Particle-resolved topological defects of smectic colloidal liquid crystals in extreme confinement
René Wittmann, Louis B. G. Cortes, Hartmut Löwen, Dirk G. A. L. Aarts
PParticle-resolved topological defects of smectic colloidal liquid crystals in extremeconfinement
Ren´e Wittmann,
1, 4, 5
Louis B. G. Cortes,
2, 3, 4
Hartmut L¨owen,
1, 5 and Dirk G. A. L. Aarts
2, 5 Institut f¨ur Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universit¨at D¨usseldorf,D-40225 D¨usseldorf, Germany Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford,South Parks Road, Oxford OX1 3QZ, United Kingdom School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA These two authors contributed equally: Ren´e Wittmann and Louis B. G. Cortes Correspondence and requests for materials should be addressed to R.W. (email: [email protected]),to H.L. (email: [email protected]) or to D.G.A.L.A. (email: [email protected])
Confined samples of liquid crystals are characterized by a variety of topological defects and can be exposedto external constraints such as extreme confinements with nontrivial topology. Here we explore the intrinsicstructure of smectic colloidal layers dictated by the interplay between entropy and an imposed externaltopology. Considering an annular confinement as a basic example, a plethora of competing states is foundwith nontrivial defect structures ranging from laminar states to multiple smectic domains and arrays ofedge dislocations which we refer to as Shubnikov states in formal analogy to the characteristic of type-IIsuperconductors. Our particle-resolved results, gained by a combination of real-space microscopy of thermalcolloidal rods and fundamental-measure-based density functional theory of hard anisotropic bodies, agree ona quantitative level.
I. INTRODUCTION
Liquid crystals consist of particles that possess bothtranslational and orientational degrees of freedom andexhibit a wealth of mesophases with partial orientationalor positional order such as nematic, smectic and colum-nar states . As such, these phases are highly suscepti-ble to external topological and geometrical influences .This opens a fascinating new research realm on the in-ternal structural response to such externally imposedconstraints with various highly relevant applications intechnology and material science . While these perspec-tives have been extensively exploited for spatially ho-mogeneous mesophases, such as nematics, there is muchyet undisclosed potential stemming from the complex in-terplay between external constraints and internal orderemerging in more complex mesophases, such as the lay-ered smectic phase.One of the main research goals in liquid crystals isfocused on topological defects. These not only repre-sent fingerprints of singularities and discontinuities inthe ordering but also naturally link topology to con-densed matter physics. The general importance of de-fects of liquid crystals is further fueled by the possibil-ity to directly visualize the inherent orientational frus-tration on the macroscopic scale through the schlierentexture between two crossed polarizers . Different orien-tational defect structures have therefore been exploreda lot in the homogeneous nematic phase with re-cent digressions to active systems . Due to their si-multaneous orientational and positional ordering, defectsin the smectic phase naturally exhibit an even higherdegree of complexity. The main emphasis has beenput hitherto on the positional layering or orienta-tional textures alone, as well as, on coarse-grained calculations and computer simulation of par-ticle models .Here we approach the defect structure of smectic liq-uid crystals from the most fundamental particle-resolvedscale and quantify both their positional and orienta-tional disorder simultaneously in theory and experim-nent. In doing so we study two-dimensional smecticscomposed of lyotropic colloidal rods whose size enablesa direct observation , while they have the advan-tage over granulates that they are fully thermallyequilibrated. The colloidal samples are exposed to ex-treme confinements possessing an annular shape and di-mensions of a few particle lengths. This combinationof curved geometry and nontrivial topology is triggeringcertain characteristic defect patterns. In our study weuniquely combine real-space microscopy of colloidal sam-ples with modern first-principles density functional the-ory (DFT) based on geometric fundamental measures which provides a full microscopic description of inho-mogeneous and orientationally disordered smectics .A plethora of different states with characteristic defecttopologies is observed in perfect agreement between the-ory and experiment up to the microscopic nuances in thedefect shape and wall alignment.Our study explores the intriguing competition betweenthe internal liquid crystal properties and the extrinsictopological and geometrical constraints. In annular con-finement this gives rise to three essential types of smecticdefect configurations. Each of these defects is characteris-tic for a unique state with a discrete rotational symmetryin the orientation field, which we refer to as follows. Inthe laminar states, the smectic layers in a large defect-free domain (bounded by two parallel disclination lines)resemble the flow lines around a circular obstacle (in-clusion). The domain states are governed by individual1 a r X i v : . [ c ond - m a t . s o f t ] F e b xperiment Density Functional Theory (DFT) coverslipTeflon tuberod solutionannular cavity a) R in R out b = R in / R out b) L DW c) p = L / D = λ d) d e n s it y & d i r ec t o r . . . . . . . .
12 1 .
14 1 . · − layer spacing λ / W F / F − λ = λ stable state e) FIG. 1.
Overview of experimental and theoretical methods. a)
Schematic illustration of the experimental cell. b) Particle-resolved bright-field microscopy snapshot imaged in direct vicinity of the cavity bottom wall. The annular geometry isdetermined by the outer radius R out and the inclusion size ratio b = R in /R out with the inner radius R in . c) A discorectangleof rectangular length L , circular diameter D , total length W = L + D and area a = LD + D π/ d) Smectic bulk phase. Left: experimental snapshot showing individual particles. Right: theoretical density profile ρ ( r , φ ) represented by a heat map of the orientationally averaged center-of-mass density ¯ ρ = a π (cid:82) π d φ ρ ( r , φ ) and green arrowsindicating the average local director orientation. The large black arrow marks the spacing λ between two layers. e) Bulksmectic free energy F as a function of the layer spacing λ , determined by density functional theory (DFT). The minimum at F = F corresponds to the optimal bulk layer spacing λ in equilibrium. smectic domains, separated by radially oriented discli-nation lines, in three sectors of the annulus. Finally,there are the Shubnikov states, named in formal anal-ogy between the typical arrays of edge dislocations andthe flux quantization in type-II superconductors. In ad-dition, we observe peculiar symmetry-breaking compos-ite states which unite different types of defects in a singlestructure. A locally adaptable layer spacing is found hereto play the key role regarding the stability and distribu-tion of defect strucutres in extreme confinement. II. RESULTSOverview.
Our complementary experimental andtheoretical strategies (see appendices A and B for moredetails) to study smectic liquid crystals on the particlescale are illustrated in Fig. 1. Experimentally, we di-rectly observe fully equilibrated silica rods at the bottomof customized confinement chambers (Fig. 1a) throughparticle-resolved bright-field microscopy (Fig. 1b). Onthe theoretical side, we analyze the microscopic densityprofiles, obtained from a free minimization of our geo-metrical DFT for two-dimensional hard discorectangles(Fig. 1c). We strive a direct comparison of experimentalsnapshots and theoretical density profiles, as illustratedfor the bulk case in Fig. 1d, where the DFT is minimized by the optimal layer spacing λ (Fig. 1e). To create therequired overlapping parameter space, we employ bothprecise lithography to create robust confinement cham-bers with dimensions of only a few rodlengths and anefficient hard-rod density functional to tackle these sys-tem sizes. The theoretical aspect ratio p = 10 is chosento closely match the effective value p eff = 10 . c h e m a t i c D e n s i t y F un c t i o n a l T h e o r y E x p er i m e n t d e f ec t s l o c a l p a c k i n g d e n s i t y & d i rec t o r o r d er p a r a m e t er b r i g h t- fi e l d i m ag e & d e f ec t s m re l a t i v e o r i e n t a t i o n o r d er p a r a m e t er e dg e d i s l o ca ti ond i s c li n a ti on li n e Bridge ( B ) Laminar ( L ) Domain ( D ) Shubnikov ( S )Composite (cid:0) C LS (cid:1) □□□ □□ □□ □□□ □□ □□ □□ □ □□ □□□□□□□□□ □□□ . . . . . . π π − π − π . . . . . . . . . . . . . . . . . . . . . χ = b = χ =
0, 0 < b < b FIG. 2.
