Formation and field-driven dynamics of nematic spheroids
FFormation and field-driven dynamics ofnematic spheroids
Fred Fu † and Nasser Mohieddin Abukhdeir ∗ , † , ‡ , ¶ † Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L3G1, Canada ‡ Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L3G1, Canada ¶ Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario N2L3G1, Canada
E-mail: [email protected]
Abstract
Emerging technologies based on liquid crystal (LC) materials increasingly leveragethe presence of nanoscale defects, unlike the canonical application of LCs – LC dis-plays. The inherent nanoscale characteristics of LC defects present both significantopportunities and barriers for the application of this fascinating class of materials.Simulation-based approaches to the study of the effects of confinement and interfaceanchoring conditions on LC domains has resulted in significant progress over the pastdecade, where simulations are now able to access experimentally-relevant micron-scaleswhile simultaneously capturing nanoscale defect structures. In this work, continuumsimulations were performed in order to study the dynamics of micron-scale nematicLC droplets of varying spheroidal geometry. Nematic droplets are one of the simplestinherently defect-containing LC structures and are also relevant to polymer-dispersed a r X i v : . [ c ond - m a t . s o f t ] F e b C-based “smart” window technology. Simulation results include nematic phase for-mation and external field-switching dynamics of droplets ranging in shape from oblateto prolate. Results include both qualitative and quantitative insight into the complexcoupling of nanoscale defect dynamics and structure transitions to micron-scale reori-entation. Dynamic mechanisms are presented and related to structural transitions inLC defects present in the droplet. Droplet-scale metrics including order parametersand response times are determined for a range of experimentally-accessible electric fieldstrengths. These results have both fundamental and technological relevance, in thatincreased understanding of LC dynamics in the presence of defects is a key barrier tocontinued advancement in the field.
Keywords liquid crystals, nematic phase, defect dynamics, polymer-dispersed liquid crystals, simulationLiquid crystals (LCs) are materials which exhibit properties characteristic of both disor-dered liquids and crystalline solids. Their anisotropic nature imparts unique optical prop-erties and makes them susceptible to external fields. These properties have resulted in awide array of electro-optical applications, such as liquid crystal displays (LCDs). However,unlike LCDs, which are designed using uniform defect-free domains, next-generation LC-based technologies are increasingly leveraging the presence of nanoscale topological defects.These emerging technologies include tunable photonics based on blue LC phases, molecularself-assembly, and bistable optical devices. Consequently, understanding and predictingdefect-enabled LC phenomena is a key barrier to continued advancements, both fundamen-tal and technological. Theoretical and computational research is necessary to overcome thisbarrier due to the nanoscale lengths and times associated with LC structure and dynamics,which are currently inaccessible via experimental methods.One of the simplest inherently defect-containing structures is an LC droplet. When LCmaterial is confined in this way, a frustrated domain with significant spatial variation in LC2rder can emerge. This so-called LC “texture” can differ depending on LC/solid anchoringconditions, domain shape, and LC material properties. LC droplets play a major role inpolymer-dispersed liquid crystal (PDLC) films, which are typically fabricated through a“bottom-up” process which results in nano-to-microscale LC domains dispersed in a polymermatrix. PDLCs are optical functional materials which exhibit an optical response whensubjected to thermal or external field actuation (Figure 1a), introducing complex dynamicsand constraints on response and relaxation times between equilibrium states. PDLC filmshave traditionally been used for optical light shutter technology, in which LC domains aremicron-sized. More recently however, PDLCs incorporating nano-sized domains have beenincorporated into novel applications such as holographic PDLC (H-PDLC) lasers and tunablemicrolens arrays. PDLC performance is governed by a variety of material and operating parameters, in-cluding LC defect-mediated structure and dynamics due to the topological constraints on LCorder resulting from spheroidal confinement. It has been more than two decades since Drzaic found that domain shape, specifically anisometry, strongly affects device performance. Sincethen, it has been demonstrated that this anisometry can be directly controlled through var-ious means, the simplest of which is by uniaxial mechanical stretching of the PDLC filmto produce highly prolate spheroidal domains (Figure 1b).
While a significant body of past mesoscale simulation work exists for cylindrical nematiccapillaries and spherical droplets, elliptic or ellipsoidal domains have been far lessstudied.
Furthermore, of this work, most use theoretical models which are unable toaccurately capture nematic defects and phase transition. Only recently have simulationsbeen performed which capture nematic dynamics, as opposed to just determining equilib-rium states. As a result, while past research has provided some insight into the nanoscaledefect structure present in these domains, as of yet there have been no simulations of thedynamics of nematic spheroids on relevant length and timescales. Thus our aim is to pre-dict the dynamic mechanisms involved in the formation, field switching, and relaxation of3ematic spheroids, such as those present in PDLC-based devices. This objective has bothfundamental and technological relevance in that these dynamic mechanisms are both poorlyunderstood and necessary for advancement of this technology. From a fundamental perspec-tive, PDLCs provide ideal templates for the study of nanoscale defect behaviour in confinedLC domains. From a technological perspective, significant improvement in the performanceof PDLCs as electro- and thermo-optical functional materials is required for their broadercommercialization. E=0 E ≠ (a)(b) Figure 1: (a) Schematic of the operation of a PDLC-based “smart” window where light isscattered by (left) randomly oriented nematic droplets in the absence of an electric field(translucent mode), which when exposed to an external field (right) are aligned in the di-rection normal to film (transparent mode). (b) SEM images of an (left) unstretched and(right) uniaxially stretched PDLC film where the resulting droplet shape is anisometric.Reproduced with permission from ref. 10.
