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Electric Freedericksz transition in nematic liquid crystals with graphene quantum dot mixture
Applied Surface Science 487 (2019) 1301–1306
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full length article
Electric Freedericksz transition in nematic liquid crystals with graphene quantum dot mixture
T
Cristina Cirtoajea, Emil Petrescua,*, Cristina Stana, Andrey Rogachevb,c a
University Politehnica of Bucharest, Department of Physics, Splaiul Independentei 313, 060042 Bucharest, Romania Joint Institute for Nuclear Research, Dubna, Russia c Moscow Institute of Physics and Technology, Dolgoprudny, Russia b
A R T I C LE I N FO
A B S T R A C T
Keywords: Freedericksz transition Quantum dots Graphene
A theoretical model to explain the Freedericksz transition for a nematic liquid crystal with graphene quantum dots adding in electric field below the instability area is proposed. A mathematical formula for the critical voltage was obtained by elastic continuum theory combined with Burilov - Zakhlevnykh anchoring model, classical electrostatic principles and temperature dependence of elastic constant. We compared the results with the experimental data recorded for three LC samples and a good agreement between them was found encouraging us to propose this model as a tool for theoretical analysis of graphene quantum dot's dispersions in liquid crystals.
1. Introduction Liquid crystals (LC) widely used for display technologies are often mixed with different compound or nanoparticles such as dyes [1-3], magnetic particles [4–6] ferroelectric particles [23], carbon based nanoparticles [7-11] or others [12,13] to improve their performances. The importance of these composite systems is revealed by the increased interest in new materials for spectroscopy, imaging and microscopy, biosensors, lenses or medicine. All these devices require precise control of sample shape, material structure and composition, as well as effective addressability and efficient energy transfer. Although many experimental results proved the nanoparticles efficiency, there are not enough theoretical studies on these systems to completely describe them. A deep understanding of physical phenomena occurring in such mixtures is highly necessary because it gives both the engineers and researchers the necessary tools to precisely control a desired parameter by chemical methods (i.e. varying the particles concentration or covering them with different surfactants) or by the action of physical factors (temperature variation, external field action or light response) In this paper, a study of the Freedericksz transition change for a mixture of nematic 5CB with different concentrations of graphene quantum dots (GQD) was made and a theoretical model to explain their behavior is proposed. The model is based on Freedericksz effect [14–17] and on the elastic continuum theory developed by Zocher [18], Oseen [22], and Frank [20] using Burylov - Zakhlevnykh model [13] for the interaction
*
of graphene quantum dot surface with host molecules and electric properties of a semiconductor nanoparticles dispersion in anisotropic dielectric host. 2. Theoretical background A simple analytical method to evaluate the Freedericksz transition threshold is based on elastic continuum theory. According to this theory, the mixture's free energy can be written as a sum:
F = FN + FE + Fint
(1)
where FN is the term related to the elastic interactions between nematic molecules, the FE term contains the electric field interaction with the system and Fint is related to the nematic molecules anchoring on nanoparticles's surface. The elastic free energy term for planar aligned cell is given by:
FN =
1 2
L
∫ [K1 θz2 cos2 θ + K3 sin2 θ] dz 0
(2)
where θ is the deviation angle inside the cell, θz = dθ/dz and K1, K3 are the splay and bend elastic constants. Burylov and Zachlevnick proposed an interaction model for ferromagnetic nanoparticles [13] that can be applied for cylinder GQD as well.
Corresponding author. E-mail address: [email protected] (E. Petrescu).
https://doi.org/10.1016/j.apsusc.2019.05.073 Received 29 January 2019; Received in revised form 25 April 2019; Accepted 7 May 2019 Available online 11 May 2019 0169-4332/ © 2019 Elsevier B.V. All rights reserved.