Defects structures in smectic colloidal liquid crystals.
The columns represent the different states (as labeled),arranged from left to right by their occurrence in circular (Euler characteristic χ = 1) and annular ( χ = 0) confinement withincreasing inclusion size ratio b = R in /R out . First row: idealized schematic representation of the mesoscopic arrangement ofsmectic layers (solid gray lines ) and defects (red squares and lines as labeled).
Rows 2-4: local packing fractionwith marked defects, density profiles with orientational director field (as in Fig. 1d) and local order parameter from theory.
Rows 5-7: typical particle-resolved snapshots with defects or color denoting the orientation relative to the nearest wall andlocal order parameter from experiment. ●● R in / R out R o u t / L Exp. structures
LDSC LD C DS C LS phenomeno-logical L - S transition phenomenological L - D transition(metastable)DFT L - S transition inclusion size ratio b R ou t / L FIG. 3.
Topological state diagram.
Shown are the stablestates for different outer radii R out and inclusion size ratios b = R in /R out . The large pie charts indicate the percentageat which each state occurs in the experiment according tothe legend. The small circles denote the theoretical laminar–Shubnikov ( LS ) transition (the numerical error is of the orderof the symbol size; filled circles indicate that no laminar statecan exist for larger inclusions). The vertical lines represent apossible scenario (see labels) predicted by a phenomenologicalmodel based on defect energies. Smectic states.
Figure 2 illustrates our central ob-servation of different competing smectic states, each cop-ing with the externally imposed constraints in a distinctway. For each experimental structure in a given geom-etry, we find a perfectly matching theoretical densityprofile. This depicted structural variety results from aconfinement with curved walls and a nontrivial topology,represented here by an annular cavity (Fig. 1b) with Eu-ler characteristic χ = 0. The typical structure of thesestates is determined by the arrangement of the smecticlayers (see microscopic details below) and the shape oftopological defects with total charge of Q = χ = 0 (seetopological details below). A detailed classification of theobserved smectic states is given in appendix C.To demonstrate the plain behavior of smectics insimply-connected domains, we first remove the inclusionand consider circular confinement. In this reference case,the only structure observed is the bridge state ( B ). It ischaracterized by a large domain of parallel layers span-ning the system and frustration of the orientational orderat the domain boundary. The latter is either due to theformation of two anti-radial disclination lines or, ifthe layers are directly adjacent to the outer wall, dueto homeotropic (perpendicular to the wall) alignment ,contrasting the preferred planar alignment at hard walls.When adding a small inclusion, the layer arrangementresembles a laminar flow field around an obstacle, whichwe refer to as the laminar state ( L ). The structure as-sociated with the large bridging domain is identical tothat of the bridge state, but the internal boundary mayadditionally disconnect some of the central layers and in-duce orientational frustration in the two tangential lay- a) d)e) b)c) f) density & director . . . . . . . bright-field image5 µm LDS
FIG. 4.
Structural details on the particle scale.
Ac-cording observations from theory (left, density plots as inFig. 1d) and experiment (right): a) bent layers adjacent tothe inclusion and b) deformed layer near curved outer wall inthe laminar state ( L ), and c) gap between two domains closeto the inclusion in the domain state ( D ) and d) planar align-ment of rods at the outer wall e) deformed layers adjacent toan edge dislocation and f ) tilted alignment of some layers atthe inclusion in the Shubnikov state ( S ). ers. The bridge state can thus be considered as a specialundeformed case of a laminar state in the limit b → D ) with three radially oriented disclinationlines, exhibiting a characteristic zig-zag pattern on theparticle scale.Following de Gennes , we refer to the smectic struc-ture at large inclusions as the Shubnikov state ( S ). It ischaracterized by layers spanning between the two discon-nected system boundaries and an array of edge disloca-tions, which stabilizes the uniform orientational bend de-formation imposed here by the confining geometry. Thisstructural response is mathematically analogous to themagnetic vortices emerging in superconductors of type IIsubject to an external magnetic field .All smectic states introduced so far possess a discreterotational symmetry. In addition, composite states ( C LD , C DS or C LS ) with two distinct regions, displaying char-acteristic order phenomena of either laminar, domain orShubnikov states emerge at inermediate inclusion sizes.A key paradigmatic example shown in Fig. 2 is thelaminar–Shubnikov composite state C LS . State diagram.
To answer the question about thestability of each state, we illustrate in Fig. 3 the proba-bility of its occurrence in our experiments. For all statepoints considered, laminar states and Shubnikov states4 ) ........ | {z } N dis .................. | {z } N con / b) . . . . R out = . LR out = . LN tot N con N dis R out λ R out − R in λ R in λ inclusion size ratio b nu m b e r o f l a y e r s FIG. 5.
Geometry dependence of the layers inthe laminar state.
We show a) a schematic definitionof connected/disconnected layer numbers N con/dis , in total N tot = N dis + N con , and b) the nonmonotonic dependenceof layer numbers (symbols at integer values) on the inclusionsize ratio b for two outer radii R out (colors as labeled). Forcomparison, the lines indicate the characteristic geometricaldimensions of the annulus divided by the bulk layer spacing λ (as labeled). clearly dominate for b < ∼ .
25 and b > ∼ .
35, respectively.This provides compelling experimental evidence for theexistence of a topological laminar–Shubnikov ( LS ) tran-sition around b ≈ .
3. The state diagram is com-plemented by the intermediate domain state and sev-eral composite states. We observe that upon shrinkingthe system for a fixed intermediate inclusion size ratio b ≈ .
3, both the laminar state and the Shubnikov statebecome less stable, while the probability to find the do-main state drops for larger systems.The described state diagram can be qualitatively un-derstood in terms of a minimalistic phenomenologicalmodel, see appendix D, accounting solely for the lengthof the disclination lines ( L and D ) or the number of edgedislocations ( S ), cf. Fig. 2. Only in the domain state thelength of the disclination lines depends on the inclusionsize ratio b , such that the laminar–domain transition canbe located at a fixed b = 1 /
3, independently of the un-known defect energy δ per unit length. The energy of λ out λ in a)b) . . . λ out > λ DFTExp. λ in < λ R wall / λ λ w a ll / λ FIG. 6.
Geometry dependence of the layers in theShubnikov state.
We show a) a schematic definition ofthe local layer spacing λ in and λ out at the inner and outerwall and b) the local layer spacing λ wall as a function of therespective radius R wall of the inner (’wall’ = ’in’) or outer(’wall’ = ’out’) wall, normalized by λ to values smaller orlarger than one, respectively, in different theoretical (circles)and experimental (squares) geometries. The dashed line rep-resents a lower bound for rods packed at the inner wall witha perfectly planar alignment. the Shubnikov state also depends on the inclusion sizeratio b , which is required to estimate for the total num-ber of edge dislocations of energy u ed . The model thuspredicts an alternative laminar-Shubnikov transition, de-pending on the ratio of u ed and δ . This fit parametercan be estimated by localizing the transition at the ob-served b = 0 .
