Bruno Zappone
email: bruno.zappone@cnr.it.

Atilla Eren Mamuk

Iryna Gryn

Valentina Arima

Alessandra Zizzari

Roberto Bartolino

Emmanuelle Lacaze

Rolfe Petschek

Analogy between periodic patterns in thin smectic liquid crystal films and the intermediate state of superconductors

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Edited by Robert J. Birgeneau, University of California, Berkeley, CA, and approved June 4, 2020 (received for review January 30, 2020) Spontaneous breaking of symmetry in liquid crystal (LC) films often reveals itself as a microscopic pattern of molecular alignment. In a smectic-A LC, the emergence of positional order at the transition from the nematic phase leads to periodic textures that can be used as optical microarrays, templates for soft lithography, and ordering matrices for the organization and manipulation of functional nanoparticles. While both 1d and 2d patterns have been obtained as a function of the LC film thickness and applied fields, the connection has not been made between pattern formation and the peculiar critical behavior of LCs at the nematic-smectic transition, still eluding a comprehensive theoretical explanation. In this article, we demonstrate that an intense bend distortion applied to the LC molecular director while cooling from the nematic phase produces a frustrated smectic phase with depressed transition temperature, and the characteristic 1d periodic texture previously observed in thin films and under applied electric fields. In light of De Gennes' analogy with the normal-superconductor transition of a metal, we identify the 1d texture as the equivalent of the intermediate state in type I superconductors. The bend distortion is analog to the magnetic field in metals and penetrates in the frustrated phase as an array of undercooled nematic domains, periodically intermixed with bend-free smectic-A domains. Our findings provide fundamental evidence for theories of the nematic-smectic transition, highlighting the deep connection between phase frustration and pattern formation, and perspectives Q: 6 on the design of functional smectic microarrays. L iquid crystals (LCs) commonly used in displays and optical devices are composed of rod-like molecules that Q: 7 tend to align along a common director n depending on external aligning fields such as surface interactions (anchoring) and electric fields [START_REF] Oswald | Smectic and Columnar Liquid Crystals[END_REF]. The molecular density is uniform in the nematic (N) phase, whereas a 1d density modulation (i.e., layering) appears in the smectic phase. There are several kinds of smectic phases with different types of molecular order and alignment, and the one showing a density modulation along the director n is called smectic-A (S) phase [START_REF] Oswald | Smectic and Columnar Liquid Crystals[END_REF]. It has long been known that conflicting aligning fields can induce defect nucleation, and formation of 1d and 2d micropatterns in the S phase (2-10). More recently, pattern formation has been recognized as a promising platform for the rapid and cost-effective bottom-up fabrication of optical microarrays [START_REF] Kim | Optically selective microlens photomasks using self-assembled smectic liquid crystal defect arrays[END_REF][START_REF] Son | Optical vortex arrays from smectic liquid crystals[END_REF][START_REF] Serra | Curvature-driven, one-step assembly of reconfigurable smectic liquid crystal "compound eye" lenses[END_REF], templates for soft lithography [START_REF] Yoon | Internal structure visualization and lithographic use of periodic toroidal holes in liquid crystals[END_REF][START_REF] Kim | Fabrication of a superhydrophobic surface from a smectic liquidcrystal defect array[END_REF][START_REF] Kim | Self-assembled periodic liquid crystal defects array for soft lithographic template[END_REF][START_REF] Kim | Smectic liquid crystal defects for self-assembling of building blocks and their lithographic applications[END_REF], and anisotropic matrices for ordering and guiding the assembly of functional nanoparticles [START_REF] Coursault | Linear self-assembly of nanoparticles within liquid crystal defect arrays[END_REF][START_REF] Coursault | Tailoring anisotropic interactions between soft nanospheres using dense arrays of smectic liquid crystal edge dislocations[END_REF][START_REF] Honglawan | Synergistic assembly of nanoparticles in smectic liquid crystals[END_REF][START_REF] Gryn | Electric-field-controlled alignment of rod-shaped fluorescent nanocrystals in smectic liquid crystal defect arrays[END_REF][START_REF] Ro | Oriented gold nanorods and gold nanorod chains within smectic liquid crystal topological defects[END_REF]. LCs also provide the advantage of a fast and large response to external stimuli, e.g., smectic patterns can change morphology as a function of an applied electric or magnetic field [START_REF] Gryn | Controlling the self-assembly of periodic defect patterns in smectic liquid crystal films with electric fields[END_REF][START_REF] Goscianski | Field induced transitions in smectic A phases[END_REF][START_REF] Parodi | A possible magnetic transition in smectics-A[END_REF].

