Density variation in the Landau theory of the cholesteric to smectic-A phase transition

Chemical Physics 374 (2010) 83–85

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Density variation in the Landau theory of the cholesteric to smectic-A phase transition Prabir K. Mukherjee * Department of Physics, Presidency College, 86/1 College Street, Kolkata 700 073, India

a r t i c l e

i n f o

Article history: Received 11 March 2010 In final form 18 June 2010 Available online 25 June 2010 Keywords: Liquid crystals Phase transition Tricritical point

a b s t r a c t The phenomenological model of the cholesteric to smectic-A phase transition with the inclusion of the density change is examined in a simple way. The density variation on the cholesteric to smectic-A phase transition is discussed in a liquid crystal mixture. We show how the density variation changes the order of the cholesteric to smectic-A phase transition with the change of concentration. It was observed from the theoretical calculations that the transition crosses over from first to second order when the concentration change in the system. Calculations based on this model agree qualitatively with experiment. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The tricritical behavior of liquid crystal phase transitions, where a line of phase transitions changes from a first to second order, has been the subject of extensive experimental and theoretical studies. So far, the tricritical point (TCP) in various liquid crystal phase transitions has been observed by changing the length of the alkyl end chains, varying the concentration in binary mixtures, increasing the pressure and varying the electric field. Although the tricritical behavior of the nematic–smectic-A (N–SmA) phase transition has been studied extensively, reports on the cholesteric–smectic (Ch–SmA) phase transition are comparatively scarce. These include experimental [1,4–6] and theoretical [7]. A number of experimental and theoretical studies are devoted to the problem of Ch–SmA transition [1–11]. The binary mixtures of two cholesteric liquid crystal systems [4,5] indicated the second order character of the Ch–SmA transition for the reduced temperature TChSmA/TICh ranging from 0.90 to 0.92, where TChSmA and TICh are referred to as the Ch–SmA and isotropic–cholesteric (I–Ch) transition temperatures. The density measurements [6] near the Ch–SmA transition for the binary mixtures of some aliphatic cholesteryl esters also reveal the second order character of the Ch–SmA transition. The first order Ch–SmA transition has been observed only when the cholesteric range is extremely narrow. These experimental studies [4–6] indicated the strong evidence of the tricritical point in binary mixtures at the Ch–SmA phase transition. Keyes et al. [1] carried out a high pressure study on the Ch–SmA phase transition and confirmed the tricritical behavior of the Ch– SmA phase transition. Huang et al. studied the Ch–SmA transition * Tel.: +91 3322411977; fax: +91 3326742108. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.06.024

in presence of impurity. They measured the divergence of helical pitch near the Ch–SmA transition in the presence of impurity. There are quite a few theoretical studies on the Ch–SmA transition [7,9–13]. Alben [2] studied the Ch–SmA transition within mean-field theory. He predicted that although both twist elastic constant K22 and the pitch increases rapidly, the ratio K22 to pitch should be constant near the transition on the SmA phase. Lee et al. [13], predicted a second order character of the Ch–SmA transition for a reduced temperature TChSmA/TICh 6 0.88. Renn and Lubensky [9,10] studied the Ch–SmA transition from the analogy with superconductors. They showed that the cholesteric phase is the analog of a normal metal in an external magnetic field. Benguigui [11] studied the direct investigation of the properties of the Ch–SmA transition without using the analogy with superconductors. However, none of the theoretical studies undertaken the tricritical behavior of the Ch–SmA transition. In a recent paper, the tricritical behavior of the Ch–SmA transition was discussed by Mukherjee [7]. This work supports the experimental work of Keyes et al. [1]. In the present work we have endeavored to ascertain the validity of the density study on the Ch–SmA phase transition in the context of Landau theory. Although, some theoretical models [14] on the density effect in the N–SmA transition are available, there is no such study on the Ch–SmA transition. It is shown that if one consider the coupling between the concentration, density and the order parameters, the tricritical properties are predicted by the Landau theory of the Ch–SmA phase transition. 2. Theory The nematic order parameter in the cholesteric phase proposed by de Gennes [12] is a symmetric, traceless tensor described by Q ij ¼ 2S ð3ni nj  dij Þ. Here n = (nx,ny,nz) and S is the magnitude of