3, which gives rise to the scenario depictedin Fig. 3a, where the domain state is only metastable.To corroborate our experimental findings in full depth,we compute the free energies corresponding to the mi-croscopic density profiles as a direct measure for theirstability. Focusing on the precise localization of the LS transition, we observe in Fig. 3 a clear trend that thetransition line shifts to larger inclusions for smaller R out in the experimentally accessible range of this parameter.This agrees well with the distribution of the observedstructures in our experiments. For even more extremeconfinements with R out ≤ . L we locate the LS tran-5 isclination line& layers disclination line & director field:from nematic to smectic end-point defects& director field stretched and annihi-lated edge dislocationboundary and interiordefect at inner wallinterior and boundarydefect at outer wall q = q = − q = + →→→ q e = − q e = + q e = − q e = − q e = + q e = + □ FIG. 7.
Topological defects in confined smectics.
Theanti-radial line disclinations at the outer (top row) and inner(middle row) wall are located in the interior or attached tothe boundary. The radial line disclinations are topologicallyequivalent to edge dislocations (bottom row) and can be in-terpreted as a stretched or annihilated defect, respectively.
Left column: schematic illustration as in Fig. 2.
Mid-dle column: orientational director field (blue lines/arrows)around line defects (red) with topological charge q accordingto the drawn closed integration path (cyan circular arrow).The disclination lines can be interpreted as stretched nematicpoint defects, shown for comparison. Right column: direc-tor field at the end-point defects (violet) of disclination lineswith charges q e . Here the integration path is not closed butrather begins and ends on two sides of the disclination. sition close to the maximal inclusion size where a lami-nar state can form. Hence, the inclusion size ratio b ofthe transition becomes smaller upon further decreasing R out below that threshold. For large confinements, thetransition seems to approach the continuum limit with b ≈ . C LS is globally stable in a small but distinct re-gion around the predicted LS transition, which we canunderstand on a microscopic level. Microscopic details.
The entropically optimalequilibrium structure of each state emerges from a com-plex balance between several competing driving forceswhich aim to (i) remove all sorts of defects, (ii) achieveplanar wall alignment, (iii) minimize the deformation en-ergy and (iv) maintain the intrinsic layer spacing λ inbulk. Our particle-resolved methods naturally provide anoptimal account of points (i)-(iv) in the course of equi-libration. The quantitative agreement of the experimen-tal and theoretical density profiles, both generated bythe subtle interplay of these fundamental principles, al-lows us to unveil in Fig. 4 the characteristic microscopic d e n s it y & d i r ec t o r . . . . . . . S ∗ , D ∗ , FIG. 8.
Equilibration of a density profile initializedas a domain structure.
The sequence of density profiles(as in Fig. 1d) indicated by the arrows shows a continuousevolution from a domain state into a Shubnikov state withthe same numbers of layers. The geometrical parameters are R out = 6 . L and b = 0 . structural details of each state.From our microscopic insights, detailed below and fur-ther elaborated in appendix E, we draw the followingconclusions regarding the state diagram in Fig. 3. In-creasing the wall curvature (by decreasing R out ) increas-ingly distorts the layer spacing in the Shubnikov state,such that it becomes destabilized compared to the lam-inar state, for which the relative energy penalty arisingfrom homeotropic wall alignment decreases. The non-monotonic behavior of the resulting LS transition line isrelated to the varying compatibility of each state withthe particular confining geometry. The indirect LS tran-sition via an intermediate composite state C LS can beexplained by the increased number of possibilities, com-pared to the L and S states, to relax the geometricalconstraints.In detail, we observe that the layers and defect linessurrounding the inclusion in the observed laminar statesare typically deformed according to the shape of the in-ner wall, where the wall alignment is homeotropic, seeFig. 4a. In contrast, due to the planar alignment at theouter wall, the disclination lines end on point defects,recognizable by the modulation of the adjacent layer,see Fig. 4b. This has not been reported for straightwalls . The optimal number of layers depends non-monotonically on the geometric parameters, as shown inFig. 5.The particularly deformed microstructure of the do-main states is due to the competing angles 2 π/ π/ .
27 0 .
28 0 .
29 0 . · − inclusion size ratio b r e l a ti v e fr eee n e r gy ( F / F m − ) S , S , C LS , , C LS , , L , S , L , FIG. 9.
Stable and metastable structures for different inclusion sizes.
Shown is the free energy F relative to that F m of the global minimum for R out = 6 . L and different inclusion size ratios b close to the laminar–Shubnikov transition (redcircle). The legend depicts the theoretical density profiles (as in Fig. 1d) of different states (color) with different microscopicstructure (symbols) for b = 0 .
3. The structures with alike symbols are created by subsequently equilibrating the density forslightly smaller inclusions, where the solid lines serve as a guide to the eye and the dashed lines indicate a structural changeregarding the number of layers in contact with the inclusion. The numerical error is of the order of the symbol size. regions, shown in Fig. 4c, in which the rods try to alignwith the wall. Taking a closer look at the outer bound-ary, however, we observe in Fig. 4d some additional layersand edge dislocations between two adjacent domains, en-suring again an overall planar wall alignment.The layers in the Shubnikov states are deformed inthe vicinity of edge dislocations, see Fig. 4e. Althoughan overall planar wall alignment is generally possible, wefrequently observe in small systems that one or more lay-ers are tilted with respect to the inner wall, as in Fig. 4f.This reflects a strongly position-dependent layer spacing,as shown in Fig. 6, which further distinguishes the Shub-nikov from the laminar and domain states. In fact, thedeviations in the local layer spacing reduce the numberof point defects, such that we even observe some extremestructures without any defects for sufficiently small dis-tances between the walls.
Topological details
Having resolved the micro-scopic details of the topological defects, emerging dueto the rigidity of the smectic layers, we are in a positionto associate in Fig. 7 a topological charge q with eachoccurring disclination line. We further identify pairs ofend-point defects to these lines, which formally carry pe-culiar quarter-integer charges q e , such that q = (cid:80) q e .Then one can easily verify from the sketches in Fig. 2that, in each state, the total charge Q = (cid:80) q equals the Euler characteristic of the confining domain, as requiredby topology . The topological protection due to chargeconservation is discussed in appendix F.The anti-radial disclination lines in the laminar (and C LS composite) state, can be understood as an expanded q = +1 / q e = +1 / q = +1 / q = − /
2, see Fig. 7 (middle). Some structures(compare, e.g., the experimental C LS in Fig. 2), displayan explicit disclination line close to the inclusion endingon two q e = − / q e = − / q e = +1 /
4, close to the inner and outerend, respectively, see Fig. 7 (bottom). Hence this typeof line defect is nothing more than an expanded edgedislocation with topological charge q = 0, which unveilsthe true topological nature of the domain states. Theypossess the same orientational topology as the Shubnikovstates into which they can evolve upon pair annihilation,compare Fig. 8. The observation of domain states is thusowed to packing effects.7 ) b) .
65 0 .
66 0 .
67 0 .
68 0 .
69 0 . · − packing fraction η r e l a ti v e fr eee n e r gy ( F / F m − ) L , L . C LS , , C LS , , S , S , S , L b = . S b = . R out = . Lb = . , p = p = , η = . L FIG. 10.
Stable and metastable structures for different intrinsic parameters.
The geometry is fixed by R out = 6 . L and b = 0 . a) Dependence of the relative free energy on the density. The data for area fraction η = 0 .
65 (at b = 0 .
