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OFFICE OF

NAVAL RESEARCH

CONTRACT NO. N00014-86-K-0772 TECHNICAL REPORT NO. 9

Light Extinction in a Dispersion of Small Nematic Droplets

by

0S.

Zumer, A. Golemme, and J. W. Doane Liquid Crystal Institute* Kent State University Kent, OH 44242 *Subcontractor to Liquid Crystalline Polymer Research Center University of Connecticut Storrs, CT 06268

k

Prepared for Publication

in1 Journal of Optical Society of America July 13,

.

LY

1988

REPRODUCTION IN WHOLE OR IN PART IS PERMITTED FOR ANY PURPOSE OF THE UNITED STATES GOVERNMENT. THIS DOCUMENT HAS BEEN APPROVED FOR PUBLIC RELEASE AND SALE; ITS DISTRIBUTION IS UNLIMITED.

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4 PE~fOAM.NG OCAjANZAON REPORT NYBER(S)

S MONITORING ORCANiZATiON REPORT NuMBLA(S3

Technical Report No. 9 G&INAME

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ADDRESS (0Iy. State, Jr~l ZiP Code)

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TITLE tiflladf Security CIallpf"cationJ

Unlssfid

Light Extinction in a Dispersion of Small Nernatic Drople*ts.12

.VORK

TASK

ON

PERSPINAL ALJTIOA(S)

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S. Lumer, A. Golemme, and J. W. Doane TYvPE QF REPORT

Technical,

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1988-07-11

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43

SJPPI~j%1EN'AAT NOA'rCN

Submitted to J. 0ptical Societ

of A erica

Research carried out at Kent State I i ersitv ~botatr (LCPRC Publication No.) Ii ~COSATI CODE% is SUIV, TERMS (COiit.Ai~e Oft re,eq~ di At'eCiary and d1 en,ty by blodk A1.mbef) 9IO

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GROUP

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W ou roi l v el

Nematic droplets, Liquid Crystalline Polymers) Light Transmission~properties, Rayleigh7Gans approximation.

of n ec essiy It'd d iitly

by blOC N Aw mviarj

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in whc of materials properties light transmission The submicron size nematic droplets are dispersed in a solid polymeric Theoretical calculations of the differential and matrix are studied. in the Rayleigh-Gans total cross sections have been performed The approximation for different nematic director configurations. effect of interdroplet interference is discussed in relation to the droplet distribution, and different models for the pair correlation An attenuation coefficient is function are taken into account. configurations. director for different and calculated defined Theoretical calculations are in excellent agreement with experimental data obtained from thin films of materials consisting of droplets of cyanobiphenyls dispersed in an epoxy resin. 10

OIIIRISUTIONIAVAILAI'LiTY OF ASSTRACI IeCLASSINIEIUNLIMITIO Q~ SAME AS APr 12& NAME Of RESPONSILE INDIVIDUJAL

Dr. Kenneth J. Wynne_ 00 RM

1473,84AA

11 AWSRACT SECLR'T C:3otic

usERs

CLASSIFICATION

Unclassified 12D riEPiHONE tIncivoit A:Id Cod.) 122c. OFFICE SYMBOL

(202) 696-4410 1 Si APR ed-uOfl ma y be wsEo wmlh,#Ahjui~d All Olne, Ced.t,ons ate Objoliet

I ONR

SECuRiTY CLASSIFICATION OF THIS PAGE

Unclassified

LIGHT EXTINCTION IN A DISPERSION OF SMALL NEMATIC DROPLETS S. Zumer, ( 1) A. Golemme, (2 ) and J.W. Doane Liquid Crystal Institute Kent State University Kent, OH 44242

ABSTRACT

The light transmission properties of materials in which submicron size nematic droplets are dispersed in a solid polymeric matrix are studied.

Theoretical

calculations of the differential and total cross sections have been performed in the Rayleigh-Gans approximation for different nematic director configurations.

The

effect of interdroplet interference is discussed in relation to the droplet distribution, and different models for the pair correlation function are taken into account. An attenuation coefficient is defined and calculated for different director configurations. Theoretical calculations are in excellent agreement with experimental data obtained from thin films of materials consisting of droplets of cyanobiphenyls dispersed in an epoxy resin.

COPY tNSPECTED

Aaoesslio 7FT'

For

r.ro.Qt

(1 Permanent address: Physics Department, E. Kardelj University of Ljubljana, d to 61000 Ljubljana, Yugoslavia. (2) Permanent address: Department of Chemistry, University of Calabria, 87030 u n Rende (CS), Italy. IDistrbu ton/ Availability Codes Avai and/or Dit Special

1

i

I. INTRODUCTION Light extinction1 in a random dispersion of small non-absorbing objects depends on their scattering properties and their distribution in space. Light scattering from a single object is a classical electromagnetic problem. 1 3 Exact solutions are known for few objects, 4,5 so that one is forced to use approximate approaches. 1 The most known simple approaches are: Rayleigh-Gans approximation 6,7 (RGA) for optically soft small objects, anomalous diffraction' (ADA) for large optically soft objects, and geometrical optics 1 for very large objects. In relative few studies optical anisotropy was taken into account. 2 ,8 -1 1 The spatial distribution of the scattering objects influences the light extinction via two effects: the first one is interference 12,13 which predominently affects the distribution of the scattered light and to a lesser degree the total scattering cross section. The second effect is multiple scattering,'

4

which is

important when the path of light in the dispersion is long enough that the contribution of the indirect (scattered) light in the direction of the incident beam becomes substantial. In the following we are going to study light extinction in materials consisting of a random dispersion of micron size nematic liquid crystal droplets in a polymeric matrix.