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the nematic order parameter. In the geometry which is considered here the cholesteric director is assumed to be ni = excosk0z + eysink0z. Here the direction of the helix axis is in the z direction. Thus the structure of the cholesteric liquid crystal is periodic with 0 a spatial period given by L = p/jk0j. The layering in the SmA phase is described [12] by the order parameter w(r) = w0exp (iW), which is a complex scalar quantity whose modulus w0 is defined as the amplitude of a one dimensional density wave characterized by the phase W. The wave vector riWis parallel to the director ni in the SmA phase, the layer spacing of which is given by d = 2p/q0 with a nonzero q0 = jrWj. Keeping homogeneous terms up to the quartic order and gradients only up to the relevant order, free energy F near the Ch–SmA phase transition can be written as [7]

1 1 1 F ¼ F 0 þ aQ ij Q ij  bQ ij Q jk Q ki þ c1 ðQ ij Q ij Þ2 2 3 4 1 1 1 1 2 þ c2 Q ij Q jk Q kl Q li þ ajwj þ bjwj4 þ cij Q ij jwj2 4 2 4 2 1 1 1 2 2 þ d1 jri wj þ d2 jDwj þ L1 ri Q jk ri Q jk 2 2 2 1 þ L2 ri Q ik rj Q jk  L3 eijk Q il rk Q jl 2

ð2:1Þ

where cij = cninj and F0 is the free energy of the isotropic phase. a,b,c1,c2,a,b,c,d1,d2,L1,L2 and L3 are material parameters. The coefficients b, c1, c2, and b are assumed to be positive. c is coupling constant. c is chosen negative to favor the SmA phase over the cholesteric phase. The isotropic gradient terms d1 and d2 in Eq. (2.1) guarantee a finite wave vector q0 for the smectic density wave. We assume d1 < 0 and d2 > 0. L1, L2 and L3 are the orientational elastic constants. eijk is the Levi-Cevita antisymmetric tensor of the third rank. The term proportional to L3 violates parity and is responsible for the formation of a helical ground state. We assume L3 > 0. It must be stressed that in this manuscript we consider only the uniaxial and type I smectics and also neglect the possibility of blue and TGB phase for simplicity. Thus our derivation applies only for type I smectics. As usual we assume a ¼ a0 ðT  T I—Ch Þ and a ¼ a0 ðT T Ch—SmA Þ. T I—Ch and T Ch—SmA are virtual transition temperatures. a0 and a0 are positive constants. Now we consider the phases in which the nematic and smectic order are spatially homogeneous, i.e.S = const. and w0 = const., and for the smectic phase a spatially constant wave vector q0 characterizing the layering. The substitution of Qij and w in Eq. (2.1) leads to the free energy

3 1 3 9 4 1 2 1 4 1 cS þ aw0 þ bw0 þ cSw20 F ¼ F 0 þ aS2  bS þ 4 4 16 2 4 2 1 1 9 2 2 9 2 2 2 2 4 þ d1 w0 q0 þ d2 w0 q0 þ LS k0  L3 S k0 2 2 4 4

ð2:2Þ

where c = c1 + c2/2 and L1 = 2L2/3  L. The free energy (2.2)describes the Ch–SmA phase transition. The basic features of the Ch–SmA transition based on this free energy was described in Ref. 7. Let us now repeat de Gennes’s argument [12] for the order of the Ch–SmA transition. Now we want to describe coupling of S and w0 to both the density q and the concentration (mole fraction) x of a binary mixture. The simplest assumption is to add the terms to Eq. (2.2) of the form g(q,x)S2 and gðq; xÞw20 . We define x = nsol/ (nsol + nlc), where the n0 are number densities of the solute and liquid crystal. The experimental phase diagrams of Durate et al. [6] showed the linear variation of the temperature with the density and concentration. Thus in accordance with experimental phase diagrams [6], we consider the linear coupling with the density and concentration. But, the addition of the higher order coupling terms will not change the structure of the results, nor any of the conclusions drawn. Further, assuming negligible vapor pressure for both solute and solvent, the concentration is constant. The cor-

rect free energy is then the Gibbs free energy G (P, T, S, w0, q, x) with an entropy of mixing term. Thus to the lowest order coupling terms, the Gibbs free energy G for the binary mixture can be written as

