final form:
23
July
2019
Accepted:
9
August
2019
Published online: 13 September 2019
Under an electric field, chiral smectic liquid crystals transit usually to the unwound SmC* phase where the helical structure is completely unrolled. Sometimes the sample transits initially towards an intermediate polar state before the total destruction of the helix. The discovery of this intermediate phase opens a promising field of application especially in the field of color displays. This gives importance to the study and the characterization of this new phase. Based on the extension of the HammanehTaylor model (H&T model), a theoretical study of these fieldinduced phase transitions was carried out. Two hypotheses of the dynamics that give rise appearance of the intermediate phase have been discussed. Numerical analysis confirms the known experimental results; the intermediate phase has a threelayer periodicity structure.
© EDP Sciences, 2019
1 Introduction
Ferroelectric liquid crystals are promising materials for fast switching electrooptical displays with wide viewing angle. Incorporating smectic liquid crystals into display devices is extremely attractive. But their widespread commercial use has not yet been realized because of orientation problems and a multitude of structures that appear as a result of a temperature change or by application of an external electric field.
In chiral smectic liquid crystals the molecules are arranged in layers with the director tilted with respect to the layer normal by a temperaturedependent tilt angle [1]. Many phases are encountered with the same tilt inside the layers but a distribution of the azimuthal direction which is periodic with a unit cell of one (SmC*) two (SmC*_{A}), three (SmC * Fi_{1}), four (SmC * Fi_{2}) or more (SmC*_{α}) layers [2–4].
A zero field phase sequence obtained during cooling process is generally in order:where Iso and Cr represent the isotropic and crystal phases respectively. Even so, one or more mesophases may miss in this table but the order of appearance remains the same [5]. Some theoretical works have shown the possibility of new phases with 5, 6, 7, etc. layers periodicity [6–8].
Liquid crystals are influenced by external effects such as surface anchoring, applied fields and impurities. This is why we always attend the appearance of new unexpected phases [9–11]. New phases can also arise by application of external fields. It is wellknown for a long time that external electric field induce transition toward the unwound SmC* phase where the helical structure is completely unrolled [12]. Sometimes the sample transits initially towards an intermediate polar state before the total destruction of the helix [13,14]. The dynamics of these transitions was the object of several theoretical and experimental works [15–19].
Several experimental work, in particular those resulting from an investigation with Xray resonant diffraction, shows the appearance of a new fieldinduced intermediate phase with 3 layers periodicity different to SmC * Fi_{1} phase. To determine the orientation of the molecules in a layer, a structural model is developed and the evolution of the structure of the ferrielectric phase during the increase in the modulus of the applied electric field is discussed [20–23].
Gleeson et al. extended their studies to the behaviors under electric field of the three layers SmC * Fi_{1} and four layers SmC * Fi_{2} phases. They present a model for the fieldinduced phase transitions. They showed, at final, the possibility of appearance of a structure noted elementary Fi_{12} of mesh of three layers even starting from the Ferri 2 phase [24].
The rich polymorphism and the possibility of discovery of new chiral smectic phases have an extremely interesting. Several theoretical models have been established to explain the variety of mesophases, their origin and structure. One of these models is that we call afterward as the HamanehTaylor (H&T) model [25,26] and its extension proposed by Dhaouadi et al. [27].
In this work we have carried out a theoretical study, based on the extended H&T model, of the fieldinduced states in chiral smectic liquid crystals, especially those obtained starting from SmC * Fi_{2} phase. Two hypotheses of the dynamics that give rise to the intermediate phase have been discussed. The molecular arrangements within the layers of two suggested structures, with 3 and 4 layers, have been studied. A numerical calculation, aiming at comparing the free energies of the suggested structures, allows knowing which is most stable.
At first, let us start with a theoretical background which describes the phase transitions between structures of the distorted clock model.
2 Theoretical background
In this section we suggest to present briefly the phenomenological model, known as the H&T model. Thereafter we present the extension brought by Dhaouadi et al. to this model. All the calculations carried out along this work are base on the results of this extension.
2.1 The HamanehTaylor model
The HT model is a phenomenological model to describe chiral smectic phases [25,26]. It is based on the balance between two interactions; one a short range twisting term trying to impose an increment α of the azimuthal angle ϕ between adjacent layers, the increment α of the azimuthal angle from layer to layer varies with temperature from 0 to π; the other a long range related to the anisotropy of curvature energy in the layers plane. They derived an order parameterJ =⟨ cos(2ϕ _{ l }) ⟩, where the average is taken on the azimuthal angle inside the unit cell. It is nonnull in the commensurate phases and associated to an energy η J ^{2} where η is a coefficient describing the strength of the long range interaction and is of order unity. The short order term reads ⟨cos(Δϕ _{ l } − α) ⟩ with Δϕ _{ l } = ϕ _{ l+1} − ϕ _{ l }; where ϕ _{ l } correspond to azimuthal angle in layer l. So the free energy can be written as:(1) F _{0} is the electrostatic energy necessary for transition towards the ferroelectric phase. F _{0} = − P _{ s } ⋅ E _{ s }, with P_{s} and E_{s} are respectively the polarization of saturation and the field threshold which induces the transition towards the unwound SmC^{*} phase.
From this expression Hamaneh and Taylor developed a numerical calculus leads to a phase diagram in the (α, η) plane showing the sequence of sub phases which can be observed in a given liquid crystals.
2.2 Extension of the H&T model
An extension of the H&T model proposed by Dhaouadi et al. [27] based on the introduction of new order parameter I = ⟨ cosϕ _{ l } ⟩ which describes the contribution of macroscopic polarization and electric field. The contribution to the free energy of polarization and electric field are expressed respectively by
where γ and δ are two parameters having for order of magnitude the unit and have as expressions: χ is the electric susceptibility of the sample. So, the expression of the free energy is written:(2)
From this expression, Dhaouadi et al. derived new phase diagrams and correlated them to their experimental results showing the phase sequence in some compound as function of temperature and electric field.
We present in Figure 1 the (α, η) diagram obtained starting from the extension of the HT model at non zero field (γ = 0.2 and δ = 0.5). This choice of the parameters γ and δ agrees well with the usual values of the applied electric field (about 1 V/µm) and spontaneous polarization of these compounds (about 100 nC cm^{−2}) [27]. It should be noted that, for this diagram, calculations are carried out by supposing that the intermediate phase has a structure with three layers similar to that of the SmC * Fi_{1} phase (Zone delimited by stars in Fig. 1).
Fig. 1
(α,η) phase diagram obtained with the extended HT model with applied field. Zone delimited by stars correspond to the domain where appears the intermediate phase. 
3 Origins of the fieldinduced intermediate phase
In this paragraph we will try to answer the following question: which is the dynamics that gives rise to the intermediate phase? To answer this question we propose a model according to which the applied electric field induces a distortion of the structure of the phase present in the sample at null field (Fig. 2a). The study of the evolution of this distortion according to the applied field modulus will allow further explaining the appearance of this intermediate phase and deducing its possible structure.
Fig. 2
(a) Distortion of the four layer structure for E < E_{c} . (b) First assumption with discontinuity of the distortion for E_{c} < E < E_{s} . (c) Second assumption appearance of a new structure with three layers for E_{c} < E < E_{s} . Red arrows represent the directions of polarization in the various layers. 
3.1 Working hypothesis
The null field phase, this study relates, is the SmC * Fi_{2} one. The fact that this phase is nonpolar facilitates enormously calculating of the distortions which deforms the structure following the application of an electric field. We suppose that the cell, into which the product is introduced, is low thickness so that the planar anchoring imposed by treated surfaces affects the orientation of the molecules even in volume. This assumption makes it possible to restrict the study to the elementary mesh of four layers.
Two assumptions describing the dynamics which leads to the transition towards the intermediate phase are considered:

The first is that of a deformation of the basic structure with conservation of the initial four layers mesh. The transition towards the intermediate phase is explained by a discontinuity of the distortion when the electric field reaches a threshold value which one notes E_{c} (Fig. 2b).

The second consists on a total rearrangement of the initial mesh. The basic four layers structure will be destroyed when the electric field exceeds the threshold value E_{c} . A new structure with three layers, identical to that described by Shtykov et al. [21], appears (Fig. 2c).
Initially we will carry out a calculation of polarization in each of the two structures described on the assumptions below. Then, a comparison of their free energies makes it possible to determine which the most probable one is.
3.2 Distortion of the elementary mesh in the SmC * Fi_{2} phase
Figure 2a shows the evolution of molecular arrangements in the elementary mesh of the SmC * Fi_{2} phase following the application of an electric field. The distortion of the basic structure appears as an angular displacement “δ _{ i }” of the molecular orientation in layer “i” (with i = 1, 2, 3, 4). For the simplicity of calculations, one suppose all the displacement “δ _{ i }” are identical. The resulting structure presents a spontaneous polarization due to the new orientations of molecular dipole moments. A calculation of the distortion “δ” and induced polarization “P” are necessary to continue the present study.
3.2.1 Expression of the free energy
In the expression of the free energy of the H&T model (Eq. (1)) it is the second term corresponding to the long range interactions W = F _{0} ηJ ^{2} which is responsible for the appearance of the commensurable molecular arrangements [27]. It is also responsible for the stability of the distorted structure described previously. In the SmC * Fi_{2} phase, the order parameter J is written at null field as:(3) W is minimal for the structure described in Figure 2a (W = W _{0} = F _{0} ηcos^{2}(υ)).
In the structure distorted by the applied electric field, the order parameter J is a function of the distortion “δ”(4)
A development to the second order, for low values of the distortion “δ”, gives for the potential the following expression:(5)
The expression of the free energy must present in addition to the potential W(δ), a term corresponding to the energy of the polarization which appears at non null field. For a linear treatment one neglects the quadratic term in P ^{2}. A justification of this approximation will be given further. The symmetry of the configuration in Figure 2a gives an induced polarization parallel to the applied field. One can write: (6)
3.2.2 Expression of the distortion “δ”, and polarization “P”
Let us start initially by expressing polarization P(δ) according to δ. This polarization is the average of all polarizations in the mesh layers after distortion.(7)
The components of polarizations following the direction of the field are written(8)
Total polarization is then written:(9)
For weak values of δ, sin(δ) ≈ δ one has then:(10)
The free energy is then written in the following form:(11)
By minimization over δ one finds:(12)
4 Experimental results
The aim of this section is to estimate the orders of magnitude of polarizations and the transition's threshold field. These values are crucial data for the interpretations of theoretical calculations carried out along this work.
An electrooptical study and microscopic observation have been carried out for the chiral smectic liquid crystal compound C12F3 belonging to the series of fluorinated product CnF_{3} synthesized by Nguyen [5,28–30] (Scheme 1).
The polymorphism obtained by dielectric spectroscopy at zero fields is the following [28]:
All measurements have been carried out on commercial planar cell (EHC. Inc, Japan; 5 μm) coated with a conductive layer of indium tin oxide. The active area is 25 mm^{2}.
Scheme 1
The structural formula of the compound C12F3. 
4.1 Microscopic observation
We have recorded the evolution of the textures of our compound as a function of the applied electric field in the range of temperature where the sample presents the phase SmC * Fi_{2}. Photomicrographs of Figure 3 are taken at the same temperature, T = 66.2 °C. For weak field, the sample presents still the structure of the SmC * Fi_{2} phase (Fig. 3a). For stronger field, superior to the threshold value E_{c} , the change of color means transition towards the intermediate phase (Fig. 3b). A second transition towards the unwound SmC^{*} phase takes place for a field higher than a new threshold value E_{s} (Fig. 3c).
Fig. 3
Photomicrographs taken at T = 66.2 °C. (a) SmC * Fi_{2} phase; (b) intermediate phase; (c) unwound SmC* phase. 
4.2 Electrooptic study