Applied Surface Science 487 (2019) 1301–1306
C. Cirtoaje, et al.
By decomposing the integral for each region inside GQD and outside GQD according to Fig. 2, we get from Eq. (5):
1 V
a
∫ (De − ε1 ε0 Ee ) dv + ∫ (Di − ε1 ε0 Ei )2πrldr = D¯ − ε0 ε1 Ē 0
ext
(6)
where De and Ee refers to the field outside the GQD while Di and Ei refers to the field parameters inside the GQD. The first integral is null because: (7)
De = ε0 ε1 Ee and
(8)
Di = ε0 ε2 Ei
The electric field inside the GQD Ei can be calculated by taking the electric potential as
A φ1 = −E0 cos θ ⎛r − ⎞ r < a r⎠ ⎝
(9)
and
φ2 = −BE0 cos θ
φ1 (a) = φ2 (a)
L
∫ 0
2wf →→ →→ P2 (α )(m n )2 [1 − ξ (m n )2] dz a
ε1
(3)
where w is the anchoring energy, a is the graphene nanoplatelet radius, → is it's axis versor, → L is its length, m n is molecular director and α is the anchoring angle of LC molecule on graphene sheet as shown in Fig. 1. ξ is a dimensionless phenomenological parameter, P2 (α ) = (3 cos2 α − 1)/2 is the second order Legendre polynomial depending on the anchoring angle. The electric field interaction free energy term can be represented just as in any other liquid crystal but with an effective dielectric anisotropy instead of the regular parameters:
FE = −
ε0 2
L
∫
r=a
∂φ2 ∂r
r=a
(12)
A=
ε2 − ε1 2 a ε2 + ε1
(13)
B=
2ε1 ε2 + ε1
(14)
Ei = − (4)
∫V (D − ε1 ε0 E ) dv = D¯ − ε1 ε0 Ē
= ε2
The electric field applied is parallel to Ox axis, so for the inside electric field, we get:
∂φ2 2ε1 E0 = ε1 + ε2 ∂x
(15)
From Eq. (6), by denoting the volumetric fraction of nanotubes with Nv f = V , (where v is the nanoparticle's volume and V the mixture's volume) and considering the applied field E0, equal to the average field inside the mixture Ē , we obtain:
where E is the applied electric field, ε⊥eff is the effective perpendicular dielectric constant and εa eff is the dielectric anisotropy of the LC+GQD mixture. The constants can be calculated using classic electrostatic principles presented by Landau in [21].
1 V
∂φ1 ∂r
(11)
Thus we obtain:
→2 → 2 [ε⊥ eff E + εa eff (E → n ) ] dz
0
(10)
where A and B are constants that can be determined from boundary conditions:
Fig. 1. Nematic molecule's anchoring on graphene nanoplatelet surface.
Fint = −
r≥a
ε0 f (5)
2ε1 (ε2 − ε1) Ē = D¯ − ε0 ε1 Ē ε1 + ε2
(16)
By replacing D¯ = εeff ε0 Ē in Eq. (16), the effective permittivity for the considered mixture is:
where V is the mixture's volume, D is the electric displacement, E is the electric field, ε0 is the vacuum electric permittivity and ε1 is the dielectric constant of the liquid crystal in which GQDs are inserted. The parameters D¯ and Ē refers to the average values in the mixture.
εeff = ε1 + f
2ε1 (ε2 − ε1) ε1 + ε2
Fig. 2. Semiconductor cylindric GQD in dielectric environment with electric field perpendicularly applied on cylinder axis. 1302
(17)
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C. Cirtoaje, et al.
Fig. 3. Electric field and graphene quantum dot orientation in planar LC cell.
Since the field was parallel to the nematic director (on OX axis), then ε1 is the parallel component of liquid crystal dielectric constant ( ε1 ≡ ε∥LC) and ε2 is the parallel component of GQD dielectric constant (ε2 ≡ ε∥GQD ), so:
ε∥ eff = ε∥ LC + f
U = U0
U0 = π (18)
∫ext (De − ε0 ε1 Ee ) dv + VN ∫int (ε0 ε2 Ei − ε0 ε1 Ei ) dv = D¯ − ε0 ε1 Ē
A=
ε1 E0 ε2
l
∫ 0
⎜
(21)
(22)
(23)
Finally we can write the total free energy of the mixture as:
ε0 ε⊥ eff 2L2
U2 −
ε0 εa eff 2L2
ε0 εa eff
)
−
2wfL2 (3 cos2 α − 1) 0 εa eff
π 2aε
(29)
3. Experimental set-up and procedure
U2
∫ sin2 θdz
Analysed samples were prepared using 5CB nematic and water dispersed GQD (1 mg/ml) graphene quantum dots from Sigma Aldrich. The open bottles are place in dust free environment at constant temperature until all the water evaporated resulting a mass concentration of 0.45% for the first sample and 1.6% for the second one. These mixtures were used to fill 15-micrometer thick planar aligned cells from Instec. Each cell was placed on a holder in the hot stage of thermostat. The holder terminals were connected to a power source from which the voltage was increased step by step with 0.01 V from 0 to 1.5 V. As mentioned before, when the external field reaches the critical Freedericksz transition threshold, the molecules change their
0
∫ wπa2f [1 − 3 cos2 α]sin2 θdz 0
+
T 2β TC
L
L
+
(28)
As it can be seen from Eq. (29), the critical transition voltage depend on the nanoparticle's electric properties that affect the effective permittivity of the sample as shown in Eq. (18) for parallel component of dielectric anisotropy and in Eq. (23) for the perpendicular one. As in the case of other nanoparticle [1,4,5,23] the volumetric fraction of inserted nanoparticles may either decrease the Freedericksz transition limit or increase it depending on the anchoring angle (α) between the nematic director and nanoparticles's surface. Since the anchoring angle evaluation can only be made by theoretical methods [5,13], we can experimentally test this model by temperature dependence of the critical voltage for the same concentration outside pretransitional range where are many instabilities.
