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Electrical Freedericksz transitions in nematic liquid crystals containing ferroelectric nanoparticles
Physica E 67 (2015) 23–27
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Electrical Freedericksz transitions in nematic liquid crystals containing ferroelectric nanoparticles Cristina Cîrtoaje, Emil Petrescu n , Victor Stoian University Politehnica of Bucharest, Department of Physics, Splaiul Independenţei 313, 060042 Bucharest, Romania
H I G H L I G H T S
A formula for Freedericksz transition voltage in a mixture of NLC and ferroelectric particles. The transiton threshold decreases when the nematic director is parallel to nanoparticle.The transiton threshold increases when the nematic director is perpendicular to the nanoparticle. The transiton threshold decreases when the nematic director is parallel to nanoparticle. We evaluated the anchoring energy of the LC molecules in a mixture of 5CB and BaTiO3.
art ic l e i nf o
a b s t r a c t
Article history: Received 16 September 2014 Received in revised form 31 October 2014 Accepted 7 November 2014 Available online 13 November 2014
A new theoretical approach was elaborated to explain the contradictions reported in many papers about the electric threshold for Freedericksz transition in nematic liquid crystal (NLC) with ferroparticles additives. The free energy density of the mixture was estimated and the contributions of the interaction energy of NLC molecules with ferroparticles surface were calculated. Experimental results for 5CB-BaTiO3 mixture are given. & 2014 Elsevier B.V. All rights reserved.
Keywords: Ferroelectric nanoparticle Liquid crystals Freedericksz transitions
1. Introduction In the last ten years, many papers were devoted to mixtures of liquid crystals with nanoparticles such as azo-dyes [1], protons [2], ferromagnetic particles [3–6], nanoparticles [7] and carbon nanotubes [8–12]. Some of the newest research were concerned on ferroelectric nanoparticles suspension in liquid crystal matrix [13– 17]. Experimental studies presented in [13,14] evidenced a decrease of the threshold values for the Freedericksz transition when ferroelectric nanoparticles are inserted into the liquid crystal. In [13] the authors have used Sn2P6S6 (thiohypodishoshate) as ferroelectric particles with an intrinsic polarisation of 14 μC/cm2, and as liquid crystal the ZLI-4901 mixture from Merck. They noticed a 50% decrease of the Freedericksz transition threshold of the mixture when nanoparticles were included. In [14] ferroelectric nanoparticles of BaTiO3 dispersed in 5CB nematic liquid crystal in a n
Corresponding author. E-mail address: [email protected] (E. Petrescu).
http://dx.doi.org/10.1016/j.physe.2014.11.004 1386-9477/& 2014 Elsevier B.V. All rights reserved.
volume percentage of 4% (the size of the particles was 5 nm to 100 nm) were used. The measurements using an alternate electric field of 1 kHz proved that the Freedericksz transition threshold decreased from 0.79 Vrms to 0.54 Vrms. Under similar conditions, in [15], the authors studied a mixture of BaTiO3 in 6CHBT liquid crystal and found an increase of the threshold voltage for Freedericksz transition from 0.7 Vrms to 1.3 Vrms when the nanoparticles were added. The size of the ferroparticles was between 30 nm and 50 nm for a BaTiO3 concentration of 0.5%wt. V. Yu. Reshetnyak and co-workers have presented the first theoretical model for the electric Freedericksz transition threshold in a couple of papers [17,18], the second one presenting a considerable theory improved from the first one. Both papers relay on the assumption that the nanoparticles do not influence the electrical properties of the liquid crystal and each particle presents a permanent polarisation (permanent electric moment) which can only be parallel or anti-parallel to the local nematic director. Theoretical aspects presented in [18] are much more complex and the average electric field inside the liquid crystal and the
24
C. Cîrtoaje et al. / Physica E 67 (2015) 23–27
ferroelectric particles respectively as well as the average electric field in the entire medium are taken into account. The authors present the dependence of the liquid crystal free energy as a function of the effective dielectric constants ε∥eff and ε⊥eff . One of the main assumptions they made, was that the particles were very small (below 10 nm) as can be proved from the condition dvβEp0 < 1 where, d ¼6 μC/cm2 is the particles polarisation density, v is average nanoparticle volume, β = 1/kBT and Ep0 is the contribution of the electric field from all the sources other than the depolarisation field due to the permanent polarisation of the considered particles. It can be noticed that it has the same order as the electric Freedericksz transition threshold EF ≃ 7, 9 × 104 V/m . Finally they obtained a dependence on effective anisotropy of the threshold voltage.
K1 ε0εaeff
U=π
(1)
where
εa eff = ε∥ eff − ε⊥ eff
(2)
It is the aim of this paper to give an improved theoretical model of [18] by taking into account the nature of the interaction between the ferroparticles and the NLC molecules. We have introduced a specific term in the free energy density equation proposed by Burylov and Zakhlevnykh [19] considering the anchoring of the nematic molecules to nanoparticles surface.
