Investigation into spatial stochastization of an optical field scattered by nematic

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1 Optic Applict, Vol. XLIV, No. 4, 14 DOI: 1.577/o1446 Investigtion into sptil stochstiztion of n opticl field scttered y nemtic PETER MAKSIMYAK, MYKHAILO GAVRYLYAK * Correltion Optics Deprtment, Chernivtsi University, Kotsyuinsky St., Chernivtsi, 581 Ukrine * Corresponding uthor: mgvrylyk@gmil.com This pper presents the investigtion results of sptil chotiztion of n opticl field scttered y liquid crystls during phse trnsition liquid liquid crystl under n electric field. Two stochstic prmeters of the field, nmely, Lypunov s mximl index nd correltion exponent were chosen for this study. It hs een estlished tht mximum vrinces of phse inhomogeneities of the nemtic liquid crystl correspond to mximum fluctutions of n order prmeter under the temperture of phse trnsition liquid liquid crystl. It hs een found tht the nlysis of the rdition field scttered during the phse trnsition process in the liquid liquid crystl llows to ccurtely determine the phse trnsition temperture nd voltge of forming Willims s domins. Keywords: liquid crystl, Willims s domins, Lypunov s mximl index, correltion exponent. 1. Introduction In liquid crystls cn occur lrge-scle mss flow emerges, which re ccompnied y highly dynmicl collective swirling nd swrming motions of ctive prticles [1, ]. The nemtic stte of ctive mtter is of prticulr interest, where genuine gint numer fluctutions re predicted y theory nd verified in experiment on driven rods [3, 4]. If you plce nemtic liquid crystl (NLC) etween the conductive pltes then the ordered stte of molecules of liquid crystl cn e destroyed y voltge etween the pltes. This electro-opticl effect ws independently discovered y severl experimenters, ut the group of HEILMEIER understood the prcticl vlue of this discovery first [5]. The prcticl vlue of this effect is in the destruction of order plcement NLC molecules, which leds to strong diffuse scttering of light nd chotiztion of the rdition field scttered y the liquid crystl. The generl explntion for this effect is proposed y HELFRIH [6]. He explined the ppernce of convective instilities in NLC under the influence of pplied field nd temperture on the liquid crystl. The mechnism of formtion nd the evolution of sptilly modulted structures re still fr from under-

2 556 P. MAKSIMYAK, M. GAVRYLYAK stnding. Therefore, n importnt tsk is the investigtion of the influence of temperture nd pplied voltge on the dynmic properties of NLC nd sptil chotiztion of the scttered field. In NLC y incresing the electric field pplied to the liquid crystl cell re oserved crystl modes, Willims s domins nd dynmic light scttering. Crystl modes re forming t low voltges (typiclly voltge ~1 V), molecules re oriented prllel to the electric field nd liquid crystl is similr to single crystl on its properties. Willims s domins re oserved when the voltge increses to certin criticl vlue (out 9 V). This periodic deformtion rrngement of molecules ws oserved y WILLIAMS [7] nd ws investigted y TEANEY nd MIGLIORI more detiled [8]. They explined the ppernce of Willims s domins y electrohydrodynmic convection in NLC under the influence of the pplied field on the liquid crystl. An increse in pplied voltge cuses trnsition to new regime. In this cse Willims s domins ecome disordered nd moving, flows of the liquid crystl ecome turulent nd long- -rnge order of NLC molecules ecomes completely destroyed. At the mcroscopic scle, this mode leds to dynmic light scttering nd the sptil chotiztion of scttered rdition [9]. Electrohydrodynmic (EHD) convection in NLC hs numer of properties which distinguishes it from other convective instilities like the thermlly driven Ryleigh Bernrd convection in simple fluids [1], which hs een studied more intensively. Firstly, the relxtion times of such system re short owing to the smll thickness of the lyers in EHD convection (usully 5 μm). Secondly, in ddition to the mplitude of the pplied voltge one hs the frequency nd temperture s esily ccessile externl control prmeters. This, together with the fcts tht the mteril couples strongly to n dditionl mgnetic field nd tht vst vriety of NLC with different mteril constnts re ville, provides for very rich scenrios. Theoreticl models were successfully creted y BODENSCHATZ et l. [11]. They presented essentilly the full three-dimensionl liner stility nlysis of the sic stte nd mjor prt of the wekly-nonliner theory of the convective stte. However, the wekly-nonliner theory is not suitle for dynmic light scttering regime, nd the influence of temperture on the process ws not tken into ccount. The chnge in temperture cuses the phse trnsition in liquid crystl. Phse trnsition liquid liquid crystl is limited in time nd it elongs to the clss of trnsition processes, which cn e descried y the theory of chotic nd stochstic oscilltions. The coherent opticl rdition scttered y liquid crystl during phse trnsition lso ecomes chotic [1]. In the present pper, we investigte of the influence of temperture nd pplied voltge on the dynmic properties of NLC nd sptil chotiztion of scttered field.. Bsic reltions The lyer of NLC is the phse-inhomogeneous oject, which descried y the model of rndom phse screen (RPS) [13]. Such model ssumes: infinite extension of the o-