Very few studies have addressed the connection between pattern formation and the peculiar critical behavior of LCs at the N-S transition [START_REF] Parodi | A possible magnetic transition in smectics-A[END_REF][START_REF] Cladis | Growth of a smectic A from a bent nematic phase and the smectic light valve[END_REF][START_REF] Gim | Morphogenesis of liquid crystal topological defects during the nematic-smectic A phase transition[END_REF], which stands out as a major theoretical challenge in statistical mechanics and condensed matter physics [START_REF] Yethiraj | Recent experimental developments at the nematic to smectic-A liquid crystal phase transition[END_REF]. Insights can be gained from an analogy between smectics and superconductors, first pointed out by De Gennes [START_REF] De Gennes | An analogy between superconductors and smectics A[END_REF]. In both materials, the Landau-Ginzburg free energy expansion involves a complex parameter Q: 8 ψ = |ψ|e iφ and a vector field n. In metals, |ψ| is the density of Cooper electron pairs and B = ∇×n is the magnetic field. At the transition to the low-temperature superconducting state, B tends to be expelled from the bulk (Meissner effect, Fig. 1). A field with amplitude B is expelled below a critical temperature T c (B) that increases as B is decreased, i.e., T c (B) ≤ T 0 with T 0 = T c (0). At a temperature T < T 0 , the field is expelled below a critical amplitude B c (T) that increases as T is decreased. At temperatures and fields immediately below the critical values, the expulsion is incomplete and inhomogeneous, and B penetrates in the superconductor, i.e., the superconductor phase is frustrated. In a type I superconductor film, finite-sized domains of undercooled normal metal state, carrying the B field, become intermixed with superconductive domains in a pattern of meandering stripes or polygonal meshes known as intermediate state [START_REF] Poole | Superconductivity[END_REF][START_REF] Prozorov | Equilibrium topology of the intermediate state in type-I superconductors of different shapes[END_REF][START_REF] Prozorov | Suprafroth in type-I superconductors[END_REF] (Fig. 1).

Significance

Microscopic patterns of molecular alignment are easily obtained in smectic LCs subject to conflicting alignment conditions and provide a promising route to the rapid bottom-up fabrication of ordered materials, notably for the organization and manipulation of nanoparticles. A convenient but puzzling feature is that the pattern type can be switched between 1d and 2d by varying the film thickness or applying an electric field. We devised a simple experiment to show that 1d patterns are due to frustration of the smectic phase under a strong distortion of the molecular director field, leading to a mixed nematic-smectic phase. Our finding confirms the long-standing prediction of an "intermediate" LC state based on an analogy with superconductors.

In type II superconductors, B penetrates via a 2d array of elementary defects (vortices) known as Abrikosov lattice [START_REF] Poole | Superconductivity[END_REF].