2. Theoretical background
(3)
1 2
-E = −
-INT = −
1 2
∫0
L
⎡ →2 → →2 → → 2⎤ ⎢K1(∇· n ) + K2( n ·∇ × n ) + K3( n × ∇ × n ) ⎥dz ⎣ ⎦
L
⎡ →2 →→ ⎤ ε0⎢ε⊥ eff E + εaeff ( n E )2⎥dz ⎣ ⎦
(5)
∫0
L
2wf →→ 2 →→ P2(m n ) [1 − ξ (m n0)] a
(6)
where f is the volumetric fraction of nanoparticles, a is the nanoparticle's diameter (considered to be a cylinder), w is the average anchoring energy density at the nematic-nanoparticle inter→ action surface, m is the unit vector of particle dipole moment, P2 = (3cos2α − 1)/2 is the second order Legendre Polynomial, where α is the angle between the NLC and the ferroelectric par→ ticle, n0 is the direction of the nematic molecular director outside the nanoparticles range of action and ξ is a dimensionless phenomenological parameter [19]. The case of a strong anchoring of nematic molecules to the nanoparticles surface and a homogeneous alignment of the cell was considered. In the absence of any external fields, the nematic director is parallel to Ox axis. The applied electric field is parallel to Oz axis and the ferroelectric particles align themselves parallel to the same axis. In this geometry (Fig. 1), the liquid crystal free energy becomes.
1 2
∫0
⎡ ⎤ 2 2 2 ⎢K1cos θ + K3sin θ ⎥θz dz ⎣ ⎦
L
(7)
where
θz =
dθ dz
(8)
The electric field energy component can be written as
1 2
-E = −
-N is the liquid crystal free energy:
-N =
∫0
where ε⊥eff and εaeff are the dielectric constant and the dielectric anisotropy of the nematic-ferroelectric nanoparticles mixture. For the interaction free energy between nanoparticles and nematic liquid crystal molecules we used the formula given by Burylov and Zakhlevnykh [19]:
-N =
A mixture of nematic liquid crystal with ferroelectric nanoparticles suspension confined between two identical solid walls parallel to the xOy plane placed at z¼ 0 and z¼L was considered (Fig. 1). The free energy of ferroelectric nanoparticles suspension in liquid crystal electric field is
- = -N + -E + -INT
-E is the electric field free energy contribution:
∫0
L
DE dz = −
1 2
∫0
L
⎛ ⎞ ε0⎜ε⊥ eff + εaeff sin2θ ⎟E2 dz ⎝ ⎠
(9)
where
(4)
where K1, K2 and K3 are the splay, twist and bend elastic constants, and L is the cell thickness.
E=
D ε⊥ eff + εa eff sin2θ
(10)
Since D is constant for a given configuration, we obtain
U=
∫0
L
Edz = D
∫0
L
dz ε0(ε⊥ eff + εaeff sin2θ)
(11)
In this case, -E becomes
-E = −
1 D 2
∫0
L
Edz = −
⎡ 1 2⎢ U 2 ⎢⎣
∫0
L
⎤−1 ⎥ + εaeff sin2θ) ⎥⎦ dz
ε0(ε⊥ eff
(12)
The interaction free energy -INT is
-INT = −
∫0
=−A
L
2wf P2sin2θ[1 − ξsin2θ] dz a
∫0
L
sin2θ[1 − ξsin2θ] dz
(13)
where
A= Fig. 1. The nematic director and ferroelectric particle orientation.
