Fréedericksz Transition in Confined Liquid Crystals: Concentration and Microgeometry Effects

VOLUME 86, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 26 MARCH 2001

2822

Fréedericksz Transition in Confined Liquid Crystals:Concentration and Microgeometry Effects

Ohad LevyDepartment of Physics, Nuclear Research Center Negev, P.O. Box 9001, Beer-Sheva 84190, Israel

(Received 20 November 2000)

The Fréedericksz transition in dispersions of liquid-crystal droplets is studied analytically by balancingthe electrostatic energy of the droplets with a strong anchoring elastic energy. Explicit dependence of thetransition threshold field on the liquid-crystal volume fraction and the spatial arrangement of the dropletsis obtained for the first time. As a result of the confined geometry, this threshold field does not dependon the thickness of the sample and splitting of the transition occurs in some situations.

DOI: 10.1103/PhysRevLett.86.2822 PACS numbers: 61.30.Gd, 64.70.Md, 77.84.Lf

The Fréedericksz transition (FT) in nematic liquid crys-tal (LC) films has long been the subject of experimentaland theoretical studies that have elucidated the natureof elastic deformations in liquid crystals and advancedour understanding of the anchoring of their molecules atsolid surfaces [1,2]. As a consequence of the dielectricanisotropy of the LC, the free energy of the film has a mini-mum at a well defined orientation of the nematic directorrelative to the applied field. The transition occurs if in theinitial state the directions of the field and the crystal do notcorrespond to that minimum. Then, at sufficiently strongfields, capable of overcoming the elastic forces of the crys-tal, a reorientation of the director occurs to a configurationof lower free energy. Similar transitions in polymer-liquid crystal composite films have attracted muchattention in recent years [3,4]. The polymer in thesecomposites is used to stabilize the liquid crystalline phasefor a wide variety of electro-optical applications. Theseapplications are based on the easy manipulation of the LCorientation by an applied electric field, thus modifyingthe electro-optical properties of the film. Two broadcategories of composites may be distinguished by theirmicrogeometry. In polymer stabilized liquid crystals smallamounts of polymer form complex networks dispersed inthe crystal. In polymer dispersed liquid crystals (PDLC)small droplets of LC are embedded in a polymer matrix.In both categories, application of a moderate electric fieldchanges the configuration of the LC and switches thesystem from one state to another. The interface betweenthe two phases has an aligning effect on the LC moleculeswhich strongly affects the switching process. Elasticforces at the interface alter the LC structure and causeits reorientation to the original configuration followingthe removal of the field. Details of these transitionsare determined by the balance between the electric fieldand interface alignments, both of which depend onmicrogeometrical characteristics of the material, e.g.,concentration and morphology of the two components.Different aspects of the switching process have been thesubject of numerous experimental studies (some recentexamples are [5–16]). In contrast, very little theoretical

0031-9007�01�86(13)�2822(4)$15.00

work has been carried out and the understanding of theseissues is still very qualitative. In particular, the role of themicrogeometry in the switching transition has not beendirectly addressed, despite experimental indications of itsimportance [6,7,15].

PDLC films of bipolar nematic LC droplets are of par-ticular current interest for optical applications [3]. Simi-larly to the FT in LC films, switching in PDLC involvesa balance between the elastic energy, which favors ori-entation of a LC droplet in a locally preferred direction,and the electrostatic energy which favors alignment withthe applied field. However, in contrast to LC films, thecomplex microgeometry makes it difficult to obtain ac-curate estimates of the switching field. The PDLC isusually described as a random dispersion of bipolar LCdroplets in a homogeneous host, and the elastic energy istreated phenomenologically assuming strong anchoring atthe droplet walls (a microscopic theory providing a moredetailed treatment is still lacking). The electrostatic energyhas been evaluated at the dilute limit, where all microge-ometrical information is lost [17], or by using a variety ofeffective medium formulations, based on the lowest order(dipole-dipole) interaction between droplets, that reflectimplicitly only the concentrations of the two components[18–20]. However, microstructural information beyondthat contained in the phase volume fractions is required toaccurately estimate the switching field and to differentiatebetween various dispersion geometries. Methods includ-ing such information have been developed for estimatingthe effective properties of solid composite materials (see,for example, the short review [21]) but have not yet beenapplied for polymer-LC mixtures. In this Letter, we extendone of those methods [22,23] to derive switching fields thatdepend explicitly on the spatial arrangement and the ini-tial orientation distribution of the droplets as well as ontheir volume fraction. It is shown that the threshold fieldof the FT in PDLC with aligned droplets should generallydecrease with increasing droplet volume fraction, with arate depending on the spatial distribution of the droplets.In PDLC with a planar initial orientation distribution theFT is split and a hysteresis loop is obtained whose width

© 2001 The American Physical Society


depends strongly on the LC volume fraction and somewhatmore weakly on the droplets spatial arrangement.

