Thermodynamic investigation of phases transition in ...

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Andika Fajr proposes that this transition is first order in lower temperature and becomes second order in the vicinity of SmA–SmCα* transition [25]. In the present.
Journal of Molecular Liquids 215 (2016) 110–114

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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Thermodynamic investigation of phases transition in antiferrolectric liquid crystals: Theory and experiment T. Soltani a,⁎, J.P. Marcerou b, T. Othman a a b

Université Tunis Elmanar, Laboratoire de physique de la Matière Molle, Faculté des Sciences de Tunis, Tunisia Centre de Recherche Paul Pascal Pessac, France

a r t i c l e

i n f o

Article history: Received 8 March 2015 Received in revised form 1 December 2015 Accepted 15 December 2015 Available online xxxx Keywords: Liquid crystal Phase diagram Model Transition enthalpy

a b s t r a c t In this work, a theoretical model is developed, to investigate in more detail the electric field-temperature phase diagram (E–T) of antiferroelectric liquid crystals. The analysis shows that it is possible to obtain a complete description of the induced electric field transition of this material to deduce the entropies, nature of transition, enthalpies versus transition temperatures and transition electric field. Therefore, a quantitative and qualitative analysis of (E–T) phase diagram for some compounds is given. In addition, the constant current technique shows a significant increase of the cell voltage above the threshold near the electric induced (SmC*A-ferrielectric) transition. This behavior, which appears to be rather similar to the supercooled liquids near glass transition, is also discussed in the present work. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Smectic liquid crystals are multifunctional materials because they posses simultaneously several so-properties such as ferroelectricity, ferrielectricity and antiferferroelasticity. The structures that allow such properties and the phase sequences that occur in the systems are the subjects of much interest. The most general phase sequence lowering temperature is: I-SmA–SmC*α-SmC*–SmC*FI2–SmC*FI1–SmC*A–K [1–8]. Some of these phases may be missing when varying the chemical formula but the order of appearance is conserved. An additional phase smectic-C*d6 having a six-layer unit cell has been discovered [7]. Usually these phases demonstrate interesting behaviors under the electric field. In fact, by applying the electric field, some induced transitions can occur and new phases can be revealed, with symmetries and polarization directions different from the bulk ones. The presence of new phases is interesting, not only because of their potential relevance, but also because they generate new phase boundaries in the electric– temperature phase diagrams (E–T). Recently, (E–T) phase diagrams for some compounds have been established by using several techniques: electro-optical properties [9–17], dielectric measurement [11,12], microscopic observation [9,10], and birefringence measurement [18,19]. At the same time, several attempts have theoretically been made to describe the properties of the phase transition under electric field in antiferroelectric liquid crystal [20–25]. The phenomenological theories have been developed so far to explain the phase sequences and ⁎ Corresponding author. E-mail address: tawfi[email protected] (T. Soltani).

http://dx.doi.org/10.1016/j.molliq.2015.12.053 0167-7322/© 2015 Elsevier B.V. All rights reserved.

dynamics properties of chiral smectic liquid crystals. In addition, the quantitative analysis of the dependence of the intermediate phase stability on some factors including spontaneous polarization, optical angle and layer spacing has been reported by Johnson et al. [15]. In this context and in order to diversify the thermal properties of liquid crystals, we develop a theoretical model. This allows the obtention of some information regarding their enthalpies and entropies of fieldinduced phase transitions and a quantitative analysis of the (E–T) phase diagram. The constant current method is well-developed in our laboratory and is expected to give more detailed information on the induced electric phase transitions [12]. It is a very powerful and useful technique to determine the phase transitions. In fact, it can get a (E–T) phase diagram in a reasonable amount of time. However, it has not been well explored yet. Therefore, our model allows us to explore more this technique in the electric field-induced phase transition area. 2. Theory In this section, we attempt to develop a simple model to explore more constant current technique in electric field transition area. A phase transition may be influenced by adjusting the external thermodynamic parameters, such as temperature, external fields, and composition. Many prior studies have been focused on the smectic phase behavior under electric field, using various experimental techniques [12–19]. The total polarization in the sample is P = ε0χE + Ps, where χ and Ps are the dielectric susceptibility and the spontaneous polarization, respectively. Moreover, when an electric field is applied in the layer