2wf P2 a
(14)
C. Cîrtoaje et al. / Physica E 67 (2015) 23–27
When reaching the threshold value, θ increases by a small amount. This is the reason why we consider only its small values to evaluate the free energy:
-N =
1 2
L
∫0
K1θz2 dz
(15)
and
-E = −
ε U2ε0ε⊥ eff ⎛ ⎜1 + aeff 2L ⎜⎝ Lε⊥ eff
∫0
L
⎞ θ 2 dz⎟⎟ ⎠
Consequently, the interaction term of NLC molecules with the nanoparticles surface may also be estimated using the same approximation and we obtain
-INT = − A
∫0
L
θ 2(1 − ξθ 2) dz
and
L = 2
(17)
∫0
2A ⎡ ⎢ K1 ⎣
(
∫0
-=−
2
L
U2 +
1 2
∫0
L
(18)
dz
The first term in Eq. (18) is constant so we define a new free energy density
F=
⎤ 1⎡ 2 2 2 2 ⎢K1θz − 2Aθ (1 − ξθ ) − Bθ ⎥⎦ 2⎣
ε0εaeff L2
U2
(28)
1
dλ 2A ⎡ ⎢1 K1 ⎣
−
ξθm2 (1
⎤ + λ )⎦⎥ + 2
1 − λ2
B K1
(29)
1
L = 2
2A K1
+
B K1
∫0
1
dλ 1 − λ2
π
= 2
2A K1
+
B K1
(30)
Replacing B with the original formula given by Eq. (20), the critical voltage is
K1 ε0εaeff
1−
2AL2 π 2K1
(31)
When a pure nematic is used, f ¼0 εaeff = εa the Freedericksz transition threshold voltage is U0 = π K1/ε0εa , so
UF = U0
εa εaeff
1−
2AL2 π 2K1
(32)
(20)
A=
2wf (3 cos 2α − 1) 2wf = a 2 a
(33)
and
d ⎛ ∂F ⎞ ∂F =0 ⎜ ⎟− dz ⎝ ∂θz ⎠ ∂θ
(21)
we get the prime integral:
∂F −F=C ∂θz
(22)
where C is a constant. Therefore
K1θz2 + 2Aθ 2(1 − ξθ 2) + Bθ 2 = C
(23)
This relationship is an identity so it is also valid in the middle of the cell (z = L/2) where the deviation angle reaches its maximum (θ = θm and θz = dθ /dz = 0) i.e.
2Aθm2 (1 − ξθm2) + Bθm2 = C
(24)
If we denote
UF = U0
εa εaeff
1−
4wL2 aπ 2K1
z(θ , θm)
(25)
(34)
Because 1 − 4wL2/aπ 2K1 < 1 it is possible for UF /U0 to be smaller than the unit. This means that when nanoparticles are added into a liquid crystal matrix, the Freedericksz transition threshold may decrease. (b) The nematic director is perpendicular to the nanoparticle (θ = 90°) and the parameter A is smaller than zero:
A=−
wf a
(35)
Then
UF = U0
εa εaeff
1+
2wf L2 >1 a π 2K1 2
dθ = dz
1
In this case, two limit situations may be considered (a) The molecules align themselves parallel to the nanoparticle surface (α = 0). Then
From Euler–Lagrange equation,
θz
) − ξ(θm4 − θ 4)⎤⎦⎥ + KB (θm2 − θ 2)
(19)
where
B=
−θ
2
At the limit of the Freedericksz transition (θm → 0) Eq. (29) becomes
UF = π
⎡ 2 ε0εaeff 2 2⎤ 2 2 Uθ ⎥ ⎢K1θz − 2Aθ (1 − ξθ ) − ⎣ ⎦ L2
θm2
1
From Eqs. (15), (16) and (17) the total free energy is
1 ε0ε⊥ eff
dθ
θm
Introducing a new parameter λ = θ /θm so dθ = θmdλ we get
L = 2 (16)
25
(36)
2
As 1 + (2wf /a)L /π K1 > 1, the Freedericksz transition threshold for the mixture can be higher than the pure liquid crystal.
where
z(θ , θm) =
⎤ ⎞ 2A ⎡ 2 B⎛ 2 ⎢ θm − θ 2 − ξ θm4 − θ 4 ⎥ + ⎜θm − θ 2⎟ K1 ⎣ ⎦ K1 ⎝ ⎠
(
)
(
)
3. Experimental set-up and materials
(26)
we obtain
∫0
L /2
dz =
∫0
θm
dθ z(θ , θm)
(27)
The liquid crystal cells with 20 μm Mylar spacers were previously prepared for a planar alignment. A mixture of 5CB and BaTiO3 ferroelectric particles was used to fill the cells. The average size of the BaTiO3 crystals used was 50 μm and the measurements were performed at a constant temperature of 25 °C. For imaging the surface topographies we used Non-contact operating mode (AFM, PARK XE-100 SPM system). The cantilever had a nominal
26
C. Cîrtoaje et al. / Physica E 67 (2015) 23–27 Table 2 Experimental Freedericksz transition threshold (UF) at 1 kHz for each volumetric fraction and the value of εa/εaeff.
Fig. 2. Experimental set-up for the study of the electric properties of 5CB nematic with BaTiO3.