NEMATIC PROPERTIES AND DYNAMIC MODEL
LCs include a wide variety of phases, referred to as mesophases, with the simplest mesophasebeing the nematic phase. Nematics exhibit not only translational disorder like a traditionalliquid but also long-range orientational order, as shown by their tendency to self-align at themolecular scale. Technological applications of nematic LCs, such as LCDs, mainly involve4omains that are at or close to hydrostatic equilibrium which is likely due to the significantcomplexity of accounting for LC hydrodynamics.
Dynamics within this regime are re-ferred to as reorientation dynamics, in which the orientation of individual LC molecules, ormesogens, evolve in response to thermodynamic or external stimuli. This LC orientation canbe described using the continuum Landau–de Gennes model of the nematic phase, whichintroduces a symmetric traceless tensor order parameter called the alignment tensor, Q ij = S ( n i n j − δ ij ) + P ( m i m j − l i l j ) (1)which approximates the local orientational distribution function of the mesogens at eachpoint in space. The alignment tensor Q may be decomposed into its eigenvalues and eigen-vectors, which describe the local orientational axis or nematic director n , the uniaxial scalarorder parameter S , and the biaxial scalar order parameter P (and its associated axes, givenby m and l ). For a nematic domain, S = P = 0 corresponds to the isotropic phase (a tradi-tional disordered liquid), while < S < and P = 0 corresponds to the (uniaxial) nematicphase where a higher value of S corresponds to greater alignment. Biaxial orientationalordering occurs when both S and P are non-zero.The majority of past simulation-based research on nematic LCs neglects variations in S ,which results in a simplified model involving only the nematic director n , f ( n , ∇ n ) = f + 12 k ( ∇ · n ) + 12 k ( n · ∇ × n ) + 12 k ( n × ∇ × n ) − k ∇ · ( n ( ∇ · n ) + n × ∇ × n ) ) (2)which includes elastic energy terms that quantify the nematic response to orientational defor-mations of splay k , twist k , bend k , and saddle-splay k . Many past simulation studiesof elliptic nematic capillaries and ellipsoidal droplets use this simplified model despiteits inability to accurately capture nanoscale defects in nematic order, called disclinations. n , and are therefore alsoregions of high biaxial nematic order ( S, P > ), as opposed to isotropic regions of disorder( S = P = 0 ). Figure 2 shows schematics of the two main types of disclinations relevant tonematic droplets: +1 line and + loop disclinations. (a) (b) Figure 2: Schematics of (a) +1 and (b) + disclination lines using a combination of hyper-streamlines to indicate nematic orientation and an isosurface indicating the nanoscale defect“core” region.In contrast, the Landau–de Gennes model (see Methods section) is able to accuratelycapture both the presence of disclinations in nematic domains as well as their dynamics. Areview of recent studies using this model to simulate nematic dynamics may be found inref. 21. However, there are two major shortcomings of past simulation studies of nematicLC droplets. The first is the widely-used single elastic constant approximation, where it isassumed that k = k = k and k = 0 , despite the fact that these constants can widelydiffer, even for commonly studied LCs and may significantly affect simulation outcomes. The second shortcoming is the sparsity of dynamic simulations, which can offer greater insightthan simply solving for equilibrium nematic textures. In this study, material parameters areused that correspond to the 4’-pentyl-4-cyanobiphenyl (5CB), a well-characterized nematicLC. The domain is assumed to be isothermal, at hydrostatic equilibrium ( v = ), andfluctuations in nematic order are neglected. These assumptions are consistent with pastsimulations except that simplifications of nematic elasticity are not made in this work.Finally, in addition to nematic elasticity, interfacial surface anchoring effects arising fromfactors such as PDLC composition must be considered. Surface anchoring may result in a6referred nematic director n at the droplet interface and also the enhancement of nematicordering S > S , where S is the value of S at thermodynamic equilibrium. In this study,the case of homeotropic anchoring is investigated, in which n (cid:107) k is energetically preferred,where k is the unit normal vector to the LC droplet surface. Several experimental studies ofPDLC dynamics have been performed under these conditions. In order to study the formation and field-driven dynamics of spheroidal nematic domainsrelevant to electro-optical applications of PDLCs, simulations were performed of nematicspheroids with fixed volume corresponding to an initial “unstretched” sphere of diameter
500 nm . To emulate stretching of the droplets, the initial sphere was consistently elongatedor contracted along a single direction. Droplet aspect ratio R is defined as the lengthratio between the axis of elongation/contraction and the remaining (equivalent) axes of thespheroid, resulting in oblate droplets for R < and prolate droplets for R > . Various aspectratio R domains were simulated in the interval [0 . , based upon experimental evidenceregarding the expected variation in droplet shape. These simulations were performed in three stages: (i) formation of the nematic phasefrom an initially disordered (high temperature) phase, (ii) application of an electric fieldcorresponding to the “on” (transparent) state of a PDLC film, and (iii) relaxation resultingfrom release of the electric field, corresponding to the “off” (translucent) state of a PDLCfilm. For the formation dynamics simulations, heterogeneous nucleation of the nematic phaseat the solid/LC interface was assumed based on recent experimental observations. For thefield dynamics simulations, a range of experimentally accessible electric field strengths up to
