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Smectic-A–smectic-C–smectic-C⁎ Lifshitz point in mixtures of chiral and achiral smectic liquid crystals Prabir K. Mukherjee Department of Physics, Government College of Engineering and Textile Technology, 12 William Carey Road, Serampore, Hooghly 712201, India

a r t i c l e

i n f o

Article history: Received 24 October 2014 Received in revised form 18 December 2014 Accepted 4 January 2015 Available online 9 January 2015 Keywords: Ferroelectric liquid crystals Lifshitz point Phase transition

a b s t r a c t A phenomenological theory to describe the smectic-A–smectic-C transition, smectic-A–smectic-C⁎ transition and smectic-C–smectic-C⁎ transition in liquid mixtures is proposed. The inﬂuence of the concentration of the chiral liquid crystal on these phase transitions is discussed by varying the coupling between various order parameters. The phase diagrams of the thermodynamic parameters, wave vector, helix pitch, tilt angle and polarization are studied and a possible smectic-A–smectic-C–smectic-C⁎ Lifshitz point is predicted. We present a detailed analysis of the different phases that can occur and analyze the question under which conditions a Lifshitz point is observed in the phase diagram. Calculations based on this model agree qualitatively with experiment. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The smectic-A (SmA) and the smectic-C (SmC) phases may be regarded as stacked layers of two dimensional orientationally ordered ﬂuids. The molecules in the former are, on average, normal to the layer and are described by a one-dimensional mass-density wave, while those of the latter are tilted. The chiral smectic C (SmC⁎) phase represents a spatially modulated structure [1]. The ferroelectric ordering in the SmC⁎ phase is usually discussed in terms of hindrances of rotation of the molecules around their long axes. However, the microscopic origin of this ordering remained obscure. The SmC⁎ phase shows an intrinsic twist of the director from layer to layer. This additional symmetry breaking (C2h → C2) allows microscopic electric dipoles to form a spontaneous electric polarization P, which lies in the smectic planes. The study of critical behavior of the smectic-A (SmA) to smectic-C (SmC) or SmC⁎ transition has been an active area of research. A large number of both experimental and theoretical studies are devoted to the problem of the SmA–SmC⁎ transition [2–15]. Early experimental studies revealed that this transition exhibits mean-ﬁeld behavior and are well described by the Landau theory which includes the sixth order term in the tilt order parameter [6,9–11]. A very interesting feature of many SmA–SmC⁎ transitions is the closeness to a tricritical point [16–19]. The smectic-A–smectic-C–smectic-C⁎ (ACC⁎) Lifshitz point in ferroelectric liquid crystals (FLCs) has also been the subject of extensive experimental and theoretical studies [20–25]. So far, the ACC⁎ Lifshitz point in FLCs has been observed by varying the magnetic and electric ﬁelds and varying the concentration in binary mixtures. It

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http://dx.doi.org/10.1016/j.molliq.2015.01.012 0167-7322/© 2015 Elsevier B.V. All rights reserved.