Table 1 Experimental Freedericksz transition threshold (UF) at 100 kHz for each volumetric fraction (f) and anchoring energy density (w). f (%)
UF(100 kHz)(V)
w (N/m)
0 0.33
0.64 0.59
3.47 × 10−7
0.50
0.56
3.57 × 10−7
f (%)
UF(1 kHz)(V)
εa εaeff
0 0.33 0.50
0.64 0.56 0.51
– 0.90 0.82
Thus, using experimental values for UF (100 kHz) and U0 we can evaluate the anchoring energy density w from Eq. (37). The results are given in Table 1 using the following values a ¼50 nm, K1 = 6, 2 × 10−12 N and L = 20 μm . The second measurements set was made using a 1 kHz frequency for which the dielectric constant of BaTiO3 is high enough to satisfy the εaeff ≠ εa condition. Just like the previous case, we noticed a decrease of the Freedericksz transition threshold:
UF (1 kHz) = U0
εa 4wL2 1− εaeff aπ 2K1
(38)
From Eqs. (37) and (38) we obtain
–
UF (1 kHz) = UF (100 kHz)
εa εaeff
(39)
which allows us to evaluate εa/εaeff for a 1 kHz voltage frequency. The results are presented in Table 2. length of 125 mm, a nominal force constant of 40 N/m, and oscillation frequencies in the range of 275–373 kHz. We used horizontal line by line flattening as planarization method [20]. The sample was prepared by evaporation on silicon substrate of a dilute solution of BaTiO3 in water. The average diameter was calculated by investigation of four different area (1 1 μm2, 3 13 μm2, 5 5 μm2and 10 110 μm2), for ten different regions of the samples. We have considered the polarisation density of BaTiO3 to be 0.15 μC/cm2. BaTiO3 nanoparticles were dispersed in different volumetric concentrations ( f1 = 0.33% and f2 = 0.50% ) in 5CB liquid crystal. The experimental set-up, presented in Fig. 2 consists of a 632.8 nm He–Ne laser used to send a beam through the LC cell fixed in a slot inside the thermal stage to keep the temperature constant. A power source able to apply an alternate voltage is connected to the electrode of the cell and a photodetector is used to record the laser beam after crossing the sample.
5. Conclusions Our theoretical estimations reveal that the molecules alignment angle on ferroelectric particles influences the Freedericksz transition threshold. We proved that an increase of this threshold value is also possible and we explained the contradictory results reported in other papers. The critical voltage for the transition strongly depends on the orientation angle of the nematic director to the ferroelectric particles surface: when they are parallel to each other, the transition threshold may decrease, but when they are perpendicular to each other, the transition threshold may increase.
Acknowledgments The work has been founded by the Sectorial Operational Program Human Resource Development 2007–2013 of the Ministry of European Funds through the Financial Agrement POSDRU 159/1.5/ S/132397.
4. Results and discussions The dielectric properties of BaTiO3 are thoroughly discussed in [21]. According to the authors, when the frequency of the applied electric field is higher than 100 kHz, the dielectric constant of BaTiO3 decreases to a constant value. In this case εaeff → εa which means that for this frequency the Freedericksz transition threshold changes only due to the interaction of the liquid crystal molecules with the ferroelectric particles surface. The experimental results proved that the Freedericksz transition threshold is lower when ferroelectric particles are added meaning that the LC molecules are oriented parallel to the nanoparticles. The critical voltage can be expressed from Eq. (34) by replacing εaeff with εa.
UF (100 kHz) = U0 1 −
4wfL2 aπ 2K1
(37)
References [1] C. Moţoc, G. Iacobescu, Modern Phys. Lett. B 20 (17) (2006) 1015. [2] C. Dascălu, Rev. Mex. Fiz. 47 (3) (2001) 281. [3] N. Tomaîdoviî¡ová, M. Timko, Z. Mitróová, M. Koneracká, M. Rajn˘ak, N. Éber, T. Tóth-Katona, X. Chaud, J. Kopî¡anský, Phys. Rev. E 87 (2013) 014501. [4] M.C. Calder, A. Desimore, D. Golovaly, A. Panchenco, SIAM J. Appl. Math. 74 (2) (2014) 237. [5] D.V. Makarov, A.N. Zakhlevnykh, J. Magn. Magn. Mater. 74 (2) (2008) 1312. [6] P. Kopănscký, N. Tomaîdoviî¡ová, M. Koneracká, M. Timko, A. Dz˘arová, J. Jadzyn, E. Beaugnon, X. Chaud, Int. J. Thermophys. 32 (4) (2011) 807. [7] Y.A. Garbovskiy, A.V. Glushchenko, Solid State Phys 62 (2011) 1. [8] C. Ci^rtoaje, E. Petrescu, C. Moţoc, Physica E: Low-dimension. Syst. Nanostruct. 54 (2013) 242. [9] V. Popa-Niţă, S. Kralj, J. Chem. Phys. 132 (2010) 024902. [10] V. Popa-Niţă, Samo Kralj, M. Cvetko, in: Proceedings of the SEM Annual Conference and Exposition on Experimental and Applied Mechanics 2009, Albuquerque, New Mexico, USA 1–4 June 2009, pp. 2762–2768.
