electro-optical phase shift in polymer dispersed liquid crystals

Eur. Phys. J. E 3, 11–20 (2000) THE EUROPEANPHYSICAL JOURNAL Ec©

EDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2000

Electro-optical phase shift in polymer dispersed liquid crystals

O. Levya

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

Received 25 November 1999 and Received in final form 20 January 2000

Abstract. An anisotropic version of the Maxwell Garnett approximation is applied for studying the electro-optical phase modulation by polymer dispersed liquid crystals (PDLC). The PDLC contain bipolar liquidcrystal droplets that can be reoriented by an external field causing a change in the optical birefringence. Thisapproach provides an explicit link between the droplet orientation distribution and the electro-optical phaseshift. For aligned droplets we find that the sharpness of the change in the birefringence may be controlledby selecting the initial orientation. For a planar distribution we find sharp transitions with a hysteresis loopwhose width depends on the droplet concentration. For a random distribution, the droplet orientation andthe optical phase shift change more gradually with the applied field. These results demonstrate that PDLCmay be suitable for a wide range of electro-optic applications based on their field-induced phase modulationproperties. In addition, it is apparent that the optical phase shift is quite sensitive to changes in dropletorientation. It should therefore be useful for studying reorientation phenomena in PDLC, overcoming theproblems due to light scattering in these materials.

PACS. 42.70.Df Liquid crystals – 78.20.Jq Electro-optical effects – 78.66.Sq Composite materials

1 Introduction

Polymer dispersed liquid crystals (PDLC) are inhomoge-neous materials in which a liquid crystal (LC) is containedinside tiny droplets embedded in a polymer matrix. PDLCfilms of bipolar nematic LC droplets are of particular cur-rent interest for optical applications, ranging from switch-able windows to active matrix projection displays (see,for example, [1,2]). Electro-optic phase modulation of theKerr or Pockels type can also be achieved by the use ofthese materials. These applications are based on the easymanipulation of the LC orientation inside the dropletsby an externally applied electric field, thus modifyingthe electro-optical properties of the material. Typically, aPDLC film is sandwiched between conducting electrodesand is electrically driven to obtain the desired effect. Thereorientation process is influenced by the microgeometryof the PDLC, its dielectric properties and the interfacialinteraction between the polymer and the LC at the dropletwalls. The dielectric properties and their dependence onexternally applied fields and the morphology of the mate-rial have been the subject of numerous experimental stud-ies (some recent examples are [3–13]). In contrast, despitetheir inherent importance, little theoretical work has beendone to understand these issues. Wu, Erdmann and Doane[14] studied the effect of an applied field on the directororientation of an isolated bipolar LC droplet. They calcu-lated the electrostatic torque on the droplet using a simpleapproximation for the field inside it. Balancing this torque

a e-mail: [email protected]

with an elastic torque, created by strong anchoring at thedroplet walls, they obtained the field-dependent orienta-tion of the director. This derivation leads to a relationbetween the switching voltage of a PDLC, in which LCdroplets are dilutely dispersed, and some of its microscopicparameters. Kelly and Palffy-Muhoray [15] introduced amodel that relates the dielectric response of PDLC filmsto external fields via their effect on a hierarchy of orien-tation order parameters. In this model the LC dropletsare approximated as isotropic inclusions with an averagedscalar dielectric coefficient which depends on an internalorder parameter. This approximation is then used to cal-culate the bulk dielectric coefficient of the film via theclassical Maxwell Garnett formula [16] for a dispersion ofisotropic inclusions in an isotropic host. This description ofa PDLC in terms of field-dependent order parameters hasbeen applied by Basile et al. [4] and Vicari [11] to studythe electro-optical phase modulation in systems where theLC droplets are initially randomly oriented and the mate-rial is optically isotropic. They showed that reorientationof the droplets, induced by an applied electric field, leadsto an electro-optical phase shift in light impinging trans-versely on the sample. To calculate this phase shift, theyused a simplified model of the microgeometry of PDLCas consisting of parallel layers of liquid crystal and poly-mer. In this model the electro-optical effects result fromchanges in the order parameter of the LC layers. Two latermodels consider explicitly the anisotropic nature of the LCdroplets. Reshetnyak, Sluckin and Cox [17] proposed an ef-fective medium approach to calculate the dielectric tensor

