CHEMPHYSCHEM ARTICLES DOI: 10.1002/cphc.201301154
Sign Inversion of the Spontaneous Polarization in a “de Vries”-Type Ferroelectric Liquid Crystal Dorothee Nonnenmacher,[a] Robert P. Lemieux,[b] Mikhail A. Osipov,*[c] and Frank Giesselmann*[a] In contrast to common ferroelectric smectic C* liquid crystals, the siloxane-terminated smectic mesogen E6 is characterized by an unusual temperature variation of the spontaneous polarization. The polarization starts to grow from nearly zero despite the first-order SmA*-SmC* transition, and increases faster than linearly over a large temperature interval while the tilt angle rapidly saturates. To study this behavior in more detail, binary mixtures of different concentrations of E6 in the achiral SmC material C8Cl, which has a similar chemical structure, were investigated. Surprisingly, all mixtures show a temperature dependent polarization sign inversion, which shifts towards
the SmC*-SmA* transition with increasing E6 concentration. For the pure E6 the inversion temperature meets the SmA*SmC* phase transition temperature. In a second binary mixture with E6 and a conventional material C9–2PhP we found out, that the dependence of the inversion temperature on the concentration of E6 changes qualitatively when the nanosegregation is partially destroyed. A molecular theory of the polarization sign inversion in smectics C* with strong polar intermolecular interactions is developed which enables one to explain the concentration dependence of the inversion temperature in both mixtures.
1. Introduction In ferroelectric smectic C* liquid crystals the macroscopic polarization does not appear self-consistently, but is induced by the tilt of the director in a chiral medium.[1] Thus, the polarization appears to be the secondary order parameter while the tilt is the primary one, and therefore the ferroelectric ordering in chiral tilted smectics may be called improper. Ferroelectric liquid crystals possess some very interesting polarization properties related to molecular chirality and polarity, and these materials also attract significant attention because of their applications in the new generation of fast electro-optic displays.[2] In recent years the interest has been focused upon the socalled de Vries type ferroelectric smectic C* liquid crystals which are characterized by anomalously weak layer contraction, large values of the electroclinic coefficient,[3] low orientational order[4, 5] and other unusual properties. The de Vries type smectic materials are particularly important for applications because they enable one to avoid the so-called “zig-zag” defects which appear due to a buckling of layers in the bookshelf geometry determined by the tilt of the director in the smectic C phase. The majority of de Vries type materials possess a silox-
[a] D. Nonnenmacher, Prof. Dr. F. Giesselmann Institute of Physical Chemistry, University of Stuttgart Pfaffenwaldring 55, 70569 Stuttgart (Germany) E-mail: [email protected] [b] Prof. Dr. R. P. Lemieux Department of Chemistry, Queen’s University Kingston, Ontario K7L3N6 (Canada) [c] Prof. Dr. M. A. Osipov Department of Mathematics, University of Strathclyde 26 Richmond Street, Glasgow G1 1XH (UK) E-mail: [email protected]
2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
ane or a fluorinated terminal chain which has a tendency to separate from the rest of the molecule. Thus the effect of nanosegregation is expected to play a significant role in these systems. Ferroelectric properties of de-Vries-like smectics are also rather unusual. In contrast to a simple phenomenological theory,[1] in some cases the spontaneous polarization of smectics C with weak layer contraction is not proportional to the tilt angle over any significant temperature interval, and it continues to grow when the tilt angle saturates.[6] Moreover, the nanosegregated siloxane-terminated mesogen with chiral (R,R)2,3-epoxyoctyloxy side chains (E6) (see Scheme 1) exhibits the
Scheme 1. Chemical structures and phase-transition temperatures (in 8C) of E6, C8Cl,[18] and C9–2PhP.
spontaneous polarization which continues to grow faster than linearly (i.e. the second derivative of the polarization with respect to temperature is positive) with the decreasing temperature throughout the whole smectic C* range (see Figure 1).[7] In this material the tilt angle rapidly saturates while the ratio of polarization and tilt possesses a very strong nonlinear temChemPhysChem 2014, 15, 1368 – 1375
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Figure 1. Temperature dependence of the spontaneous polarization Ps and the tilt angle q in the ferroelectric liquid crystal E6 (see Scheme 1). While the director tilt angle q appears stepwise in a first-order phase transition, the spontaneous electric polarization Ps grows continuously but faster than linearly with decreasing temperature T (measured on cooling).
www.chemphyschem.org cally coincides with the SmA*-SmC* phase transition. The polarization-sign inversion, induced by a change of temperature, has been observed before in few one-component smectic C* liquid crystals[11, 12] and in binary mixtures.[13] Such an inversion has originally been interpreted in terms of the coexistence of the two conformers, which make contributions of opposite signs to the total polarization.[14] Another interpretation of the polarization-sign inversion is based on the theory that describes a competition between dipolar and quadrupolar contributions to the polarization,[15] which may possess different temperature variations as described in refs. [16] and [17]. One notes, however, that in the existing theory the temperature variation of both dipolar and quadrupolar contributions to the spontaneous polarization are determined by the temperature variation of the tilt angle. However, in the pure material E6 studied in ref. [8] and in binary mixtures investigated in this paper, the tilt angle saturates rapidly and possesses a very weak temperature dependence which cannot be responsible for the sign inversion. In this paper, the polarization sign inversion in binary mixtures of E6 and the achiral SmC material C8Cl, including the concentration dependence of the inversion temperature, is explained using the molecular theory of ferroelectric ordering.[8–10]
perature dependence and increases by a factor of 10 within the SmC* phase range. Clearly, such a strong temperature variation of the polarization cannot be explained by the weak variation of the tilt. The unusual temperature variation of the spontaneous polarization in the nanosegregated mesogen E6 has been explained in the literature[8] using a recently developed molecular 2. Results and Discussion theory of ferroelectric ordering in the smectic C* phase[9, 10] 2.1. Experimental Results which takes into consideration both polar and nonpolar biaxial ordering of short molecular axes. It has been shown[8] that Two binary mixtures containing the siloxane-terminated smecfaster than linear temperature variation of the polarization/tilt tic mesogen E6, which possesses the anomalous temperature ratio is determined by strong polar intermolecular interactions. variation of the spontaneous polarization in the smectic C* These polar interactions define a transition temperature T0 into phase, were investigated using a number of experimental techthe virtual proper ferroelectric phase, in which polarization apniques. In the first mixture, the material E6 was mixed with the pears in self-consistent way and polarization is the primary achiral smectic liquid crystal C8Cl, which has a similar structure order parameter. One notes, however, that T0 appears to be including the terminal siloxane group. The second mixture lower than the crystallization temperature, and thus the contains E6 mixed with a conventional smectic material C9proper ferroelectric phase cannot be observed. In this case the 2PhP which does not contain a siloxane group. Chemical strucanomalously strong increase of the polarization with the detures and phase transition temperatures of E6, C8Cl and C9creasing temperature can be considered as a “pretransitional 2PhP are presented in Scheme 1. Figure 2 shows the phase dieffect” provided that the temperature T0 is sufficiently close to agrams of both binary mixtures possessing a wide SmC phase the smectic C* range. at a mole fraction of x(E6) 0.8. In this paper, the unusual polarization properties of a nanoAccurate measurements of smectic layer spacing as a funcsegregated de-Vries-type liquid crystal has been further investition of temperature were carried out for both mixtures by gated by studying binary mixtures of different concentrations using small-angle X-ray scattering (SAXS). The results are preof E6 in the achiral SmC material C8Cl, which has a similar chemical structure (see Scheme 1). It is shown that all mixtures exhibit a temperature-induced sign inversion of the spontaneous polarization, and the inversion temperature shifts towards the SmC*-SmA* transition temperature linearly with increasing concentration of E6. In the one component system of pure E6 Figure 2. Phase diagram of the mixtures E6/C8Cl (a) and mixture E6/C9-2PhP (b) measured on heating. The inverthe inversion temperature practi- sion temperature Ti (open circles) was determined by electro-optical measurements. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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Figure 4. Temperature dependence of the apparent tilt angle q (dotted line) and the spontaneous polarization Ps (solid line) of E6/C8Cl x = 0.56 measured on heating.