3) and theestimated numerical error are the same as in Fig. 9. Here, alike symbols denote the structures obtained by gradually increasingthe area fraction in steps of 0 . b) Stable structures (as in Fig. 1d) for shorter rods with p = 5 with inclusion size ratios b = 0 .
25 and b = 0 .
26. The laminar–Shubnikov transition is thus located in between.
Free energy landscape
As apparent from the mul-titude of observed structures in some geometries, the sys-tem does not always equilibrate towards the global energyminimum. It is thus important to understand the full freeenergy landscape generated by the described competingdriving forces. To this end, we additionally calculate thefree energy associated with various theoretical densityprofiles, to directly assess their stability. As a represen-tative example, we choose R out = 6 . L and compare inFig. 9 seven sets of structures obtained by smoothly de-creasing b in the vicinity of the laminar–Shubnikov tran-sition. We draw four important conclusions.First, we explicitly see that the laminar–Shubnikovtransition is not sharp. Instead, over a significant range∆ b ≈ .
015 of inclusion size ratios, a composite state ofboth structures is energetically favorable. Second, thefree-energy differences between two distinct structuresare extremely small and the optimal microscopic struc-ture changes multiple times upon small modifications ofthe confinement. These observations explain the largenumber of different structures observed in the experi-ment for b ≈ .
3. Third, it is important to identify theoptimal microscopic structure of each state to make aproper statement about possible topological transitions.For example, only considering for b = 0 . Dependence on density and rod length
Apartfrom the external topological and geometrical con-straints, the formation and stability of the reported statesalso depends on different intrinsic parameters, which isdetailed further in appendix H. The effect of the pre-ferred bulk layer spacing λ is illustrated in Fig. 10. Wesee that increasing the density (Fig. 10a), resulting ina smaller λ , stabilizes laminar structures compared toShubnikov structures, while decreasing the aspect ratioto p = 5 (Fig. 10b), resulting in a larger relative λ /p , hasthe opposite effect. Extending our state diagram towardsshorter rods at a fixed density, we further anticipate theemergence of stable tetratic structures , since smecticorder is generally destabilized .For long rods at lower densities, different nematicstates D n are found , classified by the number n of q = ± / D and D ∞ , respectively, whilepossessing a distinct orientational director field (compareFig. 7), imprinted by the arrangement of smectic lay-ers. The smectic analogy to D (three line disclinationof charge q = +1 / ocal packing . . . . . . . . . . . . density & director . . . . . . . . . . order parameter . . . . . . . . . . . . . FIG. 11.
Metastable smectic state with threefold sym-metry.
Shown is the theoretical prediction of a structureanalog to a D nematic state for R out = 6 . L , b = 0 . η = 0 .
75. We use the same representations as in Fig. 2 anddepict the local packing fraction (mind the differences dueto the higher overall density), the orientationally averageddensity and director field and the local orientational orderparameter. Note that the free energy is much higher than forthe other states reported in the main text using the same pa-rameters, as there is a large number of deformations requiredto fit into the given geometry.
III. DISCUSSION
We have performed a complementary particle-resolvedexperimental and theoretical study of hard colloidal rodsin annular confinement. Our observations emerge fromthe fundamental principle of globally maximizing the en-tropy subject to the constraints arising from the externalinfluences of the confinement and the internal smecticlayer structure, which depends on the particle shape anddensity. All these competing driving forces are accountedfor explicitly by our DFT data for the equilibrated struc-tures.In the future, it will be interesting to have a closerlook at the position dependence of particle diffusion be-tween the layers or the formation dynamics of the dif-ferent smectic structures, e.g., using dynamical DFT .Drawing phase-stacking diagrams will provide vital in-formation on how the coexistence of nematic and smecticstructures affects their stability in the experiment. Whilesome additional smecitc states could become stable in dif-ferent geometries, an even larger structural variety can beanticipated in more complex topologies, e.g., those withtwo holes. The next level of geometrical and topologi-cal complexity will be reached when immersing colloidalsmectic liquid crystals in random porous media andfractal confinement . On the other hand, there is also ahigh intrinsic potential for finding novel structures whenconsidering more exotic particle shapes .In conclusion, we have shown that the topology andgeometry of an externally imposed confinement largelydetermine the preferred internal structure of a smecticliquid crystal. Adjusting these screws allows to createa protocol for a guided self-assembly of a desired defectstructure. Owing to their robustness and large range ofmetastability, the described smectic states can then be smoothly transferred to any desired confining geometryand, if desired, solidified to unfold their potential for var-ious microtechnological applications . These possiblyinclude novel devices for information storage, templatesfor functional microstructured materials and channels formicro- or nanofluidics. Regarding the recently flourishingresearch realm of living or self-motile particles, a chal-lenging extension of the present work could consist ofsystematically studying the influence of activity on thepredicted equilibrium state diagram . Finally, a fasci-nating connection with biology emerges from drawing theanalogies between colloidal liquid crystals and growingcolonies of rod-shaped bacteria . Our results thus laythe foundation for a deeper microscopic understanding ofthe structures emerging and persisting along the evolu-tion dynamics when such living systems are subjected toextreme topological confinement. Appendix A: Summary of the methodsSedimentation of silica rods
To experimentallycreate confined quasi-two-dimensional smectic struc-tures, we take advantage of the phase stacking of silicarods in sedimentation equilibrium . The bare di-mensions of the rods are measured directly from scanningelectron microscopy images. The rods are dispersed intoa 1 mM NaCl water solution to ensure stability throughdouble layer repulsion. Introducing effective dimensions(see appendix B for more details) to account for the De-bye screening, our particles behave like hard rods of aneffective aspect ratio p eff = 10 . . In practice, several cham-bers are fitted in a single cell. After preparation, the rodsolution is left in the tube to sediment for at least 12hours. During sedimentation, the concentration of parti-cles gradually increases along the direction of the gravityfield leading to successive isotropic, nematic and smecticorder at the bottom. After a few hours, sedimentationdiffusion equilibrium is reached and the three phases co-exist in the cavities.The smectic structures are observed by means ofbright-field microscopy in direct vicinity of the bottomwall. We use a 1 .
42 numerical aperture apochromat oilimmersion objective mounted on an Olympus IX73 mi-croscope and coupled to a Ximea CMOS xiQ camera,which allows an optical resolution comparable to the roddiameter. Due to degenerate planar anchoring at thebottom wall, the system can be considered a quasi-two-dimensional fluid in annular confinement. We choose thetotal amount of rods to obtain an effective volume frac-tion φ eff ≈ −
50% close to the bottom. This ensuresthat there is no crystalline state and that the rods in di-rect contact with the bottom wall exhibit smectic order.To create some statistics, we repeat the measurements in9 given geometry up to twelve times.
Density functional theory (DFT)
We study byfree minimization of a DFT in two dimensions harddiscorectangles (see Fig. 1c) with an aspect ratio p = 10that well reflects the experimental parameters. The in-teraction between these particles are described by a freeenergy functional constructed as an extension of funda-mental measure theory to account for anisotropicparticle shapes . These geometrical functionals de-rived from first principles are exact in the low-densitylimit and have proven very reliable for highly packed sys-tems. The annular confinement is included as an externalhard-wall potential.The key quantity in our theory is the one-body densityprofile ρ ( r , φ ), providing the probability to find a particlewith the center-of-mass position r and its symmetry axisoriented along an angle φ . Consider now a density func-tional Ω[ ρ ( r , φ )] = F [ ρ ]+ (cid:82) d r (cid:82) π φ π ρ ( r , φ )( V ext ( r , φ ) − µ ),where F [ ρ ] is the intrinsic Helmholtz free energy func-tional, V ext ( r , φ ) is the external potential and µ the chem-ical potential (see appendix B more details). Then thedensity ρ ( r , φ ) of a (meta-) stable state is found by itera-tively solving the extremal condition δ Ω[ ρ ] /δρ = 0 start-ing from a particular initial guess for the density profile.The average area fraction η = 0 .