12 The European Physical Journal E

of a PDLC film. They too obtained an expression for theswitching voltage of a very dilute PDLC, slightly differentfrom that of [14]. In non-dilute systems, they showed thatthis approach can be used to numerically calculate the di-electric tensor for a few initial orientation distributions ofthe bipolar droplets. An alternative approach, proposed in[18], is based on a recent extension of the Maxwell Garnetapproximation for mixtures of anisotropic inclusions (inour case the LC droplets) embedded in an isotropic host(the polymer) [19]. The Maxwell Garnett (MG) approxi-mation [16], also known as the Clausius-Mossotti approx-imation, is one of the most widely used methods for cal-culating the bulk dielectric properties of inhomogeneousmaterials [20,21]. It is particularly useful when one of thecomponents can be considered as a host in which inclu-sions of the other component are embedded, as is the casein PDLC. Using this approach, it is possible to obtain anexpression for the effective dielectric tensor of a PDLCwhich depends explicitly on the distribution of directororientations of the LC droplets [18]. The field-dependentresponse of a PDLC film is calculated by relating this dis-tribution to the magnitude of an externally applied low-frequency electric field. In the dilute limit this methodleads, as expected, to results identical to those of the ef-fective medium approach of [17]. In the non-dilute case, itgives an analytic solution for two interesting types of ini-tial droplet orientation distributions, with switching fieldsthat depend on the volume fraction of the LC droplets. Forall other initial distributions it provides a simple schemefor a numerical calculation of the dielectric response. Inthis paper, the MG approach is applied to studying theelectro-optic phase shift in PDLC under various condi-tions. In this model we consider the actual microgeome-try of the PDLC and not the simplified layered geometryintroduced in [4] and [11]. For aligned droplets we findsignificant changes in the birefringence in the vicinity of athreshold field that depends on the LC concentration. Themagnitude of the change and its sharpness as a function ofapplied field is shown to be determined by the initial orien-tation. For a planar distribution we find sharp transitionswith a hysteresis loop whose width depends on the dropletconcentration. For a random distribution the droplet ori-entation and the induced phase shift change more gradu-ally with the applied field. These results demonstrate thatPDLC may be suitable for a wide range of electro-opticapplications based on their field-induced phase modula-tion properties. The optical phase shift turns out to bequite sensitive to changes in droplet orientation. It maytherefore be useful for studying reorientation phenomenain PDLC, overcoming the problems due to light scatteringin these materials.

The rest of the paper is organized as follows.The Maxwell Garnett approximation for suspensions ofanisotropic inclusions is introduced in section 2, and ex-plicit results are derived for uniaxial materials. In sec-tion 3, we discuss the reorientation of bipolar LC dropletsin the presence of applied field and its effects on theelectro-optical phase shift in PDLC films for a few differ-ent types of initial orientation distribution. Finally, somebrief conclusions are included in section 4.

2 Dielectric response of mixtures ofanisotropic inclusions

An exact calculation of the effective properties of an in-homogeneous medium is in general an intractable prob-lem. The literature on this subject therefore includes awide variety of approximate schemes, each of which is ap-propriate for different types of composite microgeometries(See for example the review papers [20,21] and referencestherein). One of those, which is particularly useful for mi-crogeometries in which a host material and isolated in-clusions of other materials are clearly identified, is theMaxwell Garnett (MG) approximation. In this paper, weuse a variation of the MG approach which is adapted formixtures where the host is an isotropic material with ascalar dielectric constant εp and the inclusions are madeof an anisotropic component with dielectric tensor εs [19].The orientation of the dielectric tensor differs from inclu-sion to inclusion, such that

εs = RεRT , (1)

where ε is a diagonal tensor and R is the inclusion-dependent rotation. The distribution of orientations sig-nificantly influences the bulk effective properties of thematerial. In particular, it is shown in reference [19] thatthe bulk effective dielectric tensor of such a mixture is

εe = εpI + 3fεp

⟨εs − εpI

εs + 2εpI

⟩R

× 1

(1− f) + 3fεp⟨(εs + 2εpI)

−1⟩

R

, (2)

where f is the volume fraction of the inclusions and 〈〉Rdenotes an average over the dielectric tensor orientationsinside the inclusions. This is the Maxwell Garnett resultfor mixtures of anisotropic inclusions. It is obtained bysolving the electrostatic problem of a spherical inclusionimmersed in a uniform field EL, which is the average fieldin the host medium. The volume-averaged field over theentire system, host and inclusions, is E0. A PDLC film istypically sandwiched between two conducting electrodesconnected to an electrical circuit. A voltage difference V isapplied between the electrodes to modify the LC orienta-tion distribution in the droplets. E0 is determined by thisvoltage difference and the width of the film d, E0 = V/d. Itis clear that the same uniform field E0 exists in the spacebetween the electrodes when it is either empty or filled bya homogeneous slab of polymer, in the absence of the liq-uid crystal droplets, under the same boundary conditions.The LC droplets modify the local field in this space butthe volume-averaged field is not changed by their presenceand remains E0 = V/d. A simple method to calculate EL,usually referred to as the excluded-volume approach, wasproposed by Bragg and Pippard [22]. The difference be-tween EL and E0 is due to the correlations between posi-tions of different spheres that arise from the prohibition ofoverlap between them [22]. Solving this electrostatic prob-lem, we find that the field inside the inclusion satisfies the

O. Levy: Electro-optical phase shift in polymer dispersed liquid crystals 13

εs = RθφεRTθφ = ε⊥I + α

cos2 φ sin2 θ cosφ sinφ sin2 θ cosφ cos θ sin θcosφ sinφ sin2 θ sin2 φ sin2 θ sinφ cos θ sin θcosφ cos θ sin θ sinφ cos θ sin θ cos2 θ

, (10)