For the electrooptic study, we used a triangular wave form. The measurement recorded at different temperatures, by varying voltages with fixed frequency 40 Hz in the 5µm cell. We record the signal on a Tektronix 340 oscilloscope. The current response is the sum of the charge of the cell capacitor, the ionic conductivity and a peak linked to the polarization which reads:
An estimation of polarization can be obtained by a simple measurement of the air of the polarization peak. Figure 4 represents evolution of the switching current at T = 66.2 °C for 4, 5 and 8 V. At 4 V, a large current peak was observed, it correspond to the ferri 2/intermediate state switching (Fig. 4a). A further increase of the voltage leads to a single current peak. It means that the switching occurs directly between the two ferroelectric states (Fig. 4b and c).
The measurement of the polarization of the sample according to the applied field modulus at various temperatures enabled us to plot the curves of Figure 5. The changes of shapes observed in the curve of Figure 5a indicate phase transitions. Thus, one determined the values of the threshold fields E_{c} and E_{s} of the SmC * Fi_{2} → intermediate phase and intermediate phase → unwound SmC* phase transitions.
In Figure 5b we present the evolution of the polarization of saturation Ps of the sample and the variation ΔP of polarization during the transition towards the intermediate phase in the range of temperature [62 °C, 67 °C]. The ratio ΔP/P_{s} is almost constant and equal to “0.4”.
Figure 5c represents the part of the (E,T) phase diagram of the compound C12F3 relating to the range of temperature [62 °C, 70.5 °C]. The variation according to the temperature of the ratios E_{s} /E_{c} and ΔP/P_{s} are represented in Figure 5d. It is to be also noticed that the ratio E_{s} /E_{c} is almost constant and equal to “1.4”.
Fig. 4
Switching current for the SmC * Fi_{2} phase at 66.2 °C, under a varying triangular electric field (frequency 40 Hz); (a) at 4 V; (b) at 5 V and (c) at 8 V. The air of the peak is proportional to the polarization of the sample. 
Fig. 5
Results of the electrooptical study; (a) polarization of the sample according to the applied field modulus at T = 66.2 °C. The changes of shapes observed in the curve indicate phase transitions. (b) Evolution with temperature of the polarization of saturation P_{s} and the variation ΔP of polarization during the transition towards the intermediate phase. (c) (E,T) phase diagram of the compound C12F3. (d) Variation according to the temperature of the ratios E_{s} /E_{c} and ΔP/P_{s} . 
5 Numerical modeling and phases diagram
5.1 Order of magnitude
To estimate the ¦order of magnitude of the angular distortion and polarization induced by this distortion it is necessary to know the order of magnitude of the characteristic angle υ of the SmC * Fi_{2} phase. Starting from resonant xray scattering measurements, Cady et al. [31] found for the SmC * Fi_{2} structure a value of υ of about 164°. Roberts et al. [32] measured the angular distortion of the SmC * Fi_{2} structure in mixtures at two temperatures; they found a value of υ of about 166° with no discernable dependence on temperature. We adopt in what follows the value (υ = 164°).
5.1.1 The distortion and the induced polarization
According to the H&T model, the SmC * Fi_{2} phase is obtained for values of η higher than 0.5. It is then possible to estimate the value of the distortion δ:(13)
Figure 5d shows that in a large range of temperature, the transition towards the intermediate phase occurs for the critical field E_{C} such as the ratio , where E_{s} is the threshold field inducing the transition towards the unwound SmC* state. Equation (13) gives then a maximum value of the distortion δ of about de rd = 0.024 rd ≈1.37°.
While replacing δ by its value in equation (10), one finds the value of the polarization which the sample, following this process of distortion, can have: P _{max} (δ) = 0.0031P _{ s }. It is to be noticed that even for rather strong fields (E ≈ 0.7E _{ s }) the values of the distortion δ and polarization P remain low in front of the characteristic angle υ and polarization of saturation P_{S} of the SmC * Fi_{2} Phase. To conclude, a simple distortion of the structure SmC * Fi_{2} cannot explain the appearance of the intermediate phase with a polarization close to half of P_{S} .
5.1.2 Discontinuity of the distortion; the four layers structure
We initially consider the four layers structure described in Figure 5b. The transition towards the intermediate phase results in a discontinuity of the distortion “δ”. The average polarization of the new structure is written according to the angle β:(14)
Evolution of the polarization of the sample according to the applied electric field at the temperature T = 66.2 (Fig. 5d) shows that the polarization of the intermediate phase is about 0.4P_{s} . Starting from equation (14), the calculation of the new characteristic angle β gives; β ≈ 101°, which corresponds to a value of the distortion δ _{0} of order 71°.