In this case, the field was perpendicularly applied on the nematic director, so we have ε1 ≡ ε⊥LC and ε2 ≡ ε⊥GQD:
F=−
(
K 01 1 − U=π
⎟
ε⊥ LC ⎞ ⎤ ⎛ ε⊥ eff = ε⊥ LC ⎡ ⎢1 + f ⎜1 − ε⊥ GQD ⎟ ⎥ ⎝ ⎠⎦ ⎣
⎟
where i refers to the order of elastic constant (i = 1, 2, 3 for splay, twist and bend elastic deformation), T is the sample's temperature, Tc is the clearing temperature of the liquid crystal and β = −0.1855 is a material constant [25] and K01 are the elastic constants at T = 273.15 K. In this case, we get the temperature dependence of the transition voltage:
Considering the applied electric field equal to the average field inN side the mixture and denoting by f = V πa2l the volumetric fraction of GQD, we obtain the effective dielectric constant as: ⎜
(27)
2β
⎟
ε εeff = ε1 ⎡1 + f ⎛1 − 1 ⎞ ⎤ ⎢ ε2 ⎠ ⎥ ⎝ ⎦ ⎣
(26)
wf (3 cos2 α − 1) 2a
⎜
(20)
⎛ε1 ε0 E0 − ε12 ε0 E0 ⎞ πa2dl = D¯ − ε0 ε1 Ē ε2 ⎠ ⎝
K1 εa ε 0
T Ki = Ki0 ⎛1 − ⎞ Tc ⎠ ⎝
Thus we have:
N V
(25)
Lin and collaborators [24] reported a temperature dependence of elastic constants:
(19)
where De and Ee are the electric displacement and the electric field outside the GQD and Ei is electric field inside GQD. The first term of the equation is null and we only have to calculate the second integral. Using the same principle as in previous case, the electric field inside the GQD is:
Ei =
AL2 π 2K1
is the Freedericksz transition threshold for liquid crystal, and
For the perpendicular dielectric constant of the mixture we shall consider the external field parallel to the cylindric GQG's axis (Fig. 3), so Eq. (5) becomes:
1 V
εa eff
1−
where
2ε∥ LC (ε∥ GQD − ε∥ LC ) ε∥ LC + εGQD
εa
L
L
0
0
∫ ⎡⎣ kBvT f ln f ⎤⎦ dz + 12 ∫ (K1 θz2 cos2 θ + K3 sin2 θ) dz
(24)
After applying the Euler-Lagrange equations and following the same procedure as presented in [23], we obtain the Freedericksz transition voltage as: 1303
Applied Surface Science 487 (2019) 1301–1306
C. Cirtoaje, et al.
Fig. 4. Figure experimental set-up for Freedericksz transition threshold voltage.
orientation tending to align with the field. Experimentally, this reorientation is revealed by an intensity variation of the emergent laser beam through the sample. A photovoltaic cell was used to record the emergent intensity (Fig. 4). Experimental laser intensity versus voltage plot obtained from reference sample filled with 5CB and for other two samples with 0.45% GQD and 1.6% GQD were recorded at various temperatures. 4. Results and discussions From the emergent intensity versus applied voltage plot, the Freedericksz transition threshold was determined as the point where a significant intensity change appears. An example of threshold voltage experimental evaluation for 5CB at 25 °C is presented in Fig. 5. The threshold voltage versus temperature presented in Fig. 6 indicates an uniform behavior at low temperature (below 299 K) and a high instability near the nematic isotropic transitions due to the increase of thermal agitation. We used Eq. (28) to fit experimental data for the temperature in stability region (bellow 299 K) and, as it can be seen from Fig. 7, the fitting is good. For the dielectric parameters we consider ε∥ = 18.65,
Fig. 6. Freedericksz transition threshold voltage versus temperature plot. Lines are guide for the eyes.
ε⊥ = 7.15, ε∥GQD = 1.8, ε⊥GQD = 3, w = 5 × 10−7 N/m, ρCL = 1.022 g/ cm3 and ρGQD = 2.02 g/cm3 [26–28,13]. A second check of the model was to calculate the elastic constant K1 at 293.15 K from Eq. (25). The results are in good agreement with the ones resulting from other paper [29,30], so we extend the calculation to the GQD containing samples as shown in Table 1. From Eq. (29) it result an anchoring angle α = 54.7° for both GQD concentrations, so we may assume that, in this concentration range we have the same anchoring mechanism and no particles clusters appears inside the sample.