A =

δ − 3αεp

⟨cos2 θ + sin2 φ sin2 θ

⟩ 3αεp

2

⟨sin 2φ sin2 θ

⟩ 3αεp

2 〈cosφ sin 2θ〉3αεp

2

⟨sin 2φ sin2 θ

⟩δ − 3αεp

⟨cos2 θ + cos2 φ sin2 θ

⟩ 3αεp

2 〈sinφ sin 2θ〉3αεp

2 〈cosφ sin 2θ〉 3αεp

2 〈sinφ sin 2θ〉 δ − 3αεp⟨sin2 θ

⟩ ,

B =

ε⊥ + 2εp + α

⟨cos2 θ + sin2 φ sin2 θ

⟩ −α2

⟨sin 2φ sin2 θ

⟩ −α2 〈cosφ sin 2θ〉

−α2

⟨sin 2φ sin2 θ

⟩ε⊥ + 2εp + α

⟨cos2 θ + cos2 φ sin2 θ

⟩ −α2 〈sinφ sin 2θ〉

−α2 〈cosφ sin 2θ〉 −α

2 〈sinφ sin 2θ〉 ε⊥+ε‖+4εp−α〈cos 2θ〉2

.

relationDs + 2εpEs = 3εpEL. (3)

The averaged field over the entire system, inside and out-side the inclusions, must still be E0. This leads to a simplerelation between the average fields in the host and in theinclusions

f 〈Es〉+ (1− f)EL = E0, (4)

where the angular brackets denote a volume average insidethe inclusions. Substituting Es from equation (3), we solvefor EL and find [18,19]

EL =E0

(1− f) + 3fεp⟨(εs + 2εpI)

−1⟩

R

. (5)

The induced dipole moment of a single spherical inclu-sion is

ps = VsDs − εpEs

4π=

Vs

4πεs − εpI

εs + 2εpI3εpE0

(1− f) + 3fεp⟨(εs + 2εpI)

−1⟩

R

, (6)

where Vs is the volume of the inclusion. Using this result,it may be shown that EL is the Lorentz local field, whichis usually calculated using the relation

EL = E0 +4π3εp

〈P 〉 , (7)

where 〈P 〉 ≡ (1/V )∑

s ps is the volume-averaged polar-ization in the medium [18,19].

The nematic LC droplets in a PDLC are usually ar-ranged in a bipolar configuration. This configuration isinsensitive to an externally applied electric field. The onlyeffect of an imposed field is to rotate the bipolar axis ofthe droplet to a new direction more closely aligned withthe field [14,17]. It is therefore reasonable to describe theLC droplet as a uniaxial dielectric material where the di-electric coefficient obtains one value along one preferreddirection and another value in all perpendicular directions.

The dielectric tensor of such a material is

ε =

ε⊥ 0 0

0 ε⊥ 00 0 ε‖

(8)

in the coordinate system defined by the dielectric axis par-allel to the droplet director. It should be noted that ε⊥ andε‖ are effective values obtained by averaging the actual LCdielectric tensor over the non-uniform local director fieldinside the droplet. The relation between the effective val-ues and the local values is determined by the droplet orderparameter S [15]. If ε⊥ and ε‖ are the elements of the localdielectric tensor then the effective elements for the dropletare given by [17]

ε⊥=ε⊥+13(1− S)

(ε‖ − ε⊥

)and ε‖=ε⊥+S

(ε‖ − ε⊥

).

(9)Analytical and numerical estimates of the order parameterof a bipolar droplet by Reshetnyak et al. lead to S ≈ 0.8[17]. Fitting with experimental results gives similarly S ≈0.7 [11].

For a mixture of many such inclusions, embedded inan isotropic host, it is convenient to define the coordi-nate system such that the applied voltage and the volume-averaged field E0 are in the positive z direction. In thiscoordinate system the dielectric tensor of each inclusionmay be written explicitly as

see equation (10) above

where α ≡ ε‖ − ε⊥ and θ and φ are the orientation anglesfrom the z-axis. Substituting this in equation (2) we findan explicit expression for the bulk effective dielectric ten-sor depending on the distribution of the orientation anglesθ and φ

εe = εpI +3fεpA

(ε⊥ + 2εp)(ε‖ + 2εp

)×

[1− f +

3fεpB(ε⊥ + 2εp)

(ε‖ + 2εp

)]−1

, (11)

wheresee the two matrices A and B above


δ ≡ (ε⊥ + 2εp)(ε‖ − εp

), and the angular brackets denote

averaging over θ and φ of all the inclusions in the mix-ture. This result is greatly simplified in cases where theaverages over θ and φ can be calculated exactly. Two suchexamples are:1) For very large fields, the dielectric axes of all the inclu-sions are aligned with the applied field (θ = 0). The MGeffective dielectric tensor is

εe =

εx 0 00 εx 00 0 εz

, (12)

where

εx = εp +3fεp (ε⊥ − εp)

(ε⊥ − εp) (1− f) + 3εp, (13)

and

εz = εp +3fεp

(ε‖ − εp

)(ε‖ − εp

)(1− f) + 3εp

. (14)

As expected, this tensor is uniaxial and its diagonal ele-ments are given by the scalar MG results for inclusionswith dielectric coefficients ε⊥ and ε‖, respectively. Similarresults are obtained when all the inclusions are oriented inthe x direction (θ = π/2, φ = 0), by exchanging the x-axisand z-axis terms, or when all the inclusions are orientedin the y direction (θ = π/2, φ = π/2), by exchanging they-axis and z-axis terms.2) Uniform orientation distribution over θ ∈ [0, π] andφ ∈ [0, 2π]. It is clear that in this case the material isisotropic, with a scalar dielectric coefficient. Carrying outthe averaging in equation (11), we indeed obtain