3 Investigtion into sptil stochstiztion ject, smoothness of inhomogeneities, nd phse vrince less thn unity. We cn write the phse correltion function of NLC s ψ ( ρ) = σ K( ρ ) (1) where σ is the phse vrince, nd K(ρ) is the phse correltion coefficient. If the phse fluctutions re sttisticlly homogeneous nd oey the Gussin sttistics, then there exists the following interreltion etween the correltion chrcteristics of n RPS nd the trnsverse coherence function of the field Γ (ρ) [14, 15]: Γ ( ρ) = exp σ K( ρ ) 1 () Scttering of coherent rdition on liquid crystls y the ction of temperture nd voltge leds to time nd sptil chotiztion of the opticl field. The theory of stochstic nd chotic oscilltions [16] provides considerle extension of conventionl light-scttering techniques. We choose for this study two stochstic prmeters of the field, nmely, Lypunov s mximl index λ 1 nd correltion exponent ν. For experimentl dt, otined t oserving dynmic systems, the vilility of the positive Lypunov exponent λ 1 cn e the proof of chos existence in the system. The correltion exponent ν chrcterizes the complexity of such systems nd determines the numer of hrmonics with incommensurle periods descriing the suject of the study [16]. Lypunov exponents ply n importnt role in studying dynmic systems. They chrcterize the verge velocity of exponentil divergence of close phse trjectories. If d is the initil distnce etween two initil points of phse trjectories, the distnce etween trjectories, coming off these points, in time t will e s follows: dt ( ) = d exp( λ t ) (3) The vlue λ is clled Lypunov exponent [17]. Ech dynmic system is chrcterized y Lypunov exponents spectrum λ i (i =1,,, n), where n is the numer of differentil equtions which re necessry for system description. Generlly speking, chotic system is chrcterized y the divergence of phse trjectories in the similr directions nd their convergence in others, i.e. there re oth positive nd negtive Lypunov exponents in the chotic system. The sum of ll the indices is negtive, i.e., the trjectory convergence degree exceeds tht of divergence. If this condition is not fulfilled, the dynmic system is instle, nd the ehvior of such system is recognized esily. Thus, in most cses, it is sufficient to clculte the lrgest Lypunov exponent only. The positive vlue of the lrgest Lypunov exponent gives the possiility of chos existence in the system, nd the vlue of this index chrcterizes chosity intensity. We determined the lrgest Lypunov exponent in opticl fields using the nlog interference method for mesurement of the trnsverse correltion function [18].

4 558 P. MAKSIMYAK, M. GAVRYLYAK To clculte the correltion dimension ν for single generlized coordinte, nmely for the oject field s intensity I(r), we construct the dynmic systems so tht ( m) I i = { I i, I i + 1,, I i + m 1 } where I i re the field intensities t the points x i. Then, the correltion integrls of the form (4) C m ( ε ) = lim N Θ ( m) ε y i N N N i = 1 j = 1 ( m) y j re evluted. Here Θ is the Heviside step function, N is the totl numer of points, C m (ε) gives the reltive numer of the pirs of points seprted y less thn ε, nd m is the numer of smpling points, while the numericl distnces re given y (5) ( m) I i ( m) I j = ( I i I j ) + ( I i + 1 I j + 1 ) + + ( I i + m I j + m ) 1/ (6) Hence, the correltion di- For smll ε, the correltion integrl ecomes mension is given y 3. Experiment C( ε ) ε ν. ln[ C m ( ε )] ν = lim (7) ε ln( ε ) where m is to e incresed until the slopes of C m (ε) versus ln(ε) curves sturte [19]. We investigted sptil chotiztion of n opticl field scttered y liquid crystl N8 during the phse trnsition liquid liquid crystl under the ction of n electric field. N8 is n eutectic mixture of MBBA nd EBBA, whose structurl formuls re shown in Fig. 1. Thermodynmic properties of N8 re well investigted in the temperture rnge of the existence of mesomorphic phse (from 63 up to 36 K t tmospheric pressure) []. The lyer of liquid crystl of thickness μm ws plced etween two glss pltes with ITO. The correltion function of the phse inhomogeneous of liquid crystl ψ (ρ), Lypunov s mximl index λ 1 nd correltion exponent ν, coherent opticl rdition CH 3 O CH N C 4 H 9 C H 5 O CH N C 4 H 9 Fig. 1. The structurl formuls of MBBA () nd EBBA ().