represents the intersection of the SmA–SmC, SmA–SmC⁎ and SmC– SmC⁎ phase boundaries. According to Michelson et al. [20], the ACC⁎ Lifshitz point is a point where a line of ﬁrst order SmC–SmC⁎ transition branches into two lines of continuous transitions, namely SmA–SmC and SmA–SmC⁎ transitions. This observation has been conﬁrmed by the number of experiments [21,22,25]. According to Musevic et al. [21] and Blinc et al. [22,23], at a high temperature and at a small magnetic ﬁeld, the system remains in the SmA phase. If the ﬁeld is large enough, it tilts the molecules and the system transforms from the SmA phase into the homogeneous tilted SmC phase. When the magnetic ﬁeld is applied to the helicoidal SmC⁎ phase, it distorts the helix and increases the pitch, but the system remains in the SmC⁎ phase. Only when the ﬁeld is strong enough, the SmC⁎ phase unwinds and transforms into the SmC phase. Above a certain ﬁeld, the SmC–SmC⁎ line changes its direction so that a reentrant SmC⁎ phase was found. Rananavare et al. [25] carried out the high-resolution temperature– concentration phase diagrams of the binary mixtures of chiral DOBAMBC [p-(n-decyloxy-benzylidene)p-amino-(2-methyl-butyl)] and achiral 10O.8 (4-decyloxybenzylidene-4-octylaniline) and observed the nature of temperature–concentration phase digram near the ACC⁎ Lifshitz point. They observed that both wave vector q and polarization P vanish discontinuously at the SmC–SmC⁎ boundary revealing its ﬁrst order nature, consistent with possible Lifshitz point. There is practically no theoretical work on the ACC⁎ Lifshitz point for a binary mixtures although some theoretical works are available on the ﬁeld dependence of the ACC⁎ Lifshitz point. The purpose of the present paper is to explain the ACC⁎ Lifshitz point in the binary mixtures within the framework of the Landau–Ginzburg theory. We discuss the concentration dependence of the ACC⁎ Lifshitz point in binary liquid crystal mixtures. We present our analysis with the goal to ﬁnd out the conditions for the SmA–SmC, SmA–SmC⁎ and SmC–SmC⁎ transitions using

P.K. Mukherjee / Journal of Molecular Liquids 204 (2015) 10–14

the Landau–Ginzburg theory. First we derive the conditions for the various phase transitions involved. This is followed by plotting and discussing the topology in the phase diagram. 2. Model The ferroelectric ordering in the SmC⁎ phase is usually described by two order parameters [12,26]: one is the molecular tilt angle θ, the other is spontaneous polarization P. The tilt angle of the long molecular axis with the smectic layer normal precesses from one smectic layer to the other forming a helicoidal structure with helical axis coincident with the layer normal. Spontaneous polarization developed in the plane of the smectic layers perpendicular to the tilt plane also precesses from layer to layer and forms in helix. The tilt angle in the SmC⁎ phase can also be described by the orientational order parameter [27,28]. The layering in the SmC⁎ phase is described [29] by the order parameter ψ(r) = ψ0exp(−iΦ), whose modulus ψ0 is deﬁned as the amplitude of a one dimensional density wave characterized by the phase Φ. The wave vector ∇iΦ is parallel to the director ni in the smectic A phase. The layer spacing is given by d = 2π/q0 with a non-zero q0 = |∇Φ|. Following [27,28], the tilt angle in the SmC⁎ phase is described by the uniaxial orientational order parameter S Qij ¼ 3ni n j −1 2

ð2:1Þ

2

ð2:5Þ

2

ð2:6Þ

T 2 ðxÞ ¼ T 2 þ u2 ðx−x0 Þ þ v2 ðx−x0 Þ

ð2:2Þ

where θ is the angle between the layer normal and the director ni. The azimuthal angle ϕ describing the average position of the molecules on the cone which changes with the coordinate z as ϕ = qz, q being the wave vector of the helix. The in-plane spontaneous polarization is deﬁned as P ¼ P 0 ð−sin ϕðzÞ; cos ϕðzÞ; 0Þ:

where dij = δ1ninj. F0 is the non-singular part of the free energy. As usual in the Landau theory we assume that a = a0(T − T1⁎(x)) and α = α0(T − T2⁎(x)). T1⁎(x) and T2⁎(x) are the virtual transition temperatures. Other material parameters are assumed to be constants. The isotropic gradient terms in (Eq. (2.4)) guarantee a ﬁnite wavelength q0 for the smectic density wave. δ1, δ2 and γ are coupling constants. L1 and L2 are the orientational elastic constants. εijk is the antisymmetric third rank tensor. The chiral character of the SmC⁎ phase results in the pseudoscalar ﬁrst order spatial derivative term in the free energy. Thus the coefﬁcient L3 is analogous to the coefﬁcient of the Lifshitzinvariant term and induces the helical modulation of the SmC⁎ phase. The gradient terms ~e and ~h involving Qij govern the relative direction of the layering with respect to the director and lead to the tilt angle of the SmC⁎ phase. The coefﬁcient g is analogous to the ﬂexoelectric coefﬁcient. This coupling term is of a chiral character and induces a transverse polar ordering. The negative values of δ1 and e favor the SmA and SmC⁎ phases. We assume δ2 N 0 and h N 0. From the experimental phase diagrams [25] one observes that the concentration vs. the temperature curve for the SmA–SmC, SmA– SmC⁎ and SmC–SmC⁎ transitions are not straight lines. We therefore assume the quadratic form of T1⁎(x) and T2⁎(x), T 1 ðxÞ ¼ T 1 þ u1 ðx−x0 Þ þ v1 ðx−x0 Þ