εe = εp + 3fεp

× (ε⊥ + 2εp)(ε‖ − εp

) − 2εp(ε‖ − ε⊥

)(1− f) (ε⊥ + 2εp)

(ε‖ + 2εp

)+ fεp

(ε⊥ + 2ε‖ + 6εp

) .(15)

3 The electro-optical phase shift

The calculation of the electro-optical phase shift may begenerally divided into two steps. In the first step, thedroplet orientation distribution function is evaluated asa function of the applied field, given the initial conditionsappropriate for the specific case of interest. In the secondstep, this distribution function is used to calculate the or-dinary and extraordinary refractive indices of the PDLCfrom which the electro-optical phase shift can be deduced.In this section we present three simple examples of thisprocedure to demonstrate the usefulness of the formalismof section 2.

In the previous section we presented a description ofthe dielectric properties of PDLC films, based on theMaxwell Garnett approximation. The results obtaineddepend explicitly on the orientation distribution of thedroplet directors. Various averages of this distributionmust be evaluated for the different elements of the bulk ef-fective dielectric tensor. To determine the field-dependent

distribution, it is necessary to consider the two opposingtorques acting on each of the LC droplets. One is the elec-trostatic torque that seeks to orient the droplet in thedirection of the applied field. This torque is derived fromthe electrostatic energy

Eelec = −12ps · E, (16)

where ps is the dipole moment of the droplet and E is theelectric field in its vicinity. In the MG approximation ps isgiven by (6) and E = EL is given by (5). The electric fieldsapplied to induce the droplet reorientation are low fre-quency (typically 1 kHz or less). The dielectric coefficientsε‖, ε⊥ and εp used in (16) should therefore be the dielec-tric coefficients of the LC droplets and polymer, respec-tively, at these low frequencies. In the following exampleswe consider a 50µm PDLC film made of a PMMA (poly-methylmethacrylate) polymer matrix and E7 LC dropletswith typical radius of 1µm. The dielectric coefficients ofthese materials are εp = 5, ε‖ = 18 and ε⊥ = 6.

The second torque is an elastic torque that seeks tohold the droplet in its original direction. It is derived froman elastic energy of the general form [14]

Eelas = −WVs (M ·N)2 , (17)

where Vs is the droplet volume, M = (cosφ0 sin θ0,sinφ0 sin θ0, cos θ0) is the original director vector in theabsence of electric field, and N = (cosφ sin θ, sinφ sin θ,cos θ) is the director in the presence of an applied field. Inperfectly spherical droplets the various orientations of thebipolar configuration are degenerate in zero field. In thiscase alignment is achieved at relatively low fields where theanisotropic component of the electrostatic energy is largerthan the thermal rotation energy. However, in practice thedroplets are usually slightly elongated and have preferredorientations to which they tend to reorient when the ap-plied field is removed. In equation (17), therefore,M is theunit vector along the long symmetry axis of an ellipsoidand N is the droplet director. The prefactorW representsthe elastic distortion required to align a bipolar dropletby an electric field to preserve the tangential anchoringof the liquid crystal at the droplet interface. It dependson the droplet shape and temperature, which affect thestrength of the elastic surface anchoring [6,14]. In the fol-lowing, we will assume for simplicity that W is similar forall droplets in the system, because the PDLC is approxi-mately monodispersed. In such slightly ellipsoidal dropletsthe elastic energy density may be approximated as

W ≈ K

R2, (18)

where K is the Frank elastic constant and R is thedroplet radius [6,14]. In the following examples of a50µm PMMA/E7 film in which the droplets have typ-ical radius of 1µm, the Frank constant is of the or-der of K = 10−6 dyne and the elastic energy density isW ≈ 100 erg/cm3. It should be noted that some experi-mental evidence exists for conditions in which the depen-dence ofW on droplet size is much weaker than in (18) [9],


Eelec = −3εpE20

8π(ε⊥ + 2εp)

(ε‖ + 2εp

) [(ε⊥ + 2εp)

(ε‖ − εp

) − 3εp(ε‖ − ε⊥

)sin2 θ

][(1− f) (ε⊥ + 2εp)

(ε‖ + 2εp

)+ 3fεp

(ε⊥ + 2εp +

(ε‖ − ε⊥

) ⟨sin2 θ

⟩)]2 . (19)

where the same switching voltage has been obtained over alarge range of droplet sizes. This suggests that under suchconditions the anchoring energies are relatively small andthat the electro-optic behavior is determined mainly bythe dielectric anisotropy.

Once the orientation distribution is known, the MG re-sults of section 2 can be used to calculate the ordinary andextraordinary refractive indices of the PDLC. In this cal-culation, the dielectric coefficients ε‖ = n2

‖, ε⊥ = n2⊥ and

εp = n2p substituted in (11), are those of the LC droplets

and polymer, respectively, at the frequency of the lightincident on the sample. n‖, n⊥ and np are the correspond-ing refractive indices. In the following examples we choosen‖ = 1.736, n⊥ = 1.511 and np = 1.55, which are typicalvalues for PMMA/E7 mixtures at 600 nm.