65 is kept fixed through-out the iteration by adapting µ in each step. Then wecompare the values of the free energy F [ ρ ] of the differentstructures to identify the global minimum and quantifythe likelihood to observe a metastable local minimum ρ ( r , φ ) in a corresponding experiment. Calculations areperformed on a quadratic spatial grid with a high enoughresolution ∆ x = ∆ y = 0 . N φ = 96 discrete orienta-tional angles. We iterate until the free energy differencesbetween different structures can be sufficiently resolved. Overlapping parameter space
Our experimentand theory are designed, such that they can both tacklehard rods with a comparable anisotropy that is suffi-ciently high to ensure that the smectic phase is stableover a large range of densities . Systems with vari-able inclusion size ratios b and the radii R out of the cir-cular outer wall ranging between 1 . L ≤ R out ≤ . L arecovered by both approaches. Data analysis and presentation
From the theo-retical data we determine a local packing fraction byweighting the density with the local particle area to high-light the particle resolution within our data (second rowof Fig. 2). As a standard representation of the full densityfield we use in the third row of Fig. 2 and for all otherillustrations a plot of the orientationally averaged den-sity with the orientational director field, represented bygreen arrows of length given by the local order parameter.Moreover, we also directly display the local order param-eter field (fourth row of Fig. 2). Note that the distortedappearance of the order parameter close to the inclusion in the laminar state reflects the very low but nonvan-ishing probability to find particles left and right of thesymmetry axis that are perfectly aligned with the wall.This underlines the similarity of interior and boundarydefects illustrated in Fig. 7. We compare the free ener-gies of the different structures in Figs. 3 and 9. Furtherdetails on numerical errors are given in appendix B.The experimental snapshots (fifth row of Fig. 2) areinspected visually and further processed using WolframMathematica computing system. This allows us to coloreach particle according to its orientation relative to thewall (sixth row of Fig. 2). From the measured center-of-mass positions and orientations we further extract a localfield of the orientational order parameter (seventh row ofFig. 2). The different smectic states and their micro-scopic structures are identified according to the criteriadescribed in appendix C.
Appendix B: Further details on the methodsSedimentation of silica rods.
To experimentallycreate confined quasi-two-dimensional smectic struc-tures, we take advantage of the phase stacking of sil-ica rods in sedimentation equilibrium . The bare di-mensions of the rods are measured directly from scan-ning electron microscopy images. The mean length is W = 5 . µ m with a standard deviation σ W = 0 . µ m.The rods are dispersed into a 1 mM NaCl water so-lution to ensure stability through double layer repul-sion, whose range is characterized by the Debye length κ − = 0 . µ m. Moreover, the phase behavior of ourcharged rods can be mapped onto the phase behaviorof hard rods by introducing the effective rod length W eff = 5 . µ m, diameter D eff = 470nm and aspect ratio p eff = 10 . . In practice, several cham-bers are fitted in a single cell. After preparation, the rodsolution is left in the tube to sediment for at least 12hours. During sedimentation, the concentration of parti-cles gradually increases along the direction of the gravityfield leading to the successive formation of isotropic, ne-matic and smectic phases. After a few hours, sedimenta-tion diffusion equilibrium is reached and the three phasescoexist in the cavities.The smectic structures are observed by means ofbright-field microscopy in direct vicinity of the bottomwall. We use a 1 .
42 numerical aperture apochromat oilimmersion objective mounted on an Olympus IX73 mi-croscope and coupled to a Ximea CMOS xiQ camera,which allows an optical resolution comparable to the roddiameter. Due to degenerate planar anchoring at thebottom wall, the system can be considered a quasi-two-dimensional fluid in annular confinement. We choose the10otal amount of rods such that there is no crystallinestate and the rods in direct contact with the bottom wallexhibit smectic order. The effective (three-dimensional)volume fraction φ eff ≈ −
50% at the bottom of the cellis measured indirectly by comparing the height of thesmectic region to the sedimentation equilibrium of ourrod system in bulk and assuming that the rods behavelike hard spherocylinders . For this setup, we measure λ ≈ . W eff for the effective two-dimensional bulk layerspacing at the bottom wall. Density functional theory (DFT).
ClassicalDFT is a powerful and versatile tool to access the struc-ture of inhomogeneous fluids on the particle scale. Here,we study by free minimization of a DFT in two dimen-sions hard discorectangles with rectangular length L , to-tal length W = L + D and unit circular diameter D , seeFig. 1c. If not denoted otherwise, we use L = 10 D ,so that the aspect ratio p = L/D = 10 well reflectsthe experimental parameters. The energy unit is set by β − := k B T , where k B and T are Boltzmann’s constantand the temperature, respectively. Note that the struc-tural transition of perfectly hard particles are completelydriven by entropy, since β only appears as a trivial scalingfactor. The key quantity in our theory is the one-bodydensity profile ρ ( r , φ ), providing the probability to find aparticle with the center-of-mass position r and its sym-metry axis oriented along an angle φ .The general form of the density functional reads Ω[ ρ ] = F [ ρ ] + (cid:90) d r (cid:90) π d φ π ρ ( r , φ )( V ext ( r , φ ) − µ ) , (B1)where the external potential V ext ( r , φ ) imposes the an-nular confinement through a hard-wall potential and µ denotes the chemical potential. The intrinsic Helmholtzfree energy functional β F [ ρ ] = (cid:90) d r (cid:90) π d φ π ρ ( r , φ ) (cid:0) ln( ρ ( r , φ )Λ ) − (cid:1) + β F ex [ ρ ](B2)consists of an ideal-gas term (Λ denotes the irrelevantthermal wave length) and the excess free energy func-tional F ex [ ρ ], which describes the interactions and corre-lations between the individual particles. Here, the latteris constructed as an extension of fundamental measuretheory to account for anisotropic particle shapes. Thesegeometrical functionals derived from first principles areexact in the low-density limit and have proven very reli-able for highly packed systems.The employed excess functional β F ex [ ρ ] = (cid:90) d r (cid:18) − n ln(1 − n ) + N − n ) (cid:19) , (B3)of the two dimensional fundamental mixed measure the-ory is constructed as a function of weighted densities n ν ( r ) = (cid:90) d r (cid:90) π d φ π ρ ( r , φ ) ω ( ν ) ( r − r , φ ) . (B4) These are calculated by convolution of the density andthe one-body geometrical measures ω ( ν ) , representing alocal area ( ν = 2), circumference ( ν = 1) and boundarycurvature ( ν = 0) of the particles. The mixed weighteddensity N ( r ) generally depends on the geometry of twobodies. To efficiently study long rods in large systems,we use the approximate representation N ≈ a π n n + a − π n ,α n ,α + 2 − a π n ,αβ n ,βα , (B5)found by an expansion introducing vectorial n ,α andtensor-valued weighted densities n ,αβ up to rank-two,where we use the convention of summation over repeatedindices α, β ∈ { , } . The correction parameter a = 4 al-lows to qualitatively describe the smectic phase over thefull range of aspect ratios .Generally, the density ρ ( r , φ ) of a (meta-) stable stateis found by iteratively solving the extremal condition δ Ω[ ρ ] /δρ = 0. In practice, an individual density pro-file ρ i ( r , φ i ) is considered for each discrete orientation φ i . After each iteration step j , the current densities ρ ( j ) i are updated according to a Picard iteration scheme .This involves a formal solution of δ Ω[ ρ ( j ) i ] δρ ( j ) i = 0 (B6)for a new ˜ ρ ( j ) i and calculating ρ ( j +1) i = (1 − γ ) ρ ( j ) i + γ ˜ ρ ( j ) i with a dynamical mixing parameter γ , initially set to γ = 0 .