where ni is not parallel to ∇iΦ. We assume the ﬂat layers in the smectic phases and take the layer normal q−1 0 ∇iΦ = ez as the z-axis. Then ni is deﬁned by ni ¼ ex sin θ cos ϕðzÞ þ ey sin θ sin ϕðzÞ þ ez cos θ

11

ð2:3Þ

Here P0 is the magnitude of the spontaneous polarization in the unwound ferroelectric state. Thus Q ij, ψ(r) and P are taken as three order parameters involved in the SmA–SmC⁎ transition. Now we consider the binary mixture of two liquid crystalline compounds chiral DOBAMBC and achiral 10O.8. The pure form of DOBAMBC shows the second order SmA–SmC⁎ transition. While the achiral compound 10O.8 shows the second order SmA–SmC transition. When mixing with the achiral 10O.8 liquid crystal, the experiment conﬁrms the existence of the ACC⁎ Lifshitz point. In the case of mixture, the free energy must be expressed in terms of the symmetry-breaking order parameters and the concentration (weight percent) Let x be the concentration of the chiral compound DOBAMBC in a mixture with DOBAMBC + 10O.8. Keeping the homogeneous terms up to a quartic, the total free energy density near the SmA–SmC⁎ transition can be written as [19,27] 2 1 1 1 1 F ¼ F 0 þ aQ i j Q i j − bQ i j Q jk Q ki þ c1 Q i j Q i j þ c2 Q i j Q jk Q kl Q li 2 3 4 4 1 1 1 2 1 1 2 4 2 2 þ α jψj þ βjψj þ P þ di j Q i j jψj þ δ2 Q i j Q i j jψj 2 4 2χ 0 2 2 1 1 þγQ i j P i P j þ L1 ∇i Q jk ∇i Q jk þ L2 ∇i Q ik ∇ j Q jk þ L3 ε i jk Q il ∇k Q jl ð2:4Þ 2 2 1 1 1 1 2 2 þ b1 j∇i ψj þ b2 jΔψj þ eQ i j ð∇i ψÞ ∇ j ψ þ gijkl P l ∇k Q i j 2 2 2 2 1 þ hQ i j Q kl ∇i ∇ j ψ ∇k ∇l ψ 2

where u1, u2, v1 and v1 are constants. x0 is the value of the concentration of DOBAMBC for which the SmC⁎ phase begins to appear. We assume that the temperature and concentration regions are not small so that the non-linear dependence of phase transitions on concentration is justiﬁed. We consider the phases in which the nematic and smectic orders are spatially homogeneous, i.e. S = const. and ψ0 = const. The substitution of the values of Qij and ψ in Eq. (2.4) leads to the free energy density expansion 3 2 1 3 9 4 1 1 4 2 cS þ αψ0 þ βψ0 F ¼ F 0 þ aS − bS þ 4 4 16 2 4 1 2 1 3 1 2 9 0 2 2 2 2 2 2 þ P 0 þ δ1 ψ0 S þ δ2 S ψ0 − γP 0 S þ L S q sin θ 2χ 0 2 4 2 4 9 9 1 1 2 2 4 2 2 2 2 2 4 − L2 S q sin θ þ L3 S qsin θ þ b1 ψ0 q0 þ b2 ψ0 q0 8 4 2 2 2 1 2 2 3 1 2 4 2 2 2 þ eψ0 q0 S 3 cos θ−1 þ gSP 0 q sin θ cos θ þ hψ0 q0 S 3 cos θ−1 4 2 8