C. Cîrtoaje et al. / Physica E 67 (2015) 23–27 [11] L.N. Lisetski, N.I. Lebovka, S.V. Naydenov, M.S. Soskin, J. Mol. Liq. 164 (2011) 143. [12] L.N. Lisetski, A.P. Fedoryako, A.N. Samilov, S. S Minenko, M.S. Soskin, N. I. Lebovka, Eur. Phys. J. E 37 (3) (2014), http://dx.doi.org/10.1140/epje/i201414068-3, article number 68. [13] Yu. Reznikov, O. Buchnev, O. Tereschenco, V. Resnetnyak, A. Glushchenko, J. West, Appl. Phys. Lett. 82 (12) (2003) 1917. [14] J.F. Blach, S. Saitzek, C. Legrand, L. Dupont, J.-F. Henniot, M. Warenghen, J. Appl. Phys. 107 (2010) 074102. [15] S.N. Paul, R. Dhar, R. Verma, S. Sharma, R. Dabrowski, Mol. Cryst. Liq. Cryst. 545 (2011) 23 105/[1329]-111[1335].
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[16] L. Scolari, S. Gauza, H. Xianyu, L. Zhai, L. Eskildsen, T.T. Alkeskjold, S.T. Wu, A. Bjarklev, Opt. Express 17 (2009) 3754. [17] Y.Yu. Reshetnyak, S.A. Shelestiuk, T.J. Sluckin, Mol. Cryst. Liq. Cryst. 454 (2006) 201/[1603]). [18] Y.Yu. Reshetnyak, S.A. Shelestiuk, T.J. Sluckin, Phys. Rev. E 83 (2011) 041705. [19] S.V. Burylov, A.N. Zakhlevnykh, Phys. Rev. E 88 (2013) 012511. [20] I. Palarie, C. Dascalu, G.E. Iacobescu, Liq. Cryst. 37 (2) (2010) 195. [21] R.H. Upadhyay, A.P. Argekar, R.R. Deshmukh, Bull. Mater. Sci. 37 (3) (2014) 481.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Electrical Freedericksz transitions in nematic liquid crystals containing ferroelectric nanoparticles Cristina Cîrtoaje, Emil Petrescu n , Victor Stoian University Politehnica of Bucharest, Department of Physics, Splaiul Independenţei 313, 060042 Bucharest, Romania
H I G H L I G H T S
A formula for Freedericksz transition voltage in a mixture of NLC and ferroelectric particles. The transiton threshold decreases when the nematic director is parallel to nanoparticle.The transiton threshold increases when the nematic director is perpendicular to the nanoparticle. The transiton threshold decreases when the nematic director is parallel to nanoparticle. We evaluated the anchoring energy of the LC molecules in a mixture of 5CB and BaTiO3.
art ic l e i nf o
a b s t r a c t
Article history: Received 16 September 2014 Received in revised form 31 October 2014 Accepted 7 November 2014 Available online 13 November 2014
A new theoretical approach was elaborated to explain the contradictions reported in many papers about the electric threshold for Freedericksz transition in nematic liquid crystal (NLC) with ferroparticles additives. The free energy density of the mixture was estimated and the contributions of the interaction energy of NLC molecules with ferroparticles surface were calculated. Experimental results for 5CB-BaTiO3 mixture are given. & 2014 Elsevier B.V. All rights reserved.
Keywords: Ferroelectric nanoparticle Liquid crystals Freedericksz transitions
1. Introduction In the last ten years, many papers were devoted to mixtures of liquid crystals with nanoparticles such as azo-dyes [1], protons [2], ferromagnetic particles [3–6], nanoparticles [7] and carbon nanotubes [8–12]. Some of the newest research were concerned on ferroelectric nanoparticles suspension in liquid crystal matrix [13– 17]. Experimental studies presented in [13,14] evidenced a decrease of the threshold values for the Freedericksz transition when ferroelectric nanoparticles are inserted into the liquid crystal. In [13] the authors have used Sn2P6S6 (thiohypodishoshate) as ferroelectric particles with an intrinsic polarisation of 14 μC/cm2, and as liquid crystal the ZLI-4901 mixture from Merck. They noticed a 50% decrease of the Freedericksz transition threshold of the mixture when nanoparticles were included. In [14] ferroelectric nanoparticles of BaTiO3 dispersed in 5CB nematic liquid crystal in a n
Corresponding author. E-mail address: [email protected] (E. Petrescu).
http://dx.doi.org/10.1016/j.physe.2014.11.004 1386-9477/& 2014 Elsevier B.V. All rights reserved.