Figure 3. Temperature dependence of the smectic layer thickness d of E6/ C8Cl (a) and E6/2PhP-C9 (b) measured on heating.
ture passing through zero and changing sign 31 K below the transition point. At the same time, the apparent tilt angle qapp drops to zero. Since the apparent tilt is measured as the (half) reorientation angle of the optic axis during ferroelectric switching, this is a direct result of the incomplete ferroelectric switching at low moduli of spontaneous polarization. At incomplete switching, the apparent tilt lacks behind the true SmC* optical tilt angle and reaches even zero at zero Ps. Figure 5 shows the textures of the E6/C8Cl mixture at x(E6) = 0.56 at different temperatures in a surface-stabilized liquid crystal cell at the border line of the electrode surface with applied DC-field. At low temperature, the polarization is positive and the texture in the electric field E is homogenous
sented in Figure 3. In the first mixture, the layer spacing increases linearly with decreasing concentration of the shorter molecule E6 according to the Diele additivity rule[19] (see Figure 3 a). In the second mixture, the molecules have the same length, but the pure material E6 form a partial double layered structure and thus the layer spacing is larger than the layer spacing of the Material 2PhP-C9, which form a monolayered smectic phase (see Figure 3 b). So in the mixture the partial double layered structure is stable until a concentration of x(E6) > 0.5 and turns to a monolayered structure at smaller concentration of E6, which is reflected by a rapid decrease of the Figure 5. Textures in the polarization microscope of E6/C8Cl, x = 0.56 in an SSFLC cell at the border line of the layer thickness. electrode surface with an applied DC field of E = 2.4 V mm1. a) At (TTAC) = 56 K, the spontaneous polarization The spontaneous polarization vector Ps, the layer normal k, and the director n form a right-handed system, and Ps has a positive sign by definition. b) At (TTAC) = 36 K, the polarization is close to zero, and c) at (TTAC) = 6 K, Ps has a negative sign. Ps and the optical tilt angle q were measured by a high-resolualigned in one tilt direction and the spontaneous polarization tion temperature-scanning technique[20] in both mixtures. SurPs, the layer normal k and director n the build a right handed prisingly, all mixtures with a mole fraction of x(E6) 0.2 showed a temperature-dependent inversion of the sign of Ps in system (Figure 5 a). At higher temperatures close to the A–C transition temperature, the polarization is negative. As Ps the smectic C* phase. The results for the mixture E6/C8Cl at x(E6) = 0.56 are presented in Figure 4. One can readily see that always points in the direction of the applied field, the tilt direcdirectly below the SmA*-SmC* transition point, the polarization tion changes because Ps, k and n build a left-handed system is negative and relatively small. The polarization then reaches (see Figure 5 c). Close to the inversion temperature, the electric a local minimum and increases with the decreasing temperafield is not able to switch between the two tilted states prop 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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erly because the polarization is close to zero. So the sample turns into many small domains of different tilt direction (see Figure 5 b). The outcome of this is the rapid apparent change of the tilt angle in the vicinity of the sign inversion temperature in Figure 6, which can be used to localize the inversion temperatures very precisely.
Figure 7. Temperature dependence of the spontaneous polarization Ps in E6/ C8Cl (a) and E6/C9-2PhP (b) measured on heating.
Figure 6. Temperature dependence of the apparent tilt qapp angle of E6/ C8Cl (a) and E6/C9-2PhP (b) measured on heating. The sign of the apparent tilt angle was changed at the inversion temperature for reasons of clarity.
Temperature variations of the spontaneous polarization and the apparent tilt angle in both mixtures at different concentrations of E6 are presented in Figures 6 and 7. One can readily see that a polarization-sign inversion is observed for all x. In the mixture E6/C8Cl, the sign-inversion temperature increases with increasing concentration of E6. A linear fitting of the reduced inversion temperature TACTi(x) for E6/C8Cl mixtures is presented in Figure 8. Extrapolating the inversion temperature to x = 1.0 indicates that for the pure E6, it meets the SmA*SmC* phase transition temperature. As also shown in Figure 8, the SmA* smectic layer spacing 4 K below the transition temperature in E6/C8Cl mixtures decreases in line with the polarization sign inversion temperature and can be fitted by a linear curve. However, in contrast to the E6/C8Cl mixtures, the dependence of the inversion temperature Ti on the concentration x(E6) in the E6/C9-2PhP mixtures is not monotonous (see Figure 9). As shown in Figure 9, for x > 0.5 the dependence TACTi(x) can be fitted by a linear curve similar to the one presented in Figure 8, that is, the inversion temperature increases with increasing x. Extrapolation of the inversion temperature 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 8. Linear fitting of the reduced inversion temperature (TACTi) (open symbols, solid line) and the smectic layer thickness d measured 4 K below the clearing point d (TTiso = 4 K) (full symbols, dashed line) versus the molar fraction x of E6 in C8Cl.
to x = 1 again results in the SmA*-SmC* phase transition temperature for pure E6 (solid line). But for x < 0.5 the inversion temperature decreases approximately linearly with increasing x. Thus the variation of the polarization inversion temperature in E6/C9-2PhP changes qualitatively around x = 0.5. Also the layer spacing increases at low x, for x < 0.5, and decreases for x > 0.5 as can be seen dashed line in Figure 9. Thus the qualitative change in the variation of the layer spacing approximately coincides with that of the polarization sign inversion temperature. This may be interpreted as a transition from the monolayer smectic structure at low concentration of E6 to the partially bilayer structure at large concentration of E6. ChemPhysChem 2014, 15, 1368 – 1375
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www.chemphyschem.org V ¼ hsin 2g cos 2i, which is proportional to the tilt angle q at small tilt with the polar and azimuthal angles g and f, and on the biaxial order parameter G ¼ hða kÞðb tÞi, where the unit vector k is the smectic layer normal and the unit vector t is in the direction of the tilt. One notes that the biaxial order parameter G is related to the magnitude of the spontaneous polarization.[3] Indeed, the spontaneous polarization is parallel to ðk tÞ and thus the modulus Ps ¼ 1Dm hðða bÞðk tÞÞi can be expressed as [Eq. (5)]: Ps ¼ 1Dm hðða kÞðb tÞÞi hðða tÞðb kÞÞi
Figure 9. Linear fitting of the reduced inversion temperature (TACTi) (open symbols, solid line) and the smectic layer thickness d measured 4 K below the clearing point (full symbols, dashed line) versus the molar fraction x of E6 in C9-2PhP.