3.1 Random initial distribution

Let us first consider the case in which in the absenceof applied field the bipolar droplet axes are uniformlydistributed in all possible orientations θ0 ∈ [0, π/2] andφ ∈ [0, 2π]. The effective dielectric tensor is uniaxial andits principal axis is perpendicular to the film plan. It re-mains perpendicular when an external field is applied. Theelectrostatic interaction between the droplets leads to acomplicated dependence of the electrostatic energy of eachdroplet on the orientations of all the other droplets


This should be minimized together with the elastic energydensity (17) to find θ as a function of θ0 and E0. In mostcases this minimization problem, although easy in princi-ple, does not have an analytic solution but can only besolved numerically. The exception to this rule is the dilutelimit in which the electrostatic interaction among the dif-ferent droplets is negligible. In this limit the θ-dependentpart of the electrostatic energy may be easily obtained bysubstituting f = 0 in (19)

Eelec =9ε2pE

20

8π

(ε‖ − ε⊥

)sin2 θ

(ε⊥ + 2εp)(ε‖ + 2εp

) . (20)

This is equivalent to considering the field in the vicinity ofeach droplet to be equal to the volume-averaged field E0.The minimization of the energy of a single droplet thenleads to

tan 2θ =sin 2θ0

cos 2θ0 + (E0/Ecr)2 (21)

for θ as a function of θ0 and E0, with the dilute limitcritical field

E2cr =

8πW (ε⊥ + 2εp)(ε‖ + 2εp

)9ε2p

(ε‖ − ε⊥

) . (22)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Θ

PΘ

0 0.05 0.1 0.150

50

100

150

Fig. 1. The distribution function Pθ of θ (in units of π) forapplied field intensities below and above (inset) the criticalfield. The different curves are for C = 0 (solid line), C = 0.5(dash-dot line), C = 1 (dotted line), C = 1.5 (circles) andC = 2 (dashed line), where C = (E0/Ecr)

2.

The angle φ remains uniformly distributed since an ap-plied field in the z direction affects only θ. The switchingvoltage of a PDLC film can be easily estimated using thisresult. For the 50µm thick PMMA/E7 film mentionedabove, we substitute the low-frequency dielectric coeffi-cients to find a threshold voltage Vth ≈ 21V. Let us startwith a uniform distribution function Pθ = 1/π over the en-tire range of orientation angles θ0 ∈ [0, π]. Because of theuniaxial symmetry of the problem, we may consider ex-plicitly only half of this range θ0 ∈ [0, π/2]. An externallyapplied field will change Pθ, concentrating it at smallerangles while preserving its total area

∫ π

0Pθdθ = 1. This

effect is demonstrated in Figure 1. At low fields E0 < Ecr,Pθ spans the entire range of possible θ values [0, π/2]. Atthe critical field E0 = Ecr the distribution is again uni-form, but over the narrower range of angles [0, π/4]. Abovethe critical field, θ as a function of θ0 derived from equa-tion (21), is no longer monotonic but has a maximum ata certain value of θ0 that decreases with increasing field.This maximum means that many droplets tend to ori-ent in directions near the largest θ, that is determined bythe applied field. Pθ, therefore, no longer spans the entirerange [0, π/2] but is concentrated at small angles, with astrong divergence at the corresponding maximum of equa-tion (21). The sharp change of Pθ at the critical field leadsto a significant change in the value of the effective refrac-tive indices.


εx =(ε⊥ + 2εp)

(ε‖ − εp

) − 3εp(ε‖ − ε⊥

) ⟨cos2 θ + (1/2) sin2 θ

⟩(1− f) (ε⊥ + 2εp)

(ε‖ + 2εp

)+ 3fεp

(ε⊥ + 2εp +

(ε‖ − ε⊥

) ⟨cos2 θ + (1/2) sin2 θ

⟩) (24)

εz =(ε⊥ + 2εp)

(ε‖ − εp

) − 3εp(ε‖ − ε⊥

) ⟨sin2 θ

⟩(1− f) (ε⊥ + 2εp)

(ε‖ + 2εp

)+ 3fεp

(ε⊥ + 2εp +

(ε‖ − ε⊥

) ⟨sin2 θ

⟩) . (25)

The effective dielectric tensor is, in this case

εe = εp + 3fεp

εx 0 00 εx 00 0 εz

, (23)

wheresee equation (24) above

andsee equation (25) above

In the dilute limit we simply set f = 0 in the denomina-tors of (24) and (25). The ordinary refractive index of thePDLC is therefore given by

no =√εp + 3fεpεx, (26)

and the extraordinary index is

ne =√εp + 3fεpεz. (27)

The optical phase shift experienced by light incidentobliquely at angle ϑ on the plane surface of the PDLCis

ϕ =2πd

λ cosϑδn , (28)

whereδn =

none√n2

e cos2 ϑ+ n2o sin

2 ϑ− no, (29)