5.1.3 The three layers structure
This new structure is similar to that observed in experiments [20–23] and showed in Figure 2c. We carry out in the continuation a calculation of the characteristic angle “µ” of this new phase. One expresses the variation of polarization ΔP as function of the angular parameter μ and the polarization of saturation P_{S} .(15)
To estimate the order of magnitude of the angle μ, one takes the value of ΔP = 0.4P_{s} mentioned previously. One find a value of μ of about 84.26°.
In what follows we carry out a numerical calculation allowing a comparison of the free energies of the two suggested structures of the intermediate phase. A Maple program was elaborated to carry out this task.
The expression of the free energy equation (2), written for these two structures, is used to plot the phase diagrams in the plan (α, η). The program ensures a numerical calculation which compares the absolute minima of the free energies associated to the two structures and indicates which is most stable for different values of α and η. The construction of the diagrams is carried out by an interface to create graphs starting from matrices containing 10^{6} elements.
For a couple of selected values of the characteristic angles β and μ, the obtained phase diagram shows the domains of stability of the two structures suggested for the intermediate phase. The diagrams of Figure 6a–d correspond to choices of the angles β and μ with values equal or slightly different from the optimal values β = 101° and μ = 84.26° reported in the section order of magnitude. The thin blue band in bottom of these diagrams corresponds to a domain where the free energy of the four layers structure is weaker than that of three layers.
Even with a choice of the couple (β, μ) very different from the optimal values, the blue band changes thickness but the domain of stability of the three layers structure always remains dominating Figure 6e and f.
The first conclusion to be made, according to the diagrams of Figure 6, is that the three layers phase is the most stable in the zone of the plan (α, η) delimited by stars, zone corresponding to the domain where appears the intermediate phase in diagrams H&T of Figure 1.
Finally the results of numerical calculation confirm the experimental results according to which the intermediate phase admits a structure with 3 layers.
Fig. 6
(α, η) phase diagrams drawn up numerically with different values of the parameters μ and β. They show the domains of stability of the two suggested structures for the intermediate phase. In blue the domain of four layers phase and in pink that of the three layers one. The zone delimited by stars correspond to the domain where appears the intermediate phase in diagrams H&T of Figure 1. 
6 Conclusion
Following the application of an electric field, a chiral smectite liquid crystal undergoes modifications of structure. Certain modifications constitute true phase transitions similar to those induced by variation in the temperature [33]. These transitions are similar to those observed in some ferromagnetic materials for which the multiferroic cycloidal phase becomes a ground state. In these materials either continuous or abrupt field induced reorientations of the cycloidal magnetic structures were observed at lower magnetic fields [34].
It is wellknown for a long time that under an electric field, chiral smectic liquid crystals transit usually to the unwound SmC* phase where the helical structure is completely unrolled. Sometimes the sample transits initially towards an intermediate polar state (ferrielectric), before the total destruction of the helix. One can thought that we are observing a discontinuous destruction of the helix which occurs by increasing the modulus of the applied field [15]. In a previous experimental work we showed that it is indeed a transition from first order [14].
As we have seen in the introduction, the widespread commercial use of chiral smectics has not yet been realized because of orientation problems. The discovery of electric fieldinduced intermediate phase opens the way for promising applications especially in the field of color displays. Indeed, experience shows that we always see color changes after each transition. The hue presented by the sample depends on the optical and dielectric characteristics of the product and the thickness of the cell in which it is introduced. This makes it possible to change, at will, the color of a pixel by a simple change of the applied field modulus.
In this work we tried, using a simple theoretical study, to understand the process which gives rise to this intermediate phase in the case of a sample presenting the SmC * Fi_{2} phase at null field. Two possible structures of this phase are proposed.