14 V µ m − were applied. Further details of the nematic dynamics model, numerical methods,and auxiliary conditions used in these simulations may be found in the Methods section.7 ORMATION FROM DISORDERED PHASE
Formation dynamics simulations were initially performed for oblate spheroids of aspect ratio R ∈ [0 . , . This geometry can be considered a rotational extrusion of a two-dimensionalellipse about its minor axis. It is therefore comparable to previous simulations of nematicelliptic capillaries, in which a sequence of three different growth regimes were identifiedduring droplet formation: (i) free growth, (ii) defect formation, and (iii) bulk relaxation. Thefree growth regime consists of the stable nematic “shell” growing into an unstable isotropicphase, with the bulk nematic orientation being commensurate with the homeotropic surfaceanchoring conditions. Next, the defect formation regime involves the impingement of thenematic-isotropic interface on itself. This resultes in the simultaneous formation of a pairof + disclination lines along the major axis of the elliptic cross-section of the capillary.Finally, during bulk relaxation, the domain as a whole relaxes to its equilibrium state throughsimultaneous disclination motion and bulk reorientation.The simulation results of the formation process for a R = 0 . oblate droplet are shownin Figure 3. The same set of growth regimes can be identified, starting with the initialfree growth of the stable nematic boundary layer into the central unstable isotropic region(Figures 3a–b). As free growth proceeds, the curvature of the isotropic/nematic interfaceincreases in the focal regions of the spheroid and simultaneously the interface velocity de-creases. This critical slowing down of the nematic/isotropic interface may be explained byan approximation of the interface velocity v , βv = C − ∆ F (3)where β is an effective viscosity term, ∆ F is the difference in energy between the nematic andisotropic phases, and C the capillary force. For an isothermal domain, ∆ F is constant and,in the absence of curvature of the interface ( C = 0 ), the model predicts constant interfacevelocity v . While the isotropic/nematic interfaces in the equatorial region of the droplet8re able to grow inwards with minimal increase in interface curvature, this is not the casefor focal regions of the interface. As the radii of curvature of the interfaces in this regionapproach the nematic coherence length λ n ≈
10 nm , the capillary force C approaches thedifference in free energy resulting from the transition ∆ F and v → . At this point, the freegrowth regime transitions to the defect formation regime.Figures 3c–d show the defect formation regime dynamics. Simultaneously, a + disclina-tion loop forms in the focal region through a interface-driven defect “shedding” mechanism and the isotropic/nematic fronts in the equatorial region impinge. This is followed by thebulk relaxation regime where the droplet texture relaxes through bulk reorientation and thedisclination loop expands towards the focal boundaries. As expected, the formation processof oblate nematic droplets is analogous to that of nematic elliptic capillaries due to theirgeometric similarities. a) 0.38 μ s b) 1.61 μ s c) 2.95 μ s e) 57.7 μ s f) >0.8 msd) 3.55 μ s Figure 3: Simulation visualizations of the formation process of an oblate ( R = 0 . ) ne-matic droplet from an initially isotropic (disordered) state. Hyperstreamlines colored by themagnitude of the uniaxial nematic scalar order parameter S are used to visualize nematicorientation (alignment tensor) and isosurfaces indicate nanoscale defect “core” regions.However, prolate nematic droplets behave differently. While prolate spheroids can alsobe generated by extruding an ellipse, the homeotropic surface anchoring conditions distortthe symmetric nature of the system. Figure 4 shows the formation dynamics of a prolate9ematic droplet, which is found to exhibit the same general regimes as the oblate droplet:free growth (Figure 4a), defect formation (Figures 4b–d), and bulk relaxation (Figures 4e,f).Despite being topologically equivalent to the oblate droplet, the defect formation mech-anism for a prolate droplet is substantially more complex. First, a pair of +1 point defect-like structures form as the high-curvature focal regions impinge (Figure 4b). These struc-tures are not true point defects in that the nematic phase within the droplet is not fully-formed. The defect formation mechanism proceeds through the continued impingement ofthe isotropic/nematic interface along the droplet equator. This results in the point-like de-fects growing into the center of the droplet where they impinge to form a high-energy +1 disclination line. As expected based on past two-dimensional simulation results anddefect energy scaling analysis, this defect line then splits into a + disclination loop forthe prolate and spherical droplet (not shown) cases. Notably, the simulations predict thedynamic mechanism through which this transition occurs. Figures 4c,d show that there is adegeneracy in the direction in which the +1 disclination line splits, which results in this split-ting direction varying along its length. The resulting + disclination loop then undergoesan elastic relaxation process driven by defect line tension, bending, and torsion.Figure 5 shows the uniaxial S and biaxial P nematic order parameters in the vicinityof the central region of the prolate droplet during the disclination splitting process. Thisprocess is similar to disclination line-loop dynamics observed by Shams and Rey, forwhich they developed a nematic elastica model for defect dynamics which captures linetension and bending of disclinations. In the presently observed defect splitting process, linetorsion, in addition to tension and bending, would need to be accounted for which couldbe accomplished through incorporating higher order terms in the nematic elastica model.Figures 5a–b show the formation of an unstable +1 disclination line