These materials have been recently developed for use in optical and

electrooptical devices. 1 1 , 15 , 16 Nematic droplets in a polymer are optically soft but anisotropic scattering objects. In most cases their indices of refraction ne - 1.7 and n o -1.5 are close to the index of polymer nri-1.50-1.55.

The anisotropic properties are

related to the nematic configuration in the droplets and can therefore be influenced by external electric or magnetic fields. To focus our attention on the effects of the optical anisotropy we shall limit our treatment to cases where multiple scattering can be neglected.

In case of submicron nematic droplets this is not a severe

limitation because they have relatively small scattering cross sections.

2

For our starting point we use the results obtained in RGA approximation for differential and total cross section in our recent paper"1 . In Section II we describe our system of droplets and possible director configurations.

In Section III we

introduce effective cross sections per droplet and the structure factor for the description of the interdroplet interference. Section IV is devoted to the study of pair correlations and their effect on the structure factor. In Section V we calculate effective scattering cross sections in RGA for different structures. In Section VI we define and calculate the attenuation coefficient. In Section VII theoretical results are compared with experimental data.

II. NEMATIC DISPERSION These materials are the result of polymerization induced phase separation from initially homogeneous solution of liquid crystals and monomers. The dispersions can have relatively uniform droplet sizes, 15 ,17 with a mean size ranging from 0.1 to 10pm, depending on the concentration of the liquid crystal and polymerization rate. 17 A typical example of an electron microscope picture of a cut-through is shown in Fig. 1. The equilibrium director configuration of a nematic liquid crystal in a spherical droplet is well known only in large droplets where it can be studied with an optical microscope. 18,19 For submicron droplets there are only theoretical predictions based on the strong anchoring conditions.20-22 In this paper we will deal with director configurations calculated in the approximation of a single elastic constant K, =K 2 =K 3 =K and strong surface anchoring which can be either parallel or orthogonal to the polymer-liquid crystal interface. Further, we neglect a possible spatial dependence of the nematic order parameter. The minimization of the free energy (elastic and external field contributions) results in a nonlinear partial differential equation which is solved using the nonlinear over-relaxation method. 23

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The degree of the field induced director alignment in the droplet depends on the strength of the external field. The correlation length 2 4 = (k/Ac 0 ) /E is usually used as a measure of the range of the surface induced ordering in the presence of the external field E. Here Ac is the anisotropy of the dielectric constant. The solution for parallel anchoring has two point defects in the bipolar structure (see schematic representation in Fig. 2). We introduce the term droplet director N for the direction of the axis of the cylindrical symmetry. In the case of normal surface alignment the solution is a star structure for low fields and configurations with an equatorial disclination

1

line for strong fields.

Ill. STRUCTURE FACTOR

Neglecting multiple scattering we can write the total scattered electric field at an arbitrary observation point r, where the far field approximation is valid, as a simple sum:

idk-k),

E (r)= S

Here E

r.

-

S' E .(r)e

(1)

Si

is the contribution of the jth droplet located at rj (see Fig. 3).

We are

assuming that the coherence length of the incident light is very large in comparison to the inter droplet distances. To get a measure of the scattering power we introduce the effective differential cross section per droplet as

do r2 d = -

dQ

N

E(r)

2 (2)

E 0

4I V

-e.~~~

where Eo is the amplitude of the incoming planar wave and N the number of droplets. Substituting Eq. (1) into definition (2) one finds:

do

r2

Y

.t(r,

1E

E .(r) i(k.-k-).

E jr).

2

O

dQ N

~t

+

E-

(r.r

*

e

2 0

-

Further, we substitute the summation over different droplets by averaging over possible orientations of aiiisotropic droplets and as well over different droplet sizes:

N

J

and get

do

do._=)

d!Q

dQ

+

ik . (r.-r

do _ dQ

e

N

itj

where do( /dQ stands for the cross section corresponding to the average scattered field < E > and doj/dQ for the differential cross section ofj t h droplet. We use k s for k-k'. We have taken into account that there is no correlation between droplet 25 position, orientation, and size. Introducing positional pair correlation function

g(r) we can write for the case of N> > 1:

N irj

~-

ik" S

I

=

N

5

Nr-r ikr' V- g(rl e __

V

dV',

(5)

where NIV is the number of droplets per unit volume. The Fourier transform of g(r) given by Eq. (5) has a strong forward (k s = 0) part, which in fact does not contribute to the scattering and is usually subtracted. 26 Therefore, we can rewrite Eq. (4) as:

do =

d-

do. =-)+

d

dou -G(k)

ss

dQ

(6)

where ik .r' r dV'. jg(r) e

N f Gs(k S

VJ

S

(7)

2 6,27 It is convenient to introduce the structure factor

do( F(k )I S

do.

+ G (k s sd

d(2

(8)

so that effective differential cross section is given by:

do

do ).

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