The first is a four layers structure, which rises from a discontinuity of the distortion of the basic structure.

The second is a three layers structure, similar to those observed in experiments by the technique of Xray resonant diffraction.
A theoretical study allows, starting from experimental data, to estimate the characteristic parameters of the suggested structures. A numerical calculation allows, by comparison of their free energies, to determine which of both is most stable. The results of this numerical study give favor to the three layers structure.
Author contribution statement
Hassen Dhaouadi (main author: idea and organization of work) wrote most of the article and carried out all calculations and directed the experiments. Oussama Riahi contributed to experimental work and numerical calculations. Rihab Zgueb contributed to experimental work and treatment of the curves. Fedia Trabelsi contributed to experimental work and treatment of the curves. Tahar Othman is head of the laboratory.
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Cite this article as: Hassen Dhaouadi, Oussama Riahi, Rihab Zgueb, Fedia Trabelsi, Tahar Othman, Study of the orientational ordering in the fieldinduced intermediaite phase from the SmC * Fi_{2} phase in chiral smectic liquid crystals, Eur. Phys. J. Appl. Phys. 87, 10201 (2019)
All Figures
Fig. 1
(α,η) phase diagram obtained with the extended HT model with applied field. Zone delimited by stars correspond to the domain where appears the intermediate phase. 