ð2:7Þ where L′ = (L1 + L2/2) and c = c1 + c2/2. The values of the temperature dependence of the spontaneous polarization and the wave vector of the helix in the SmC⁎ can be expressed as 3 P 0 ¼ − gSqχ 0 M sin 2θ 4 q ¼ −

L 3 2L0 −g 2 χ 0 M − L2 −g 2 χ 0 M sin2 θ

ð2:8Þ

ð2:9Þ

where M = (1 − γχ0S)−1. Here the temperature enters through the temperature dependent tilt angle and through S in M. For a non-chiral material, L3 = 0, hence q = 0. Eq. (2.9) shows that L3 b 0 since the denominator is always positive. The pitch of the helix in the SmC⁎ phase is written as

Z¼−

i 2π h 0 2 2 2 2L −g χ 0 M − L2 −g χ 0 M sin θ : L3

ð2:10Þ

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P.K. Mukherjee / Journal of Molecular Liquids 204 (2015) 10–14

Eq. (2.10) shows that the pitch does not diverge at the ﬁrst order SmA–SmC⁎ or the ﬁrst order SmC–SmC⁎ transitions. Eq. (2.8) shows that a positive value of P0 can only be obtained for g b 0. The values of the smectic ordering and the wave vector of the density modulation in the SmC⁎ phase can be expressed as 2

ψ0 ¼ −

1 3 2 3 3 9 2 2 2 4 α − e −δ1 S þ δ2 S þ sin θ e S− h0 S þ h0 S sin θ β 2 2 4 16

ð2:11Þ 2

q0 ¼ −

1 3 9 2 4 2 2 2 b1 þ eS−h S −sin θ eS−3h S − h S sin θ ð2:12Þ 2b2 2 4

where α* = α − (b21/4b2), e* = eb1/2b2, h* = hb1/b2, h⁎ ⁎ = hb21/b22, h0 = h⁎ ⁎ − (e2/b2) and δ2⁎ = δ2 − (e2/6b2) + (h⁎ ⁎/6). It is clear that a non-zero real value of ψ0 exists only when the term within the bracket in Eq. (2.11) is less than zero. Since there is a small temperature range, α⁎ N 0, δ1 b 0 and δ2 N 0 in this region. The variation of the tilt angle θ with the temperature in the SmC⁎ phase can be calculated after minimizing Eq. (2.7) with respect to θ and inserting the values of ψ0, P0, q0 and q. This results in a complicated algebraic equation for sin2θ, 2

sin θ ¼

½ð3e α =4βÞ þ mS 2nS

condition for the SmA phase to appear and consequently for the SmC phase to disappear is S ≤ S0. When the high value of the concentration of DOBAMBC is added to the achiral compound 10O.8, it distorts the helix and increases the pitch and the system remains in the SmC⁎ phase. The SmC⁎ phase begins to appear either from the SmA phase or SmC phase. Only when the concentration is small, the SmC⁎ phase unwinds and transforms into the SmC phase. Then a ﬁrst order SmC–SmC⁎ phase transition takes place. Both q and P vanish discontinuously at the SmC–SmC⁎ transition. The line of the SmA–SmC⁎ transition starts at an ACC⁎ Lifshitz point. The SmA–SmC⁎ transition is the second order. This is a consequence of the symmetry of the tensor order parameter. The existence ranges of the SmA, SmC and SmC⁎ phases generally overlap. The phase with the lowest free energy is the stable one. The free energy of the SmA, SmC and SmC⁎ phases can be expressed as ð2:20Þ

F SmC ¼ F 0 −

α 2 α δ 3 2 1 3 9 4 2 − 2 1 S þ a2 S − b2 S þ c S 4β 2β 4 4 16 2

ð2:21Þ

ð2:13Þ

n¼−

þ

9L2 uv0 9e2 9α 2 − h0 þ e − 3 0 16β 32β 16L

ð2:22Þ 2

1 where F 0 ¼ F 0 − α4β − 4z .