5. Conclusion We deduced a mathematical formula for the Freedericksz transition threshold voltage of a nematic liquid crystal mixed with graphene quantum dots taking into account the influence of the GQD amount and the temperature. The results indicated a decrease of the threshold value with the increase of GQD concentration for the first half of the nematic range, in good agreement with experimental data. This behavior is a result of the nematic anchoring on GQD surface. In the vicinity of GQD the planar alignment is disturbed and some of the nematic molecules
Fig. 5. Experimental evaluation of Freedericksz transition threshold from intensity versus voltage plot. 1304
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C. Cirtoaje, et al.
Fig. 7. Critical voltage versus temperatures for three different concentration and fitting curve using Eq. (29): a) for 5CB+0.45% GQD, b) for 5CB+1.6% GQD, c) for 5CB. Table 1 Expenditure share and nucleolus. CQD concentration mass fraction (%)
K0 × 1012 [N]
0 0.45 1.6
11.946 12.024 11.990
[4] D. Manaila Maximean, C. Cirtoaje, O. Danila, D. Donescu, Novel colloidal system: magnetite - polymer particles lyotropic liquid crystal under magnetic field, J. Magn. Magn. Mater. 438 (2017) 132–137, https://doi.org/10.1016/j.jmmm.2017.02.034. [5] A.N. Zakhlevnykh, M. Lubnin, D.A. Petrov, A simple model of liquid-crystalline magnetic suspension of anisometric particles, J. Magn. Magn. Mater 431 (2017) 62–65, https://doi.org/10.1016/j.jmmm.2016.09.044. [6] A.N. Zakhlevnykh, D.A. Petrov, Magnetic field induced orientational transitions in soft compensated ferronematics, Phase Transit. 88 (1) (2014) 1–18, https://doi. org/10.1080/01411594.2012.752085. [7] M.K. Massey, A. Kotsialos, D. Volpati, E. Vissol-Gaudin, C. Pearson, L. Bowen, B. Obara, D.A. Zeze, C. Groves, M.C. Petty, Evolution of electronic circuits using carbon nanotube composites, Sci, Rep. 6 (2016) 32197, https://doi.org/10.1038/ srep32197. [8] S.P. Yadav, S. Singh, Carbon nanotube dispersion in nematic liquid crystals: an overview, Prog, Mater. Sci. 80 (2016) 38–76, https://doi.org/10.1016/j.pmatsci. 2015.12.002. [9] C. Denktas, H. Ocak, M. Okutan, A. Yildiz, B. Bilgin Eranc, O. Köysal, Effect of multi wall carbon nanotube on electrical properties 4-[4-((S)-Citronellyloxy)benzoyloxy] benzoic acid liquid crystal host, Compos. Part B- Eng. 82 (2015) 173–177, https:// doi.org/10.1016/j.compositesb.2015.08.021. [10] V. Popa-Nita, S. Kralj, Liquid crystal-carbon nanotubes mixtures, J. Chem. Physi. 132 (2) (2010) 024902, https://doi.org/10.1063/1.3291078. [11] L.N. Lisetski, A.N. Samoilov, S.S. Minenko, A.P. Fedoryako, T.V. Bidna, A novel photoelectroooptical effect based on Freedericksz-type transition in nematic mixtures of azoxy compounds and cyanobiphenyls, Functional Materials 25 (4) (2018) 681–683, https://doi.org/10.15407/fm25.04.081. [12] C. Rosu, D.M. Maximean, V. Circu, Y. Molard, T. Roisnel, Differential negative resistance in the current - voltage characteristics of a new palladium(II) metallomesogen, Liq. Cryst 38 (6) (2011) 757–765, https://doi.org/10.1080/02678292.2011. 573585. [13] S.V. Burylov, A.N. Zakhlevnykh, Orientational energy of anisometric particles in liquid-crystalline suspensions, Phys, Rev. E 88 (2013) 012511, https://doi.org/10. 1103/PhysRevE.88.012511. [14] E. Petrescu, R. Bena, C. Cirtoaje, Polarization gratings using ferronematics - an elastic continuum theory, J. Magn. Magn. Mater. 336 (2013) 44–48, https://doi.
are already lifted making an anchoring angle α = 54.7° angle with the substrate support. Acknowledgments The authors acknowledge the financial support through the grant of Romanian Governmental Representative at JINR Dubna and scientific project No322/21.05.2018item 13. References [1] G. Iacobescu, A.L. Paun, C. Cirtoaje, Magnetically induced Freedericksz transition relaxation phenomena in nematic liquid crystals doped with azo-dyes, J. Magn. Magn Mater. 62 (2008) 2180–2184, https://doi.org/10.1016/j.jmmm.2008.03. 050. [2] C. Motoc, C. Cirtoaje, A. Stoica, V. Stoian, A. Albu, Relaxation phenomena in nematic-polymer mixtures, U.P.B. Sci. Bull. Series A 75 (3) (2013) 181–186. [3] L. Tao, X. Ying, L. Yi-Kun, W. Jian, Y. Shun-Lin, Transient reorientation of a doped liquid crystal system under a short laser pulse, Chinese Phys Lett, 26(8) 086108 (2009), https://doi.org/10.1088/0256-307X/26/8/086108.