volume percentage of 4% (the size of the particles was 5 nm to 100 nm) were used. The measurements using an alternate electric field of 1 kHz proved that the Freedericksz transition threshold decreased from 0.79 Vrms to 0.54 Vrms. Under similar conditions, in [15], the authors studied a mixture of BaTiO3 in 6CHBT liquid crystal and found an increase of the threshold voltage for Freedericksz transition from 0.7 Vrms to 1.3 Vrms when the nanoparticles were added. The size of the ferroparticles was between 30 nm and 50 nm for a BaTiO3 concentration of 0.5%wt. V. Yu. Reshetnyak and co-workers have presented the first theoretical model for the electric Freedericksz transition threshold in a couple of papers [17,18], the second one presenting a considerable theory improved from the first one. Both papers relay on the assumption that the nanoparticles do not influence the electrical properties of the liquid crystal and each particle presents a permanent polarisation (permanent electric moment) which can only be parallel or anti-parallel to the local nematic director. Theoretical aspects presented in [18] are much more complex and the average electric field inside the liquid crystal and the
24
C. Cîrtoaje et al. / Physica E 67 (2015) 23–27
ferroelectric particles respectively as well as the average electric field in the entire medium are taken into account. The authors present the dependence of the liquid crystal free energy as a function of the effective dielectric constants ε∥eff and ε⊥eff . One of the main assumptions they made, was that the particles were very small (below 10 nm) as can be proved from the condition dvβEp0 < 1 where, d ¼6 μC/cm2 is the particles polarisation density, v is average nanoparticle volume, β = 1/kBT and Ep0 is the contribution of the electric field from all the sources other than the depolarisation field due to the permanent polarisation of the considered particles. It can be noticed that it has the same order as the electric Freedericksz transition threshold EF ≃ 7, 9 × 104 V/m . Finally they obtained a dependence on effective anisotropy of the threshold voltage.
K1 ε0εaeff
U=π
(1)
where
εa eff = ε∥ eff − ε⊥ eff
(2)
It is the aim of this paper to give an improved theoretical model of [18] by taking into account the nature of the interaction between the ferroparticles and the NLC molecules. We have introduced a specific term in the free energy density equation proposed by Burylov and Zakhlevnykh [19] considering the anchoring of the nematic molecules to nanoparticles surface.
2. Theoretical background
(3)
1 2
-E = −
-INT = −
1 2
∫0
L
⎡ →2 → →2 → → 2⎤ ⎢K1(∇· n ) + K2( n ·∇ × n ) + K3( n × ∇ × n ) ⎥dz ⎣ ⎦
L
⎡ →2 →→ ⎤ ε0⎢ε⊥ eff E + εaeff ( n E )2⎥dz ⎣ ⎦
(5)
∫0
L
2wf →→ 2 →→ P2(m n ) [1 − ξ (m n0)] a
(6)
where f is the volumetric fraction of nanoparticles, a is the nanoparticle's diameter (considered to be a cylinder), w is the average anchoring energy density at the nematic-nanoparticle inter→ action surface, m is the unit vector of particle dipole moment, P2 = (3cos2α − 1)/2 is the second order Legendre Polynomial, where α is the angle between the NLC and the ferroelectric par→ ticle, n0 is the direction of the nematic molecular director outside the nanoparticles range of action and ξ is a dimensionless phenomenological parameter [19]. The case of a strong anchoring of nematic molecules to the nanoparticles surface and a homogeneous alignment of the cell was considered. In the absence of any external fields, the nematic director is parallel to Ox axis. The applied electric field is parallel to Oz axis and the ferroelectric particles align themselves parallel to the same axis. In this geometry (Fig. 1), the liquid crystal free energy becomes.
1 2
∫0
⎡ ⎤ 2 2 2 ⎢K1cos θ + K3sin θ ⎥θz dz ⎣ ⎦
L
(7)
where
θz =
dθ dz
(8)
The electric field energy component can be written as
1 2
-E = −
-N is the liquid crystal free energy:
-N =
∫0
where ε⊥eff and εaeff are the dielectric constant and the dielectric anisotropy of the nematic-ferroelectric nanoparticles mixture. For the interaction free energy between nanoparticles and nematic liquid crystal molecules we used the formula given by Burylov and Zakhlevnykh [19]:
-N =
A mixture of nematic liquid crystal with ferroelectric nanoparticles suspension confined between two identical solid walls parallel to the xOy plane placed at z¼ 0 and z¼L was considered (Fig. 1). The free energy of ferroelectric nanoparticles suspension in liquid crystal electric field is
- = -N + -E + -INT
-E is the electric field free energy contribution:
∫0
L
DE dz = −
1 2
∫0
L
⎛ ⎞ ε0⎜ε⊥ eff + εaeff sin2θ ⎟E2 dz ⎝ ⎠
(9)
where
(4)
where K1, K2 and K3 are the splay, twist and bend elastic constants, and L is the cell thickness.