08. If the value of the local packing fraction n ( r ),see Eq. (B4), exceeds one for any coordinate r , the stepis rejected and γ is decreased by a factor of 0 .
2. Aftereach 200 steps, the program tries to gradually increase γ by factors of 2 to speed up the iteration. The averagearea fraction η = N LD + D π πR (1 − b ) (B7)is kept fixed throughout the iteration by adapting µ ineach step. Hence, the stable state corresponds to aglobal minimum of the free energy F [ ρ ].The minimization is initialized by creating some appro-priate random density profiles with different symmetries.If the value of n ( r ) exceeds one for an initial guess, thedensity profiles are renormalized to obtain a valid profile.In the course of the minimization, the packing fraction η is then gradually increased back to its input value. Weiterate until the free energy differences between differentstructures, which typically are of the order 10 − in unitsof the thermal energy, can be sufficiently resolved. Thisis usually the case when the free energy changes by lessthan 5 · − in the last 1000 iteration steps. We thusassume that the numerical error in the free energy is ofthe order 5 · − . Since the true free energy is alwayssmaller for each structure, most of this error occurs as asystematic shift, irrelevant when comparing relative freeenergies as we do here. In some cases, as for R out = 6 . L b = 0 .
29 or b = 0 . − andsmaller, making it difficult to unambiguously determinethe global minimum. However, the systematic errors dueto the approximations in the density functional can belarger.After equilibration of multiple structures, we comparethe values of the free energy F [ ρ ] to distinguish betweenlocal and global minima and quantify the likelihood toobserve a particular state in a corresponding experiment.Calculations are performed on a quadratic spatial gridwith a high enough resolution ∆ x = ∆ y = 0 . N φ =96 orientational angles (for p = 10). Some structureswere created with a smaller resolution and then furtherequilibrated for the given parameters.The obtained density profiles ρ ( r , φ ) are usually graph-ically represented by a color scheme on a range from zeroto three denoting the dimensionless orientationally aver-aged density¯ ρ ( r ) := (cid:18) LD + D π (cid:19)(cid:90) π d φ π ρ ( r , φ ) . (B8)Here we also display lines of length given by the orien-tational order parameter and orientation indicating thelocal orientational director field. These quantities canbe extracted from the local order tensor Q ( r , φ ) = 2 ρ ( r , φ ) (cid:82) π d φ ρ ( r , φ ) (cid:18) cos φ − cos φ sin φ cos φ sin φ sin φ − (cid:19) (B9)as its largest Eigenvalue and the corresponding Eigen-vector, respectively. An alternative graphical represen-tation, used in Fig. 2, depicts the local packing frac-tion n ( r ). As this quantity corresponds to the densityweighted with the particle area, it directly illustrates thelocations of the particles, while ranging between zero andone. Since the behavior is fluid-like within smectic lay-ers, the local packing fraction is generally smeared-out,while the silhouettes of individual particles in the packedregions close to the walls are directly visualized.Our benchmark calculation in bulk using the describedDFT approximation locates the nematic–smectic transi-tion around η ≈ .
62. As this two-dimensional area frac-tion is not directly comparable to the experimentally ac-cesible volume fraction φ , our theoretical calculations inthe confined system are generally carried out for η = 0 . λ = 12 . D = 1 . W .To determine the inclusion size ratio b t at which astructural transition occurs in annular confinement, weselect two values b and b , usually differing by b − b =0 .
01, and check whether the sign of the free-energy dif-ference between two states of interest is different. If so,we determine b < b t < b as the point where the linearlyinterpolated free energies are equal. Appendix C: Classification and identification ofthe different smectic states
In general, we use the following criteria to distinguishthe different smectic states through occuring defects andthe arrangement of smectic layers. This is most eas-ily done by observing the number and orientation ofline disclinations and identifying connected layers, whichspan between two opposite sites of the outer boundary.Usually, there is a typical number of edge dislocations as-sociated with Shubnikov structures, but a certain numberof edge dislocations can also occur in other states. Micro-scopic details are not relevant for this first classifications.Laminar (or bridge) state L (or B ): there is at leastone connected layer at each side of the inclusion. Usu-ally, this is equivalent to observing two anti-radial discli-nation lines close to the outer boundary. Domain state D : there are exactly three radially oriented disclinationlines, separated by 120 degrees; the layers from differentdomains meet at an angle of 90 degrees. Shubnikov state S : there is neither a connected layer nor any disclinationline; all layers span between the inner and outer bound-ary. Laminar–domain composite state C LD : there are oneor two radial disclination lines and at least one connectedlayer (or anti-radial disclination line) at only one side ofthe inclusion. Laminar–Shubnikov composite state C LS :there are no radial disclination lines and at least one con-nected layer (or anti-radial disclination line) at only oneside of the inclusion. Domain–Shubnikov composite state C DS : there are one or two radial disclination lines andno connected layer.For each intact experimental chamber, we identifythe dominant state by visual inspection of our particle-resolved images according to the above criteria. In thesmallest experimental chambers considered it is not al-ways possible to clearly associate an observed structurewith one of these states. Strongly deformed theoreticalstructures are not taken into further consideration, sinceit is clear from their large free energy that they are irrel-evant in the search for the most stable mesoscopic state. Appendix D: Phenomenological model for defectenergies
The existence of the laminar–Shubnikov transition inFig. 3 can be understood in the light of a phenomeno-logical model. Making some minimal assumptions, weestimate the energy penalty resulting from the character-istic defects in each state. Let us first denote the energyof a disclination line with unit length by δ and of eachedge discolation by u ed . We then assume that the twoanti-radial disclination lines in the laminar state have thelength R out and that each radial domain boundary is astraight line of length R out − R in . The number N out − N in ≈ πR out − πR in λ (D1)12f edge dislocations in the Shubnikov state is assumed asthe difference of the outer and inner wall perimeters, di-vided by the bulk layer spacing λ . The resulting energiesread U L = 2 δ R out ,U D = 3 δ R out (1 − b ) ,U S = 2 πλ R out (1 − b ) u ed (D2)for the laminar, domain and Shubnikov state, respec-tively.For all choices of the fit parameters δ and u ed , thestate diagram predicted by this simple model is indepen-dent of the size of the annulus, because all energies inEq. (D2) scale with R out . Moreover, there is exactly one stable transition for increasing b , namely from a laminarstate, at small inclusions, to either a domain or Shub-nikov state at large inclusion. One of the two latterstates is always metastable, since both U D and U S areproportional to (1 − b ). These energies further decreaseto zero for an infinitely thin annulus ( b → U L assumed for the laminar state remains con-stant when increasing b . Therefore, the laminar statealways becomes metastable for large inclusion sizes. Astable laminar–domain transition may occur at b = if U D < U S , which means that 3 δλ < πu ed . In the oppo-site case, we expect a stable laminar–Shubnikov transi-tion at b = 1 − λδπu ed < , while the domain state is onlymetastable.Although this phenomenological model does not cap-ture all theoretical and experimental observations, itnicely reflects the competition between domain- andShubnikov states, which possess the same director topol-ogy at the outer wall, and rationalizes the laminar–Shubnikov transition, which is one of our main results.Using as an input to the presented model that this tran-sition can be roughly observed at b = 0 .