According to the general molecular field theory of ferroelectric ordering in the SmC*-phase,[9, 10] the spontaneous polarization is given by the following expression [Eq. (1)]: Ps ¼ 1hm? i¼ 1
Z
m? f1 ða; bÞdadb Z ¼ 1Dm ða bÞf1 ða; bÞdadb
One can readily see that the first term in Equation (5) is proportional to the order parameter G and thus G can be written in the form [Eq. (6)]: G¼
2.2. Molecular Theory of Polarization Sign Inversion
1 VG þ Ps 11 D1 m 2
Now the mean-field potential can be expressed as [Eq. (8)]:
ð2Þ
Here, Z is the normalisation constant and UMF ða; bÞ is the mean-field potential which can be written in the form [Eq. (3)]: UMF ða; bÞ ¼ 1
Z
f1 ða2 ; b2 ÞU1;2 d 2 Rda1 db2
ð3Þ
where U1;2 is the total interaction potential between biaxial molecules. A simple expression for the mean-field potential has been obtained using the dipole–dipole interaction model considered in ref. [10] [Eq. (4)]: h pffiffiffi i UMF ða; bÞ ¼ UMF ðaÞ þ w5 AV þ 2G ða kÞðb tÞ
ð4Þ
where the first term depends only on the orientation of the long molecular axis a and the constant w5 determines the strength of the dipole-dipole interaction. The mean field potential depends on the uniaxial tilt order parameter 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
ð7Þ
ð1Þ
UMF ða; bÞ ¼ UMF ðaÞ þ w5 ½Aq þ BPs þ CVG ða kÞðb tÞ
1 U ða; bÞ exp MF Z kB T
ð6Þ
where VG is the biaxial tilt order parameter introduced in ref. [21] [Eq. (7)]: VG ¼ ðhða kÞðb tÞi hðða tÞðb kÞÞiÞ
where Dm ¼ jm? ða bÞj, 1 is the number density of molecules per unit volume, and f1(a,b) is the orientational distribution function (ODF) of biaxial molecules with long and short axes a and b, respectively. The ODF can be expressed as [Eq. (2)]:[10, 11] f ða; bÞ ¼
ð5Þ
ð8Þ
pffiffiffi pffiffiffi 1 1 where A = cota, B ¼ 2=2 1 Dm and C ¼ 2=2.[10] It should be noted that the mathematical form of the mean-field potential [Eq. (8)] is not necessarily related to the electrostatic dipole-dipole intermolecular interaction. In fact it can be obtained taking into account any polar interaction including the steric one between bent molecules. Assuming that the coupling constant w5 is small compared with kT the exponent in Eq.(2) can be expanded in powers of w5 taking into account only the linear term. As a result the polarization can approximately be expressed as [Eq. (9)]: Ps ¼
Wk ½Aq þ BPs þ CVG kB T
ð9Þ
where W ¼ w5 1Dm and the constant k ¼ hða kÞðb tÞi0 is a function of the nematic order parameter.[10] As shown in ref. [10], the coupling constant can be estimated as w5 / 1m2 r1 D, where 1 is the 2D molecular number density in the smectic layer, m is the molecular dipole, r is the average intermolecular distance within the layer and D is the numerical coefficient of order 1. For typical values of the transverse dipole m 1–2 Debye this constant is much smaller then one. In fact, values between 0.05 and 0.2 have been used in modelling of real mixtures in ref. [10].[10] From Equation (9) one readily obtains the following explicit expression for the spontaneous polarization [Eq. (19)]: ChemPhysChem 2014, 15, 1368 – 1375
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WkAV=kB WkCVG =kB þ T T0 T T0
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where we have introduced the characteristic temperature T0 ¼ WkAV=kB , which has the meaning of the temperature of the transition into the virtual proper ferroelectric phase.[8] The biaxial tilt-order parameter VG can be obtained in a similar way by expanding the orientational distribution function in the averages in Equation (7) [Eq (11)]: VG
Wk ½AV þ BPs þ CVG kB T
ð11Þ
and hence [Eq. (12)]: k
VG
W kB ½AV þ BPs T T0
ð12Þ
Substituting Equation (12) into Equation (10) and expanding the resulting expression to the second order in WkB/T one obtains the approximate expression for the spontaneous polarization of the smectic C* with strong dipolar interactions derived in ref. [3] [Eq. (13)]: Ps A0 B0 þ q T T0 ðT T0 Þ2
ð13Þ
where A0 ¼ WkA=kB and B0 ¼ ðWk=kB Þ2 AC. It is interesting to note that Equation (13) contains naturally the polarization sign inversion. Indeed, Equation (13) can be rewritten in the form [Eq. (14)]:
Ps A0 B0 ðT T0 Þ þ q T T0 A0
ð14Þ
One can readily see that according to Equation (14), the polarization vanishes and changes sign at the temperature [Eq. (15)]: Ti ¼ T0
B0 kWAC ¼ T0 A0 kB
ð15Þ
In the previous paper,[3] the parameters A0, B0 and T0 have been determined by fitting the experimental temperature variation of the spontaneous polarization Ps(T). It has been found that T0 100 K, B0/A0 300 K. One notes that the parameters A0 and B0 possess opposite signs and thus the inversion temperature Ti is larger than T0. Using these values one obtains, that is, the inversion temperature appears to be very close to the experimentally observed SmA-SmC transition temperature in E6. This explains why the polarization starts to grow approximately from zero despite the first order character of the SmASmC transition in E11 (which manifests itself in a significant jump of the tilt angle). Now let us consider the dependence of the inversion temperature on the concentration of non-chiral molecules in the 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
binary mixtures of E6 and the non-chiral material C8Cl. One notes that in Equation (15) T0 ¼ WkB=kB and therefore the inversion temperature is proportional to the parameter W which, in turn, is proportional to the number density of molecules 1 as W = w51Dm, that is, Ti / 1. The non-chiral molecules added to a chiral smectic C* do not contribute directly to the polarization. In the first approximation they just dilute the system, in that they decrease the number density of chiral molecules which can now be expressed as 1 = 10·x, where x is the volume fraction of chiral molecules. Thus the polarization inversion temperature is proportional to 1(x): that is, it increases linearly with increasing concentration of chiral molecules according to the experiment. If the polar interaction between chiral and non-chiral molecules cannot be neglected, the total mean-field potential contains a contribution from an interaction between chiral molecules and another contribution from an interaction between chiral and non-chiral molecules in the mixture. Then the meanfield potential in the mixture can be expressed as [Eq. (16)]: UMF ða; bÞ ¼UMF ðaÞþ ð1ch wcc þ 1nc wcn Þ½AV þ BPs þ CVG ða bÞðb tÞ
ð16Þ
where 1ch, 1cn are the number densities of chiral and non-chiral molecules per unit volume, respectively, wcc is the interaction constant which characterises the strength of polar interaction between two chiral molecules and wcn is the corresponding polar interaction constant between chiral and non-chiral molecules. In a incompressible fluid 1ch ¼ 10 ð1 x Þ and 1nc ¼ 10 x, where x is the molar fraction of non-chiral molecules and 10 is the average number density. Assuming that wcn > wcc one concludes that the mean-field potential for a chiral molecule in the binary mixture is approximately proportional to the factor 10 ðwcn þ x ðwcn wcc ÞÞ which increases with increasing x. For relatively small w values, the spontaneous polarization of a mixture of chiral and non-chiral molecules is given by the same Equations (9) and (10) as for the one-component system but with the parameter w = w(x) which now depends on the concentration of chiral molecules: wðxÞ ¼ 10 ðwcc þ x ðwcn wcc ÞÞ. Using all the same arguments as above one obtains [Eqs. (17) and (18)]: T0 ¼
WðxÞkB kB10 ðwcc þ x ðwcn wcc ÞÞ ¼ kB kB
B0 WðxÞkAC kAC10 ðwcc þ x ðwcn wcc ÞÞ ¼ ¼ kB kB A0
ð17Þ ð18Þ
Then, the polarization sign inversion temperature is also proportional to the factor 10 ðwcc þ x ðwcn wcc ÞÞ which is a growing function of x if wcn > wcc . Thus, the polarization sign inversion temperature increases approximately linearly with increasing concentration of chiral molecules if the corresponding polar interactions between chiral molecules (i.e. between E6 molecules in our mixture) are stronger than that between chiral and non-chiral molecules. Such a concentration dependence of the inversion temperaChemPhysChem 2014, 15, 1368 – 1375
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CHEMPHYSCHEM ARTICLES ture is observed experimentally at high molar fractions of E6 in both of E6/C8Cl and of E6/C9-2PhP binary mixtures (see Figs. 8 and 9). In the E6/C9-2PhP mixture, however, the concentration dependence of the inversion changes dramatically at some intermediate molar fraction of E6 as presented in Figure 9. At low molar fractions of E6, the inversion temperature decreases approximately linearly with the increasing concentration of E6. This behavior can be understood qualitatively if one takes into account that the ratio of the effective interaction constants wcn =wcc may change due to the change of structure from the partially bilayer to the monolayer one because monomer-monomer interactions should generally be different from monomer-dimer interactions. Such a change of structure apparently occurs in the E6/C9-2PhP mixture at the molar fraction of E6, which corresponds to the change of the type of concentration dependence of the inversion temperature.