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[23] C. Cirtoaje, E. Petrescu, V. Stoian, Electrical Freedericksz transitions in nematic liquid crystals containing ferroelectric nanoparticles, Physica E 67 (2015) 23–27, https://doi.org/10.1016/j.physe.2014.11.004. [24] Y.Q. LinS, Temperature effect on threshold voltage and optical property of twisted nematic liquid crystal with applied different voltages, Optik 121 (18) (2010) 1693–1697, https://doi.org/10.1016/j.ijleo.2009.04.001. [25] J. Li, S. Gauza, S. Wu, Temperature effect on liquid crystal refractive indices, J. APppl. Phys 96 (1) (2004) 19–24, https://doi.org/10.1063/1.1757034. [26] P.G. Cummins, D.A. Dunmur, D.A. Laidler, The dielectric properties of nematic 44’ n-pentylcyanobiphenyI, Mol. Cryst. Liq. Cryst 30 (1975) 109–121, https://doi.org/ 10.1080/15421407508082846. [27] L.A. Karamysheva, S.I. Torgova, E.I. Kovshev, Liquid-crystalline esters of trans-4alkylcyclohexanecarboxylic acids, Zh. Org. Khim 15 (5) (1979) 1013–1017. [28] E. Santos, E. Kaxiras, Electric-field dependence of the effective dielectric constant in graphene, Nano. Lett 13 (3) (2013) 898–902, https://doi.org/10.1021/nl303611v. [29] D.A. Balzarini, D.A. Dunmur, P. Palffymuhoray, High-voltage birefringence measurements of elastic-constants, Mol. Cryst. Liq. Cryst 102 (2) (1984) 35–41, https:// doi.org/10.1080/01406568408247037. [30] vol. T. Inukai, K. Miyarova, Outine of the development of nematic liquid crystals compound for LCD, pag. Ekisho 1 (1994) 9–22.
org/10.1016/j.jmmm.2013.02.018. [15] P. Pieranski, F.E.G. Brochard, Static and dynamic behavior of a nematic liquid crystal in a magnetic field. Part II : dynamics, J. Phys-Paris 34 (1973) 35–48, https://doi.org/10.1051/jphys:0197300340103500. [16] V. Freedericksz, A. Repiewa, Theoretisches und Experimentelles zur Frage nach der Natur der anisotropen Flussigkeiten, Zeitschrift fur Physik, 42 (7) (1927) 532–546, https://doi.org/10.1007/BF01397711. [17] V. Freedericksz, V. Zolina, Forces causing the orientation of an anisotropic liquid, Trans. Faraday Soc. 29 (1933) 919–930, https://doi.org/10.1039/TF9332900919. [18] H. Zocher, The effect of a magnetic field on the nematic state, Trans, Faraday Soc. 29 (1933) 945–957, https://doi.org/10.1039/TF9332900945. [19] C. Ossen, The theory of liquid crystals, Trans. Faraday Soc. 29 (1933) (1933) 883–889, https://doi.org/10.1039/TF9332900883. [20] F.C. Frank, I. Liquid crystals. On the theory of liquid crystals, Discuss Faraday Soc 25 (1958) 19–28, https://doi.org/10.1039/DF9582500019. [21] L. Landau, E. Lifchitz, Electrodinamique des Milieux Continues, Editions MIR, Moscou, 1969. [22] W. Lewandowski, M. Fruhnert, J. Mieczkowski, C. Rockstuhl, E. Gorecka, Dynamically self-assembled silver nanoparticles as a thermally tunable metamaterial, Nat Commun. 6 (2015) 65, https://doi.org/10.1038/ncomms7590.
1306
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full length article
Electric Freedericksz transition in nematic liquid crystals with graphene quantum dot mixture
T
Cristina Cirtoajea, Emil Petrescua,*, Cristina Stana, Andrey Rogachevb,c a
University Politehnica of Bucharest, Department of Physics, Splaiul Independentei 313, 060042 Bucharest, Romania Joint Institute for Nuclear Research, Dubna, Russia c Moscow Institute of Physics and Technology, Dolgoprudny, Russia b
A R T I C LE I N FO
A B S T R A C T
Keywords: Freedericksz transition Quantum dots Graphene
A theoretical model to explain the Freedericksz transition for a nematic liquid crystal with graphene quantum dots adding in electric field below the instability area is proposed. A mathematical formula for the critical voltage was obtained by elastic continuum theory combined with Burilov - Zakhlevnykh anchoring model, classical electrostatic principles and temperature dependence of elastic constant. We compared the results with the experimental data recorded for three LC samples and a good agreement between them was found encouraging us to propose this model as a tool for theoretical analysis of graphene quantum dot's dispersions in liquid crystals.