In LCs, |ψ| is the amplitude of the 1d smectic density modulation and the rotor field ∇×n is associated with twist and bend director distortions [START_REF] Oswald | Smectic and Columnar Liquid Crystals[END_REF][START_REF] De Gennes | An analogy between superconductors and smectics A[END_REF]. The phase φ is constant on surfaces (known as smectic "layers") that are normal to the wave-vector q = ∇φ of the density modulation. In an undistorted smectic LC, the spacing 2π/q between layers is constant (q = q 0 ) and q || n. In analogy with superconductors, ∇×n tends to be expelled at the transition from the N phase to the low-temperature S phase, which is equivalent to imposing constant spacing or confocality of (curved) smectic layers [START_REF] De Gennes | An analogy between superconductors and smectics A[END_REF]. De Gennes' analogy famously led to the discovery of twist grain boundary phases in chiral thermotropic LCs, where the twist field penetrates as a set of periodically spaced screw dislocations analog to the Abrikosov vortex lattice [START_REF] Renn | Abrikosov dislocation lattice in a model of the cholesteric-to-smectic-A transition[END_REF][START_REF] Nguyen | TGBA and TGBC phases in some chiral tolan derivatives[END_REF]. On the other hand, very few studies have focused on type I behavior [START_REF] Dozov | Quantized grain boundaries in bent smectic-A liquid crystals[END_REF][START_REF] Wang | Bend-induced melting of the smectic-A phase: Analogy to a type-I superconductor[END_REF][START_REF] Ruan | Coexisting nematic and smectic-A phases in a twisted liquid-crystal cell[END_REF], even though the molecular structure of common thermotropic LCs used in displays and optics produces this behavior [START_REF] Oswald | Smectic and Columnar Liquid Crystals[END_REF][START_REF] Renn | Abrikosov dislocation lattice in a model of the cholesteric-to-smectic-A transition[END_REF]. In particular, the "intermediate" LC state predicted by De Gennes has never been clearly identified [START_REF] De Gennes | An analogy between superconductors and smectics A[END_REF].

Results

The common thermotropic LC compound 4-cyano-4ʹ-octylalkylbiphenyl (8CB), showing a bulk N-S transition at 32.5 °C, was confined between two parallel glass surfaces. One surface was coated with a thin layer of polymer and gently rubbed along the x direction to align the LC molecules along x, i.e., induce unidirectional (nondegenerate) planar anchoring at the surface. The other surface was coated with a monolayer of surfactant to align the molecules along the surface normal z, i.e., induce homeotropic (normal) anchoring. Such boundary conditions are known as unidirectional (nondegenerate) hybrid anchoring conditions. In the N phase, they induce a splay-bend distortion of the director field n in the xz plane, i.e., with components n x and n z depending on z, and perpendicular component n y = 0. Pattern formation was studied as a function of the LC film thickness h and temperature T in regions showing a sharp step-like variation of h, obtained by etching the glass surfaces with a corrosive acid solution. Fig. 2 shows the textures formed across a step in a region with large thickness, h ≥ 2.0 μm, as the temperature was decreased from the N phase. At high temperatures, the N phase did not show any texture and appeared completely dark between crossed polarizers when the planar anchoring direction x was set parallel or perpendicular to the polarizers, showing that n was indeed distorted in the xz plane with n y = 0. When x was at 45°with the crossed polarizers, the optical transmission slightly decreased as the temperature was decreased toward the S phase. This shows that n rotated slightly toward the surface normal but remained distorted, in contrast with the theoretical prediction that n should become uniform [START_REF] Barbero | Critical thickness of a hybrid aligned nematic liquid crystal cell[END_REF] due to the divergence of the bend elastic constant K 3 approaching the N-to-S transition [START_REF] De Gennes | An analogy between superconductors and smectics A[END_REF][START_REF] Davidov | High-resolution x-ray and light-scattering study of critical behaviour associated with the nematic-smectic-A transition in 4-cyano-4′-octylbiphenyl[END_REF][START_REF] Bradshaw | The Frank constants of some nematic liquid-crystals[END_REF]. Instead, longitudinal stripes appeared approximately parallel to the anchoring direction x (L-texture) and acquired a transverse striation (sL-texture) as the temperature was decreased (Fig. 2). This texture continuously evolved into strings of fan-shaped focal conic domains (FCDs) as the temperature was further decreased and eventually into a close-packed "honeycomb" FCD pattern (S-texture, Fig. 2). Such texture is a hallmark of the S phase confined under hybrid anchoring conditions (2-5, 8, 24, 25, 39). Indeed, the constitutive feature of a FCD is that curved smectic layers keep a constant interlayer spacing or, equivalently, ∇×n = 0. The centers of curvature of the layers describe two singular lines, a hyperbola and a confocal ellipse [START_REF] Oswald | Smectic and Columnar Liquid Crystals[END_REF]. Polarized fluorescent confocal microscopy revealed that the FCDs were of the upon cooling an 8CB film with large thickness, h ≥ 2.0 μm, under unidirectional hybrid anchoring conditions. The step was etched on one of the glass cell boundaries. The polarizers were oriented at 45°with the planar anchoring direction x. The thicker region was more birefringent and appeared brighter between crossed polarizers than the thinner region. N is the untextured nematic phase, L is a texture with stripes approximately parallel to x, whereas stripes of the sL texture show a transverse striation. The S-texture is a honeycomblike assembly of focal conic domains (FCDs) characteristic of the smectic phase. "square" type already observed in 8CB films under nondegenerate hybrid anchoring conditions, i.e., with the ellipse at 45°from both the substrate and the asymptotes of the hyperbola (SI Appendix, Fig. S1) [START_REF] Zappone | Periodic lattices of frustrated focal conic defect domains in smectic liquid crystal films[END_REF]. Notably, the S-texture expanded from thick to thin regions of the film as the temperature was decreased (Fig. 2).