3, we can deter-mine the ratio λδu ed = π (1 − b ) ≈ . Appendix E: Optimal microscopic arrangementof layers
Associating an observed structure with a particularsmectic state does not account for the full microscopic in-formation. In particular, different structures can mainlybe distinguished by counting the explicit numbers ofsmectic layers. The optimal microscopic structure whichis most stable for a given state itself depends on the com-petition between different driving forces, which we detailbelow.
Laminar state L N con ,N dis . In a given laminar state L N con ,N dis , there are precisely N con connected layers span-ning throughout the confinement and N dis layers discon-nected by the inclusion, summing up to a total number N tot = N con + N dis of layers (where N tot = N con in thebridge state), see Fig. 5. Depending on the geometry,there may further be two anti-radial disclination linesclose to the boundary. We consider the shape of thedisclination line and the number of layers in the sepa-rated domains as structural details of next order, sincethe theory automatically finds the optimal structure cor-responding to L N con ,N dis in a given confinement.In general, the theory predicts only relatively small de-viations of the laminar layer spacing from the bulk value λ . As a consequence, the total number N tot of smecticlayers in the parallel domain, fluctuates as a function ofthe inclusion size ratio b , as shown in Fig. 5, which alsoapplies to the length and shape of the disclination lines.The optimal microscopic laminar structure is then deter-mined by the integer multiples of λ which are closest tothe respective dimensions of the confinement. For exam-ple, N dis is usually bounded by the diameter 2 R in /λ ofthe inclusion to avoid additional defects or strong defor-mations near the wall. Shubnikov state S N in ,N out . the most significantstructural property of a given Shubnikov state S N in ,N out is the number N out − N in of point defects, which is is de-termined by the numbers N in and N out of layers in directcontact with the inner and outer wall, respectively. Theexact radial (and relative) location of these defects arestructural details of next order, which can be accountedfor by creating and comparing a large number of differentstructures. As indicated in Fig. 6, we further consider alocal layer spacing λ in = π (2 R in + D ) N in (E1)at the inner and λ out = π (cid:112) (2 R out − D ) − L N out (E2)13t the outer wall, explicitly calculated here from the cir-cumference of the line connecting the particle centers indirect vicinity of each wall. The effective layer spacingassociated with the closest planar packing (assuming thatall rods are aligned perfectly tangential to the wall) cor-responds to the length of the arc through the particlecenter between the two radial lines tangential to the par-ticle and reads λ ∗ in = (2 R in + D ) (cid:18) arctan (cid:18) L R in + D (cid:19) + arcsin (cid:32) D (cid:112) (2 R in + D ) + L (cid:33)(cid:33) . (E3)This will serve as a reference value.To quantify the variation in the local layer spacing, wedetermine λ wall (the subscript ’wall’ either stands for ’in’or ’out’) for several theoretical and experimental struc-tures. According to Fig. 6, we find that λ in is muchsmaller and λ out is much larger than the bulk value λ in both theory and experiment. This means that the op-timal microscopic Shubnikov structure results from thecompetition between optimizing the layer spacing andminimizing the number N out − N in of point defects. Thisdeviation is most significant in extreme confinement. Forsmall inclusions, N in can even become larger than themaximal number of rods that can be packed around thewall with a planar orientation, i.e., λ in < λ ∗ in , which re-sults in the tilt shown in Fig. 4f. Moreover, there areextreme structures without any defects for sufficientlysmall distances between the walls, e.g., the theory pre-dicts a stable S , for R out = 4 . L and b = 0 . Laminar–Shubnikov composite state C LS N con ,N in ,N out . In a given Laminar–Shubnikov com-posite state C LS N con ,N in ,N out , there is a precise number N con of connected layers, reflecting the characteristicsof the laminar state. The number of perpendicularlayers separated by the adjacent disclination line (ifpresent) constitutes here an important detail, since thelocation of the connected layers relative to the centerof the annulus is not fixed by symmetry, in contrastto the laminar state. Therefore, this composite stateis characterized by two further numbers N in and N out ,i.e., the total number of contacts with the two walls,reflecting the characteristics for the Shubnikov state.The stability of such a laminar–Shubnikov compositestate can be explained by a mutual relaxation of the ex-ternal constraints, which we understand as follows. Theinnermost connected layer in the laminar part can takethe optimal position relative to the inclusion that min-imizes the penalty arising from deformations and falsewall alignment. In contrast, the arrangement of the lay-ers in an ordinary laminar state is dictated by its axialsymmetry. The same is true in view of the optimal layerspacing at the boundaries in the Shubnikov part of thestructure, which is restricted to discrete values in the ordinary Shubnikov state. Further note that the inter-section of the two half-structures is smooth, i.e., it doesnot introduce additional domains or deformations. Appendix F: Topological protection ofmesoscopic defect structures
Here we make some more statements regarding the sta-bility of the different states deduced from their topolog-ical details. As elaborated in the main text, the totaltopological charge Q , which is the sum of the charge ofall occurring defects, compare Fig. 7, reflects the Eulercharacteristic χ of the bounding domain. Hence, in annu-lar confinement we have Q = χ = 0. Due to the typicallydifferent types of defects and their spatial distribution(oppositely charged defects are separated by rigid lay-ers), the L , C LS and S states are topologically protected,i.e., they cannot be transformed into one another upon asmooth variation of the particle distribution.In contrast, we find that the theoretically created do-main structures are not topologically protected. In fact,they are unstable with respect to topologically equiva-lent Shubnikov structures, as depicted in Fig. 8. In thecourse of the minimization of DFT, the end-point de-fects of the radial disclination lines gradually annihilate,while the layers at the boundary become larger, eventu-ally resulting in a uniform orientational order. The cho-sen annular geometry thus prefers the Shubnikov state.However, keeping in mind the experimental observations,we suspect that a domain state can become stable, e.g.,by increasing the density or adding some polydispersity.In general, it should become more likely to observe a do-main state in other confining geometries respecting thethreefold symmetry.An alternative and somewhat more insightful interpre-tation of the topologically protected states can be madeby considering the inclusion as a topological defect it-self, which possesses an own winding number k . In theShubnikov state S , we have k = 1 representing the uni-formly bent director field parallel to the wall. When-ever the rods are misaligned along the integration patharound the inclusion k is reduced by − /
2. Thus we have k = 1 / C LS and k = 0 for L . Thereby, the nega-tive half-integer boundary charges described in the maintext are absorbed into k and do not contribute to thetotal charge ˜ Q , which is therefore different in these threestates when choosing the current interpretation. Here,the charge conservation resulting in topological protec-tion becomes directly apparent. However, the conserva-tion law between ˜ Q and the geometric quantity χ has tobe generalized to χ = ˜ Q + k − h , where h denotes thenumber of holes in the confining domain ( h = 0 in cir-cular and h = 1 in annular confinement) and thus stillholds. Further note that for the domain state, it is notpossible to unambiguously define the winding number k due to the presence of the radial line disclinations, whichunderlines its topological instability.14 , → → → →S , , b = 0 . S , , b = 0 . S , , b = 0 . S , , b = 0 . S , , b = 0 . S , → → → →S , , b = 0 . S , , b = 0 . S , , b = 0 . S , , b = 0 . S , , b = 0 . S , → → → →S , , b = 0 . S , , b = 0 . S , , b = 0 . S , , b = 0 . S , , b = 0 . Hysteresis in the Shubnikov state I.