3. Conclusions We have studied the ferroelectric properties of binary mixtures of the siloxane-terminated smectic material E6 and two different achiral smectic materials. The structure of the first achiral liquid crystal C8Cl is similar to that of E6 while the second material C9-2PhP has a conventional structure without a siloxane group. It has been shown that in both mixtures, one observes the sign inversion of the spontaneous polarization induced by a change of temperature. In the E6/C8Cl mixture, the interval between the sign inversion temperature and the SmA-SmCtransition temperature decreases approximately linearly with the increasing molar fraction of E6. Extrapolation of the reduced inversion temperature to x(E6) = 1 indicates that for the pure E6 the inversion temperature is approximately equal to the SmA*-SmC* phase transition temperature. In contrast, in E6/C9-2PhP mixtures, the reduced inversion temperature increases with increasing x for x(E6) > 0.5 and decreases for x(E6) < 0.5. Simultaneously, the dependence of the smectic layer spacing on the molar fraction of E6 also changes qualitatively, approximately at x(E6) = 0.5, while in the E6/C8Cl mixture, the layer spacing decreases linearly with increasing x(E6), in line with inversion temperature. This qualitative change of behavior in E6/C9-2PhP mixtures can be attributed to a possible transition from the monolayer smectic structure (characteristic for pure C9-2PhP) to the bilayer structure typical for mixtures with large fraction of E6. The sign inversion of the spontaneous polarization in these mixtures has been explained theoretically using the recently developed molecular-statistical theory of ferroelectric ordering in chiral smectics C*.[8–10] It has been shown in our previous paper[8] that the faster than linear growth of the spontaneous polarization in pure E6 is determined by the “pre-transitional effect” which resembles the Curie–Weiss behavior of the dielectric constant above the virtual transition into the proper ferroelectric phase. According to Equations (10) and (13), the spontaneous polarization is proportional to the factor (TT0)1, where T0 is the characteristic temperature determined by polar intermolecular interactions which has the meaning of the virtual transition temperature into the proper ferroelectric phase. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
www.chemphyschem.org One notes that T0 appears to be lower than the crystallization temperature, and thus the proper ferroelectric phase cannot be observed. At the same time, if T0 is not too low, the factor (TT0)1 makes a significant contribution to the temperature variation of the spontaneous polarization and is responsible for unusual convex curvature of the function Ps(T) in E6.[8] According to Equation (10), the spontaneous polarization is a sum of two terms which have different origins. The first term is proportional to the tilt order parameter V, which describes the tilt of long molecular axes, and which is approximately equal to the primary tilt angle at low tilt. The second term is proportional to the biaxial order parameter VG which also vanishes in the SmA phase. The biaxial order parameter is also proportional to the pre-transitional factor (TT0)1 and as a result the two terms in Equation (10) for the spontaneous polarization are characterized by different temperature variation. Moreover, fitting of the temperature variation of Ps in pure E6 with Equation (13) results in opposite signs of the coupling constants A and B that is, the two terms possess opposite signs. This enables one to explain the polarization sign inversion in mixtures of E6 with other smectic materials. As discussed above, in mixtures of E6 with non-chiral molecules, the second term in Equation (13) for the spontaneous polarization is smaller than the first one. However, the second term grows faster with decreasing temperature due to the additional factor of (TT0)1. As a result, the two terms compensate each other at some polarization inversion temperature Ti which appears to be an increasing function of the concentration of chiral molecules in the mixture provided the polar interaction between two chiral molecules is stronger than the interaction between chiral and non-chiral ones. This explains why the inversion temperature increases linearly with increasing concentration of E6 in the binary mixtures E6/C8Cl for all molar fractions of E6 and for large concentration of E6 in the mixtures E6/C9-2PhP. The linear decrease of the inversion temperature observed for small concentration of E6 in mixtures of E6 with C9-2PhP may be explained by the change from the partially bilayer to the monolayer structure which may result in an increase of effective interactions between chiral and non-chiral molecules. In principle, one cannot completely exclude other explanations of the polarization sign inversion including the one based on the model of a temperature-dependent balance between two or more conformations of the chiral E6 molecules which make contributions of opposite signs to the total spontaneous polarization.[11] This assumption, however, requires an additional experimental confirmation. One notes also that from our point of view, it is not straightforward at all to use the conformation model to explain the concentration dependence of the inversion temperature in binary mixtures as it requires too many assumptions concerning the effect of non-chiral molecules on the balance between chiral conformations. Another explanation of the polarization sign inversion in ferroelectric smectic C* is based on the model which takes into consideration a competition between dipolar and quadrupolar contributions to the polarization.[13, 15–17] The dipolar contribution is proportional to the tilt angle q while the quadrupolar contribuChemPhysChem 2014, 15, 1368 – 1375
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CHEMPHYSCHEM ARTICLES tion, which may possess the opposite sign, is proportional to the product of q and the biaxial (“quadrupole”) order parameter which increases like q2 at small tilt. Thus in this model the polarization sign inversion is determined by the temperature variation of the tilt angle. One notes, however, that in pure E6 and in the corresponding binary mixtures the tilt angle rapidly saturates below the SmA*-SmC* transition and then possesses a very weak temperature dependence. On the other hand, the theory of the polarization sign inversion presented in this paper is based on the same general ideas as the one developed in ref. [15]. Indeed, the spontaneous polarization is also expressed as a sum of two terms where the first term is proportional to the tilt order parameter while the second term is proportional both to the tilt and to the biaxial order parameter VG. The difference from the previous models is determined by the origin of the temperature variation of the biaxial order parameter which is not related to that of the tilt angle (outside a narrow interval below the SmA*SmC* transition) but is determined by the “pre-transitional” effect discussed above. Thus in the context of the present model the polarization sign inversion is intimately related to the unusual faster than linear temperature variation of the spontaneous polarization in E6 and in the corresponding binary mixtures. It is interesting to note that a similar but less pronounced temperature variation of the polarization has also been observed in another smectic C* material with polarization sign inversion.[11]
Experimental Section A Mettler Toledo DSC822 was used for differential scanning calorimetric measurements, on cooling and on heating at 5 K min1. Polarizing optical microscopy (POM) was conducted with an Olympus BH 2 polarizing microscope combined with a Linkam LTS 350. Xray experiments were performed on a SAXSess small angle x-ray scattering system from Anton Paar GmbH. The unaligned samples (filled into Hilgenberg Mark capillary tubes of 0.7 mm diameter) were kept in a temperature-controlled sample-holder unit (TSC 123). The X-ray beam coming from a ceramic tube generator was focused by a bent-multilayer mirror and shaped by a line collimation block. The scattered radiation were recorded with a CCD detector (KAF 2084 2083 SCX) and processed and analysed automated with a SAXSquant 3.5 software. The surface-stabilized ferroelectric liquid crystal (SSFLC) films were prepared in indium tin oxide (ITO) glass cells with single-side rubbed nylon (1.6 mm spacing, 0.25 cm2 addressed area, WAT PPW, Poland) alignment layers. Temperature-dependent measurements of the spontaneous polarization Ps and the director tilt angle q were performed with an electro-optic setup which was described previously in ref. [22], using a 70 Hz AC field with 18.75 V mm1 amplitude.