In regions with small thickness, h ≤ 1.5 μm, the transverse striation of the L-texture rapidly expanded as the temperature was decreased and completely replaced the L-texture with a highly regular array perpendicular to x (P-texture, Fig. 3 C and D, Right side). Again, the transition proceeded from large to small thickness as the temperature was decreased (SI Appendix, Fig. S2). Both the S-and P-texture persisted when the temperature was decreased several degrees below the undistorted bulk transition temperature T 0 .

The L-texture could not be extinguished between crossed polarizers due to a modulation of the in-plane director components (n x , n y ) along the y-direction (Fig. 4 A, B, E, andF). In contrast, the P-texture could be extinguished when the stripes were parallel or perpendicular to the polarizer (Fig. 4 C, D, G, andH), showing that n lay in the xz plane, with n x and n z periodically modulated along the x-direction, and n y = 0.

For intermediate film thicknesses, h = (1.5 to 2.0) μm, two different texture sequences were observed. In most cases the L-texture was replaced by a P-texture punctuated by clusters of FCDs with a string-like arrangement or patches of S-texture (Fig. 4 D andH). Only in a few samples the L-texture evolved into a continuous S-texture (Fig. 3 B and C, Left side). In all cases, the P-texture and S-texture never transformed into each other as the temperature was changed.

Calling T c the temperature at which the L-texture changed into the P-texture, different values T c,1 and T c,2 were measured in region with different film thickness, respectively h 1 and h 2 (SI Appendix, Fig. S2). The temperature T c increased as h increased and followed the relation (T c,2 -T c,1 )/T 0 = (h 0 /h 1 ) γ -(h 0 /h 2 ) γ , where h 2 > h 1 , h 0 = (0.60 ± 0.03) nm was determined by fitting (Fig. 5) and γ = 0.98 is calculated below.

When the temperature was decreased below T c at one sample position with fixed h, the period d of the P-texture increased from the critical value d c = d(h,T c ) (Fig. 3 6B). A linear dependence of d vs. h was previously observed also at room temperature, but with a larger slope α ∼ 2 [START_REF] Gryn | Controlling the self-assembly of periodic defect patterns in smectic liquid crystal films with electric fields[END_REF][START_REF] Gryn | Electric-field-controlled alignment of rod-shaped fluorescent nanocrystals in smectic liquid crystal defect arrays[END_REF].

We also conducted a survey of pattern formation in smectic compounds differing from 8CB in molecular composition, properties, and phase sequences, using various aligning interfaces (e.g., solid/fluid, crystalline/amorphous, smooth/rough). The survey showed that P-texture invariably appears upon cooling the LC under nondegenerate hybrid anchoring conditions (SI Appendix, Figs. S3 andS4), clearly indicating that Ptexture formation is a general phenomenon related to the N-S phase transition.