Shown are the structures obtained by first decreasing the inclusion sizeratio b from b = 0 .
29 down to b = 0 .
27 and then increasing b up to b = 0 .
29, in steps of 0.01. The initial structures for b = 0 . b = 0 .
3, shown in Fig. 9 and labeled on the left (see Fig. 13 for the transition from S , to S , omitted in the first row). Color bar and arrows denote the orientationally averaged density and the director field, compareEqs. (B8) and (B9), respectively. Appendix G: Response of microscopic layeringto geometric changes
The identified topologically protected states ( L , C LS and S ) even remain metastable over a large range of in-clusion sizes as analyzed in Fig. 9. However, we are fur-ther interested in the particular microscopic layer struc-ture corresponding to a certain state, whose stability isdiscussed below in more detail.An initial intrinsic laminar structure L N con ,N dis re-mains invariant upon smoothly changing b until the lay-ers do not fit the geometry any more, e.g., if N con λ >R out − R in , which then results in an irreversible forma-tion of defects. Hence, there exist some (metastable)laminar–laminar transitions, as also observed in Fig. 9.In contrast, the number of edge dislocations in a givenShubnikov structure S N in ,N out is not topologically pro-tected and thus adapts to the geometrical changes assoon as the local layer spacing near a boundary differstoo much from the bulk value. . This comes along withsome hysteresis effects upon first decreasing b and thenreturning to its original value. In particular, Fig. 12depicts that, for all structures considered in Fig. 9, the number N in of layers in contact with the inclusion de-creases from N in = 11 to N in = 10 upon decreasing theinclusion size ratio from b = 0 .
28 to b = 0 .
27, Upon asubsequent reversal from b = 0 .
27 back to b = 0 .
28 thecontact number N in = 10 remains the same for two ofthe structures considered (shown in the two top rows),whose energy is then clearly larger than at the beginning( b = 0 . N in = 11. Further increasing the sizeof the inclusion to b = 0 .
29, all structures have N in = 11contacts, which closes all hysteresis loops. In fact, theirenergy lies slightly below that of the original structuresfor b = 0 .
29, which indicates that a fluctuating confine-ment aids the equilibration. On the other hand, startingwith the metastable structure S , for b = 0 .
3, the hys-teresis loop (shown in Fig. 13) via b = 0 .
29 includes ina new structure S , for b = 0 . b = 0 . N in = 11.The composite states C LS N con ,N in ,N out do not show anymicroscopic changes in the number of edge dislocationsover the range of b considered in Fig. 9, since the value of15 , → →S , , b = 0 . S , , b = 0 . S , , b = 0 . Hysteresis in the Shubnikov state II.
Shown are the structures obtained from S , by first decreasing theinclusion size ratio b from b = 0 . b = 0 .
29 and increasing b back to b = 0 .
3. Color bar and arrows denote theorientationally averaged density and the director field, compare Eqs. (B8) and (B9), respectively. the local layer spacing λ in at the inclusion in the Shub-nikov part of the structure can be balanced by a defor-mation of the adjacent laminar layer. If a hysteresis loopsimilar to the Shubnikov state exists here, it thus mustbe much larger. Appendix H: Dependence on intrinsicparameters
Apart from the external topological and geometricalconstraints, the formation of smectic structures can alsobe controlled by indirectly tuning the preferred layerspacing λ in bulk. This intrinsic structural propertyof the smectic liquid crystal depends on the size and as-pect ratio of the particles, as well as, the total density. Inthe following, we briefly explore this extended parameterspace within DFT in more detail. Changes in density and particle shape.
Increas-ing the density, the bulk layer spacing decreases andpacking effects become more important. To explore theeffect on the smectic states, we consider again the den-sity profiles for R out = 6 . L and b = 0 . N out becomes more favorable.The dependence of the state diagram on the absoluteparticle size can be extracted from the scaling used inFig. 3. Decreasing now the aspect ratio of the rods to p =5 for a fixed density and relative system size R out = 6 . L ,we find that the Shubnikov state is still stable for b =0 .
26, cf. Fig. 10b, although the absolute system size R out has accordingly decreased by a factor two. The laminar–Shubnikov transition occurs thus at a smaller value b ≈ .
259 than for more elongated rods. Hence, we concludethat the Shubnikov state is most stable when consideringsystems of small and short rods at a low density withinthe smectic regime.
Relation to nematic states in an annulus.
Asdiscussed in the main text, some of the observed smecticstates possess topologically equivalent nematic states atlower density. Here, we elaborate on two further ques-tions. Can we infer the existence of additional smecticstates from nematic ones with the same symmetry? Howdoes the transition from nematic to smectic states occurupon increasing the density?To answer the first question, we undertook some effortsto stabilize a three-fold symmetric smectic structure thatbehaves to the nematic D state like the laminar stateto the nematic N . This means that there are three anti-radial disclination lines of charge q = 1 / q = − / k = − /
2, compare ap-pendix F. We show in Fig. 11 that such a state indeedexists. However, it is highly metastable as the layers aresubject to a strong bend. Even with this expensive defor-mation, there is not enough space for the layers hostingthe rods oriented tangentially to the inclusion, to extendtowards the outer wall without forming additional de-fects. We can only speculate that such a structure canbe stabilized in different geometries possessing a threefoldsymmetry, as, for example, one with hexagonal walls.The second question is addressed in Fig. 14, where wecreated five distinct nematic states at an area fraction η = 0 . η = 0 . D and D would destabilize at the low nematic densi-ties. This procedure allows us to mimic, in a rough way,the nonequilibrium densification observed during the sed-imentation process in the experiment. We make two mainobservations. First, the density at which we observe anonset of smectization depends on the nematic state. Moreprecisely, it appears that this density is lower if the de-formations of the nematic director field are stronger, asin D and D . Second, the initially emerging smecticpatterns still follow the orientational director field of thenematic states and thus become frustrated in their posi-tional order. This nicely underlines that the distinct di-rector field of the equilibrated smectic structures is favor-able and underlines our suspicion that, in experimental16 → → → → D → → → → D → → → → D → → → → D → → → → η = 0 . η = 0 . η = 0 . η = 0 . η = 0 . Onset of smectic order emerging from nematic states.
Shown are the density profiles initialized for R out = 6 . L and b = 0 . D n and those emerging upon increasing the density in discrete steps untilthe bulk transition density η = 0 .
62 is reached (see labels). Not all structures are fully equilibrated, such that the otherwiseunstable structures with n = 4 and n = 5 defect pairs could also be included. Color bar and arrows denote the orientationallyaveraged density and the director field, compare Eqs. (B8) and (B9), respectively. Note that the color bar only has half therange as in the all other density plots throughout the manuscript. sedimentation equilibrium, the structure of the nematicstates is influenced by the smectic states below and notvice versa. ACKNOWLEDGEMENTS
We thank Christoph E. Sitta for implementing largeparts of the DFT code, Paul A. Monderkamp for help-ful discussions and providing a plotting tool, and Axel Voigt for helpful discussions. This work was supported bythe German Research Foundation (DFG) within projectLO 418/20-2. This project has received funding fromthe European Union’s Horizon 2020 research and in-novation programme under the Marie Sk(cid:32)lodowska-CurieGrant Agreement No 641839.17
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