1 1 1 1 1 ðS  S0 Þ2 þ aw20  bw40 þ dw60 þ d1 w20 q20 2v 2 4 6 2 1 1 9 9 1 1 2 þ d2 w20 q40 þ cSw20 þ LS2 k0  L3 S2 k0 þ g1 S2 q þ g2 w20 q 2 2 4 4 2 2 1 1 2 2 þ k1 S x þ k2 w0 x þ RTð1  xÞlnðx  1Þ þ RTxlnx ð2:3Þ 2 2

G ¼ GCh ðS0 Þ þ

where FCh(S0) is the free energy of the cholesteric phase at the Ch–SmA phase transition temperature, S0 is the nematic order parameter value at the transition point and v is the response function which decreases with decreasing temperature. g1, g2, k1 and k2 are coupling constants. The sixth order coefficient d is added for the stability of the free energy. For a first order Ch–SmA phase transition, b is negative and for a second order Ch–SmA transition b is positive. The coupling constants k1 and k2 are assumed to be positive (this yields a suppression of the transition temperature for x > 0). After minimizing the free energy (2.3) with respect to k0 and q0, the wave vector of the helix in the cholesteric phase and the wave vector q0 of the smectic density wave and substitution of k0 and q0, we obtain

G ¼ GCh ðS0 Þ þ

1 1 1 1 ðS  S0 Þ2 þ a1 w20  bw40 þ dw60 2v 2 4 6

1 9 L23 2 1 1 1 þ cSw20  S þ g1 S2 q þ g2 w20 q þ k1 S2 x 2 16 L 2 2 2 1 2 þ k2 w0 x þ RTð1  xÞlnðx  1Þ þ RTxlnx 2

ð2:4Þ

2

where a1 ¼ a  d1 =4d2 . The minimization of Eq. (2.4) with respect to S gives

S¼

!   1 9 L23 1þ S0  cvw20 v  k1 vx  g 1 vq 2 8 L

ð2:5Þ

The substitution of Eq. (2.5) into Eq. (2.4) gives

1 1 1 1 1 G ¼ GCh þ a w20  b w40 þ dw60 þ g2 w20 q þ c2 v2 g1 w40 q 2 4 6 2 8

ð2:6Þ

where GCh ¼ GCh ðS0 Þ þ ðS20 =2Þðk1 x þ g1 q  ð9L23 =8LÞÞ þ RTð1  xÞlnðx 1Þ þ RTxlnx. The renormalized coefficients are

9 L23 cvS0 þ k2 x 16 L 1 9 L23 2 2 1 2 2 c v  c v k1 x b ¼ b þ c2 v þ 2 16 L 2 g2 ¼ g2  g1 cvS0

a ¼ a1 þ cS0 þ

k2 ¼ k2  k1 cvS0

ð2:7Þ ð2:8Þ ð2:9Þ ð2:10Þ

The cholesteric to SmA phase boundary can be obtained from the expression of the pressure P = q2(@F/@ q)x,T and the chemical potential l = (@F/@x)q,T. Setting P for the SmA phase equal to those of the cholesteric phase defines the phase boundary. From the equality of P and assuming (@Fch/@ q) = (@FSmA/@ q) = a1 at TChSmA, one obtains the difference qSmA  qch between the SmA and cholesteric densities

qSmA  qch ¼ qSmA

    1  2 1 2 2 g2 w0 þ c v g1 w40 =a1 1 þ ðqtch =qtSmA Þ 2 8 ð2:11Þ

where qtch and qtSmA are densities of the cholesteric and SmA phases at the transition point.

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The pressure at the SmA phase can be expressed as