Discussion

To explain these results, we first note that the amplitude B of the bend distortion in the N phase is B = |∇×n| ∼ 1/h [START_REF] De Gennes | An analogy between superconductors and smectics A[END_REF] and texture variations can be viewed as transition lines in a temperature-bend (T, B) phase diagram (Fig. 7A) analog to that of a superconductor (Fig. 1). The S-texture appears at low B and T values, when the S phase is fully established and the rotor field is expelled, analog to superconductor state of a metal. The observation that the S-texture grows from the L-texture upon cooling, expanding from thick to thin film regions (Fig. 2), indicates that the transition line into the S phase has a negative slope dB/dT < 0 (Fig. 7). Since for B = 0 this transition occurs at the undistorted bulk transition temperature T 0 , for B > 0 it must occur at a lower temperature. Therefore, the L-texture persists below T 0 and is either a depressed undercooled N phase or a frustrated smectic phase with penetration of the rotor field ∇×n.

The latter possibility can be excluded based on the following arguments. Longitudinal stripes similar to the L-texture have been observed in LC films with hybrid anchoring conditions both near the N-S transition [START_REF] Cladis | Growth of a smectic A from a bent nematic phase and the smectic light valve[END_REF][START_REF] Gim | Morphogenesis of liquid crystal topological defects during the nematic-smectic A phase transition[END_REF][START_REF] Liang | Towards tunable defect arrangements in smectic liquid crystal shells utilizing the nematic-smectic transition in hybridaligned geometries[END_REF] and in the N phase of LCs that do not have a smectic phase [START_REF] Lavrentovich | Stripe domain phase of a thin nematic film and the K13 divergence term[END_REF]. They can be theoretically explained for both degenerate and nondegenerate hybrid anchoring conditions as the result of an imbalance of elastic constants, leading to a mechanical instability with respect to small deviations of n from the xz-plane defined by the anchoring directions [START_REF] Gim | Morphogenesis of liquid crystal topological defects during the nematic-smectic A phase transition[END_REF][START_REF] Sparavigna | Saddle-splay and periodic instability in a hybrid aligned nematic layer subjected to a normal magnetic-field[END_REF][START_REF] Pergamenshchik | Surfacelike-elasticity-induced spontaneous twist deformations and long-wavelength stripe domains in a hybrid nematic layer[END_REF][START_REF] Sparavigna | Periodic stripe domains and hybridalignment regime in nematic liquid crystals: Threshold analysis[END_REF][START_REF] Manyuhina | Thin nematic films: Anchoring effects and stripe instability revisited[END_REF]. Longitudinal textures have also been observed under quasi-homeotropic (i.e., slightly hybrid) anchoring conditions [START_REF] Pergamenshchik | Stripe domains in a nearly homeotropic nematic liquid crystal: A bend escaped state at a nematic-smectic-A transition[END_REF] and occur in lieu of the Freedericksz's instability when the aligning field is perpendicular to a uniform director [START_REF] Goscianski | Field induced transitions in smectic A phases[END_REF][START_REF] Gooden | Simultaneous magnetic-deformation and light-scattering study of bend and twist elastic-constant divergence at the nematic-smectic-A phase transition[END_REF][START_REF] Lonberg | New ground state for the Splay-Fréedericksz transition in a polymer nematic liquid crystal[END_REF]. In all cases, stripes appear in the N phase and approaching the S phase, if present. For 8CB, the bend K 3 and twist K 2 constants diverge at the N-to-S transition, while the splay constant K 1 stays finite: K 3 >> K 2 >> K 1 [START_REF] Bradshaw | The Frank constants of some nematic liquid-crystals[END_REF]. The L-texture allows to partially exchange bend for twist and splay through a modulation of the director in the horizontal xy-plane, which cannot be extinguished between crossed polarizers (Fig. 4 A, B, E, andF). Importantly, stripe formation in the N phase can be explained without invoking any positional (smectic) order and can depress the N phase below T 0 [START_REF] Pergamenshchik | Stripe domains in a nearly homeotropic nematic liquid crystal: A bend escaped state at a nematic-smectic-A transition[END_REF]. We therefore identify the L-texture as a modulated N phase that persists below T 0 with an incomplete relaxation of the bend distortion (i.e., incomplete expulsion of ∇×n). The evolution of the L-texture into the S-texture has been studied for both degenerate [START_REF] Gim | Morphogenesis of liquid crystal topological defects during the nematic-smectic A phase transition[END_REF] or nondegenerate ( 24) hybrid anchoring conditions in the limit of small amplitudes B (i.e., large h). Since the N phase, L-texture, and S-texture can be mapped into the normal metal, undercooled metal, and superconductor state, respectively, the analogy with superconductors suggests that the P-texture must be a frustrated smectic phase (Fig. 1). The type I or type II character of a rod-like thermotropic LC such as 8CB is determined by its molecular structure and intermolecular interactions [START_REF] Renn | Abrikosov dislocation lattice in a model of the cholesteric-to-smectic-A transition[END_REF]. Common LC compounds optimized for electro-optics applications, including 9CB [START_REF] Dozov | Quantized grain boundaries in bent smectic-A liquid crystals[END_REF] and 8OCB [START_REF] Wang | Bend-induced melting of the smectic-A phase: Analogy to a type-I superconductor[END_REF], that are closely related to 8CB for molecular structure, order, and phase sequence, are known to be type I (1). Therefore, we identify P-texture formation in 8CB with the transition from the undercooled N phase to the intermediate phase of a type I LC. The analogy with type I superconductors is highlighted in Fig. 7A reproducing the typical features of the metal-superconductor phase diagram (Fig. 1). The intermediate state appears at temperatures T and amplitudes B immediately below a critical line with negative slope in the (T, B) phase diagram, and is eventually replaced by the S phase at low values of T and B, i.e., ∇×n is expelled (Fig. 7A) [START_REF] Poole | Superconductivity[END_REF]. The P-texture comprises periodically spaced S-phase domains (without bend) in equilibrium with regions of undercooled N phase (with bend) (Fig. 7B). Supporting the idea of a nematic-to-intermediate LC phase transition, first we note that evidence of a N phase persisting below T 0 in equilibrium with the S phase has emerged in studies of LCs subject to strong splay-bend [START_REF] Michel | Structure of smectic defect cores: X-ray study of 8CB liquid crystal ultrathin films[END_REF][START_REF] Coursault | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF] or twist [START_REF] Ruan | Coexisting nematic and smectic-A phases in a twisted liquid-crystal cell[END_REF] distortions. Second, our survey of P-texture formation shows that it is a general phenomenon, independent from the choice of LC compound or substrates (SI Appendix, Figs. S3 andS4). Last, but most important, the critical temperature T c of the transition from the undercooled N phase (L-texture) to the intermediate LC state (P-texture) decreases as the amplitude B increases (Fig. 5) in quantitative agreement with De Gennes' predictions, as shown below. In the N phase close to the S phase, the free energy density has the following form:

f = a|ψ| 2 + (b=2)|ψ| 4 + (C ||/ 2) ⃒ ⃒ ∇ || ψ -iq 0 n ⃒ ⃒ 2 + (C ⊥ =2)|∇ ⊥ ψ| 2 + (K 3 =2)B 2 + (K 1 =2)(∇ • n) 2 , [1]
where || and ⊥ indicate the direction parallel and perpendicular to n, respectively [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF]. Eq. 1 is formally identical to the Ginzburg-Landau free energy functional of a superconductor, except for the last term. The director n plays the role of the magnetic vector potential, B = ∇×n is equivalent to the magnetic field and q 0 to the charge -2e of a Cooper electron pair. The first and second energy terms are required to create a uniform order parameter ψ, whereas the third and fourth term are required to create order gradients ∇ψ different from the ground-state gradient iq 0 n. The fifth term is the bend elastic energy of the LC that is analog to the magnetic energy in superconductors. The last term is specific to LCs and represents the splay elastic energy. Also, the coefficients C ||,⊥ of an LC are a priori different due to anisotropy. The key argument made by De Gennes in analogy with the normal-to-superconductor transition, is that the N-to-S transition occurs when the energy (K 3 /2)B 2 equals the condensation energy -a 2 /2b. Introducing the longitudinal coherence length ξ = (C || /2a) 1/2 and bend penetration length λ = (K 3 b/ C || a) 1/2 q 0 (1, 27, 50), the equality can be written as h = 1/B ∼ 2 1/2 q 0 ξλ. In the N phase, ξ = ξ 0 t -ν , where ξ 0 q 0 ∼ 0.8, t = T/T 0 -1 is the reduced temperature and ν ∼ 2/3 is the critical exponent of the 3d-XY universality class [START_REF] Davidov | High-resolution x-ray and light-scattering study of critical behaviour associated with the nematic-smectic-A transition in 4-cyano-4′-octylbiphenyl[END_REF][START_REF] Litster | High resolution x-ray and light scattering studies of bilayer smectic A compounds[END_REF][START_REF] Sprunt | Equality of X-ray and light scattering measurements of coherence lengths at the nematic-smectic-a phase transition[END_REF][START_REF] Primak | Critical behavior at the nematic-to-smectic-A transition in a strong magnetic field[END_REF]. The penetration length can also be written as λ = (K 3 /β) 1/2 , where β = C || (q 0 |ψ 0 |) 2 is the layer compressibility modulus and ψ 0 is the undistorted bulk order parameter. K 3 diverges as ξ in the N phase [START_REF] Davidov | High-resolution x-ray and light-scattering study of critical behaviour associated with the nematic-smectic-A transition in 4-cyano-4′-octylbiphenyl[END_REF][START_REF] Bradshaw | The Frank constants of some nematic liquid-crystals[END_REF]. Since the ratio of bare correlation lengths below (ξ 0 -) and above (ξ 0 + ) the transition temperature T 0 is ξ 0 -/ξ 0 + = 3 in a 3d-XY system the bend constant in the smectic phase is K 3 -= 3K 3 [START_REF] Wang | Bend-induced melting of the smectic-A phase: Analogy to a type-I superconductor[END_REF]. On the other hand, the compressibility in the S phase shows a nonuniversal behavior: β = β 0 + β 1 |t| ϕ with β 0 ∼ 1 × 10 6 J/m 2 and β 1 ∼ 6 × 10 6 J/m 2 , and an anomalous exponent ϕ ∼ 0.47 [START_REF] Fisch | Absolute measurement of the critical behavior of the smectic elastic constant of bilayer and monolayer smectic-A liquid crystals on approaching the transition to the nematic phase[END_REF][START_REF] Benzekri | Nonvanishing of the layer compressional elastic constant at the smectic-A-to-nematic phase transition: A consequence of Landau-Peierls instability?[END_REF]. Using the values of K 3 and β from the literature [START_REF] Davidov | High-resolution x-ray and light-scattering study of critical behaviour associated with the nematic-smectic-A transition in 4-cyano-4′-octylbiphenyl[END_REF][START_REF] Bradshaw | The Frank constants of some nematic liquid-crystals[END_REF][START_REF] Fisch | Absolute measurement of the critical behavior of the smectic elastic constant of bilayer and monolayer smectic-A liquid crystals on approaching the transition to the nematic phase[END_REF][START_REF] Benzekri | Nonvanishing of the layer compressional elastic constant at the smectic-A-to-nematic phase transition: A consequence of Landau-Peierls instability?[END_REF], the penetration length is found to be λ = λ 0 t -η , where λ 0 = 1.1 nm and η = 0.31. The condensation condition h ∼ 2 1/2 q 0 ξλ can therefore be written as t c = (h 0 /h) γ , where γ = ν + η ∼ 0.98 and h 0 is a characteristic length. We also note that the Ginzburg parameter is κ = λ -/ξ -= 3 1/2 λ/3ξ = (1/3) 1/2 (λ 0 /ξ 0 )t ν-η = 1.57t 0.35 < 6 × 10 -2 over the entire temperature range of the smectic phase (t < 4 × 10 -2 ). Therefore, the condition κ < 1/ 2 1/2 for type I character of 8CB is verified.