To simplify the discussion, we consider the theory ofthe FT in PDLC under the following common assump-tions: The LC droplets are bipolar, monodispersed, andnearly spherical. Experiments indicate that the bipolarconfiguration is stable to the presence of applied field,i.e., the LC configuration inside the droplet rotates rigidlyto a good approximation [3]. The dielectric tensor ofsuch a droplet es is uniaxial with eigenvalues ek ande�. The orientation �u, f� of the droplet principal di-electric axis relative to the applied field varies from onedroplet to another. There is strong anchoring of the LCmolecules at the droplet surfaces and all the Frank elasticconstants are equal. Under these assumptions the elas-tic energy density is Eelas � 2W�M ? N�2, where M ��cosf0 sinu0, sinf0 sinu0, cosu0� is the droplet director inthe absence of field, N � �cosf sinu, sinf sinu, cosu� isthe director in the presence of an applied field, and Wrepresents the elastic distortion required to align a bipo-lar droplet by the field. In slightly ellipsoidal dropletsW � K�R2, where K is the Frank elastic constant andR is the droplet radius [5,17]. Typically K � 1026 dyne,R � 1 mm, and the elastic energy density is of the orderof 100 erg�cm3. The electric conductivity is disregardedsince the fields applied to induce droplet reorientationare low frequency (�1 kHz) and both components of thePDLC are in the dielectric regime [5]. The FT in LC filmsis usually discussed in configurations with initial planararrangement of the nematic directors [1,2]. The corre-sponding configuration in PDLC is such that the dropletdirectors are initially oriented along the x axis or uniformlydistributed on the �x, y� plane and the electric field is ap-plied along the z axis. Initial alignment may be obtainedby using a small orienting field or by slightly shearing thePDLC film during its formation. The planar distribution istypical of films formed by encapsulation methods [3].

The LC droplets are embedded in a polymer of dielectriccoefficient ep and are subject to a uniform applied fieldE0. The solution of the electrostatic equations gives theLorentz local field

EL�r� � E0 1Z

dr 0 T �r 2 r 0� ? P�r 0� , (1)

where P�r� is the induced polarization relative to themedium in the absence of the spherical inclusions,T �r 2 r 0� is the dipole-dipole interaction tensor, andthe integration is carried out with the exclusion of aninfinitesimal sphere centered at r [24]. The inducedpolarization is

P�r� � w�r�k�r�EL�r� , (2)

where k�r� � 3ep�es�r� 2 epI��4p�es�r� 1 2epI�is the local polarizability tensor, I is the identity matrix,and w�r� is a function of position equal to 1 inside thedroplets and to 0 in the polymer. In this, and in subse-

quent equations, the denominators should be understoodto represent inverse matrices. Relations between theensemble averages of P and EL and the applied fieldE0 are obtained following Refs. [22,23]: An integralequation for P�r�, obtained from (1) and (2), is solved bysuccessive substitutions leading to an expansion of P�r�in powers of k�r�. This relation, up to third order, isensemble averaged to give

E0 �

∑1

�k�f2 gI 2 �k�l

∏�P� , (3)

where triangular parentheses denote ensemble average,f � �w�r�� is the volume fraction of the LC droplets,

g �Z f

�2�12

f2 T �r1 2 r2� dr2 , (4)

l �ZZ "

f�3�123

f2 2f

�2�12 f

�2�23

f3

#T �r1 2 r2�

3 T �r2 2 r3� dr2 dr3 , (5)

and f�n�12,...,n � �w�r1�w�r2�, . . . , w�rn�� is the n-point cor-

relation function of the LC droplets. The averaging is car-ried out assuming, as mentioned above, that the dropletsare either aligned or initially oriented at the �x, y� planesuch that all have the same u and �kn� � �k�n. The sec-ond term in (3) is the Lorentz field intensity at r1 producedby the polarization � f

�2�12 �f2� �P� at r2. In a macroscopic

specimen, i.e., a film much larger than the individualdroplets, f

�2�12 �f2 � 1 for all but microscopically small val-

ues of r1 2 r2 and the Lorentz field intensity is the sameas for a uniform polarization �P�. Therefore transferringg to the left side of (3) we find

�EL� �

∑1

�k�f2 �k�l

∏�P� . (6)