E=
D ε⊥ eff + εa eff sin2θ
(10)
Since D is constant for a given configuration, we obtain
U=
∫0
L
Edz = D
∫0
L
dz ε0(ε⊥ eff + εaeff sin2θ)
(11)
In this case, -E becomes
-E = −
1 D 2
∫0
L
Edz = −
⎡ 1 2⎢ U 2 ⎢⎣
∫0
L
⎤−1 ⎥ + εaeff sin2θ) ⎥⎦ dz
ε0(ε⊥ eff
(12)
The interaction free energy -INT is
-INT = −
∫0
=−A
L
2wf P2sin2θ[1 − ξsin2θ] dz a
∫0
L
sin2θ[1 − ξsin2θ] dz
(13)
where
A= Fig. 1. The nematic director and ferroelectric particle orientation.
2wf P2 a
(14)
C. Cîrtoaje et al. / Physica E 67 (2015) 23–27
When reaching the threshold value, θ increases by a small amount. This is the reason why we consider only its small values to evaluate the free energy:
-N =
1 2
L
∫0
K1θz2 dz
(15)
and
-E = −
ε U2ε0ε⊥ eff ⎛ ⎜1 + aeff 2L ⎜⎝ Lε⊥ eff
∫0
L
⎞ θ 2 dz⎟⎟ ⎠
Consequently, the interaction term of NLC molecules with the nanoparticles surface may also be estimated using the same approximation and we obtain
-INT = − A
∫0
L
θ 2(1 − ξθ 2) dz
and
L = 2
(17)
∫0
2A ⎡ ⎢ K1 ⎣
(
∫0
-=−
2
L
U2 +
1 2
∫0
L
(18)
dz
The first term in Eq. (18) is constant so we define a new free energy density
F=
⎤ 1⎡ 2 2 2 2 ⎢K1θz − 2Aθ (1 − ξθ ) − Bθ ⎥⎦ 2⎣
ε0εaeff L2
U2
(28)
1
dλ 2A ⎡ ⎢1 K1 ⎣
−
ξθm2 (1
⎤ + λ )⎦⎥ + 2
1 − λ2
B K1
(29)
1
L = 2
2A K1
+
B K1
∫0
1
dλ 1 − λ2
π
= 2
2A K1
+
B K1
(30)
Replacing B with the original formula given by Eq. (20), the critical voltage is
K1 ε0εaeff
1−
2AL2 π 2K1
(31)
When a pure nematic is used, f ¼0 εaeff = εa the Freedericksz transition threshold voltage is U0 = π K1/ε0εa , so
UF = U0
εa εaeff
1−
2AL2 π 2K1
(32)
(20)
A=
2wf (3 cos 2α − 1) 2wf = a 2 a
(33)
and
d ⎛ ∂F ⎞ ∂F =0 ⎜ ⎟− dz ⎝ ∂θz ⎠ ∂θ
(21)
we get the prime integral:
∂F −F=C ∂θz
(22)
where C is a constant. Therefore
K1θz2 + 2Aθ 2(1 − ξθ 2) + Bθ 2 = C
(23)
This relationship is an identity so it is also valid in the middle of the cell (z = L/2) where the deviation angle reaches its maximum (θ = θm and θz = dθ /dz = 0) i.e.
2Aθm2 (1 − ξθm2) + Bθm2 = C
(24)
If we denote
UF = U0
εa εaeff
1−
4wL2 aπ 2K1
z(θ , θm)
(25)
(34)
Because 1 − 4wL2/aπ 2K1 < 1 it is possible for UF /U0 to be smaller than the unit. This means that when nanoparticles are added into a liquid crystal matrix, the Freedericksz transition threshold may decrease. (b) The nematic director is perpendicular to the nanoparticle (θ = 90°) and the parameter A is smaller than zero:
A=−
wf a
(35)
Then
UF = U0
εa εaeff
1+
2wf L2 >1 a π 2K1 2
dθ = dz
1
In this case, two limit situations may be considered (a) The molecules align themselves parallel to the nanoparticle surface (α = 0). Then
From Euler–Lagrange equation,
θz
) − ξ(θm4 − θ 4)⎤⎦⎥ + KB (θm2 − θ 2)
(19)
where
B=
−θ
2
At the limit of the Freedericksz transition (θm → 0) Eq. (29) becomes
UF = π
⎡ 2 ε0εaeff 2 2⎤ 2 2 Uθ ⎥ ⎢K1θz − 2Aθ (1 − ξθ ) − ⎣ ⎦ L2
θm2
1
From Eqs. (15), (16) and (17) the total free energy is
1 ε0ε⊥ eff
dθ
θm
Introducing a new parameter λ = θ /θm so dθ = θmdλ we get
L = 2 (16)
25
(36)
2
As 1 + (2wf /a)L /π K1 > 1, the Freedericksz transition threshold for the mixture can be higher than the pure liquid crystal.