originating from thejoining of a pair of +1 disclination lines growing into the unstable isotropic center of theprolate droplet.Figure 5b shows that the central region of the fully-formed +1 disclination line is uni-10 ) 0.71 μ s b) 3.48 μ s c) 6.89 μ s e) 32.5 μ s f) >1 msd) 11.0 μ s Figure 4: Simulation visualizations of the formation process of a prolate ( R = 2 ) nematicdroplet from an initially isotropic (disordered) state. Hyperstreamlines colored by the mag-nitude of the uniaxial nematic scalar order parameter S are used to visualize nematic orien-tation (alignment tensor) and isosurfaces indicate nanoscale defect “core” regions.axial, in agreement with past theoretical predictions. Figure 5c shows the initial distorted + disclination loop immediately following the splitting process. The loop has significantbending and torsion resulting from the degeneracy in the splitting process. It eventuallyrelaxes into a loop with no torsion (Figure 5d) where the central region of the droplet iswell-aligned with little distortion of the nematic director.Finally, following the complex defect formation regime, the bulk relaxation regime isobserved where the fully-formed nematic texture of the droplet relaxes through bulk reorien-tation and expansion of the disclination loop. Comparing the equilibrium nematic textures ofthe oblate (Figure 3f) and prolate droplets (Figure 4f), the oblate droplet exhibits a relativelyuniform nematic texture due to the surface exhibiting commensurate anchoring conditionswith the bulk elasticity. The prolate droplet exhibits a more non-uniform texture, and ismore similar to the radial-like textures often observed in spherical nematic droplets.11igure 5: Plot of uniaxial S and biaxial P nematic order parameters versus position alongthe major axis of the R = 2 droplet (illustrated in red) showing the progression of +1 disclination formation and splitting process. EXTERNAL FIELD-DRIVEN REORIENTATION ANDRELAXATION
Electric-field driven reorientation of nematic droplets is a key process in the operation ofPDLC-based technology. Past experimental research has shown that droplet shape has asignificant effect on the electro-optical switching process and can result in shorter switchingtimes.
Subsequently, simulations were performed for both oblate and prolate spheroidsusing the equilibrium states resulting from the formation process (Figures 3f and 4f, respec-tively). Electric fields ranging from −
14 V µ m − were applied in the direction parallel tothe major axis of the droplets ( x -axis), which, for the case of film stretching, correspondsto the direction orthogonal to the optical axis of the droplet at equilibrium. Since 5CB isa positive dielectric anisotropy LC, nematic orientation parallel to the electric field is ener-getically favored. Thus, imposition of the electric field orthogonal to the optical axis resultsin the maximum amount of field-induced reorientation, leading to more complex and inter-12sting dynamics. This corresponds to an in-plane switching mode that has been exploredexperimentally for PDLC-based devices. Two different field-switching regimes were observed depending on the magnitude of theelectric field, corresponding to a Fredericks-like transition. For electric fields strengths E below a critical value E c , the nematic texture changes only slightly without undergoing bulkreorientation in the field direction. In contrast, for E above E c , a complex reorientationprocess occurs with defect dynamics and a reorientation of the nematic texture to a field-aligned state. Overall, a general sequence of three dynamic regimes, consistent with nematiccapillaries, can be identified during this process, consisting of: Regime I.
Bulk growth and recession , involving growth of the field-aligned focal regionsand recession of the misaligned central region;
Regime II-A.
Disclination and bulk rotation , involving rotation of the disclination looporthogonal to the field direction; and
Regime II-B.
Bulk relaxation , involving expansion of the disclination loop until the forcefrom the applied field equilibrates with the elastic and anchoring forces in the system.In all cases it was observed that upon release of the electric field, the nematic texture wasrestored to the initial equilibrium texture resulting from the earlier formation process.Figure 6 shows simulation results of the field-driven switching dynamics of a R = 0 . oblate nematic droplet where E > E c . Initially, the disclination loop contracts along the x -axis (Figures 6b–c, dynamic regime I) as the field-aligned regions grow. This process endsonce the defect loop is “compressed” sufficiently into an elliptic shape such that the elasticenergy penalty resulting further shape dynamics of the defect loop approaches that of theapplied field. For oblate droplet simulations where E < E c (not shown), dynamic regime Iwas the only dynamic regime observed.Figures 6d–e show the disclination/bulk rotation regime that follows. Unlike the dynamicmechanism for field-switching of nematic capillaries, the rotation of the loop is accompanied13y both expansion of the loop and bulk rotation of the nematic director throughout thedroplet. This corresponds to a combination of dynamic regimes II-A and II-B. a) 0.10 μ s b) 107 μ s c) 312 μ s e) 487 μ s f) >1 msd) 431 μ s Figure 6: (a-f) Simulation visualizations of the electric field-switching process for E =14 V µ m − > E c applied along the x -axis of an oblate ( R = 0 . ) nematic droplet start-ing from (a) the equilibrium texture (following formation) and proceeding to the (f) thefield-driven equilibrium texture. Hyperstreamlines colored by the magnitude of the uniax-ial nematic scalar order parameter S are used to visualize nematic orientation (alignmenttensor) and isosurfaces indicate nanoscale defect “core” regions.Figure 7 shows the field-driven switching dynamics of a R = 2 prolate droplet where E > E c . The dynamic regimes observed here are more