5 Investigtion into sptil stochstiztion scttered y liquid crystl were determined from mesurements of the trnsverse function of the coherence of field [18, 19]. The dvntge of this pproch for chrcteriztion of the liquid crystl is tht the mximum vlue of the correltion function is determined y the vrince of phse fluctutions (chrcterized y scttering ility NLC) nd the hlf-width of the correltion function depends on the trnsverse scle of phse fluctutions of the liquid crystl. As for the importnce of the stochstic field prmeters, then λ 1 is criterion of chos, nd ν determines the complexity of diffused field Fig.. Opticl scheme for mesurement of the trnsverse function of coherence of field: 1 He-Ne lser, telescope, 3 cell with nemtic liquid crystl, 4 trnsverse sher interferometer, 5 lens, 6 perture, 7 photodetector, 8 nlog-to-digitl converter, 9 personl computer. The scheme of the experiment for mesurement of the trnsverse function of the coherence of the field is shown in Figure. A single-mode He-Ne lser 1 (λ 1 = =.638 μm) is used s the source of coherent opticl rdition. The telescopic system forms plne wve incident on the cell 3 with NLC. Rdition scttered y the oject is divided into two components of equl mplitudes y using polriztion interferometer 4. Then, the reltive trnsversl shift of these components nd colliner mixing of them t the interferometer output re provided. The ojective 5 imges ny cross-section of the scttered field into the field-of-view diphrgm 6 nd photodetector 7. The signl from the photodetector is sent through the nlog to digitl converter nd then to the computer 9 for further processing. The opticl scheme of the polriztion interferometer 4 is shown in Fig. 3. It consists of two identicl wedges 3 nd 4, which form plne-prllel plte nd re situted h ψ e ϕ ϕ ψ o e o ρ Fig. 3. Bundles pth in n experimentl setup interferometer.

6 56 P. MAKSIMYAK, M. GAVRYLYAK etween crossed polrizers 1 nd 5. Principl opticl xes of wedges 3 nd 4 re prllel nd form 45 ngles with plne of polriztion of polrizers 1 nd 5. Smple is situted etween the polrizer 1 nd the wedge 3. In Fig. 3, ordinry (o) nd extrordinry (e) undles pths in such opticl scheme re shown. Spce division of undles hppens on the wy out from the first wedge 3. At norml incoming undle incidence on the wedge 3 surfce, refrction ngles of ordinry ψ o nd extrordinry ψ e undles could e written s: sin( ψ o ) n o = sin( ϕ ), sin( ψ n e ) = n e sin( ϕ ) n where ϕ incident ngle, which is equl to prism ngle; n o nd n e refrctive indexes of ordinry nd extrordinry undles ccordingly; n refrctive index of surrounding medium. Trnsverse displcement etween undles ρ is ssigned y the distnce etween wedges h nd depends on the wedge ngle nd irefringent properties of wedge mteril. From the geometricl construction in Fig. 3, we cn get (8) ρ = h tn( ψ o ) tn( ψ e ) cos( ϕ ) = h (9) As you see, ρ is linerly dependent only on h (prmeters ϕ, ψ o, ψ e re constnt for specific scheme reliztion). So, for trnsverse displcement definition, it is necessry to know the dependence ρ = f (h) =h. In the scheme in Fig. 3, the trnsverse displcement ρ is ccompnied y the longitudinl ρ, which is defined from y the following eqution: 1 1 ρ = n cos( ψ o ) cos( ψ e ) e tn( ψ o ) tn( ψ e ) sin( ϕ ) h = h (1) The longitudinl displcement etween undles in the interferometer cuses the modultion of totl field intensity, which dependence, while chnging the distnce etween wedges from zero till the defined vlue, is represented in Fig. 3. As the longitudinl displcement etween undles is linerly ound with trnsverse, it is convenient to clirte the trnsverse displcement with extreme vlues of totl field intensity for longitudinl displcements (Fig. 4). The distnce etween extremes (mximum nd minimum) mounts to λ/. Such clirtion could e provided in cse of sustntil scle excess of longitudinl field modultion over the trnsverse modultion scle. So, even for irregulr chnges in the distnce etween wedges (Fig. 4), trnsverse displcement vlue is known y extremes of totl field intensity t the output of the interferometer. In our experiment, the distnce etween neighoring extremes corresponded to the trnsverse displcement in 3.36 μm.