T. Soltani et al. / Journal of Molecular Liquids 215 (2016) 110–114

planes, χ and ε are assumed to be χ = χ⊥ and ε = ε⊥ (Fig. 1). Then the polarization in the sample is given as: P ¼ ε0 χ⊥ E þ Ps ;

ð1Þ

and the electric displacement D, in an isotropic medium, can be written in the following form: D ¼ ε 0 E þ P ¼ ε 0 ðχ ⊥ þ 1ÞE þ P s ¼ ε0 εr⊥ E þ P s ¼ ε⊥ E þ P s :

ð2Þ

111

It is important to note that this equation, which provides relationship between the slope of the saturation line (dE / dT), the changes in the polarization, and the transition enthalpy (ΔHtrs) is equivalent to the Clausius–Clapeyron equation. In addition, one notes from this expression that the slopes of boundaries in phase diagrams are connected quantitatively and qualitatively to thermodynamic properties, such as the sign and the value of the enthalpy depend on those of the slope. The polarization changes at the transition point, where E1 = E2 = Etrt, can be calculated by the following expression:

From the thermodynamics point of view, the temperature (T) and E are the intensive variables. The Gibbs free energy (G) of a liquid crystal in the presence of electric field is given as:

P 2 −P 1 ¼ ε0 ðχ ⊥2 −χ ⊥1 ÞEtrt þ ðP s2 −ðP s1 Þ ¼ ðε⊥2 −ε ⊥1 ÞEtrt þ ðP s2 −ðP s1 Þ; ðΔε ⊥ ¼ ε0 Δχ ⊥ Þ

G ¼ U−T  S−E  P:

It is important to note that this latter (ΔP) can be determined by the constant current technique. A brief overview of this technique. It has been successfully used to study induced phase transition for some compounds of liquid crystal [9,12]. It consists in connecting the planar sample to a generator delivering a constant current in the nanoampere range and measuring the cell voltage. The local field in the cell, which is given by the cell voltage, starts to increase linearly versus time with a slope inversely in proportion to the dielectric constant of the initial phase (Fig. 2). The ε⊥ can be extracted from the following expression:

ð3Þ

Since the differential of internal energy dU = T · dS + E · dP, one finds: dG ¼ −S  dT−P  dE:

ð4Þ

From Eq. (2) one readily obtains for entropy (S) and polarization in the following general form:   ∂G E ∂T   ∂G T: P¼− ∂E S¼−

ð5Þ

For the phase boundary line (equilibrium between two smectic phases at constant temperature and electric field), we have:         ∂G1 ∂G1 ∂G2 ∂G2 dE þ dT ¼ dE þ dT: ∂E T ∂T E ∂E T ∂T E We can rearrange the terms as     ∂G2 ∂G1 ∂G1 ∂G2 dE ¼ dT: − − ∂E ∂E ∂T ∂T

ð6Þ

Including both polarization and entropy (Eq. (5)), the transition entropy change can be expressed as:

ΔStrs

ðP 1 –P 2 ÞdE ¼ ðS2 –S1 ÞdT dE ¼ ðS2 −S1 Þ ¼ ðP 1 −P 2 Þ : dT

ð7Þ



V Q it ; ¼ ¼ e Ce ε⊥ S

ð10Þ

ð11Þ

where e is the cell thickness, S is the active surface, i is the current, t is the time and Q is the charge at t. At the transition between two tilted smectic phases (φ1 and φ2), the local field stays fixed until the transition is completed and re-increases in the second phase until the saturation (upper plateau) (Fig. 2). Then the value of the field at the phase transition can be determined and the difference of spontaneous polarizations (ΔPS) between the two phases can be calculated by: ΔP S ¼

iΔt ; S

ð12Þ

where Δt is the plateau duration. In this case the equation becomes formally applicable and the induced transition enthalpy can be determined. It is interesting to mention that in the case of two induced phase transitions, one can see two successive plateaux [9–12]. It is important to signal that the last plateau is not relevant; it reflects only the upper voltage limit set for the generator.