similar to those for nematic capillariesthan for oblate spheroids. In particular, the transition between dynamic regime II-A and II-Bis more distinct. This result can be attributed to the difference in disclination loop structurebetween the two cases, which is imposed by their geometries. For the prolate droplet, asthe size of the disclination loop decreases following application of the electric field, the loopbecomes circular and its size is nanoscale. In contrast, the oblate droplet disclination looptransitions from circular to elliptic after application of the field and the major axis of theelliptic loop maintains the micron-scale size of the overall droplet. Next, dynamic regimeII-A proceeds (Figures 7c–d) with a minimal increase in the defect loop diameter, unlikein the oblate case. Following this, dynamic regime II-B is observed which the disclinationloop diameter transitions from nanoscale to micron-scale, corresponding to the length scale14mposed by the droplet geometry. a) 0.10 μ s b) 26.7 μ s c) 90.5 μ s e) 166 μ s f) >1 msd) 127 μ s Figure 7: (a-f) Simulation visualizations of the electric field-switching process for E =14 V µ m − > E c applied along the x -axis of a prolate ( R = 2 ) nematic droplet starting from(a) the equilibrium texture (following formation) and proceeding to the (f) the field-drivenequilibrium texture. Hyperstreamlines colored by the magnitude of the uniaxial nematicscalar order parameter S are used to visualize nematic orientation (alignment tensor) andisosurfaces indicate nanoscale defect “core” regions.Upon release of the external field, the nematic texture at equilibrium while the field wasapplied is now a high-energy state. Relaxation of the texture back to the original equilibriumstate is due to a so-called “restoring” force which arises from a combination of confinementgeometry and surface anchoring conditions. Figures 8–9 show simulation results of thesedynamic mechanisms for R = 0 . oblate and R = 2 prolate droplets, respectively. Here,the relaxation process is, qualitatively, the reverse of the field-on process. One significantdifference was observed for the oblate droplet, in which the shape of the disclination loopduring the relaxation process is different than that for the field-on case.Focusing on the oblate droplet, the disclination loop shape during field-on conditions(Figure 6c) is elliptic while during release conditions (Figure 8b) it is circular. For the field-on case, the elliptic disclination loop has a minor axis parallel to the field direction. Thiselliptic shape is initially driven by the growth of the field-aligned regions and recession of the15naligned central region within the droplet. As dynamic regime II-A proceeds, the ellipticcharacter of the disclination loop is enhanced due to its proximity to the droplet’s ellipticcross-section. For the release case, the disclination loop is circular at the beginning of therotation regime, and continues to maintain this shape throughout the rotation process. Asthe disclination loop recedes from the elliptic part of the nematic/solid interface, it contin-uously transitions toward a state of minimum mean curvature which results in a circularshape. As the loop rotates, this circular character of disclination loop is enhanced due to itsproximity to a circular cross-section of the nematic/solid interface. Additionally, unlike inthe field-driven case there is a distinct transition from dynamic regime II-A (Figures 8b-d)to the dynamic regime II-B (Figures 8e–f). a) 0.10 μ s b) 134 μ s c) 208 μ s e) 411 μ s f) >1 msd) 241 μ s Figure 8: (a-f) Simulation visualizations of the field-off relaxation process after applying afield E = 14 V µ m − > E c along the x -axis of an oblate ( R = 0 . ) nematic droplet startingfrom (a) the field-on equilibrium texture and proceeding to the (f) the field-off equilibriumtexture. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar orderparameter S are used to visualize nematic orientation (alignment tensor) and isosurfacesindicate nanoscale defect “core” regions.Figure 9 shows simulation results of the relaxation of a R = 2 prolate droplet. Forthis case, the difference in the disclination loop shape between the field-on (Figure 7d) andrelease (Figure 9c) is more subtle, but similar to the oblate case. For the field-on case, shownin Figure 7d, the disclination loop is slightly elliptic with minor axis parallel to the field16irection. As it rotates the disclination loop transitions to a circular shape resulting from itsproximity to a circular cross-section of the nematic/solid interface, shown in Figure 7f. Uponrelaxation of the field (Figure 9c), the disclination loop adopts a circular shape throughoutthe rotation process which is followed by transition to an elliptic shape due to its proximityto the elliptic cross-section of the droplet. a) 0.10 μ s b) 41.1 μ s c) 107 μ s e) 657 μ s f) >10 msd) 207 μ s Figure 9: (a-f) Simulation visualizations of the field-off relaxation process after applying afield E = 14 V µ m − > E c along the x -axis of a prolate ( R = 2 ) nematic droplet startingfrom (a) the field-on equilibrium texture and proceeding to the (f) the field-off equilibriumtexture. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar orderparameter S are used to visualize nematic orientation (alignment tensor) and isosurfacesindicate nanoscale defect “core” regions. DROPLET-SCALE DYNAMICS