δ ðδ −eÞ2 2 α − 1 ; β 3β ð δ −e Þδ 2 ; b1 ¼ b þ 3 1 β 2 c1 ¼ c−δ2 =β; δ α δ2 a2 ¼ a− 2 2 − 1 ; β 2β δ1 δ2 b2 ¼ b þ 3 ; β 2 δ c2 ¼ c− 2 ; β 0 z2 δ2 α e2 δ1 0 − þ 2e −δ1 − ð1− 3h0 α =4βz a ¼ a− β 2β 3β 3z 2 þ 9 h0 þ e =32βz Þ;

a1 ¼ a−

2 2 0 2 0 þ h:o: v ¼ g χ 0 =2L 1 þ 2g χ 0 =2L

ð2:17Þ

3 0 0 2 0 2 0 2 02 v ¼ L2 =2L 1 þ g χ 0 1=2L −ð1=L2 Þ 1 þ g χ 0 =2L − g χ 0 =4L L2 þ h:o:

ð2:18Þ Eq. (2.13) shows that the behavior of the tilt angle θ in the SmC⁎ phase is completely determined by the behavior of the orientational order parameter S. The value of S can be calculated from Eq. (2.22). The tilt angle changes with the change of concentration. Eq. (2.13) further shows that the tilt angle is inﬂuenced by the chirality of the system. Eqs. (2.8)–(2.10) and (2.13) show that P0, q, Z and θ are functions of temperature and concentration. The values of q, P0 and θ decrease with the decrease of the concentration of DOBAMBC. At a high temperature and at a small concentration of the chiral compound DOBAMBC, the systems remain in the SmA phase. If the concentration increases, it tilts the molecules and the system transforms from the SmA phase into the homogeneous tilted SmC phase. Then a second order SmA– SmC transition takes place. The tilt angle θ in the SmC phase can be expressed as 2 ðS−S0 Þ 3 S

9 4 c S 16

The renormalized coefﬁcients are

ð2:16Þ

2

3e α z0 α 3 2 1 3 e −δ1 − Sþ a S − b S 2β 8βz 4 4

ð2:15Þ

2 0 2 0 2 u ¼ 1 þ g χ 0 =2L þ g χ 0 =2L þ h:o:

sin θ ¼

F SmC ¼ F 0 þ

ð2:14Þ

α α 3 2 1 3 9 4 þ e −δ1 S þ a1 S − b1 S þ c S 4β 2β 4 4 16 1

where 9e δ1 3e2 3α h0 9L23 u − − þ m¼− 8β 4β 8β 16L0

2

F SmA ¼ F 0 −

α 2 ¼ α−

where S0 = eb2/2b1h. Eq. (2.19) shows that the non-zero value of the tilt angle θ exists for S N S0. As temperature increases, the orientational order parameter S decreases and the tilt angle θ decreases. This is possible only if e b 0 (since b1 b 0). As long as S N S0, there is no SmA phase, Thus S N S0 is found to be the necessary condition for which a SmC phase exists. The

b21 e2 − ; 4b2 4h

δ2 ¼ δ2 − z¼−

ð2:19Þ

e δ2 eα h 3δ δ δ h þ þ 1 2þ 1 0; β βb2 β 2β δ2 4 4eh δ 2α 2 2 1 þ h −he ; c ¼ c− 2 þ ee h − β 9βb2 9βb2 9

b ¼ b−3

0

e2 hb2 þ 12 ; 6b2 6b2

2

2

0

9e 9L uv − 3 0 ; 16β 16L

z ¼−

3e2 9L23 u 9e δ1 þ : − 4β 8β 16L0

The conditions for the second order SmA–SmC transition can be obtained as 0

F SmA ðSÞ ¼ 0

ð2:23Þ

P.K. Mukherjee / Journal of Molecular Liquids 204 (2015) 10–14

Fig. 1. The wave vector (q) as a function of concentration (x) of DOBAMBC. The measured data are from Ref. [25] and the line is the best ﬁt of Eq. (2.9).