PSmA

  h i 1  2 1 2 2 ¼ q2SmA g2 w0 þ c v g1 w40 = 1  ðqtSmA =qtch Þ2 2 8

ð2:12Þ

From the thermodynamic point view, a first order transition is associated to a discontinuity in the density values at the transition, whereas a second order is related to a continuous change in the density change. Eq. (2.11) shows that density change Dq(qSmA  qch) at the Ch–SmA phase transition is directly related to the change of the translational order parameter w0. Thus the continuous or discontinuous variation of Dq at the transition follows from the continuous or discontinuous variation of w0. The change of the order of the Ch–SmA transition via the renormalization of b is accounted for by the x dependent term in Eq. (2.4). Now we notice from Eqs. (2.7) and (2.8)that taking into account of couplings (S,w0), (S,x) and (w0,x) lead to the renormalization of the of the coefficients a and b. For a pure material (x = 0), b* is positive and the transition is first order. Thus for the weak coupling and the low concentration x of the second compound to the the pure material, b* > 0. Then the Ch–SmA phase transition becomes first order. So the density and the translational order parameter changes discontinuously. In this case both the Ch and SmA phases can coexist i.e. a two phase region appears. The Ch–SmA phase transition is accompanied by a jump of Dw0 and Dq. In addition, there is also a jump in DS and Dq0. The jump of the translational order parameter wtr0 at the transition temperature

wtr0 ¼ ð3b =4dÞ1=2

ð2:13Þ

For the first order Ch–SmA phase transition, Ch–SmA transition temperature is given by

T trCh—SmA ¼ T  þ

3b2 k g  2 x  2 qtSmA 16a0 d a0 a0

ð2:14Þ

d21

where T  ¼ T Ch—SmA þ 4d2 a0  caS00  8a90 L S0 vcL23 . Substituting the value of qtSmA from Eq. (2.12) into Eq. (2.14) we obtain the transition temperature as

T trCh—SmA ¼ T  þ 

3b2 k g  2 x  2 ðPSmA =bÞ1=2 16a0 d a0 a0

ð2:15Þ

 h i where b ¼ 12 g2 ðwtr0 Þ2 þ 18 c2 v2 g1 ðwtr0 Þ4 = 1  ðqtSmA =qtch Þ2 . Eq. (2.14) shows that the Ch–SmA phase transition temperature decreases with the increase of concentration. In the same way, the entropy and latent heat discontinuities are 

1 3a0 b Ds ¼ a0 ðwtr0 Þ2 ¼ 2 8d 3a0 b tr DL ¼ T trCh—SmA Ds ¼ T 8d Ch—SmA

tr

g2 1=2

a0 b

tr dðlnT Ch—SmA Þ=dðlnVÞ

¼

P1=2 SmA

g2 qt2 SmA a0 T trCh—SmA

T CCh—SmA ¼ T  

k2

a0

x

g2 t q a0 SmA

ð2:20Þ

Eq. (2.20) shows that second order Ch–SmA transition temperature decreases with the increase of the concentration to the pure compound. The variation of the Ch–SmA transition temperature with the concentration x is obtained as C

dT Ch—SmA =dx ¼ 

k2

a0

ð2:21Þ

3. Conclusions We have presented here density variation in the Landau-like theory of the Ch–SmA phase transition for binary mixture. We have examined the effect of density variation near the transition. We have seen that the coupling between the order parameters with the concentration and the density is found to play a crucial role in determining the phase behavior and the order of the transition. In a binary mixture, the Ch–SmA transition which is first order in pure forms, becomes second order with the increase of concentration of the second compound. This leads to a crossover from first to second order transition behavior via a TCP. Although we have made some progress qualitatively to compare the theoretical results with available experimental results, there is still a lack of basic data which would make possible a more complete quantitative comparison with the theory. Acknowledgment The author thanks the Alexander von Humboldt-Foundation for book grant.

ð2:16Þ References

ð2:17Þ

Eqs. (2.16) and (2.17) show that the transition entropy changes with concentration which agrees well with experiment [5]. When b* change, then DL changes. The variation of T trCh—SmA with the pressure and volume can be expressed as

dT Ch—SmA =dP ¼ 

second order Ch–SmA transition occurs. Then the translational order parameter becomes zero i.e. the jump of Dq becomes zero and the variation of the density becomes continuous. For a particular value of the concentration xtcp, b* = 0 and the first order transition goes to a second order transition. So there is a crossover from first to a second order transition and a TCP appears as observed by experiments [4–6]. On the other hand since the coupling between w20 and either q or x makes a negative contribution contribution to b it also decreases the tendency to first order behavior. Also, since (@ l/@x)1 / x one can reasonably expect that the effect of adding a second component to a pure material may be to induce a TCP at some finite x. From the condition a* = 0, one finds the second order Ch–SmA transition temperature

ð2:18Þ ð2:19Þ

With changing the concentration, b* can become zero, then the entropy change Ds = 0, then the two phase region disappears i.e. a

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