Dielectric permittivity in the isotropic phase above the isotropic to smectic-A phase transition

Physics Letters A 377 (2013) 2436–2439

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Dielectric permittivity in the isotropic phase above the isotropic to smectic-A phase transition 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 7 February 2013 Received in revised form 29 May 2013 Accepted 16 July 2013 Available online 22 July 2013 Communicated by A.R. Bishop Keywords: Liquid crystal Dielectric permittivity Phase transition

a b s t r a c t We use a phenomenological model to find theoretically the temperature dependence of the static dielectric permittivity at the isotropic to smectic-A phase transition. The temperature dependence of the static dielectric permittivity is presented for the isotropic and smectic-A phases of the transition. Eventually, a comparison is made with experimental data available in the isotropic phase above the isotropic to smectic-A phase transition. We finally conclude that the model provides a good description of all known features of the static dielectric permittivity in the isotropic phase above the phase transition. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The isotropic to smectic-A (I-SmA) phase transition is a prototype symmetry breaking phase transition. Studies of the electrooptical effect i.e. the Kerr effect, nonlinear dielectric effect (NDE) and dielectric permittivity in isotropic melts of smectic liquid crystals has increased the interest considerably during the past few years. In particular, electro-optical study of the isotropic to smectic-A mesophase transition is interesting for contributions to the expansion of fundamental phase transition knowledge. The I-SmA phase transition appears to be moderately first order transition compared to the isotropic to nematic (I-N) phase transition. Coles and Strazielle [1] measured the Kerr effect in the isotropic phase and confirmed the first order character of the I-SmA phase transition. Polushin et al. [2,3] studied the electro-optical properties in the vicinity of the isotropic phase above the I-SmA phase transition. Sinha et al. [4] measured the real part of the third order nonlinear susceptibility from electro-optic Kerr effect studies above the I-SmA phase transition. Drozd-Rzoska et al. [5–8] measured the temperature and pressure dependence of the static dielectric permittivity and NDE in the isotropic phase above the I-SmA phase transition. The I-SmA phase transition is associated with a pronounced pretransitional fluctuation in static dielectric permittivity and NDE since the aligning electric field E couples to the critical fluctuations. Further they observed the same critical-like behavior with exponent α = 0.5 occurs for the I-SmA phase transition similar to the I-N phase transition. In our previous papers [9,10]

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we calculated the temperature, pressure and frequency dependence of the NDE in the isotropic phase above the I-SmA phase transition. In this publication we derived the temperature dependent static dielectric permittivity in the isotropic and SmA phases of the I-SmA phase transition. The purpose of the present Letter is to explain the static dielectric permittivity in both the phases of the I-SmA phase transition within the framework of Landau fluctuation theory. 2. Theory A phenomenological theory that describes the direct I-SmA phase transition was proposed by Mukherjee et al. [11]. The layering in the SmA phase is characterized [12] by the order parameter ψ(r) = ψ0 exp(−i Ψ ), which is a complex scalar quantity whose modulus ψ0 is defined as the amplitude of a one-dimensional density wave characterized by the phase Ψ . The nematic order parameter proposed by de Gennes [12] is a symmetric, traceless tensor described by Q i j = 2S (3ni n j − δi j ). The quantity S defines the strength of the nematic ordering. All nCB materials are very polar due to the same cyano group located at the terminal position. So one can concludes that the permanent dipole moment is parallel to the long rod-like axis. So we consider the SmA liquid crystals composed of strongly polar molecules. The external electric field induces a macroscopic polarization P in the SmA phase. Then the free energy of the polar SmA phase is a function of the orientational order parameter Q i j , translational order parameter ψ and the polarization P. Taking into account the relatively small value of the induced polarization, one can expand the free energy in powers of Q i j , ψ and P

P.K. Mukherjee / Physics Letters A 377 (2013) 2436–2439

F=

1

 