Sc ¼ S0 ¼

eb2 2b1 h

″

F SmC ðSÞ≥0:

ð2:24Þ ð2:25Þ

The conditions for the second order SmA–SmC⁎ transition are given by 0

F SmA ðSÞ ¼ 0

3e α þ mS ¼ 0 4β ″

F SmC ðSÞ≥0:

ð2:26Þ ð2:27Þ ð2:28Þ

Finally, the conditions for the ﬁrst order SmC–SmC⁎ transition read F SmC ðSÞ ¼ F SmC ðSÞ

ð2:29Þ

0

ð2:30Þ

″

ð2:31Þ

F SmC ðSÞN0 F SmC ðSÞN0:

Solving Eqs. (2.23)–(2.31) simultaneously will determine the various phase transition lines. 3. Comparison with experiment In this section we compare our theoretical results on the concentration effect on the SmA–SmC–SmC⁎ phase sequence with the measurement of Rananavare et al. [25] on the DOBAMBC-10O.8 mixture. This

Fig. 2. The helical pitch (Z) as a function of concentration (x) of DOBAMBC. The measured data are from Ref. [25] and the line is the best ﬁt of Eq. (2.10).

13

Fig. 3. The polarization (P0) as a function of concentration (x) of DOBAMBC. The measured data are from Ref. [25] and the line is the best ﬁt of Eq. (2.8).

is, to the best of our knowledge, the only experimental concentration dependence study of the ACC⁎ Lifshitz point available in the literature. According to Eq. (2.9), the wave vector q changes with the change of concentration of the chiral compound. The concentration dependence of q was reported by Rananavare et al. [25]. Eq. (2.9) is ﬁtted with the measured data [25] of q as a function of the concentration of DOBAMBC taking x0, h, L3, γ, δ1, u2 and v2 as ﬁt parameters. The other parameter values are taken from previous literature values. These values are L1 = L2 = 1.24 × 10− 12 N, b1 = 0.03 N/m2, b2 = 0.7 N/m2, e = −0.1 N/m2, χ0 = 2.0, g = −0.23 × 10−6 N/m, α0 = 0.1 N/m2 °C and β = 2.07 N/m2. The ﬁt (solid) line and the measured data (closed circles) are shown in Fig. 1. The ﬁt yields L3 = − 0.41 × 10− 6 N/m, h = 15.27 N/m2, δ1 = − 102.63 N/m2, γ = 38.04 N/m2, u2 = − 2418.74 °C wt.%−1, v2 = 1635.74 °C wt.%− 2, and x0 = 0.36. The ﬁtted value of x0 = 0.36 coincides with the experimental value x0 = 36. The concentration dependence of the helical pitch given by Eq. (2.10) can easily be veriﬁed with Fig. 2 of Rananavare et al. [25]. We have, therefore, ﬁtted Eq. (2.10) with the measured Z vs. x data using h, L3, γ, δ1, u2 and v2 as ﬁt parameters while keeping the other parameter values ﬁxed at the values listed for Fig. 1. The ﬁt (solid) line and the measured data (closed circles) are shown in Fig. 2. The ﬁt yields L3 = − 0.3 × 10−6 N/m, h = 17.25 N/m2, δ1 = − 64.16 N/m2, γ = 7.65 N/m2, u2 = − 6328.59 °C wt.%− 1, v2 = 354.17 °C wt.%− 2. To check Eqs. (2.8) and (2.13), the measured P0 vs. x and θ vs. x of Rananavare et al. [25] are plotted in Figs. 3 and 4 (closed circles). The line in Fig. 3 is the ﬁt of Eq. (2.8) to the data resulting in parameter values L3 = − 0.41 × 10−6 N/m, δ1 = − 0.25 N/m2, h = 17.07 N/m2, γ = 0.308 N/m2 u2 = −204.18 °C wt.%−1, v2 = −3.01 °C wt.%−2. The line in Fig. 4 is a plot of Eq. (2.13) using the parameter values L3 = − 0.41 × 10−6 N/m, h = 70.97 N/m2, δ1 = − 39.62 N/m2, u2 = − 0.40 °C wt.%−1, v2 = 0.0028 °C wt.%−2. The other parameter

Fig. 4. The tilt angle (θ) as a function of concentration (x) of DOBAMBC. The measured data are from Ref. [25] and the line is the best ﬁt of Eq. (2.13).