1. Introduction Liquid crystals (LC) widely used for display technologies are often mixed with different compound or nanoparticles such as dyes [1-3], magnetic particles [4–6] ferroelectric particles [23], carbon based nanoparticles [7-11] or others [12,13] to improve their performances. The importance of these composite systems is revealed by the increased interest in new materials for spectroscopy, imaging and microscopy, biosensors, lenses or medicine. All these devices require precise control of sample shape, material structure and composition, as well as effective addressability and efficient energy transfer. Although many experimental results proved the nanoparticles efficiency, there are not enough theoretical studies on these systems to completely describe them. A deep understanding of physical phenomena occurring in such mixtures is highly necessary because it gives both the engineers and researchers the necessary tools to precisely control a desired parameter by chemical methods (i.e. varying the particles concentration or covering them with different surfactants) or by the action of physical factors (temperature variation, external field action or light response) In this paper, a study of the Freedericksz transition change for a mixture of nematic 5CB with different concentrations of graphene quantum dots (GQD) was made and a theoretical model to explain their behavior is proposed. The model is based on Freedericksz effect [14–17] and on the elastic continuum theory developed by Zocher [18], Oseen [22], and Frank [20] using Burylov - Zakhlevnykh model [13] for the interaction
*
of graphene quantum dot surface with host molecules and electric properties of a semiconductor nanoparticles dispersion in anisotropic dielectric host. 2. Theoretical background A simple analytical method to evaluate the Freedericksz transition threshold is based on elastic continuum theory. According to this theory, the mixture's free energy can be written as a sum:
F = FN + FE + Fint
(1)
where FN is the term related to the elastic interactions between nematic molecules, the FE term contains the electric field interaction with the system and Fint is related to the nematic molecules anchoring on nanoparticles's surface. The elastic free energy term for planar aligned cell is given by:
FN =
1 2
L
∫ [K1 θz2 cos2 θ + K3 sin2 θ] dz 0
(2)
where θ is the deviation angle inside the cell, θz = dθ/dz and K1, K3 are the splay and bend elastic constants. Burylov and Zachlevnick proposed an interaction model for ferromagnetic nanoparticles [13] that can be applied for cylinder GQD as well.
Corresponding author. E-mail address: [email protected] (E. Petrescu).
https://doi.org/10.1016/j.apsusc.2019.05.073 Received 29 January 2019; Received in revised form 25 April 2019; Accepted 7 May 2019 Available online 11 May 2019 0169-4332/ © 2019 Elsevier B.V. All rights reserved.
Applied Surface Science 487 (2019) 1301–1306
C. Cirtoaje, et al.
By decomposing the integral for each region inside GQD and outside GQD according to Fig. 2, we get from Eq. (5):
1 V
a
∫ (De − ε1 ε0 Ee ) dv + ∫ (Di − ε1 ε0 Ei )2πrldr = D¯ − ε0 ε1 Ē 0
ext
(6)
where De and Ee refers to the field outside the GQD while Di and Ei refers to the field parameters inside the GQD. The first integral is null because: (7)
De = ε0 ε1 Ee and
(8)
Di = ε0 ε2 Ei
The electric field inside the GQD Ei can be calculated by taking the electric potential as
A φ1 = −E0 cos θ ⎛r − ⎞ r < a r⎠ ⎝
(9)
and
φ2 = −BE0 cos θ
φ1 (a) = φ2 (a)
L
∫ 0
2wf →→ →→ P2 (α )(m n )2 [1 − ξ (m n )2] dz a
ε1
(3)
where w is the anchoring energy, a is the graphene nanoplatelet radius, → is it's axis versor, → L is its length, m n is molecular director and α is the anchoring angle of LC molecule on graphene sheet as shown in Fig. 1. ξ is a dimensionless phenomenological parameter, P2 (α ) = (3 cos2 α − 1)/2 is the second order Legendre polynomial depending on the anchoring angle. The electric field interaction free energy term can be represented just as in any other liquid crystal but with an effective dielectric anisotropy instead of the regular parameters:
FE = −
ε0 2
L
∫
r=a
∂φ2 ∂r
r=a
(12)
A=
ε2 − ε1 2 a ε2 + ε1
(13)
B=
2ε1 ε2 + ε1
(14)
Ei = − (4)
∫V (D − ε1 ε0 E ) dv = D¯ − ε1 ε0 Ē
= ε2
The electric field applied is parallel to Ox axis, so for the inside electric field, we get:
∂φ2 2ε1 E0 = ε1 + ε2 ∂x
(15)
From Eq. (6), by denoting the volumetric fraction of nanotubes with Nv f = V , (where v is the nanoparticle's volume and V the mixture's volume) and considering the applied field E0, equal to the average field inside the mixture Ē , we obtain:
where E is the applied electric field, ε⊥eff is the effective perpendicular dielectric constant and εa eff is the dielectric anisotropy of the LC+GQD mixture. The constants can be calculated using classic electrostatic principles presented by Landau in [21].