The average polarization contribution to the Lorentz fieldhas a known value dependent on the sample shape [24].For the sake of consistency with previous work, we use theclassical relation g � 4p�3ep generally employed in ef-fective medium theories, assuming implicitly a sphericallysymmetric isotropic sample. The scalar l is a shape inde-pendent microgeometrical parameter widely used in theo-ries of composite materials to obtain estimates and boundson their effective properties [21]. The related parame-ter z � fl�2�1 2 f� has been calculated for three di-mensional dispersions of spheres, e.g., simple cubic (SC),bcc, fcc, and random, and has been tabulated in Ref. [23].Substituting (3) in (6) we find a relation between �EL�and E0 from which the induced dipole moment, p, of aLC droplet may be estimated by considering it again asa spherical inclusion of dielectric tensor es embedded ina homogeneous host ep in the presence of a uniform lo-cal field �EL�. The solution of this electrostatic problemgives p �

3ep �es2epI�4p�es12epI� ? �EL� and the electrostatic energy

density of the droplet is Eelec � 2�1�2�p ? �EL�.

2823


The total energy density Eelas 1 Eelec may now be usedto study the FT in a PDLC film. Let us first assume thatall the droplets are initially aligned in the same direction�u0, f0�. The polarizability tensors of all droplets are thenidentical and the u-dependent part of the electrostatic en-ergy is

Eelec � 23epE2

0

16p

"3ep�ek 2 e��

�ek 1 2ep� �e� 1 2ep�1 C� f, l�

#

3 cos2u , (7)

where C� f, l� is a function of the dielectric coefficientsof the components, the droplet volume fraction f, andthe microgeometrical parameter l. Minimizing the totalenergy as a function of u (f does not change when thefield is applied along the z axis), we find

tan2u �sin2u0

cos2u0 1 �E0�Eth�2 , (8)

where

E2th �

8pW

3ep� 3ep �ek2e��ek12ep � �e�12ep� 1 C� f, l��

(9)

is the threshold field for the FT in PDLC with aligneddroplets. In contrast to the case in LC films, this thresholdfield is independent of the film thickness and the thresholdvoltage will thus depend on it linearly. This is a result ofthe confinement of the LC in small cavities that determineits deformation, whereas in LC films a given deviation ofthe director at the center of the film requires smaller de-formation and a lower field to overcome the elastic forcesin thicker films [1,2].

In the dilute limit C� f � 0, l� � 0 the microgeometri-cal information is lost and Eth depends only on the dielec-tric coefficients. For l � 0 the threshold field is reduced tothe corresponding result of the Maxwell-Garnett approxi-mation [20]. In Fig. 1, threshold voltages are shown as afunction of f for three different microgeometries, using thevalues of z given in [23] (the results for bcc and fcc geome-tries are almost identical to those of the Maxwell-Garnettapproximation). If u0 � p�2, then u � u0 for low fields.At the threshold field a sharp transition occurs to the highfield state u � 0. This is one of the effects of confinement,since in LC films the alignment with applied field abovethe threshold is more gradual. Similarly to the situation inLC films the threshold of the FT disappears if u0 , p�2,i.e., the alignment of the droplets with the field starts atinfinitesimally small fields and is much more gradual thanfor u0 � p�2. Figure 2 shows this behavior for two val-ues of u0 , p�2 and the three microgeometries of Fig. 1.

An interesting situation arises when the droplet directorsare initially confined to the �x, y� plane, i.e., u0 � p�2and f0 is uniformly distributed over �0, 2p�. In this caseaveraging over f leads to diagonal polarizability tensorsand the electrostatic energy of a LC droplet is

2824

0 0.1 0.2 0.3 0.4 0.520

25

30

35

40

45

f

Vth

(V)

FIG. 1. Threshold voltages (in volts) of a 100 mm thick PDLCfilm with ek � 30, e� � 6, and ep � 8, for a SC geometry(dash-dotted lines), random dispersion (dashed lines), and theMaxwell-Garnett approximation l � 0 (solid lines). Resultsare shown for aligned droplets Eq. (9) (middle triplet) and forthe split FT obtained with initial planar orientation distribution(upper and lower triplets). The curves of the upper triplet areindistinguishable at this scale.