14

P.K. Mukherjee / Journal of Molecular Liquids 204 (2015) 10–14

Fig. 5. Typical temperature (T) vs. concentration (x) phase diagram in the vicinity of the SmA–SmC–SmC⁎ Lifshitz point (LP). The solid line represents a line of ﬁrst order transition while the dashed lines represent the second order transition.

values for Figs. 3 and 4 are ﬁxed at the values listed for Fig. 1. The ﬁt to the measured values are good in Figs. 1–4. The agreement of the theory with the experiment is very good considering the scattered experimental data. Fig. 2 shows that pitch Z disappears discontinuously at the SmC–SmC⁎ transition at x0 = 36 of DOBAMBC. Fig. 3 shows that polarization P0 decreases non-linearly and disappears completely at x0 = 36. Fig. 5 summarizes the topology of the phase diagram associated with the free energy F, the conditions (2.23)–(2.31) and the different values of the phenomenological parameter. Fig. 5 shows a typical phase diagram using conditions (2.23)–(2.31). To draw Fig. 5 we chose the model parameters used for the ﬁgures Figs. 1–4. There are three phases: SmA, SmC and SmC⁎. For this system there are only four possible phase sequences: SmA–SmC, SmA–SmC⁎, SmC–SmC⁎ or SmA–SmC–SmC⁎. The SmC and SmC⁎ phases are separated by the ﬁrst order transition line followed by two direct second order SmA–SmC and SmA–SmC⁎ transition lines. All three phases meet at the Lifshitz point. As can be seen from Fig. 5, the SmC and the SmC⁎ phases arise from the SmA phase along the curves SmA–SmC and SmA–SmC⁎ or along the curve SmC–SmC⁎ respectively. For the second order SmA–SmC transition, the line of the SmA–SmC transition starts at the ACC⁎ Lifshitz point as shown in Fig. 5. When the temperature or the concentration of the SmA–SmC, SmA–SmC⁎ and of SmC–SmC⁎ transitions coincide, a Lifshitz point appears. The region of the SmC⁎ phase shrinks and ﬁnally disappears when the SmA–SmC transition takes place. Thus we conclude that for the particular values of the coupling constants, the SmA, SmC and SmC⁎ phases coexist at a particular temperature and concentration i.e. the system is at its Lifshitz point. The topology of the phase diagram (Fig. 5) is in complete agreement with the experimental result [25]. 4. Conclusions In conclusion, we have developed a simple phenomenological free energy function to describe the SmA–SmC transition, SmA–SmC⁎

transition and SmC–SmC⁎ transition. We have derived the conditions for the SmA–SmC, SmA–SmC⁎ and SmC–SmC⁎ transitions. The SmC– SmC⁎ transition must always be ﬁrst order, even in the mean-ﬁeld theory. This is a direct consequence of the symmetry of the tensor order parameter. The latent heat of the SmC–SmC⁎ transition should vanish at the Lifshitz point. The T–x phase diagram shown in Fig. 5 encompass the topology that can be constructed from the three phases (SmA, SmC and SmC⁎) under consideration. The topology of the phase diagram is found to be in good agreement with published experimental result. Our results are in qualitative agreement with the experimental results of Rananavare et al. [25]. Finally, we point out that the theoretical conﬁrmation of SmA–SmC– SmC–SmC⁎ Lifshitz point for the mixture of DOBAMBC-10O.8 would suggest the existence of the possibility of the Lifshitz point for other mixtures of FLCs. Although we focused on the SmA–SmC–SmC–SmC⁎ Lifshitz point for the mixture of DOBAMBC-10O.8, a similar calculation can also be applied for the other mixtures of FLCs where the Lifshitz point is observed. We hope that the present work will stimulate further experimental studies in this direction.

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