λ is the wavelength and d is the thickness of the film.The reorientation process and its effect on the distribu-

tion Pθ may be easily calculated, either analytically in thedilute limit or numerically for non-dilute systems, by sub-stituting the low-frequency dielectric coefficients in equa-tions (21) and (22) or in equations (17) and (19), respec-tively. The refraction indices and the electro-optical phaseshift can then be calculated by substituting the opticalfrequency dielectric coefficients in (26), (27) and (28). Re-sults of this calculation, for the example of a PMMA/E7film mentioned above, are shown in Figure 2. The phaseshift is shown as a function of applied voltage for two dif-ferent values of LC concentration, two incidence angles ϑand two choices of the order parameter S. In the absence offield the phase shift is zero in all of those examples, as ex-pected from the isotropy of the sample. It increases grad-ually with applied field and saturates at fields much largerthan the critical one. Its variation is relatively rapid in thevicinity of the concentration-dependent critical field. Forf = 0.01 this occurs near the dilute limit threshold volt-age, estimated above at 21V. For f = 0.5 it is about 5V

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

7

8

V

0 5 10 15 20 25 30 350

50

100

150

200

250

300

350

400

V

a)

b)

Fig. 2. The electro-optical phase shift ϕ (degrees) as a functionof applied voltage V (volts) in samples with a uniform initialorientation distribution for (a) f = 0.01 and (b) f = 0.5. Theincidence angles are ϑ = 10◦ (dash-dot lines) and ϑ = 20◦

(solid lines), and the order parameter is S = 0.7. The dottedlines are the corresponding results for S = 1.

lower. As expected, the phase shift also increases with theincidence angle as is apparent from comparing the resultsfor ϑ = 10◦ and ϑ = 20◦. The choice of order parameteralso changes the results by affecting both the orientationprocess and the magnitude of the different elements ofthe dielectric tensor. In Figure 2, results are shown forS = 0.7, which corresponds to strong elastic anchoring


at the droplet walls [11,17], and for S = 1 which maycorrespond to very weak anchoring [6].

3.2 Planar orientation distribution

Let us now consider the case in which in the initial stateall the bipolar droplet axes are in the film plane (thex-y plane), θ0 = π/2, with φ distributed uniformly over[0, 2π]. Such a distribution is an approximate descriptionof PDLC films formed by encapsulation methods, whichtypically have their major axes distributed near the filmplane [15]. The application of an external field in the z di-rection does not change the distribution of the azimuthalangle φ. The effective dielectric tensor is therefore uni-axial and the optical axis remains perpendicular to thefilm plane throughout the switching process. The angle θ,on the other hand, will decrease as the field is increased.The electrostatic energy density can again be derived fromequations (6) and (5)

Eelec = −3εpE20

8πX

(1− fX)2, (30)

where

X = a− b sin2 θ ,

a =ε‖ − ε⊥ε‖ + 2εp

,

b =3εp

(ε‖ − ε⊥

)(ε⊥ + 2εp)

(ε‖ + 2εp

) . (31)

The elastic energy density is

Eelas = −W sin2 θ. (32)

The first derivative of the total energy density is(3εpbE2

0

8πW1 + fX

(1− fX)3− 1

)W sin 2θ . (33)

This derivative is zero either at θ = π/2 which is a mini-mum when the term in the parentheses is negative, θ = 0which is a minimum when this term is positive, or whenthis term itself is zero. At the latter situation, the en-ergy is degenerate and the droplet may be directed in anyorientation. However, the term in the parentheses itselfdepends on θ through the variable X. This means that itwill vanish at different values of applied field dependingon the initial orientation. If we start with θ = π/2 at zerofield, then increasing the field gradually will have no effecton θ until we reach the point

E20 =

8πW3εpb

(1− f (a− b))3

1 + f (a− b)(34)

at which the term in the parentheses of (33) vanishes. Atthis point θ will abruptly jump from π/2 to zero and will

stay there at all higher fields. On the other hand if westart at very high fields with θ = 0 then decreasing thefield gradually will have no effect on θ until we reach thepoint

E20 =

8πW3εpb

(1− fa)3

1 + fa(35)

at which the term in the parentheses of (33) again van-ishes. For f = 0 both transition points are reduced to thedilute limit critical field (22). In the non-dilute case, theinteraction between the different droplets causes splittingin the transition point with two critical fields instead ofone. The width of the resulting hysteresis loop depends onthe volume fraction of the droplets and on the dielectriccoefficients of the components. A more detailed discussionof this hysteresis appears in [18]. It was also previouslyobtained from numerical calculations by Reshetnyak et al.[17].

The effects of an externally applied field on εe mayagain be easily deduced from the above results. The ef-fective dielectric tensor of equation (11) is again of thegeneral form (23). As explained above, starting from aplanar distribution, θ has only two states. The low-fieldstate θ = π/2 and the high-field state θ = 0. Therefore,the effective dielectric tensor also has only two possiblevalues corresponding to these states, the transition be-tween which is determined by the critical fields of (34)and (35). At θ = π/2 we get

εx =2 (ε⊥+2εp)

(ε‖ − εp

) − 3εp(ε‖ − ε⊥

)2 (1− f) (ε⊥+2εp)

(ε‖+2εp

)+3fεp

(ε⊥+ε‖+4εp

)(36)

and

εz =(ε⊥ − εp)

(1− f) (ε⊥ − εp) + 3εp. (37)

The z element of the dielectric tensor is equal, as expected,to (13) which is the scalar MG result for inclusions withdielectric coefficient ε⊥. For θ = 0, the diagonal elementsof εe are given by the scalar MG results (13) and (14),