7 Investigtion into sptil stochstiztion Normlized intensity.5. 1 Numer of points, 1 3 Fig. 4. Intensity chnges, mesured y longitudinl displcements of interferometer wedges. From Fig. 4, the visiility of the interference pttern V ws determined: V = I mx I min + I mx I min (11) Here, I mx nd I min re the intensities of the resulting field for opticl pth differences of the mixed components mλ/ nd (m +1)λ/ (m is the integer), respectively. If the interfering ems re of equl intensities, the visiility equls to the coherence degree of the resulting field. The dependence Γ (ρ) otined in such mnner is used for determining phse vrince σ nd the phse correltion coefficient K(ρ). Two stochstic prmeters of the field, nmely, Lypunov s mximl index λ 1, nd correltion exponent ν, we determined from the intensity structure function D I (ρ) [18, 19, 1], which ws connected with trnsverse coherent functions Γ (ρ) y reltion D I ( ρ ) = 8Γ ( ) 1 Γ ( ρ ) (1) where Γ () is the trnsverse coherent function for zero displcement. 4. Results nd discussion As result of the experiment, the trnsverse correltion functions of phse fluctutions of the NLC depending on the temperture (the dependences for voltges, 6, 9 nd 1 V re shown in Fig. 5) were otined. Correltion functions show the structure dynmics of the NLC during phse trnsition s well s for different voltges pplied to the liquid crystl cell. When pplied voltge is equl to zero, the mximl vlues of the correltion functions correspond to the phse trnsition temperture of the liquid liquid crystl (Fig. 5).

8 56 P. MAKSIMYAK, M. GAVRYLYAK 1 ψ ψ 4 8 r [μm] r [μm] c d 4 ψ ψ 4 8 r [μm] r [μm] Fig. 5. The temperture dependence of the correltion function of phse t pplied voltge: V (), 6V(), 9 V (c), nd 15 V (d). This is due to the fct tht the light scttering is minly cused y fluctutions of the orienttion in NLC. If the temperture of NLC is close to the phse trnsition temperture T c, the mplitude of fluctution increses ccording to the lw (T T c ) 1. It leds to the destruction of the liquid crystl structure nd chotiztion of scttered rdition []. Voltge pplied to the liquid crystl leds to convective instility nd fter certin vlue (for our cse, 9 V) to dynmic light scttering. It is shown in the form of correltion functions. The intensity of phse fluctutions in NLC ws evluted y their vrinces of phse σ (the mximum vlue of the correltion function of the phse) Fig. 6. The vrince of phse inhomogeneities in the NLC σ does not chnge with n increse in temperture up to 45 C for voltges less thn 1 V. Before the temperture of phse trnsition, the vrinces of phse inhomogeneities σ shrply increse to mximum nd then decrese to zero. This indictes tht the mximum vrince of phse inhomogeneities in the NLC corresponds to mximl fluctutions of the order prmeter of the NLC nd the temperture of the phse trnsition liquid liquid crystl. For volt-

9 Investigtion into sptil stochstiztion σ 1 V 3 V 6 V 9 V 1 V 15 V Fig. 6. The dependences of the vrince of phse inhomogeneities in the nemtic liquid crystl on temperture () nd voltge () σ 1 55 C 47 C 45 C 4 C 3 C U [V].16.1 λ λ 1.8 V 6 V 9 V 15 V.4. 3 V 6 V.4 9 V 15 V Fig. 7. The temperture dependence of Lypunov s mximl indexes of the sptil distriution of the scttered field () nd the sptil distriution of phse inhomogeneities in the nemtic liquid crystl () t different voltges λ C 47 C 45 C 4 C 3 C.1 λ C 47 C 45 C 4 C 3 C U [V] U [V] Fig. 8. The voltges dependence of Lypunov s mximl indexes of the sptil distriution of the scttered field () nd the sptil distriution of phse inhomogeneities in the nemtic liquid crystl () on temperture.