In the text 
Fig. 2
(a) Distortion of the four layer structure for E < E_{c} . (b) First assumption with discontinuity of the distortion for E_{c} < E < E_{s} . (c) Second assumption appearance of a new structure with three layers for E_{c} < E < E_{s} . Red arrows represent the directions of polarization in the various layers. 

In the text 
Scheme 1
The structural formula of the compound C12F3. 

In the text 
Fig. 3
Photomicrographs taken at T = 66.2 °C. (a) SmC * Fi_{2} phase; (b) intermediate phase; (c) unwound SmC* phase. 

In the text 
Fig. 4
Switching current for the SmC * Fi_{2} phase at 66.2 °C, under a varying triangular electric field (frequency 40 Hz); (a) at 4 V; (b) at 5 V and (c) at 8 V. The air of the peak is proportional to the polarization of the sample. 

In the text 
Fig. 5
Results of the electrooptical study; (a) polarization of the sample according to the applied field modulus at T = 66.2 °C. The changes of shapes observed in the curve indicate phase transitions. (b) Evolution with temperature of the polarization of saturation P_{s} and the variation ΔP of polarization during the transition towards the intermediate phase. (c) (E,T) phase diagram of the compound C12F3. (d) Variation according to the temperature of the ratios E_{s} /E_{c} and ΔP/P_{s} . 

In the text 
Fig. 6
(α, η) phase diagrams drawn up numerically with different values of the parameters μ and β. They show the domains of stability of the two suggested structures for the intermediate phase. In blue the domain of four layers phase and in pink that of the three layers one. The zone delimited by stars correspond to the domain where appears the intermediate phase in diagrams H&T of Figure 1. 

In the text 
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