In order to analyze the external field-switching and relaxation dynamics quantitatively, avolume-averaged droplet uniaxial scalar order parameter S d and director n d can be deter-mined through eigendecomposition of the volume-averaged alignment tensor Q d : Q d,ij = V − (cid:90) V Q ij dV (4)17here V is the volume of the domain. The droplet scalar order parameter S d is analogousto nematic scalar order parameter S in Equation 5, where S d → corresponds to a nematicdroplet with no preferred alignment and S d → corresponds to uniform aligned along n d .The case where S d → may correspond to two possible states of the nematic droplet: afully isotropic (disordered) state or a symmetrically radial nematic texture. In this work, allanalysis is performed for fully-formed nematic droplets and thus S d → corresponds to thelatter state. From a general optical applications perspective, lower values of S d correspondto nematic droplets which scatter light, while higher values of S d correspond to nematicdroplets with improved optical transparency. Figure 10 shows the evolution of S d and n d for the field-switching and relaxation sim-ulations of oblate (Figures 6 and 8), prolate (Figures 7 and 9), and (not shown) sphericalnematic droplets which were presented in the previous section. For the field-on dynamics,evolution of S d for E < E c exhibits a single bulk growth/recession regime. For the E > E c cases, however, the evolution of S d for spherical and prolate droplets is found to involvethree dynamic regimes, while the oblate droplet involves only two. These quantitative find-ings support the qualitative observations from the previous section, where for oblate dropletsdynamic regimes II-A and II-B occur simultaneously, whereas for the prolate droplets theyare distinct. Furthermore, the spherical droplet field-switching dynamics are found to becomparable to that of the prolate droplet, except for that the field-alignment of the dropletdirector n d occurs very early in the field-on process for the spherical case.These trends also indicate that dynamic regime I for spherical and prolate droplets occursin two stages, unlike for the oblate droplet case. In Figures 10a and 10c (spherical and prolatedroplets), S d initially decreases during regime I, followed by a rotation of the droplet director n d and a simultaneous increase in S d . This is more pronounced for the prolate droplet thanthe spherical droplet. In Figure 10b, S d does not exhibit this nonmonotonic evolution duringdynamic regime I.The difference in field-on dynamics between prolate/spherical and oblate droplets may18e explained through qualitative comparison of the disclination dynamics of the prolate andoblate droplets during the initial bulk growth/recession regime. Focusing on the prolatedroplet case, during the first stage of this dynamic regime, the field-aligned regions of thedroplet grow while simultaneously the disclination loop diameter decreases. The decrease isdisclination loop diameter does not initially result in interaction of adjacent regions of theloop, which would result in a high-energy elastic interaction of the nanoscale defect “core”regions. During the second stage of this regime, the droplet scalar order parameter evolutiondecreases resulting from an overall slowing of the reorientation dynamics. This is due to aslowing down of the macroscale field-alignment in the bulk domain as the disclination loopdiameter approaches a critical value where adjacent defect core regions interact. Followingthis, the domain transitions to dynamic regime II-A which occurs rapidly followed by a longtimescale regime II-B. For the oblate droplet case, the dynamic regime I is not observedto have two stages, implying different dynamics of the disclination loop during this regime.Referring back to Figure 6, as the disclination loop reduces in size, it forms an elliptic shapewhich results in the focal segments of the loop having high curvature. These high-energyregions preclude the possibility of adjacent disclination cores approach each other, and thusdynamic regime I for the oblate droplet does not involve interaction of adjacent defect coreregions of the loop, unlike in the prolate case.As mentioned in the previous section, the field-off/release dynamics, also shown in Figure10, are inherently different from the field-on dynamics due to the absence of an external field.The restoring force resulting from the frustration of the field-on nematic texture with respectto the combination of the geometry, surface anchoring conditions, and nematic elasticity issubstantially different for the spherical droplet case compared to both the oblate and prolatedroplets in that there is only a very weakly imposed droplet director due to the geometrybeing essentially isometric. Thus the release dynamics for this droplet involve only a bulkrelaxation of the nematic texture. As was described in the previous section, oblate andprolate droplets exhibit dynamics qualitatively similar to the field-on case, except in reverse.19 .0 0.2 0.4 0.6 0.8 1.0 t (ms) S d t (ms) V /µ m V /µ m V /µ m V /µ m V /µ m V /µ m V /µ m (a) t (ms) S d a b c d e t (ms) a b c d e (b) t (ms) S d a b c d e t (ms) a b c d e (c) Figure 10: Droplet-scale order evolution plots for (a) R ≈ spherical (not shown), (b) R = 0 . oblate (Figures 6 and 8), and (c) R = 2 prolate (Figures 7 and 9) nematic dropletsresulting from application (left column) and release (right column) of electric fields withstrengths ranging from E = 2 − µ m − . Curves represent the droplet scalar order param-eter S d with solid/dotted lines corresponding to the droplet director n d orthogonal/parallelto the electric field direction. Vertical bars with labels indicate the simulation time at whichthe corresponding simulation snapshots were taken for the oblate (Figures 6 and 8) andprolate (Figures 7 and 9) switching dynamics.20nalysis of the droplet order parameter evolution for the field-off case shown in Figure 10indicates that the dynamics are qualitatively similar, but both prolate and oblate dropletsexhibit only two dynamic regimes with dynamic regimes II-A and II-B combined.Equilibrium droplet scalar order parameter values and response times for a range ofelectric field strengths were also determined from simulations, which are of interest for PDLC-based devices and other technological applications. Figure 11 shows simulation results ofdroplet order parameter S d at equilibrium, field-on response times τ on , and field-off responsetimes τ off for