Eelec � 23epE2

0X

8p

∑lX 2 1�fX

1 1 lX 2 1�fX

∏, (10)

where X � a 2 b sin2u, a �ek2ep

ek12ep, b �

3eP �ek2e��ek12ep� �e�12ep � , and l � l� 3ep

4p �2. The elastic energydensity is Eelas � 2W sin2u. The first derivative of thetotal energy density is therefore∑µ

E0

Eth

∂2

2 1

∏W sin2u , (11)

where the threshold field Eth is a function of f, l, andX. This derivative is zero either at u � p�2 which is a

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

E0/E

cr(f=0)

θ (π

rad)

FIG. 2. The orientation angle u (in units of p) as a functionof reduced field for u0 � 0.4p (upper curves) and u0 � 0.1p(lower curves) for the PDLC film of Fig. 1 with f � 0.5.


minimum when E0 , Eth, at u � 0 which is a minimumwhen E0 . Eth, or when E0 � Eth. However, Eth itselfdepends on u through the variable X. This means that theFT is split. It occurs at different fields depending on theinitial orientation. If we start with u � p�2 at zero fieldthen X � a 2 b and increasing the field gradually willhave no effect until we reach the corresponding thresholdfield. At this point u will abruptly jump from p�2 to zeroand will stay there at all higher fields. On the other hand,if we start at u � 0 at a very high field then X � a andthe jump occurs at a different point. The width of theresulting hysteresis loop depends on f, l, and the dielectriccoefficients. Typical results for the two threshold fieldsare shown in Fig. 1. In general, the dependence on fis stronger than the dependence on l. PDLC for opticalapplications are fabricated with e� matched as closely aspossible with ep . In perfect matching we get X � 0 foru0 � p�2 and one of the threshold fields (the upper onefor b . 0) is independent of f and l. For f � 0 bothtransition points are reduced to the dilute limit result E2

th �8pW�3epb as in Eq. (9). For l � 0 we again obtain thecorresponding Maxwell-Garnett result of Ref. [20].

The FT in PDLC films exhibits a few significant differ-ences from the FT in LC films which are due to the con-finement of the LC in small droplets: The threshold fieldis independent of the film thickness; the transition to thehigh field configuration is sharp and is split in some situa-tions. These effects have been studied here using a pertur-bation method originally developed for solid dispersions ofisotropic inclusions. In particular, dependence of the tran-sition field on the spatial distribution of the droplets andtheir volume fraction is easily studied within this approach.Additional effects, e.g., anchoring strength, droplet shapeand size, and dynamics of the transition, can also be in-cluded in the same general framework. This theory shouldtherefore serve as a useful tool for the interpretation ofexperimental studies of the FT, either by light intensitytransmission or electro-optical phase shift measurements,

and for the prediction of PDLC properties under variousconditions.

I thank Y. Rosenfeld for useful comments.

[1] L. M. Blinov, Electro-Optical and Magneto-Optical Prop-erties of Liquid Crystals (Wiley-Interscience, New York,1983).

[2] P. G. DeGennes and J. Prost, The Physics of Liquid Crys-tals (Clarendon Press, Oxford, 1993), 2nd ed.

[3] P. S. Drazaic, Liquid Crystal Dispersions (World Scientific,Singapore, 1995), and references therein.

[4] Liquid Crystals in Complex Geometries, edited by G. P.Crawford and S. Zumer (Taylor and Francis, London,1996).

[5] K. Amundson, Phys. Rev. E 53, 2412 (1996).[6] I. Dierking et al., J. Appl. Phys. 81, 3007 (1997).[7] G. A. Held et al., Phys. Rev. Lett. 79, 3443 (1997).[8] K. Amundson et al., Phys. Rev. E 55, 1646 (1997).[9] M. Ambrozic et al., Phys. Rev. E 56, 1825 (1997).

[10] S. A. Carter et al., J. Appl. Phys. 81, 5992 (1997).[11] J. D. LeGrange et al., J. Appl. Phys. 81, 5984 (1997).[12] M. Copic and A. Mertelj, Phys. Rev. Lett. 80, 1449 (1998).[13] K. Amundson and M. Srinivasarao, Phys. Rev. E 58, R1211

(1998).[14] R. Bartolino et al., J. Appl. Phys. 85, 2870 (1999).[15] R.-Q. Ma and D.-K. Yang, Phys. Rev. E 61, 1567 (2000).[16] A. Mertelj and M. Copic, Phys. Rev. E 61, 1622 (2000).[17] B. G. Wu et al., Liq. Cryst. 5, 1453 (1989).[18] J. R. Kelly and P. Palffy-Muhoray, Mol. Cryst. Liq. Cryst.

243, 11 (1994).[19] V. Y. Reshetnyak et al., J. Phys. D 30, 3253 (1997).[20] O. Levy, Phys. Rev. E 61, 5385 (2000).[21] S. Torquato, Appl. Mech. Rev. 44, 37 (1991), and refer-

ences therein.[22] W. F. Brown, J. Chem. Phys. 23, 1514 (1955).[23] S. Torquato, J. Appl. Phys. 58, 3790 (1985).[24] J. D. Jackson, Classical Electrodynamics (Wiley, New

York, 1975), 2nd ed.

2825

Documents

Fréedericksz Transition in Confined Liquid Crystals: Concentration and Microgeometry Effects