where
z(θ , θm) =
⎤ ⎞ 2A ⎡ 2 B⎛ 2 ⎢ θm − θ 2 − ξ θm4 − θ 4 ⎥ + ⎜θm − θ 2⎟ K1 ⎣ ⎦ K1 ⎝ ⎠
(
)
(
)
3. Experimental set-up and materials
(26)
we obtain
∫0
L /2
dz =
∫0
θm
dθ z(θ , θm)
(27)
The liquid crystal cells with 20 μm Mylar spacers were previously prepared for a planar alignment. A mixture of 5CB and BaTiO3 ferroelectric particles was used to fill the cells. The average size of the BaTiO3 crystals used was 50 μm and the measurements were performed at a constant temperature of 25 °C. For imaging the surface topographies we used Non-contact operating mode (AFM, PARK XE-100 SPM system). The cantilever had a nominal
26
C. Cîrtoaje et al. / Physica E 67 (2015) 23–27 Table 2 Experimental Freedericksz transition threshold (UF) at 1 kHz for each volumetric fraction and the value of εa/εaeff.
Fig. 2. Experimental set-up for the study of the electric properties of 5CB nematic with BaTiO3.
Table 1 Experimental Freedericksz transition threshold (UF) at 100 kHz for each volumetric fraction (f) and anchoring energy density (w). f (%)
UF(100 kHz)(V)
w (N/m)
0 0.33
0.64 0.59
3.47 × 10−7
0.50
0.56
3.57 × 10−7
f (%)
UF(1 kHz)(V)
εa εaeff
0 0.33 0.50
0.64 0.56 0.51
– 0.90 0.82
Thus, using experimental values for UF (100 kHz) and U0 we can evaluate the anchoring energy density w from Eq. (37). The results are given in Table 1 using the following values a ¼50 nm, K1 = 6, 2 × 10−12 N and L = 20 μm . The second measurements set was made using a 1 kHz frequency for which the dielectric constant of BaTiO3 is high enough to satisfy the εaeff ≠ εa condition. Just like the previous case, we noticed a decrease of the Freedericksz transition threshold:
UF (1 kHz) = U0
εa 4wL2 1− εaeff aπ 2K1
(38)
From Eqs. (37) and (38) we obtain
–
UF (1 kHz) = UF (100 kHz)
εa εaeff
(39)
which allows us to evaluate εa/εaeff for a 1 kHz voltage frequency. The results are presented in Table 2. length of 125 mm, a nominal force constant of 40 N/m, and oscillation frequencies in the range of 275–373 kHz. We used horizontal line by line flattening as planarization method [20]. The sample was prepared by evaporation on silicon substrate of a dilute solution of BaTiO3 in water. The average diameter was calculated by investigation of four different area (1 1 μm2, 3 13 μm2, 5 5 μm2and 10 110 μm2), for ten different regions of the samples. We have considered the polarisation density of BaTiO3 to be 0.15 μC/cm2. BaTiO3 nanoparticles were dispersed in different volumetric concentrations ( f1 = 0.33% and f2 = 0.50% ) in 5CB liquid crystal. The experimental set-up, presented in Fig. 2 consists of a 632.8 nm He–Ne laser used to send a beam through the LC cell fixed in a slot inside the thermal stage to keep the temperature constant. A power source able to apply an alternate voltage is connected to the electrode of the cell and a photodetector is used to record the laser beam after crossing the sample.
5. Conclusions Our theoretical estimations reveal that the molecules alignment angle on ferroelectric particles influences the Freedericksz transition threshold. We proved that an increase of this threshold value is also possible and we explained the contradictory results reported in other papers. The critical voltage for the transition strongly depends on the orientation angle of the nematic director to the ferroelectric particles surface: when they are parallel to each other, the transition threshold may decrease, but when they are perpendicular to each other, the transition threshold may increase.
Acknowledgments The work has been founded by the Sectorial Operational Program Human Resource Development 2007–2013 of the Ministry of European Funds through the Financial Agrement POSDRU 159/1.5/ S/132397.
4. Results and discussions The dielectric properties of BaTiO3 are thoroughly discussed in [21]. According to the authors, when the frequency of the applied electric field is higher than 100 kHz, the dielectric constant of BaTiO3 decreases to a constant value. In this case εaeff → εa which means that for this frequency the Freedericksz transition threshold changes only due to the interaction of the liquid crystal molecules with the ferroelectric particles surface. The experimental results proved that the Freedericksz transition threshold is lower when ferroelectric particles are added meaning that the LC molecules are oriented parallel to the nanoparticles. The critical voltage can be expressed from Eq. (34) by replacing εaeff with εa.
UF (100 kHz) = U0 1 −
4wfL2 aπ 2K1
(37)
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