The change in enthalpy at phase transition is ΔH trs ¼ T trs ΔStrs :

ð8Þ

By substituting Eq. (7) into the above expression, one can obtain: ΔH trs ¼ −T trs ðP 1 –P 2 Þ

dE ; dT

ð9Þ

dE are respectively the transition temperature and the where Ttrs and dT slope of the electric–temperature phase transition line.

Fig. 1. Schematic diagram of the planar alignment of liquid crystal. Electric field is parallel to the smectic layer planes. The plates are fixed.

Fig. 2. Typical curve showing the evolution of the cell voltage with the time under a constant current of 5 nA. The last plateau corresponds to the upper voltage limit set for the generator.

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In our case, since approximation:

ðε ┴2 −ε ┴1 ÞE ðP S2 −P S1 Þ

is about 1%, we deduce with a good

P 2 −P 1 ¼ ΔP S Finally, the enthalpy of transition under electric field can be expressed by the following equation: ΔH trs ¼ −T trs ΔP S

dE dT

ð13Þ

3. Application We first present both phase diagrams for C12HH (Fig. 3) and C10F3 (Fig. 4) compounds studied experimentally by our group [12,13,26]. These phase diagrams have been obtained by several procedures including microscopic observation, dielectric measurement, electrooptic properties and constant current method. The C12HH compound shows the following ground phase sequence: K-(78 °C)–SmCA*(96.5)–SmCFI1*-(98)–SmCFI2*-(100)–SmC*-(119)–SmA. Under electric field, the SmC* phase shows two distinct behaviors: near SmA*–SmC* phase transition, the direct transition from helical SmC* to the unwound SmC* phase was detected. At lower temperature, this transition occurs via an induced intermediate phase. This latter is found to be ferrielectric phase with polarization (noted here “SmC*FIind”) [9,10,12,15–18]. In fact the mechanism of unwinding SmC* has been detailed in Ref. [9,13]. The anticlinic (SmC*A) and unwound SmC* are separated by the SmCFIind phase (L1 and L2). Then, the slopes of line boundaries (dE / dT) are negative. Using equations (Eqs. (12) and (13)) and the constant current technique, the enthalpy change at the field induced transition can be determined. In order to obtain the molar transition enthalpy (ΔHmtrs), converting ΔHtrs from (J·m−3) to (J·mol−1), the density of the studied compound is determined using the capillary technique. The obtained value is ρ = 1.18 g/cm3, which is of the same order of magnitude as that obtained for certain liquid crystals [27]. The obtained ΔHmtrs at several temperatures are listed in Table 1. In addition, the molar entropy change at phase transitions can be calculated by the ΔHmtrs = T · ΔSmtrs formula based on the enthalpy (Table 1). One can clearly see the transitions at lower thresholds from SmC*, SmC*FI1 and SmC*A to the induced SmC*FIind phase are first order. On the other hand, our finding demonstrates the physical meaning of the shape of the line transition at lower threshold. Indeed, the slope of this latter is linked to the sign of ΔHmtrs and in the nature of the corresponding transition (exothermic or endothermic). In the SmC*A phase, the slope which is negative

Fig. 4. (E, T) phase diagram of the compound C10F3.