εx =(ε⊥ − εp)

(1− f) (ε⊥ − εp) + 3εp(38)

and

εz =

(ε‖ − εp

)(1− f)

(ε‖ − εp

)+ 3εp

. (39)

The ordinary and extraordinary refractive indices of thePDLC are again given by equations (26) and (27), re-spectively. The optical phase shift experienced by lightincident obliquely at angle ϑ on the plane surface of thePDLC is, as in the previous example, given by equations(28) and (29). Figure 3 shows the electro-optical phaseshift for two values of LC concentration and two incidenceangles ϑ. The sharp transition between the two states de-scribed above and the hysteresis loop clearly appears inall of those examples. The width of the loop increases andit is shifted towards lower applied voltages with increas-ing f . The phase shift is again larger for larger incidenceangles and for larger LC concentrations. It can be either


Eelec = −9ε2pE20

8π

(ε‖ − ε⊥

) [(ε⊥ + 2εp)

(ε‖ + 2εp

) − f2(ε2p − εp

(ε‖ + ε⊥

)+ ε‖ε⊥

)]cos 2θ

(ε⊥ + 2εp − f (ε⊥ − εp))2 (ε‖ + 2εp − f

(ε‖ − εp

))2 (40)

E2cr =

8πW (ε⊥ + 2εp − f (ε⊥ − εp))2 (ε‖ + 2εp − f

(ε‖ − εp

))2

9ε2p(ε‖ − ε⊥

) [(ε⊥ + 2εp)

(ε‖ + 2εp

) − f2(ε2p − εp

(ε‖ + ε⊥

)+ ε‖ε⊥

)] . (43)

0 5 10 15 20 25 30 35 40 45-3

-2

-1

0

1

2

3

4

5

V

0 5 10 15 20 25 30 35-150

-100

-50

0

50

100

150

200

250

V

a)

b)

Fig. 3. The electro-optical phase shift ϕ (degrees) as a functionof applied voltage V (volts) in samples with a planar initialorientation distribution for (a) f = 0.01 and (b) f = 0.5. Theincidence angles are ϑ = 10◦ (dash-dot lines) and ϑ = 20◦

(solid lines), and the order parameter is S = 0.7.

negative, at the low-field state, or positive, at the high-field state, in contrast to the previous example where itvanished at zero applied field.

3.3 Perfectly aligned droplets

One of the simplest distribution functions is the deltafunction. In this case all droplets are aligned in parallel

in a given initial direction with orientation angles θ0 andφ0. Such an initial δ function distribution may be obtainedby using a small orienting field or by slightly shearing thePDLC film during its formation. In this case it is easyto derive the θ-dependent part of the electrostatic energydensity from equations (6) and (5),


Here we took into account the fact that φ = φ0, sinceφ does not change when the field is applied in parallelto the z-axis. In this case we can choose, without loss ofgenerality, φ0 = 0. The elastic energy density is

Eelas = −W cos2 (θ − θ0) . (41)

Minimizing the total energy Etot = Eelec + Eelas, we find

tan 2θ =sin 2θ0

cos 2θ0 + (E0/Ecr)2 , (42)

with a volume fraction-dependent critical field


In dilute systems, which are simply the f = 0 limit, thisis reduced to equation (22). The effect of the electrostaticinteraction between the inclusions is simply to decreasethe critical field (assuming that ε‖, ε⊥ > εp) as the vol-ume fraction of the inclusions increases. As expected, θdecreases as E0 increases. However, the form of the transi-tion strongly depends on the initial orientation. For smallθ0, θ decreases smoothly to zero. The behavior at largeθ0 is less intuitive. For θ0 = π/2, we find as in the previ-ous section, θ = θ0 for small fields and θ = 0 for higherfields. At the critical field E0 = Ecr the energy is inde-pendent of θ and the droplet orientation is unstable. Thiscauses a very sharp transition between the low-field stateθ0 = π/2 and the high-field state θ = 0. Similar, butslightly smoother, transitions occur at all large θ0. Thisleads to a non-monotonic dependence of θ on θ0 for fieldslarger than the critical one.

The effect of an externally applied field on the re-fractive indices of the sample is easily deduced from theabove results. Since all the droplets are aligned, the ma-terial is uniaxial and its optical axis is also aligned withthe droplets. The ordinary and extraordinary refractiveindices are therefore given by (13) and (14), respectively,

n2o = εx = εp +

3fεp (ε⊥ − εp)(ε⊥ − εp) (1− f) + 3εp

(44)


0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

V

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

V

a)

b)

Fig. 4. The electro-optical phase shift ϕ (radians) as a functionof applied voltage V (volts) in samples with aligned dropletsfor (a) f = 0.01 and (b) f = 0.5. The initial orientation anglesare θ0 = π/2 (solid line), θ0 = 0.4π (dash-dot line), θ0 = π/4(dashed line) and θ0 = 0.1π (dotted line).

and

n2e = εz = εp +

3fεp(ε‖ − εp

)(ε‖ − εp

)(1− f) + 3εp

, (45)

where ε‖ = n2‖, ε⊥ = n2

⊥ and εp = n2p.