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www.chemphyschem.org Acknowledgements This work has been undertaken in the framework of the Materials World Network. M.A.O. is grateful to EPSRC (UK) (Grant EP/ H046941/1) for funding, R.P.L. acknowledges the support from NSERC (Canada) and F.G. and D.N. acknowledges support from the Deutsche Forschungsgemeinschaft (Germany). M.A.O. is also grateful to Leverhulme Trust for financial support which has enabled him to stay in Stuttgart. Keywords: chirality · liquid crystals · phase transitions · polarization · statistical mechanics
[1] N. A. Clark, S. T. Lagerwall, Appl. Phys. Lett. 1980, 36, 899; N. A. Clark, S. T. Lagerwall, Ferroelectrics 1984, 25, 59. [2] J. P. F. Lagerwall, F. Giesselmann, ChemPhysChem 2006, 7, 20. [3] N. A. Clark, T. Bellini, R. Shao, D. Coleman, S. Bardon, D. R. Link, J. Maclennan, X. H. Chen, M. D. Wand, D. M. Walba, P. Rudquist, S. T. Lagerwall, Appl. Phys. Lett. 2002, 80, 4097. [4] J. P. F. Lagerwall, F. Giesselmann, M. D. Radcliffe, Phys. Rev. E 2002, 66, 031703. [5] N. Hayashi, T. Kato, A. Fukuda, J. K. Vij, Yu. Panarin, J. Naciri, R. Shashidar, S. Kawada, S. Kondoh, Phys. Rev. E 2005, 71, 041705. [6] H. F. Gleeson, Y. Wang, S. Watson, D. Sahagun-Sanchez, J. W. Goodby, M. Hird, A. Petrenko, M. A. Osipov, J. Mater. Chem. 2004, 14, 1480. [7] Y. Takanishi, Y. Ouchi, H. Takezoe, et al., Jpn. J. Appl. Phys. Lett. 1990, 29, L984. [8] D. Nonnenmacher, M. A. Osipov, J. C. Roberts, R. P. Lemieux, F. Giesselmann, Phys. Rev. E 2010, 82, 031703. [9] M. A. Osipov, M. V. Gorkunov, Phys. Rev. E 2008, 77, 031701. [10] M. V. Gorkunov, M. A. Osipov, H. F. Gleeson, S. Jaradat, Eur. Phys. J. E 2008, 26, 395. [11] J. W. Goodby, E. Chin, J. M. Geary, J. S. Patel, P. L. Finn, J. Chem. Soc. Faraday Trans. I 1987, 83, 3429. [12] R. Eidenschink, T. Geelhaar, G. Andersson, A. Dahlgren, K. Flatischler, F. Gouda, S. T. Gagerwall, K. Skarp, Ferroelectrics 1988, 84, 167. [13] M. A. Osipov, H. Stegemeyer, A. Sprick, Phys. Rev. E 1996, 54, 6387. [14] J. S. Patel, J. W. Goodby, J. Phys. Chem. 1987, 91, 5838. [15] Z. Raszewski, J. Kedzierski, P. Perkowski, J. Rutkowska, W. Piecek, J. Zielinski, J. Zmija, R. Dabrowski, Mol. Cryst. Liq. Cryst. 1999, 328, 255. [16] H. Stegemeyer, F. Stockel, R. Meister, Liq. Cryst. 1998, 24, 531. [17] H. Stegemeyer, R. Meister, U. Hoffman, W. Kuczynski, Liq. Cryst. 1991, 10, 295. [18] J. C. Roberts, N. Kapernaum, Q. Song, D. Nonnenmacher, K. Ayub, F. Giesselmann, R. P. Lemieux, J. Am. Chem. Soc. 2010, 132, 364. [19] S. Diele, Ber. Bunsen-Ges. 1993, 97, 1326. [20] J. P. F. Lagerwall, A. Saipa, F. Giesselmann, R. Dabrowski, Liq. Cryst. 2004, 31, 1175. [21] M. V. Gorkunov, M. A. Osipov, J. Phys. A 2008, 41, 295001. [22] A. Langhoff, F. Giesselmann, ChemPhysChem 2002, 3, 424. Received: December 4, 2013 Published online on March 25, 2014
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Sign Inversion of the Spontaneous Polarization in a “de Vries”-Type Ferroelectric Liquid Crystal Dorothee Nonnenmacher,[a] Robert P. Lemieux,[b] Mikhail A. Osipov,*[c] and Frank Giesselmann*[a] In contrast to common ferroelectric smectic C* liquid crystals, the siloxane-terminated smectic mesogen E6 is characterized by an unusual temperature variation of the spontaneous polarization. The polarization starts to grow from nearly zero despite the first-order SmA*-SmC* transition, and increases faster than linearly over a large temperature interval while the tilt angle rapidly saturates. To study this behavior in more detail, binary mixtures of different concentrations of E6 in the achiral SmC material C8Cl, which has a similar chemical structure, were investigated. Surprisingly, all mixtures show a temperature dependent polarization sign inversion, which shifts towards
the SmC*-SmA* transition with increasing E6 concentration. For the pure E6 the inversion temperature meets the SmA*SmC* phase transition temperature. In a second binary mixture with E6 and a conventional material C9–2PhP we found out, that the dependence of the inversion temperature on the concentration of E6 changes qualitatively when the nanosegregation is partially destroyed. A molecular theory of the polarization sign inversion in smectics C* with strong polar intermolecular interactions is developed which enables one to explain the concentration dependence of the inversion temperature in both mixtures.
1. Introduction In ferroelectric smectic C* liquid crystals the macroscopic polarization does not appear self-consistently, but is induced by the tilt of the director in a chiral medium.[1] Thus, the polarization appears to be the secondary order parameter while the tilt is the primary one, and therefore the ferroelectric ordering in chiral tilted smectics may be called improper. Ferroelectric liquid crystals possess some very interesting polarization properties related to molecular chirality and polarity, and these materials also attract significant attention because of their applications in the new generation of fast electro-optic displays.[2] In recent years the interest has been focused upon the socalled de Vries type ferroelectric smectic C* liquid crystals which are characterized by anomalously weak layer contraction, large values of the electroclinic coefficient,[3] low orientational order[4, 5] and other unusual properties. The de Vries type smectic materials are particularly important for applications because they enable one to avoid the so-called “zig-zag” defects which appear due to a buckling of layers in the bookshelf geometry determined by the tilt of the director in the smectic C phase. The majority of de Vries type materials possess a silox-
[a] D. Nonnenmacher, Prof. Dr. F. Giesselmann Institute of Physical Chemistry, University of Stuttgart Pfaffenwaldring 55, 70569 Stuttgart (Germany) E-mail: [email protected] [b] Prof. Dr. R. P. Lemieux Department of Chemistry, Queen’s University Kingston, Ontario K7L3N6 (Canada) [c] Prof. Dr. M. A. Osipov Department of Mathematics, University of Strathclyde 26 Richmond Street, Glasgow G1 1XH (UK) E-mail: [email protected]
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ane or a fluorinated terminal chain which has a tendency to separate from the rest of the molecule. Thus the effect of nanosegregation is expected to play a significant role in these systems. Ferroelectric properties of de-Vries-like smectics are also rather unusual. In contrast to a simple phenomenological theory,[1] in some cases the spontaneous polarization of smectics C with weak layer contraction is not proportional to the tilt angle over any significant temperature interval, and it continues to grow when the tilt angle saturates.[6] Moreover, the nanosegregated siloxane-terminated mesogen with chiral (R,R)2,3-epoxyoctyloxy side chains (E6) (see Scheme 1) exhibits the
Scheme 1. Chemical structures and phase-transition temperatures (in 8C) of E6, C8Cl,[18] and C9–2PhP.
spontaneous polarization which continues to grow faster than linearly (i.e. the second derivative of the polarization with respect to temperature is positive) with the decreasing temperature throughout the whole smectic C* range (see Figure 1).[7] In this material the tilt angle rapidly saturates while the ratio of polarization and tilt possesses a very strong nonlinear temChemPhysChem 2014, 15, 1368 – 1375
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Figure 1. Temperature dependence of the spontaneous polarization Ps and the tilt angle q in the ferroelectric liquid crystal E6 (see Scheme 1). While the director tilt angle q appears stepwise in a first-order phase transition, the spontaneous electric polarization Ps grows continuously but faster than linearly with decreasing temperature T (measured on cooling).
www.chemphyschem.org cally coincides with the SmA*-SmC* phase transition. The polarization-sign inversion, induced by a change of temperature, has been observed before in few one-component smectic C* liquid crystals[11, 12] and in binary mixtures.[13] Such an inversion has originally been interpreted in terms of the coexistence of the two conformers, which make contributions of opposite signs to the total polarization.[14] Another interpretation of the polarization-sign inversion is based on the theory that describes a competition between dipolar and quadrupolar contributions to the polarization,[15] which may possess different temperature variations as described in refs. [16] and [17]. One notes, however, that in the existing theory the temperature variation of both dipolar and quadrupolar contributions to the spontaneous polarization are determined by the temperature variation of the tilt angle. However, in the pure material E6 studied in ref. [8] and in binary mixtures investigated in this paper, the tilt angle saturates rapidly and possesses a very weak temperature dependence which cannot be responsible for the sign inversion. In this paper, the polarization sign inversion in binary mixtures of E6 and the achiral SmC material C8Cl, including the concentration dependence of the inversion temperature, is explained using the molecular theory of ferroelectric ordering.[8–10]
perature dependence and increases by a factor of 10 within the SmC* phase range. Clearly, such a strong temperature variation of the polarization cannot be explained by the weak variation of the tilt. The unusual temperature variation of the spontaneous polarization in the nanosegregated mesogen E6 has been explained in the literature[8] using a recently developed molecular 2. Results and Discussion theory of ferroelectric ordering in the smectic C* phase[9, 10] 2.1. Experimental Results which takes into consideration both polar and nonpolar biaxial ordering of short molecular axes. It has been shown[8] that Two binary mixtures containing the siloxane-terminated smecfaster than linear temperature variation of the polarization/tilt tic mesogen E6, which possesses the anomalous temperature ratio is determined by strong polar intermolecular interactions. variation of the spontaneous polarization in the smectic C* These polar interactions define a transition temperature T0 into phase, were investigated using a number of experimental techthe virtual proper ferroelectric phase, in which polarization apniques. In the first mixture, the material E6 was mixed with the pears in self-consistent way and polarization is the primary achiral smectic liquid crystal C8Cl, which has a similar structure order parameter. One notes, however, that T0 appears to be including the terminal siloxane group. The second mixture lower than the crystallization temperature, and thus the contains E6 mixed with a conventional smectic material C9proper ferroelectric phase cannot be observed. In this case the 2PhP which does not contain a siloxane group. Chemical strucanomalously strong increase of the polarization with the detures and phase transition temperatures of E6, C8Cl and C9creasing temperature can be considered as a “pretransitional 2PhP are presented in Scheme 1. Figure 2 shows the phase dieffect” provided that the temperature T0 is sufficiently close to agrams of both binary mixtures possessing a wide SmC phase the smectic C* range. at a mole fraction of x(E6) 0.8. In this paper, the unusual polarization properties of a nanoAccurate measurements of smectic layer spacing as a funcsegregated de-Vries-type liquid crystal has been further investition of temperature were carried out for both mixtures by gated by studying binary mixtures of different concentrations using small-angle X-ray scattering (SAXS). The results are preof E6 in the achiral SmC material C8Cl, which has a similar chemical structure (see Scheme 1). It is shown that all mixtures exhibit a temperature-induced sign inversion of the spontaneous polarization, and the inversion temperature shifts towards the SmC*-SmA* transition temperature linearly with increasing concentration of E6. In the one component system of pure E6 Figure 2. Phase diagram of the mixtures E6/C8Cl (a) and mixture E6/C9-2PhP (b) measured on heating. The inverthe inversion temperature practi- sion temperature Ti (open circles) was determined by electro-optical measurements. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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Figure 4. Temperature dependence of the apparent tilt angle q (dotted line) and the spontaneous polarization Ps (solid line) of E6/C8Cl x = 0.56 measured on heating.