oblate, spherical, and prolate droplets for a range of electric field strengths.Measurements for τ on and τ off were estimated based on the time for S d to reach steady-statein order to be more comparable to experimental measurements, which are based on changesin optical film transmission. As shown in Figure 11, equilibrium S d values varied significantly depending on bothdroplet shape and field strength. Spherical droplets, which exhibit the lowest E c , lack astrongly preferred droplet director, meaning that even relatively weak electric fields are ef-fective for field-aligning the nematic texture. Furthermore, the droplet order parameter S d increases monotonically with increasing field strength. In contrast, for both oblate and pro-late droplets, S d is nonmonotonic with respect to electric field strength, initially decreasingfor E < E c and then increasing as E > E c . For the cases where E < E c , oblate and prolatedroplet responses do not involve reorientation of the droplet director. Instead, S d decreasescorresponding to decreased nematic alignment about the intrinsic droplet director resultingfrom geometry and anchoring conditions. For the cases where E > E c , full droplet directorreorientation occurs in both prolate and oblate droplets, but to differing degrees. The criti-cal field strength for the oblate droplet reorientation is relatively high ( –
12 V µ m − ), dueto the large portion of the nematic/solid interface promoting alignment along the intrinsicdroplet director. In contrast, the critical field strength for the prolate droplet rorientation isrelatively low ( – µ m − ) for the opposite reason. Once reorientation occurs, the dropletscalar order parameter increases linearly with E as the electric field influence overcomes21urface anchoring forces. S d R = 0.5 R = 1.05 R = 2.0 − τ o n ( m s ) E (V µ m − ) − τ o ff ( m s ) Figure 11: (top) Equilibrium droplet scalar order parameter S d versus electric field strength.(middle) Response times to reach field-driven equilibrium τ on versus electric field strength.(bottom) Response times to reach field-release equilibrium τ off versus electric field strength.Unfilled points correspond to droplet textures that are not field-aligned, while filled pointscorrespond to those which are.The results for field-on response times for both prolate and spherical droplets are compa-rable to experimental results for spherical droplets under similar conditions ( ≈ ). Bothfield-on and field-off response times for oblate droplets are significantly lower, on the orderof . , which is due to their negligible change in texture in response to an applied field, asindicated by very little change in the droplet order parameter between field-on and field-offstates. However, simulation results for field-off response times for both prolate and spherical22roplets are somewhat lower than experimental results, ≈
10 ms versus ≈
30 ms , respec-tively. This can be attributed to the significantly larger length scale of nematic dropletsstudied experimentally, − µ m , which results in a decreased ratio of restoring to viscousforces, slowing down droplet dynamics. CONCLUSIONS
In this work continuum simulations were performed in order to predict the dynamic mech-anisms involved in the formation, field switching, and relaxation of nematic LC dropletswith varying spheroidal geometry. The presented simulation results have both fundamentaland technological relevance in that formation and field-switching dynamic mechanisms werepreviously poorly understood and of significant relevance to the performance of PDLC-basedoptical functional materials. The key feature of these nematic domains is the presence ofnanoscale defect structures which contribute to the dynamics of the micron-scale domain incomplex ways.Simulations of formation dynamics from an initially unstable isotropic phase predict in-trinsically different defect formation mechanisms in anisometric droplets (oblate and prolate)compared to spherical ones. Defect loop structures, which are topologically imposed by do-main geometry and anchoring conditions, are observed to form through the combination ofdefect shedding and splitting dynamic mechanisms. A degeneracy in the splitting of a +1 disclination line structure into a + disclination loop is predicted to result in an “unraveling”of the nanoscale loop structure, similar to the nematic elastica behavior observed in nematiccapillaries.Simulations of electric field-driven reorientation and relaxation dynamics reveal the mech-anisms of the reorientation process, which are highly dependent on domain shape and ex-ternal field strength. Both oblate and prolate spheroidal droplets are found to have quali-tatively similar dynamic reorientation mechanisms, with the critical (reorientation) electric23eld strength E c being significantly higher than for spherical droplets. For electric fields E < E c , the nematic texture of anisometric droplets becomes increasingly frustrated be-tween the orientation imposed by the external field and that preferred by the geometry andanchoring conditions. This corresponds to an optical state that is increasingly light scat-tering. For electric fields E > E c , the nematic texture transitions to a field-aligned statethrough a series of complex and distinct dynamic mechanisms involving both micron-scalereorientation and nanoscale defect dynamics.In summary, the presented results provide both qualitative and quantitative insight intothe dynamics of nematic spheroids with resolution of the nanoscale length and timescales in-herent to LC domains which include defects. These simulations include the dynamic regimesrelevant to PDLC-based devices and thus could be used to guide the design and optimiza-tion of their performance as optical functional materials. Additionally, these results providefundamental insight into the effects of nanoscale defect dynamics on confined LC domains. METHODS
Nematic Reorientation Dynamics Model.