1 V
∂φ1 ∂r
(11)
Thus we obtain:
→2 → 2 [ε⊥ eff E + εa eff (E → n ) ] dz
0
(10)
where A and B are constants that can be determined from boundary conditions:
Fig. 1. Nematic molecule's anchoring on graphene nanoplatelet surface.
Fint = −
r≥a
ε0 f (5)
2ε1 (ε2 − ε1) Ē = D¯ − ε0 ε1 Ē ε1 + ε2
(16)
By replacing D¯ = εeff ε0 Ē in Eq. (16), the effective permittivity for the considered mixture is:
where V is the mixture's volume, D is the electric displacement, E is the electric field, ε0 is the vacuum electric permittivity and ε1 is the dielectric constant of the liquid crystal in which GQDs are inserted. The parameters D¯ and Ē refers to the average values in the mixture.
εeff = ε1 + f
2ε1 (ε2 − ε1) ε1 + ε2
Fig. 2. Semiconductor cylindric GQD in dielectric environment with electric field perpendicularly applied on cylinder axis. 1302
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Fig. 3. Electric field and graphene quantum dot orientation in planar LC cell.
Since the field was parallel to the nematic director (on OX axis), then ε1 is the parallel component of liquid crystal dielectric constant ( ε1 ≡ ε∥LC) and ε2 is the parallel component of GQD dielectric constant (ε2 ≡ ε∥GQD ), so:
ε∥ eff = ε∥ LC + f
U = U0
U0 = π (18)
∫ext (De − ε0 ε1 Ee ) dv + VN ∫int (ε0 ε2 Ei − ε0 ε1 Ei ) dv = D¯ − ε0 ε1 Ē
A=
ε1 E0 ε2
l
∫ 0
⎜
(21)
(22)
(23)
Finally we can write the total free energy of the mixture as:
ε0 ε⊥ eff 2L2
U2 −
ε0 εa eff 2L2
ε0 εa eff
)
−
2wfL2 (3 cos2 α − 1) 0 εa eff
π 2aε
(29)
3. Experimental set-up and procedure
U2
∫ sin2 θdz
Analysed samples were prepared using 5CB nematic and water dispersed GQD (1 mg/ml) graphene quantum dots from Sigma Aldrich. The open bottles are place in dust free environment at constant temperature until all the water evaporated resulting a mass concentration of 0.45% for the first sample and 1.6% for the second one. These mixtures were used to fill 15-micrometer thick planar aligned cells from Instec. Each cell was placed on a holder in the hot stage of thermostat. The holder terminals were connected to a power source from which the voltage was increased step by step with 0.01 V from 0 to 1.5 V. As mentioned before, when the external field reaches the critical Freedericksz transition threshold, the molecules change their
0
∫ wπa2f [1 − 3 cos2 α]sin2 θdz 0
+
T 2β TC
L
L
+
(28)
As it can be seen from Eq. (29), the critical transition voltage depend on the nanoparticle's electric properties that affect the effective permittivity of the sample as shown in Eq. (18) for parallel component of dielectric anisotropy and in Eq. (23) for the perpendicular one. As in the case of other nanoparticle [1,4,5,23] the volumetric fraction of inserted nanoparticles may either decrease the Freedericksz transition limit or increase it depending on the anchoring angle (α) between the nematic director and nanoparticles's surface. Since the anchoring angle evaluation can only be made by theoretical methods [5,13], we can experimentally test this model by temperature dependence of the critical voltage for the same concentration outside pretransitional range where are many instabilities.