corresponds to ΔHmtrs b 0, while in the SmC* the slope which is positive corresponds to ΔHmtrs b 0; such observations (slope sign) are common for some (E–T) phase diagrams [9–16]. In the C12HH compound, the SmC*FI2 is the subject of coexistence phenomenon, and the SmC*α is not present. To go further, we have studied these both phases in C1OF3 compound which presents at zero field the following phase sequence: SmA–SmC*α–SmC*FI2-K. This compound is not the subject of coexistence phenomenon because it does not exhibit polar phase. We note that the coexistence phase and the thermal hysteresis in smectic phases have been detailed [28]. In the SmCα*, there is a first order transition from this latter to unwound SmC* phase. The change enthalpy corresponding to this latter at T = 68 °C is calculated to be ΔHmtrs = 2.354 kJ·mol−1. Two research groups have tried to analyze this transition and determine its nature using phenomenological model [21,25]. Andika Fajr proposes that this transition is first order in lower temperature and becomes second order in the vicinity of SmA–SmCα* transition [25]. In the present compound, the unwinding of SmC*FI2 exhibits two thresholds. At the first (lower) threshold, to ΔHmtrs decreases with decreasing temperadE ture and tends to zero (because dT →0). This behavior suggests that the transition from SmC*FI2 to the induced SmC*FIind with small ΔHmtrs (− 0.8326 kJ·mol− 1 at 59 °C), compared to the other transition enthalpies, is weakly first order. It is interesting to note that the point of inflection in the (E–T) plane corresponds to the second order transition. dE Indeed, in this case dT ¼ 0 gives the latent heat of transition ΔHtrs = 0, on the basis of the Eq. (9). In order to compare the thermal behavior of the field induced ferriélectric phase (SmC*FIind) to that of the helical ferrielectric phase (SmC*FI1) obtained when the temperature changes without electric field, we investigate the differential scanning calorimetry (DSC) measurement. In the present work, we focus our attention on the thermal behavior of ferrielectric phase (SmC*FI1), which exists between SmC*FI1 and SmC*A phases. By cooling, the SmC*FI1 can be obtained from the Table 1 Molar enthalpy (ΔHmtrs) and molar entropy ΔSmtrs changes at some field induced transitions in C12HH compound. T (°C)

Transition

ΔHmtrs (kJ·mol−1)

ΔSmtrs (J·mol−1·K−1)

90

SmC*A–SmC*FIind SmC*FIind–SmC*und SmC*–SmC*FIind SmC*FIind–SmC*und SmC*–SmC*und

3.37 7.03 −0.59 1.94 2.41

9.28 19.63 1.54 5.09 6.19

108 Fig. 3. (E, T) phase diagram of the compound C12HH.

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electric field induced ferrielectric phase SmCFIind is qualitatively similar to that of the temperature driven one. Qualitatively, the values of ΔHmtrs obtained by DSC at driven temperature transition are much smaller than that obtained in the case of electric field induced one. For example, we find that at driven temperature SmC*A–SmC*FI1, ΔHmtrs = 0.3 kJ·mol−1 for C12HH and ΔHmtrs = 3.37 kJ·mol−1 at SmC*A–SmC*FIind (Table 1). As shown in the present paper and in other work [9–16], it seems that the medium electric field favors the intermediate induced ferrielectric phases. In addition, the field electric effect is qualitatively similar to that of the temperature in the phase transition in AFLCs. 4. Irreversible field-induced transition from SmC*A to ferrielectric phase

Fig. 5. DSC cooling (a) and heating (b) thermograms of C12HH.

SmC*FI2 phase, and the corresponding transition is exothermic. By heating, it can be obtained from SmC*A and the transition is endothermic. Fig. 5 illustrates DSC graphics obtained via continuous cooling (Fig. 5(a)) and continuous heating (Fig. 5(b)) of the C12HH compound. As seen in Fig. 5(a), the sample shows two exothermic peaks during continuous cooling; they correspond to the phase transitions from SmC* to the SmC*FI2 and from SmC*FI1 to SmC*A. The phase transition SmC*FI2–SmC*FI1 is not detected by the DSC measurement, but it has been detected by optical rotatory power [13]. In other work, this transition has been observed by the DSC instrument and it has displayed an exothermic peak [29]. Fig. 5(b) also demonstrates an endothermic peak during continuous heating corresponding to the transition from SmC*A to SmC*FI1, which is in agreement with that reported in Ref. [30]. This behavior is similar to that observed here in the induced electric field transitions from these phases to the SmCFIind phase; both induced phase transitions SmC*FI2–SmC*FIind in C10F3 and SmC*A–SmC*FIind in C12HH are found to be exothermic and endothermic, respectively. From this comparison, we conclude that the thermal behavior of the