Simulations are performed using the Landau–de Gennes continuum model for the nematic phase, which uses an alignment tensor, or Q -tensor, order parameter to quantify nematic order: Q ij = S ( n i n j − δ ij ) + P ( m i m j − l i l j ) (5)where S and P are uniaxial and biaxial nematic scalar order parameters, n i is the nematicdirector, and m i , l i are the biaxial orientation vectors. The Helmholtz free energy density of24he domain is: f b − f iso = 12 a ( Q ij Q ji ) + 13 b ( Q ij Q jk Q ki ) + 14 c ( Q ij Q ji ) + 12 L ( ∂ i Q jk ∂ i Q kj ) + 12 L ( ∂ i Q ij ∂ k Q kj ) + 12 L ( Q ij ∂ i Q kl ∂ j Q kl ) + 12 L ( ∂ k Q ij ∂ j Q ik ) − (cid:15) ◦ π (cid:20)(cid:18) (cid:15) (cid:107) + 2 (cid:15) ⊥ δ ij + ( (cid:15) (cid:107) − (cid:15) ⊥ ) Q ij (cid:19)(cid:21) E j E i (6)where f iso is the free energy of the isotropic phase, which is assumed to be constant. Allthree second-order terms in Q ij are used, while the third-order L term is used in order toresolve splay-bend anisotropy, and L is used to quantify saddle-splay elasticity. The L term is also referred to as L or L , depending on the reference source. Additionally, a contribution to the free energy from the solid/nematic interface corre-sponding to homeotropic surface anchoring is used: f s = αk i Q ij k j (7)where k i is the surface unit normal and α is the surface anchoring strength. A value of α = − . × − J / m was used, which is corresponds moderately strong surface anchoringwith a surface extrapolation length ξ s = L α ≈
100 nm) . The total free energy of thedomain includes both bulk and surface contributions: F [ Q ij ] = (cid:90) V f b dV + (cid:90) S f s dS. (8)Nematic reorientation dynamics are modelled using the time-dependent Ginzburg-Landaumodel: ∂Q ij ∂t = − Γ (cid:20) δFδQ ij (cid:21) ST (9)where Γ = µ − r where µ r is the rotational viscosity of the nematic phase, and [] ST indicatesthe symmetric-traceless component. 25umerical solution of the resulting system of nonlinear partial differential equations wasperformed using the finite element method with the software package FEniCS on meshesof spheroid geometries or “droplets” with aspect ratio R = ca , where c and a correspond tothe lengths of the major and minor axes of the spheroid. Droplet volume was maintainedconstant for each geometry and set to be equivalent to the volume of a perfectly sphericaldroplet with diameter
500 nm with mesh spacing less than the nematic coherence length inorder to accurately resolve the defect structure.The governing equations were nondimensionalized before solving, which gives rise to atime scale t s : ˜ t = tt s , t s = µ r a T ni (10)An estimate for t s can be calculated using the parameters given in Table 1. Simulation Method Conditions.
The model parameters used approximate the liquidcrystal 4-cyano-4’-pentylbiphenyl (5CB) and are given in Table 1. Values were chosen ac-cording to experimental data for a temperature of T = 307 K . Table 1: Material parameters for 5CB. T ni .
35 K a . × J / m K b . × J / m c . × J / m L . × − J / m L . × − J / m L . × − J / m L . × − J / m (cid:15) (cid:107) . (relative) (cid:15) ⊥ . (relative) µ r .
055 Ns / m The values of the elastic constants L to L were derived from the Frank elasticconstants k = 2 . × − J / m , k = 1 . × − J / m , and k = 3 . × − J / m , whichwere determined using known empirical models. The saddle-splay constant k , which26as been difficult for researchers to measure consistently for 5CB, was chosen such thatthe elastic energy penalty term L remained positive ( k = 0 . k ). However, this is nota strict condition and negative L is possible as long as the Frank elastic constants satisfyEricksen’s inequalities. The simulation for determining field-off equilibrium droplet textures was initialized usinga uniaxial boundary layer with scalar order parameter S = S eq . The boundary is alignedperpendicular to the surface in accordance with the surface boundary conditions ref. 20.Simulations of electric field switching were conducted using these equilibrium textures asinitial conditions for each field strength studied. Visualization.
Three-dimensional visualizations of the droplets were generated using hy-perstreamline seeding of the Q-tensor field. Hyperstreamlines are used to represent theorientational order tensor Q ij ( x, t ) . These structures are an extension of streamlines andorient along the director field n i ( x, t ) , with varying width in order to visualize the additionaldegrees of freedom associated with tensorial data. Hyperstreamlines are colored accordingto the scalar order parameter S . Disinclination lines are indicated the blue contour surfaceswhich were computed for a fixed biaxial scalar order parameter P > . Acknowledgement
This work was supported by the Natural Sciences and Engineering Research Council ofCanada and Compute Canada.
Supporting Information Available
The following files are available free of charge. • sphere_formation.mpg : Video of formation dynamics for a R ≈ spherical dropletsimulation. Hyperstreamlines colored by the magnitude of the uniaxial nematic scalar27rder parameter S are used to visualize nematic orientation (alignment tensor) andisosurfaces indicate nanoscale defect “core” regions (refer to Methods section). Time isgiven as a dimensionless quantity (see eqn. 10). • oblate_formation.mpg : Video of formation dynamics for a R = 0 . oblate dropletsimulation, corresponding to Figure 3. • prolate_formation.mpg : Video of formation dynamics for a R = 2 prolate dropletsimulation, corresponding to Figure 4. • sphere_fieldon_14Vum.mpg : Video of field-switching dynamics of a R ≈ sphericaldroplet for E = 14 V µ m − > E c applied along the x -axis. • oblate_fieldon_14Vum.mpg : Video of field-switching dynamics of a R = 0 . oblatedroplet for E = 14 V µ m − > E c applied along the x -axis, corresponding to Figure 6. • prolate_fieldon_14Vum.mpg : Video of field-switching dynamics of a R = 2 prolatedroplet for E = 14 V µ m − > E c applied along the x -axis, corresponding to Figure 7. • sphere_fieldrelease_from_14Vum.mpg : Video of field-off relaxation dynamics for a R ≈ spherical droplet simulation. • oblate_fieldrelease_from_14Vum.mpg : Video of field-off relaxation dynamics for a R =0 . oblate droplet simulation, corresponding to Figure 8. • prolate_fieldrelease_from_14Vum.mpg : Video of field-off relaxation dynamics for a R = 2 prolate droplet simulation, corresponding to Figure 9. References
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