1

F0 +

V

3

aQ ij Q ij −

4

9 1

1

b Q i j Q jk Q ki +

1 9

1

4

2 χ0

+ c 2 Q i j Q jk Q kl Q li + α1 |ψ|2 + β|ψ|4 + 9 1

2

δ ∗2 , β λe ∗ η∗ = η − , β δ∗ λ γ∗ =γ − , β

c∗ = c −

c 1 ( Q i j Q i j )2

1

P2

1

+ δ Q i j Q i j |ψ|2 + η Q i j P i P j 2 1

2

1

+ γ1 P k P n Q kl Q nl + γ2 P m P m Q i j Q i j 3 1

α1∗ = α1 −

3

1

2 2

2

+ λ|ψ| P + d1 |∇i ψ| 2 1

2 1

2

3

+ d2 |ψ|2 +

L ∇i Q jk ∇i Q jk +



1 2



e Q i j (∇i ψ) ∇ j ψ ∗



where F 0 is the free energy of the isotropic phase. χ0 is the polarizability of the isotropic phase. The parameters η , γ1 , γ2 and λ presents the anisotropy of the polarizability in the smectic phase. δ is the coefficient characterizing the interaction of two order parameters. The isotropic gradient terms in Eq. (2.1) guarantee a finite wavelength q0 for the smectic density wave. The gradient term ∼ e involving Q i j governs the relative direction of the layering with respect to the director. Here ζi jkl = ζ (δik δ jl + δil δ jk ). This term has already been introduced and discussed by Brand [13] for the I-N phase transition. L is the orientational elastic constant. The material parameters a and α1 can be assumed as a = a0 ( T − T 1∗ ) and α1 = α0 ( T − T 2∗ ). T 1∗ and T 2∗ are virtual transition temperatures. All other coefficients, as well as a0 and α0 are assumed to be temperature independent. Now we consider the phases in which the smectic order parameter is spatially homogeneous, i.e. ψ0 = const for the simplicity of the calculation. Further, Eq. (2.1) can be simplified if one assumes that the polarization is aligned along the nematic director n i.e. P = (0, 0, P ). We choose E = (0, 0, E ). The substitution of Q i j , ψ and P in Eq. (2.1) leads to the free energy

F=

1

 

1

F0 +

V

2

aS 2 −

1 3

bS3 +

1 4

c S4 +

1 2

1 1 1 P 2 − η P 2 S + γ S2 P 2 2 χ0 2 2 1 1 1 + λψ02 P 2 + d1 ψ02 q20 + d2 ψ02 q40 + e ψ02 q20 S 2 2 2 2  1 + L (∇ S )2 + ζ P ∇ S − P E dV , 2 2 1

F=

V

(2.2)

e ∗ α1∗

1 1 1 1 F0 − S + a∗ S 2 − b ∗ S 3 + c ∗ S 4 + P2 2β 2 3 4 2 χ0



1 1 1 − η∗ P 2 S + γ ∗ S 2 P 2 + L (∇ S )2 + ζ P ∇ S − P E dV . 2 2 2 (2.3) The renormalized coefficients are

F 0∗ = F 0 − a∗ = a −

α1∗2 4β

e ∗2 2β

b∗ = b − 3

δ ∗ e∗ 2β

F=

 1  ∗  a0 T − T ∗ + Lq2 + γ ∗ P 2 S q S −q 2V q 

+ iζ

P qS +

q

P2 2 χ0

− P E,

(2.4)

where S q is the Fourier space S (r) =





V

q →

(2π )3

a∗0 = a0 −



d3 q.



1 V 1/ 2

q

S (q) exp(iq · r) and

δ ∗ α0 , β

T ∗ = a0 T 1∗ −

 d2 δ ∗ α0 ∗ e ∗2 /a∗0 . T2 − 1 + β 4d2 2β



− kV fT

=

B

e

− kV FT B

dS q .

(2.5)

After integration Eq. (2.5) we get

f =

P2 2 χ0



1

− P E + kB T 2

d3 q

ln G −1 (q, P ),

(2π )3

(2.6)

where G −1 (q, P ) = a∗0 (

T − T ∗ ) + Lq2 + γ ∗ P 2 . We have neglected the imaginary term i.e. i ζ P q qS term in F while evaluating Eq. (2.6) since we are interested only the real value of the dielectric permittivity. Minimizing the free energy (2.6) with respect to the polarization P leads to the equation of state for the polarization

P

χ0

 − E + kB T

γ ∗P

d3 q

(2π

)3

(a∗ ( T 0

−

T ∗ ) + Lq2

+ γ ∗ P 2)

= 0.

(2.7)

In the linear approximation (i.e. low field), the effective susceptibility χ I in the isotropic phase can be calculated from Eq. (2.7) as

,

−

We now proceed to calculate the dielectric permittivity in the isotropic phase above the I-SmA phase transition. The nontrivial polarization must be derived by integrating out the thermal fluctuations of S in the presence of the applied field E. In this case free energy in Fourier space can be expressed to a first approximation (b∗ = 0 and c ∗ = 0) as

e

where c = c 1 + 12 c 2 and γ = γ1 + 12 γ2 . The elimination of the equilibrium values of ψ0 and q0 leads to free energy as ∗

.

Now we define the effective free energy, which can be obtained by integrating out S q fluctuations in Eq. (2.4). Under the Gaussian approximation, the effective free energy f is just the S q -independent part plus the free energy for the S q -dependent part, which can be written as

4

1

 

ed1

,

,

1

α1 ψ02 + βψ04

+ δψ02 S 2 +

1

4d2

4d2

2d2

(2.1)

d21

e ∗2

δ∗ = δ − e∗ =

+ ζi jkl P i ∇ j Q kl − P i E i dV ,

2437

δ ∗ α1∗ , β

χ I = χ0 + U T + V 1 T T − T ∗

,

where



1/2

tan−1



, ( T − T ∗ )1/2 W

(2.8)

2438

P.K. Mukherjee / Physics Letters A 377 (2013) 2436–2439

U = −4πγ ∗ k B χ02 q, ∗1/2

4πγ ∗ χ02 a0

V1 =

kB

L 3/2

,

qL 1/2 W = ∗1/2 . a0 Hence the dielectric permittivity in the isotropic phase above the I-SmA phase transition can be expressed as

ε I = 1 + χ0 + U T  1/2 + V1T T − T ∗ tan−1



W

( T − T ∗ )1/2

(2.9)

.