10 564 P. MAKSIMYAK, M. GAVRYLYAK ν ν V 6 V 9 V 1 15 V V 6 V 9 V 15 V Fig. 9. The temperture dependence of the correltion exponent of the sptil distriution of the scttered field () nd the sptil distriution of phse inhomogeneities in the nemtic liquid crystl () t different voltges ν C 47 C 45 C 4 C 3 C ν C 47 C 45 C 4 C 3 C U [V] U [V] Fig. 1. The voltges dependence of the correltion exponent of the sptil distriution of the scttered field () nd the sptil distriution of phse inhomogeneities in the nemtic liquid crystl () on temperture. ges higher thn 1 V, the dependence of the vrince phse on the temperture does not hve mximum. Lypunov s mximl indexes (Figs. 7, 8) nd correltion exponents (Figs. 9, 1) from the structure functions were clculted y the known technique [17, 18]. All temperture dependences of Lypunov s mximl index nd the correltion exponent of coordinte distriutions of the scttered intensity tend to zero ruptly t the temperture of phse trnsition NLC liquid. This is due to the fct tht the rdius of correltions decreses t the temperture phse trnsition, which increses the homogeneity of the ner field of NLC. The rdius of correltions of the isotropic phse of the liquid crystl is r c = r [T c /(T T c )] 1/, where r intermoleculr distnce. Its vlue is out 5 7 Å t the isotropic phse nd 5 1 Å t the point of phse trnsition. In the isotropic stte, the threshold field is homogeneous nd therefore rndomness nd complexity of the field go to zero.

11 Investigtion into sptil stochstiztion The dependences of Lypunov s mximl index hve minimum t the voltge 9 V, which cuses the Willims s domins formtion. Willims s domins look like nds with sptil frequency pproximtely equl to the thickness of the NLC. The sptil periodicity of NLC cuses reduction of the sptil chotiztion in the ner field. Correltion exponent does not hve similr minimum. This suggests tht the formtion of Willims s domins does not led to decrese in the complexity of the scttered field. The temperture dependence of Lypunov s mximl index nd correltion exponent of the sptil distriution of phse inhomogeneities hs mximum t the temperture of phse trnsition (Figs. 7, 9) nd the dependences of the intensity distriution of the scttered field (Figs. 9, 1) do not hve mximum. 5. Conclusions Bsed on the results, we cn conclude tht n increse in therml motion of molecules cuses destruction of the structure of NLC t the temperture of phse trnsition, which is due to decrese in the order prmeter (incresing chotiztion of orienttion of NLC domin). In turn, this leds to greter complexity nd rndomiztion of the ner field. The scttered field is result of the coherent summtion of rys from ll points of NLC. As result of the verging of ll work re of NLC, the sptil chotiztion nd complexity of the scttered field t the temperture of phse trnsition re not rising. Thus, the nlysis of the rdition field scttered during the phse trnsition process in the liquid liquid crystl llows to ccurtely determine the phse trnsition temperture nd voltge of forming Willims s domins. Also, the temperture of the phse trnsition cn e determined most effectively y the mximum vlue of vrince of phse inhomogeneities in NLC, nd formtion of Willims s domins is etter determined y the stochstic prmeters of the scttered field. References [1] VICSEK T., ZAFEIRIS A., Collective motion, Physics Reports 517(3 4), 1, pp [] MARCHETTI M.C., JOANNY J.F., RAMASWAMY S., LIVERPOOL T.B., PROST J., MADAN RAO, ADITI SIMHA R., Hydrodynmics of soft ctive mtter, Reviews of Modern Physics 85(3), 13, pp [3] NARAYAN V., RAMASWAMY S., MENON N., Long-lived gint numer fluctutions in swrming grnulr nemtic, Science 317(5834), 7, pp [4] RAMASWAMY S., ADITI SIMHA R., TONER J., Active nemtics on sustrte: gint numer fluctutions nd long-time tils, Europhysics Letters 6(), 3, pp [5] HEILMEIER G.H., ZANONI L.A., BARTON L.A., Dynmic scttering: A new electrooptic effect in certin clsses of nemtic liquid crystls, Proceedings of the IEEE 56(7), 1968, pp [6] HELFRICH W., Conduction-induced lignment of nemtic liquid crystls: sic model nd stility considertions, The Journl of Chemicl Physics 51, 1969, pp [7] WILLIAMS R., Domins in liquid crystls, The Journl of Chemicl Physics 39(), 1963, pp [8] TEANEY D., MIGLIORI A., Current- nd mgnetic-field-induced order nd disorder in ordered nemtic liquid crystls, Journl of Applied Physics 41(3), 197, pp

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