In this case, the field was perpendicularly applied on the nematic director, so we have ε1 ≡ ε⊥LC and ε2 ≡ ε⊥GQD:
F=−
(
K 01 1 − U=π
⎟
ε⊥ LC ⎞ ⎤ ⎛ ε⊥ eff = ε⊥ LC ⎡ ⎢1 + f ⎜1 − ε⊥ GQD ⎟ ⎥ ⎝ ⎠⎦ ⎣
⎟
where i refers to the order of elastic constant (i = 1, 2, 3 for splay, twist and bend elastic deformation), T is the sample's temperature, Tc is the clearing temperature of the liquid crystal and β = −0.1855 is a material constant [25] and K01 are the elastic constants at T = 273.15 K. In this case, we get the temperature dependence of the transition voltage:
Considering the applied electric field equal to the average field inN side the mixture and denoting by f = V πa2l the volumetric fraction of GQD, we obtain the effective dielectric constant as: ⎜
(27)
2β
⎟
ε εeff = ε1 ⎡1 + f ⎛1 − 1 ⎞ ⎤ ⎢ ε2 ⎠ ⎥ ⎝ ⎦ ⎣
(26)
wf (3 cos2 α − 1) 2a
⎜
(20)
⎛ε1 ε0 E0 − ε12 ε0 E0 ⎞ πa2dl = D¯ − ε0 ε1 Ē ε2 ⎠ ⎝
K1 εa ε 0
T Ki = Ki0 ⎛1 − ⎞ Tc ⎠ ⎝
Thus we have:
N V
(25)
Lin and collaborators [24] reported a temperature dependence of elastic constants:
(19)
where De and Ee are the electric displacement and the electric field outside the GQD and Ei is electric field inside GQD. The first term of the equation is null and we only have to calculate the second integral. Using the same principle as in previous case, the electric field inside the GQD is:
Ei =
AL2 π 2K1
is the Freedericksz transition threshold for liquid crystal, and
For the perpendicular dielectric constant of the mixture we shall consider the external field parallel to the cylindric GQG's axis (Fig. 3), so Eq. (5) becomes:
1 V
εa eff
1−
where
2ε∥ LC (ε∥ GQD − ε∥ LC ) ε∥ LC + εGQD
εa
L
L
0
0
∫ ⎡⎣ kBvT f ln f ⎤⎦ dz + 12 ∫ (K1 θz2 cos2 θ + K3 sin2 θ) dz
(24)
After applying the Euler-Lagrange equations and following the same procedure as presented in [23], we obtain the Freedericksz transition voltage as: 1303
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Fig. 4. Figure experimental set-up for Freedericksz transition threshold voltage.
orientation tending to align with the field. Experimentally, this reorientation is revealed by an intensity variation of the emergent laser beam through the sample. A photovoltaic cell was used to record the emergent intensity (Fig. 4). Experimental laser intensity versus voltage plot obtained from reference sample filled with 5CB and for other two samples with 0.45% GQD and 1.6% GQD were recorded at various temperatures. 4. Results and discussions From the emergent intensity versus applied voltage plot, the Freedericksz transition threshold was determined as the point where a significant intensity change appears. An example of threshold voltage experimental evaluation for 5CB at 25 °C is presented in Fig. 5. The threshold voltage versus temperature presented in Fig. 6 indicates an uniform behavior at low temperature (below 299 K) and a high instability near the nematic isotropic transitions due to the increase of thermal agitation. We used Eq. (28) to fit experimental data for the temperature in stability region (bellow 299 K) and, as it can be seen from Fig. 7, the fitting is good. For the dielectric parameters we consider ε∥ = 18.65,
Fig. 6. Freedericksz transition threshold voltage versus temperature plot. Lines are guide for the eyes.
ε⊥ = 7.15, ε∥GQD = 1.8, ε⊥GQD = 3, w = 5 × 10−7 N/m, ρCL = 1.022 g/ cm3 and ρGQD = 2.02 g/cm3 [26–28,13]. A second check of the model was to calculate the elastic constant K1 at 293.15 K from Eq. (25). The results are in good agreement with the ones resulting from other paper [29,30], so we extend the calculation to the GQD containing samples as shown in Table 1. From Eq. (29) it result an anchoring angle α = 54.7° for both GQD concentrations, so we may assume that, in this concentration range we have the same anchoring mechanism and no particles clusters appears inside the sample.
5. Conclusion We deduced a mathematical formula for the Freedericksz transition threshold voltage of a nematic liquid crystal mixed with graphene quantum dots taking into account the influence of the GQD amount and the temperature. The results indicated a decrease of the threshold value with the increase of GQD concentration for the first half of the nematic range, in good agreement with experimental data. This behavior is a result of the nematic anchoring on GQD surface. In the vicinity of GQD the planar alignment is disturbed and some of the nematic molecules
Fig. 5. Experimental evaluation of Freedericksz transition threshold from intensity versus voltage plot. 1304
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Fig. 7. Critical voltage versus temperatures for three different concentration and fitting curve using Eq. (29): a) for 5CB+0.45% GQD, b) for 5CB+1.6% GQD, c) for 5CB. Table 1 Expenditure share and nucleolus. CQD concentration mass fraction (%)
K0 × 1012 [N]
0 0.45 1.6
11.946 12.024 11.990
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