Figure 3. Temperature dependence of the smectic layer thickness d of E6/ C8Cl (a) and E6/2PhP-C9 (b) measured on heating.
ture passing through zero and changing sign 31 K below the transition point. At the same time, the apparent tilt angle qapp drops to zero. Since the apparent tilt is measured as the (half) reorientation angle of the optic axis during ferroelectric switching, this is a direct result of the incomplete ferroelectric switching at low moduli of spontaneous polarization. At incomplete switching, the apparent tilt lacks behind the true SmC* optical tilt angle and reaches even zero at zero Ps. Figure 5 shows the textures of the E6/C8Cl mixture at x(E6) = 0.56 at different temperatures in a surface-stabilized liquid crystal cell at the border line of the electrode surface with applied DC-field. At low temperature, the polarization is positive and the texture in the electric field E is homogenous
sented in Figure 3. In the first mixture, the layer spacing increases linearly with decreasing concentration of the shorter molecule E6 according to the Diele additivity rule[19] (see Figure 3 a). In the second mixture, the molecules have the same length, but the pure material E6 form a partial double layered structure and thus the layer spacing is larger than the layer spacing of the Material 2PhP-C9, which form a monolayered smectic phase (see Figure 3 b). So in the mixture the partial double layered structure is stable until a concentration of x(E6) > 0.5 and turns to a monolayered structure at smaller concentration of E6, which is reflected by a rapid decrease of the Figure 5. Textures in the polarization microscope of E6/C8Cl, x = 0.56 in an SSFLC cell at the border line of the layer thickness. electrode surface with an applied DC field of E = 2.4 V mm1. a) At (TTAC) = 56 K, the spontaneous polarization The spontaneous polarization vector Ps, the layer normal k, and the director n form a right-handed system, and Ps has a positive sign by definition. b) At (TTAC) = 36 K, the polarization is close to zero, and c) at (TTAC) = 6 K, Ps has a negative sign. Ps and the optical tilt angle q were measured by a high-resolualigned in one tilt direction and the spontaneous polarization tion temperature-scanning technique[20] in both mixtures. SurPs, the layer normal k and director n the build a right handed prisingly, all mixtures with a mole fraction of x(E6) 0.2 showed a temperature-dependent inversion of the sign of Ps in system (Figure 5 a). At higher temperatures close to the A–C transition temperature, the polarization is negative. As Ps the smectic C* phase. The results for the mixture E6/C8Cl at x(E6) = 0.56 are presented in Figure 4. One can readily see that always points in the direction of the applied field, the tilt direcdirectly below the SmA*-SmC* transition point, the polarization tion changes because Ps, k and n build a left-handed system is negative and relatively small. The polarization then reaches (see Figure 5 c). Close to the inversion temperature, the electric a local minimum and increases with the decreasing temperafield is not able to switch between the two tilted states prop 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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erly because the polarization is close to zero. So the sample turns into many small domains of different tilt direction (see Figure 5 b). The outcome of this is the rapid apparent change of the tilt angle in the vicinity of the sign inversion temperature in Figure 6, which can be used to localize the inversion temperatures very precisely.
Figure 7. Temperature dependence of the spontaneous polarization Ps in E6/ C8Cl (a) and E6/C9-2PhP (b) measured on heating.
Figure 6. Temperature dependence of the apparent tilt qapp angle of E6/ C8Cl (a) and E6/C9-2PhP (b) measured on heating. The sign of the apparent tilt angle was changed at the inversion temperature for reasons of clarity.
Temperature variations of the spontaneous polarization and the apparent tilt angle in both mixtures at different concentrations of E6 are presented in Figures 6 and 7. One can readily see that a polarization-sign inversion is observed for all x. In the mixture E6/C8Cl, the sign-inversion temperature increases with increasing concentration of E6. A linear fitting of the reduced inversion temperature TACTi(x) for E6/C8Cl mixtures is presented in Figure 8. Extrapolating the inversion temperature to x = 1.0 indicates that for the pure E6, it meets the SmA*SmC* phase transition temperature. As also shown in Figure 8, the SmA* smectic layer spacing 4 K below the transition temperature in E6/C8Cl mixtures decreases in line with the polarization sign inversion temperature and can be fitted by a linear curve. However, in contrast to the E6/C8Cl mixtures, the dependence of the inversion temperature Ti on the concentration x(E6) in the E6/C9-2PhP mixtures is not monotonous (see Figure 9). As shown in Figure 9, for x > 0.5 the dependence TACTi(x) can be fitted by a linear curve similar to the one presented in Figure 8, that is, the inversion temperature increases with increasing x. Extrapolation of the inversion temperature 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 8. Linear fitting of the reduced inversion temperature (TACTi) (open symbols, solid line) and the smectic layer thickness d measured 4 K below the clearing point d (TTiso = 4 K) (full symbols, dashed line) versus the molar fraction x of E6 in C8Cl.
to x = 1 again results in the SmA*-SmC* phase transition temperature for pure E6 (solid line). But for x < 0.5 the inversion temperature decreases approximately linearly with increasing x. Thus the variation of the polarization inversion temperature in E6/C9-2PhP changes qualitatively around x = 0.5. Also the layer spacing increases at low x, for x < 0.5, and decreases for x > 0.5 as can be seen dashed line in Figure 9. Thus the qualitative change in the variation of the layer spacing approximately coincides with that of the polarization sign inversion temperature. This may be interpreted as a transition from the monolayer smectic structure at low concentration of E6 to the partially bilayer structure at large concentration of E6. ChemPhysChem 2014, 15, 1368 – 1375
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www.chemphyschem.org V ¼ hsin 2g cos 2i, which is proportional to the tilt angle q at small tilt with the polar and azimuthal angles g and f, and on the biaxial order parameter G ¼ hða kÞðb tÞi, where the unit vector k is the smectic layer normal and the unit vector t is in the direction of the tilt. One notes that the biaxial order parameter G is related to the magnitude of the spontaneous polarization.[3] Indeed, the spontaneous polarization is parallel to ðk tÞ and thus the modulus Ps ¼ 1Dm hðða bÞðk tÞÞi can be expressed as [Eq. (5)]: Ps ¼ 1Dm hðða kÞðb tÞÞi hðða tÞðb kÞÞi
Figure 9. Linear fitting of the reduced inversion temperature (TACTi) (open symbols, solid line) and the smectic layer thickness d measured 4 K below the clearing point (full symbols, dashed line) versus the molar fraction x of E6 in C9-2PhP.