While examining Eq. (2.9) we see that dε I /dT ∝ C P ∝ ( T − T ∗ )−1/2 . This shows that the critical exponent α = 1/2. We will now calculate the dielectric permittivity in the SmA phase below the I-SmA phase transition. In this case, the contribution of the cubic and the quartic terms in the free energy is to be taken into account. The contribution of the cubic and the quartic terms in the free energy can be obtained using perturbation theory, with the Gaussian model taken as the zeroth-order perturbation. Then the effective free energy can be obtained by integrating out S q fluctuations as

F=

P2

1



− P E + kB T

2 χ0

d3 q

2

(2π 3 )





1 ln G − 1 (q , P )

(2.10)

where

2







1 ∗ ∗ 3 2 ∗ G− 1 (q , P ) = a + L + k B T b / L ξ0 f 1 q + k B T c / L ( f 2 /ξ0 )

1

f1 =

16π 2



 2 − k B T b ∗ / L ξ0 f 3 + γ ∗ P 2 ,

tan−1 (ξ0 q) +

ξ0 q(ξ02 q2 − 1) (1 + ξ02 q2 )2

 ,

3 

ξ0 q − tan−1 (ξ0 q) , 2π   ξ0 q 1 −1 , f3 = tan (ξ q ) − 0 4π 2 (1 + ξ02 q2 ) 1/2  ξ0 = L /a∗ . f2 =

3. Results and discussion The temperature dependence of the static dielectric permittivity of 12CB was reported by Drozd-Rzoska et al. [6]. The temperature dependence of the static dielectric permittivity in the isotropic phase given by Eq. (2.9) can easily be verified with Fig. 2 of Drozd-Rzoska et al. [6]. In Fig. 1, we have plotted the calculated temperature dependence of the static dielectric permittivity (Eq. (2.9)) in the isotropic phase above the I-SmA transition. The form of Eq. (2.9) shows that there are several unknown parameters. It is unphysical to take all these parameters as fit parameters. We have, therefore, fitted Eq. (2.9) with the measured ε I ( T ) data using χ0 , U , V , W and T ∗ as fit parameters. The fit (solid line) and the measured data (solid circles) are shown in Fig. 1. The fit yields χ0 = 16.43, U = −0.024 K−1 , V = 5.94 × 10−4 K−3/2 , W = 5.8 K1/2 and T ∗ = 325.7 K. The fit to the measured values are good in Fig. 1. From Fig. 1, we obtain the maximum value of εmax = 9.82 which is close to the value of εI-SmA = 9.8. Utilizing the fitted value of T ∗ = 325.7 K and the experimental value of T I-SmA = 331 K, we obtain T I-SmA − T ∗ = 5.3 K. The high value of T I-SmA − T ∗ indicates that the I-SmA phase transition is more strongly first order than the I-N (T I-N − T ∗ = 1 K) phase transition.

2

4. Conclusion

In a similar approach, in the linear approximation (i.e. low field), the effective susceptibility χ Sm A in the SmA phase can be calculated as

χ Sm A = χ0 + U T + V 2 T ξ1 tan−1



q

(2.11)

ξ1

where

V2 =

ξ1 =

Fig. 1. The behavior of the static dielectric permittivity in the isotropic phase above the I-SmA phase transition of 12CB. The measured data are from Ref. [6] and the line is the best fit of Eq. (2.9).

4πγ ∗k B χ02

, [ L + k B T (b∗ / L )2 ξ03 f 1 ] 1/2  ∗ (a + k B T (c ∗ / L )( f 2 /ξ0 ) − k B T (b∗ / L )2 ξ0 f 3 ) ( L + k B T (b∗ / L )2 ξ03 f 1 )

. References

Hence the dielectric permittivity in the SmA phase below the I-SmA phase transition can be written as

ε Sm A = 1 + χ0 + U T + V 1 T ξ1 tan−1



q

ξ1

We have presented here a Landau fluctuation theory analysis to describe the temperature dependence of the static dielectric permittivity in the isotropic phase above the I-SmA phase transition. The analysis presents the first theoretical support for experimental observation. Although the agreement between experiment and theory is reasonable, Eq. (2.9) is sensitive to the values of U , V and W which depends on a∗0 and L. In this Letter we have shown how our simple Landau model can explain the essential feature of the static dielectric permittivity in the isotropic phase above the I-SmA phase transition. The critical exponent α = 0.5 which is same as in the case of I-N phase transition indicates the fluid-like analogy in the isotropic phase above the I-SmA phase transition. The same pretransitional phenomena is observed in the isotropic phase above the I-SmA transition similar to the I-N transition. Thus our results are in qualitative and quantitative agreement with the experimental results.

.

(2.12)

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P.K. Mukherjee / Physics Letters A 377 (2013) 2436–2439

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