Another important phenomenon can occur at the phase transition, is the supercooling phenomenon. This latter has been observed and discussed near nematic–isotropic phase transition [31,32] and in the chiral smectic phases when the sample exhibits a polar phase [13,19, 28,33,34]. In the present work, we demonstrate a similar behavior of the supercooling near the electric field-induced phase transition. Fig. 5 illustrates the evolution of the cell voltage with time under a constant current of 15 nA. In the SmC*A phase, we observe two successive plateaux which correspond to the phase transitions SmC*A–SmC*FIind and SmC*FIind–SmC* respectively. At low temperature (T b 86 °C) of this phase, the first plateau is preceded by a small peak, due to the increase of the electric field above the electric field at which transition occurs (threshold) (Fig. 6a). This indicates that the SmC*A phase is maintained above to the threshold value. As the temperature further decreases, the peak grows and the phenomenon becomes more pronounced (Fig. 6b). It is important to mention that this behavior is observed only at lower temperature (T b 86 °C) due to the higher viscosity and polarization of the sample at lower temperatures. This result is consistent with the electro-optic measurements. At this transition, the hysteresis is observed during increasing and decreasing the electric field. Indeed, the threshold (Einc) during increasing is higher than that (Edec) during decreasing. In addition the width of the hysteresis (Einc − Edec) increases with decreasing the temperature. This means that at lower temperature, the viscosity and polarization should become higher and have a significant effect on the induced phase transitions. Such behavior is consistent

Fig. 6. Time variation of electric field under a constant current of 12 nA.

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with the work of Jaradat who reported the irreversible field-induced transition from an antiferroelectric phase to the ferrielectric phase, where the latter is more stable than the antiferroelectric phase due to the high spontaneous polarization [35]. On the other hand, the important role of the coupling between the spontaneous polarization and the discrete flexoelectric polarization in the transition phase has been discussed by Emelyanenko and Osipov [36]. We think that the viscosity, spontaneous polarization, flexoelectric and the great stability of the SmC*A (78–96.5 °C) probably are the principal factors could explain this behavior in the present case. The effect of these factors on irreversible transition will be studied in detail. Similarly, the physical constraints which could be realized using this model will be taken into consideration in a future publication. 5. Conclusion In this paper, a successful method for analyzing the phase transitions under the electric field in antiferroelectric liquid crystals is reported. In fact, it seems that the theory presented above qualitatively and quantitatively describes well the (E–T) phase diagrams in liquid crystals. Accordingly, we can calculate the transition enthalpy for the electric field induced transition, precisely the nature of phase transition, and analysis the transition lines established in these phase diagrams. We think that this technique can be used to study nematic phase and extended to describe other systems like ferroelectric compounds. Finally, the irreversible field-induced transition from the anticlinic SmC*A to the ferrielectric phase is demonstrated. References [1] F. Bibonne, J.P. Parneix, H.T. Nguyen, Eur. Phys. J. Appl. Phys. 3 (1998) 237–241. [2] S. Essid, M. Manai, A. Gharbi, J.P. Marcerou, H.T. Nguyen, J.C. Rouillon, Liq. Cryst. 31 (2004) 1185–1193. [3] A.D.L. Chandani, E. Gorecka, Y. Ouchi, H. Takezoe, A. Fukuda, Jpn. J. Appl. Phys. 2 Lett. 28 (1989) 1265–1268. [4] A.V. Emelyanenko, K. Ishikawa, Soft Matter 9 (2013) 3497–3508. [5] L.S. Hirst, S.J. Watson, H.F. Gleeson, P. Cluzeau, P. Barois, R. Pindak, J. Pitney, A. Cady, P.M. Johnson, C.C. Huang, A.M. Levelut, G. Srajer, J. Pollmann, W. Caliebe, A. Seed, M.R. Herbert, J.W. Goodby, M. Hird, Phys. Rev. E 65 (2002) 041705–041710.

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