The optical phase shift experienced by normally inci-dent light is

ϕ =2πdλ

δn , (46)

whereδn =

none√n2

e cos2 θ + n2o sin

2 θ− no, (47)

λ is the wavelength and θ is the orientation angle of thedroplets and the optical axis.

The electro-optical phase shift is shown in Figure 4,for a few different initial orientations and two LC concen-trations. It is evident that the sharpness of the transition

at the critical field and the contrast between the two ex-tremal states, at zero field and very large fields, can becontrolled by determining the initial orientation angle θ0.The phase shift is larger, at low fields, for larger θ0. Itreduces to zero at high fields regardless of the original ori-entation because of the alignment of the optical axis withthe applied field perpendicular to the film plane. As inthe previous examples, ϕ is larger in samples with largerLC concentration, however, its magnitude for every valueof f is much larger than in the previous examples. This isexpected because the droplets are aligned. Their dielectricanisotropy therefore affects the overall birefringence of thefilm in a much stronger way than in samples where theyare more uniformly oriented. In Figure 4, ϕ is presentedin radians, since at high concentrations the optical pathdifference between the ordinary and extraordinary rays isa few times larger than the wavelength. The phase shifttherefore goes through a few full cycles of 360◦.

4 Conclusions

In this paper, the reorientation of bipolar LC droplets inPDLC films and the consequent changes in their bire-fringence are studied, using an extension of the clas-sical Maxwell Garnett approximation to materials withanisotropic inclusions. As in previous studies of this prob-lem, the essential physics involves a balance between theelastic energy, which favors orientation of a droplet in alocally preferred direction, and the electrostatic energywhich favors alignment with the applied field. The MGapproach provides a convenient framework for the evalu-ation of the electrostatic effects. It can handle non-dilutesystems and is therefore more general than most previ-ously published studies. It provides analytical results forsystems with some initial orientation distributions and avery simple numerical scheme for all the other cases. Thetreatment of the elastic energy, as in all previous studies ofthis subject, is phenomenological. A detailed microscopictheory of this energy is still lacking.

Most experimental studies of the optical propertiesof PDLC have been restricted to light intensity trans-mission and light scattering. Phase shift measurementsavoid the problems due to scattering and, as demonstratedin section 3, should be very sensitive to changes in theorientation distribution of the droplets. Electro-opticalphase modulation may therefore offer a better approachfor studying the reorientation process and the physicalparameters that influence it, e.g. the interfacial elastic in-teraction and other properties of the LC that may dependon its confinement in small cavities. A recent analysis ofphase modulation measurements [4,11] (which to the bestof our knowledge is the only one published on the subject)is based on the order parameter approach of [15]. As men-tioned in the introduction this work considered systemswhere the LC droplets are initially randomly oriented andused a simplified model of the microgeometry as consist-ing of parallel layers of liquid crystal and polymer. In thismodel the electro-optical effects result from changes in theorder parameter of the LC layers. Vicari obtained good


agreement between this theory and experimental resultsby introducing a complex field-dependent droplet orderparameter [11]. This approach is somewhat difficult to jus-tify considering the rough approximations applied in themodel regarding the electrostatic effects and the microge-ometry of the PDLC. Our approach avoids some of thesedifficulties by considering the actual microgeometry of thesample and treating the electrostatics more carefully, viaan approximation that is exact to second order in the vol-ume fraction of the LC droplets. It should therefore proveuseful in analyzing phase modulation experiments and ad-dressing questions pertinent to the reorientation process,such as the nature of the interfacial elastic anchoring andthe ordering of the LC molecules inside the droplets undervarious conditions.

In addition to these fundamental questions, electro-optical phase modulation in PDLC is also interesting fromthe applied point of view. Like any birefringent materialof the Kerr or Pockels type, they may be used in electro-optical shutters and other devices. Their potential advan-tage over other materials is the variety of electro-opticalbehaviors under various conditions, demonstrated in sec-tion 3. Using PDLC we may select the operation voltage,the magnitude of the phase shift and the sharpness of thetransitions between two or more operating states of thedevice by varying the LC concentration and the initialorientation distribution.

In conclusion, we studied the reorientation process andthe electro-optical phase shift in PDLC films. We showedthat once an electric field is applied, the droplets bipolaraxes start to rotate, in accordance with their initial orien-tations and the magnitude of the field. This reorientationchanges the electro-optical phase shift experienced by atransversely impinging light. For initially aligned droplets,the sharpness of the reorientation depends on the initialorientation in a quite dramatic way. For orientations closeto the film plane the process is threshold-like with a crit-ical field that depends on the droplet concentration. Forinitial orientations far from this plane it is much moregradual. For a planar initial orientation distribution thereorientation process is threshold-like with some hystere-sis. The width of the hysteresis depends on the LC con-centration and can be calculated analytically within ourmodel. This behavior may be used for some types of opti-cal switching and should be quite interesting to verify ex-perimentally. Systems with random orientation distribu-tions exhibit reorientation processes that are much moregradual.

I thank P. Palffy-Muhoray for introducing me to this subject.I also thank R. V. Kohn for useful conversations and for sup-porting my work through NSF grant DMS-9402763 and AROgrant DAAH04-95-1-0100.

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electro-optical phase shift in polymer dispersed liquid crystals

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