According to the general molecular field theory of ferroelectric ordering in the SmC*-phase,[9, 10] the spontaneous polarization is given by the following expression [Eq. (1)]: Ps ¼ 1hm? i¼ 1
Z
m? f1 ða; bÞdadb Z ¼ 1Dm ða bÞf1 ða; bÞdadb
One can readily see that the first term in Equation (5) is proportional to the order parameter G and thus G can be written in the form [Eq. (6)]: G¼
2.2. Molecular Theory of Polarization Sign Inversion
1 VG þ Ps 11 D1 m 2
Now the mean-field potential can be expressed as [Eq. (8)]:
ð2Þ
Here, Z is the normalisation constant and UMF ða; bÞ is the mean-field potential which can be written in the form [Eq. (3)]: UMF ða; bÞ ¼ 1
Z
f1 ða2 ; b2 ÞU1;2 d 2 Rda1 db2
ð3Þ
where U1;2 is the total interaction potential between biaxial molecules. A simple expression for the mean-field potential has been obtained using the dipole–dipole interaction model considered in ref. [10] [Eq. (4)]: h pffiffiffi i UMF ða; bÞ ¼ UMF ðaÞ þ w5 AV þ 2G ða kÞðb tÞ
ð4Þ
where the first term depends only on the orientation of the long molecular axis a and the constant w5 determines the strength of the dipole-dipole interaction. The mean field potential depends on the uniaxial tilt order parameter 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
ð7Þ
ð1Þ
UMF ða; bÞ ¼ UMF ðaÞ þ w5 ½Aq þ BPs þ CVG ða kÞðb tÞ
1 U ða; bÞ exp MF Z kB T
ð6Þ
where VG is the biaxial tilt order parameter introduced in ref. [21] [Eq. (7)]: VG ¼ ðhða kÞðb tÞi hðða tÞðb kÞÞiÞ
where Dm ¼ jm? ða bÞj, 1 is the number density of molecules per unit volume, and f1(a,b) is the orientational distribution function (ODF) of biaxial molecules with long and short axes a and b, respectively. The ODF can be expressed as [Eq. (2)]:[10, 11] f ða; bÞ ¼
ð5Þ
ð8Þ
pffiffiffi pffiffiffi 1 1 where A = cota, B ¼ 2=2 1 Dm and C ¼ 2=2.[10] It should be noted that the mathematical form of the mean-field potential [Eq. (8)] is not necessarily related to the electrostatic dipole-dipole intermolecular interaction. In fact it can be obtained taking into account any polar interaction including the steric one between bent molecules. Assuming that the coupling constant w5 is small compared with kT the exponent in Eq.(2) can be expanded in powers of w5 taking into account only the linear term. As a result the polarization can approximately be expressed as [Eq. (9)]: Ps ¼
Wk ½Aq þ BPs þ CVG kB T
ð9Þ
where W ¼ w5 1Dm and the constant k ¼ hða kÞðb tÞi0 is a function of the nematic order parameter.[10] As shown in ref. [10], the coupling constant can be estimated as w5 / 1m2 r1 D, where 1 is the 2D molecular number density in the smectic layer, m is the molecular dipole, r is the average intermolecular distance within the layer and D is the numerical coefficient of order 1. For typical values of the transverse dipole m 1–2 Debye this constant is much smaller then one. In fact, values between 0.05 and 0.2 have been used in modelling of real mixtures in ref. [10].[10] From Equation (9) one readily obtains the following explicit expression for the spontaneous polarization [Eq. (19)]: ChemPhysChem 2014, 15, 1368 – 1375
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WkAV=kB WkCVG =kB þ T T0 T T0
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where we have introduced the characteristic temperature T0 ¼ WkAV=kB , which has the meaning of the temperature of the transition into the virtual proper ferroelectric phase.[8] The biaxial tilt-order parameter VG can be obtained in a similar way by expanding the orientational distribution function in the averages in Equation (7) [Eq (11)]: VG
Wk ½AV þ BPs þ CVG kB T
ð11Þ
and hence [Eq. (12)]: k
VG
W kB ½AV þ BPs T T0
ð12Þ
Substituting Equation (12) into Equation (10) and expanding the resulting expression to the second order in WkB/T one obtains the approximate expression for the spontaneous polarization of the smectic C* with strong dipolar interactions derived in ref. [3] [Eq. (13)]: Ps A0 B0 þ q T T0 ðT T0 Þ2
ð13Þ
where A0 ¼ WkA=kB and B0 ¼ ðWk=kB Þ2 AC. It is interesting to note that Equation (13) contains naturally the polarization sign inversion. Indeed, Equation (13) can be rewritten in the form [Eq. (14)]:
Ps A0 B0 ðT T0 Þ þ q T T0 A0
ð14Þ
One can readily see that according to Equation (14), the polarization vanishes and changes sign at the temperature [Eq. (15)]: Ti ¼ T0
B0 kWAC ¼ T0 A0 kB
ð15Þ
In the previous paper,[3] the parameters A0, B0 and T0 have been determined by fitting the experimental temperature variation of the spontaneous polarization Ps(T). It has been found that T0 100 K, B0/A0 300 K. One notes that the parameters A0 and B0 possess opposite signs and thus the inversion temperature Ti is larger than T0. Using these values one obtains, that is, the inversion temperature appears to be very close to the experimentally observed SmA-SmC transition temperature in E6. This explains why the polarization starts to grow approximately from zero despite the first order character of the SmASmC transition in E11 (which manifests itself in a significant jump of the tilt angle). Now let us consider the dependence of the inversion temperature on the concentration of non-chiral molecules in the 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
binary mixtures of E6 and the non-chiral material C8Cl. One notes that in Equation (15) T0 ¼ WkB=kB and therefore the inversion temperature is proportional to the parameter W which, in turn, is proportional to the number density of molecules 1 as W = w51Dm, that is, Ti / 1. The non-chiral molecules added to a chiral smectic C* do not contribute directly to the polarization. In the first approximation they just dilute the system, in that they decrease the number density of chiral molecules which can now be expressed as 1 = 10·x, where x is the volume fraction of chiral molecules. Thus the polarization inversion temperature is proportional to 1(x): that is, it increases linearly with increasing concentration of chiral molecules according to the experiment. If the polar interaction between chiral and non-chiral molecules cannot be neglected, the total mean-field potential contains a contribution from an interaction between chiral molecules and another contribution from an interaction between chiral and non-chiral molecules in the mixture. Then the meanfield potential in the mixture can be expressed as [Eq. (16)]: UMF ða; bÞ ¼UMF ðaÞþ ð1ch wcc þ 1nc wcn Þ½AV þ BPs þ CVG ða bÞðb tÞ
ð16Þ
where 1ch, 1cn are the number densities of chiral and non-chiral molecules per unit volume, respectively, wcc is the interaction constant which characterises the strength of polar interaction between two chiral molecules and wcn is the corresponding polar interaction constant between chiral and non-chiral molecules. In a incompressible fluid 1ch ¼ 10 ð1 x Þ and 1nc ¼ 10 x, where x is the molar fraction of non-chiral molecules and 10 is the average number density. Assuming that wcn > wcc one concludes that the mean-field potential for a chiral molecule in the binary mixture is approximately proportional to the factor 10 ðwcn þ x ðwcn wcc ÞÞ which increases with increasing x. For relatively small w values, the spontaneous polarization of a mixture of chiral and non-chiral molecules is given by the same Equations (9) and (10) as for the one-component system but with the parameter w = w(x) which now depends on the concentration of chiral molecules: wðxÞ ¼ 10 ðwcc þ x ðwcn wcc ÞÞ. Using all the same arguments as above one obtains [Eqs. (17) and (18)]: T0 ¼
WðxÞkB kB10 ðwcc þ x ðwcn wcc ÞÞ ¼ kB kB
B0 WðxÞkAC kAC10 ðwcc þ x ðwcn wcc ÞÞ ¼ ¼ kB kB A0
ð17Þ ð18Þ
Then, the polarization sign inversion temperature is also proportional to the factor 10 ðwcc þ x ðwcn wcc ÞÞ which is a growing function of x if wcn > wcc . Thus, the polarization sign inversion temperature increases approximately linearly with increasing concentration of chiral molecules if the corresponding polar interactions between chiral molecules (i.e. between E6 molecules in our mixture) are stronger than that between chiral and non-chiral molecules. Such a concentration dependence of the inversion temperaChemPhysChem 2014, 15, 1368 – 1375
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CHEMPHYSCHEM ARTICLES ture is observed experimentally at high molar fractions of E6 in both of E6/C8Cl and of E6/C9-2PhP binary mixtures (see Figs. 8 and 9). In the E6/C9-2PhP mixture, however, the concentration dependence of the inversion changes dramatically at some intermediate molar fraction of E6 as presented in Figure 9. At low molar fractions of E6, the inversion temperature decreases approximately linearly with the increasing concentration of E6. This behavior can be understood qualitatively if one takes into account that the ratio of the effective interaction constants wcn =wcc may change due to the change of structure from the partially bilayer to the monolayer one because monomer-monomer interactions should generally be different from monomer-dimer interactions. Such a change of structure apparently occurs in the E6/C9-2PhP mixture at the molar fraction of E6, which corresponds to the change of the type of concentration dependence of the inversion temperature.
3. Conclusions We have studied the ferroelectric properties of binary mixtures of the siloxane-terminated smectic material E6 and two different achiral smectic materials. The structure of the first achiral liquid crystal C8Cl is similar to that of E6 while the second material C9-2PhP has a conventional structure without a siloxane group. It has been shown that in both mixtures, one observes the sign inversion of the spontaneous polarization induced by a change of temperature. In the E6/C8Cl mixture, the interval between the sign inversion temperature and the SmA-SmCtransition temperature decreases approximately linearly with the increasing molar fraction of E6. Extrapolation of the reduced inversion temperature to x(E6) = 1 indicates that for the pure E6 the inversion temperature is approximately equal to the SmA*-SmC* phase transition temperature. In contrast, in E6/C9-2PhP mixtures, the reduced inversion temperature increases with increasing x for x(E6) > 0.5 and decreases for x(E6) < 0.5. Simultaneously, the dependence of the smectic layer spacing on the molar fraction of E6 also changes qualitatively, approximately at x(E6) = 0.5, while in the E6/C8Cl mixture, the layer spacing decreases linearly with increasing x(E6), in line with inversion temperature. This qualitative change of behavior in E6/C9-2PhP mixtures can be attributed to a possible transition from the monolayer smectic structure (characteristic for pure C9-2PhP) to the bilayer structure typical for mixtures with large fraction of E6. The sign inversion of the spontaneous polarization in these mixtures has been explained theoretically using the recently developed molecular-statistical theory of ferroelectric ordering in chiral smectics C*.[8–10] It has been shown in our previous paper[8] that the faster than linear growth of the spontaneous polarization in pure E6 is determined by the “pre-transitional effect” which resembles the Curie–Weiss behavior of the dielectric constant above the virtual transition into the proper ferroelectric phase. According to Equations (10) and (13), the spontaneous polarization is proportional to the factor (TT0)1, where T0 is the characteristic temperature determined by polar intermolecular interactions which has the meaning of the virtual transition temperature into the proper ferroelectric phase. 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
www.chemphyschem.org One notes that T0 appears to be lower than the crystallization temperature, and thus the proper ferroelectric phase cannot be observed. At the same time, if T0 is not too low, the factor (TT0)1 makes a significant contribution to the temperature variation of the spontaneous polarization and is responsible for unusual convex curvature of the function Ps(T) in E6.[8] According to Equation (10), the spontaneous polarization is a sum of two terms which have different origins. The first term is proportional to the tilt order parameter V, which describes the tilt of long molecular axes, and which is approximately equal to the primary tilt angle at low tilt. The second term is proportional to the biaxial order parameter VG which also vanishes in the SmA phase. The biaxial order parameter is also proportional to the pre-transitional factor (TT0)1 and as a result the two terms in Equation (10) for the spontaneous polarization are characterized by different temperature variation. Moreover, fitting of the temperature variation of Ps in pure E6 with Equation (13) results in opposite signs of the coupling constants A and B that is, the two terms possess opposite signs. This enables one to explain the polarization sign inversion in mixtures of E6 with other smectic materials. As discussed above, in mixtures of E6 with non-chiral molecules, the second term in Equation (13) for the spontaneous polarization is smaller than the first one. However, the second term grows faster with decreasing temperature due to the additional factor of (TT0)1. As a result, the two terms compensate each other at some polarization inversion temperature Ti which appears to be an increasing function of the concentration of chiral molecules in the mixture provided the polar interaction between two chiral molecules is stronger than the interaction between chiral and non-chiral ones. This explains why the inversion temperature increases linearly with increasing concentration of E6 in the binary mixtures E6/C8Cl for all molar fractions of E6 and for large concentration of E6 in the mixtures E6/C9-2PhP. The linear decrease of the inversion temperature observed for small concentration of E6 in mixtures of E6 with C9-2PhP may be explained by the change from the partially bilayer to the monolayer structure which may result in an increase of effective interactions between chiral and non-chiral molecules. In principle, one cannot completely exclude other explanations of the polarization sign inversion including the one based on the model of a temperature-dependent balance between two or more conformations of the chiral E6 molecules which make contributions of opposite signs to the total spontaneous polarization.[11] This assumption, however, requires an additional experimental confirmation. One notes also that from our point of view, it is not straightforward at all to use the conformation model to explain the concentration dependence of the inversion temperature in binary mixtures as it requires too many assumptions concerning the effect of non-chiral molecules on the balance between chiral conformations. Another explanation of the polarization sign inversion in ferroelectric smectic C* is based on the model which takes into consideration a competition between dipolar and quadrupolar contributions to the polarization.[13, 15–17] The dipolar contribution is proportional to the tilt angle q while the quadrupolar contribuChemPhysChem 2014, 15, 1368 – 1375
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CHEMPHYSCHEM ARTICLES tion, which may possess the opposite sign, is proportional to the product of q and the biaxial (“quadrupole”) order parameter which increases like q2 at small tilt. Thus in this model the polarization sign inversion is determined by the temperature variation of the tilt angle. One notes, however, that in pure E6 and in the corresponding binary mixtures the tilt angle rapidly saturates below the SmA*-SmC* transition and then possesses a very weak temperature dependence. On the other hand, the theory of the polarization sign inversion presented in this paper is based on the same general ideas as the one developed in ref. [15]. Indeed, the spontaneous polarization is also expressed as a sum of two terms where the first term is proportional to the tilt order parameter while the second term is proportional both to the tilt and to the biaxial order parameter VG. The difference from the previous models is determined by the origin of the temperature variation of the biaxial order parameter which is not related to that of the tilt angle (outside a narrow interval below the SmA*SmC* transition) but is determined by the “pre-transitional” effect discussed above. Thus in the context of the present model the polarization sign inversion is intimately related to the unusual faster than linear temperature variation of the spontaneous polarization in E6 and in the corresponding binary mixtures. It is interesting to note that a similar but less pronounced temperature variation of the polarization has also been observed in another smectic C* material with polarization sign inversion.[11]
Experimental Section A Mettler Toledo DSC822 was used for differential scanning calorimetric measurements, on cooling and on heating at 5 K min1. Polarizing optical microscopy (POM) was conducted with an Olympus BH 2 polarizing microscope combined with a Linkam LTS 350. Xray experiments were performed on a SAXSess small angle x-ray scattering system from Anton Paar GmbH. The unaligned samples (filled into Hilgenberg Mark capillary tubes of 0.7 mm diameter) were kept in a temperature-controlled sample-holder unit (TSC 123). The X-ray beam coming from a ceramic tube generator was focused by a bent-multilayer mirror and shaped by a line collimation block. The scattered radiation were recorded with a CCD detector (KAF 2084 2083 SCX) and processed and analysed automated with a SAXSquant 3.5 software. The surface-stabilized ferroelectric liquid crystal (SSFLC) films were prepared in indium tin oxide (ITO) glass cells with single-side rubbed nylon (1.6 mm spacing, 0.25 cm2 addressed area, WAT PPW, Poland) alignment layers. Temperature-dependent measurements of the spontaneous polarization Ps and the director tilt angle q were performed with an electro-optic setup which was described previously in ref. [22], using a 70 Hz AC field with 18.75 V mm1 amplitude.
2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
www.chemphyschem.org Acknowledgements This work has been undertaken in the framework of the Materials World Network. M.A.O. is grateful to EPSRC (UK) (Grant EP/ H046941/1) for funding, R.P.L. acknowledges the support from NSERC (Canada) and F.G. and D.N. acknowledges support from the Deutsche Forschungsgemeinschaft (Germany). M.A.O. is also grateful to Leverhulme Trust for financial support which has enabled him to stay in Stuttgart. Keywords: chirality · liquid crystals · phase transitions · polarization · statistical mechanics
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