Fig. 5 shows that, indeed, the critical temperature T c depends on the film thickness h in accord with the predicted dependence:

t c = 1 -T c /T 0 = (h 0 /h) γ . A fit of the temperature differences ΔT c = T c,2 -T c,1 = T 0 h 0 γ (1/h 1 γ -1/h 2 γ
) provides the characteristic length h 0 = 0.60 nm, from which the bare penetration length λ 0 = h 0 /2 1/2 q 0 ξ 0 = 1.0 nm can be calculated. This value is in remarkable agreement with the predicted value λ 0 = 1.1 nm and confirms that the L-to-P texture transition is indeed analog to the metal-to-intermediate state transition of type I superconductors.

The analogy also provides important insights into the (T, B) phase diagram of LCs. Transition lines become asymptotically parallel to the horizontal T-axis as temperature is decreased, so that the intermediate-to-superconductor and normal-to-intermediate transition lines intersect the vertical B axis, respectively, at values B 0 and B 1 in the limit T → 0 (Fig. 1) [START_REF] Poole | Superconductivity[END_REF]. Therefore, either the metal-to-intermediate transition is observed as T is decreased with B 1 > B > B 0 or the metal stays in the undercooled normal state for B > B 1 , without undergoing any transition. Indeed, only the transition from the undercooled nematic phase (i.e., L-texture) to the P-texture was observed in films thinner than h 0 = 1.5 μm, i.e., for bend amplitudes larger than B 0 = 1/h 0 . The P-texture persisted for thicknesses as small as 0.6 μm (Fig. 6B) rather than being replaced by the untextured N phase, indicating that the limit field B 1 was not reached.

The analogy with superconductors (Fig. 1) also suggests that a P-to-S transition should occur for B < B 0 . Therefore, the sequence L → P → S should be observed as T is decreased at a fixed and sufficiently low value of B (Fig. 7A). This suggests that the sL-texture (Figs. 2 and3 B and C), Left side) is an incomplete P-texture growing within the L-texture, whereas isolated FCD clusters (Fig. 4 D andH) are due to the incomplete growth of the S-texture within the P-texture. Nevertheless, the P-to-S transition was not clearly resolved in our experiments. A possible explanation is that the P-to-S transition line is almost horizontal for temperatures reaching values close to T 0 , in analogy with thin metal films with a high depolarization factor [START_REF] Poole | Superconductivity[END_REF]. Therefore, either the P-texture or S-texture is created when B is above or below the horizontal line, respectively. Alternatively, the growth of the S-texture from the P-texture could be hindered by large energy barriers, i.e., the P-texture could persist as a metastable state below the "ghost" P-to-S transition line. It has been shown that applying an electric field while cooling from the N phase induces P-texture formation in thick 8CB, instead of the Stexture expected in the absence of fields [START_REF] Gryn | Controlling the self-assembly of periodic defect patterns in smectic liquid crystal films with electric fields[END_REF]. When the field is switched off, the P-texture persists as a metastable state at temperatures and thicknesses where the S-texture should be observed. This behavior is due to different arrays of topological defects in the P-texture and S-texture that cannot be transformed into each other without breaking strong topological constraints. Therefore, topological defects can effectively prevent the development of the S-texture from the P-texture as temperature is decreased. Predicting the formation or geometry of patterns in the intermediate state of superconductors is known to be a theoretical challenge that cannot be solved solely by minimizing the Ginzburg-Landau free energy (first five terms in Eq. 1) [START_REF] Prozorov | Equilibrium topology of the intermediate state in type-I superconductors of different shapes[END_REF][START_REF] Prozorov | Suprafroth in type-I superconductors[END_REF]. Theoretical modeling of the P-texture in LCs is further complicated by the additional splay energy term in Eq. 1 [START_REF] Oswald | Smectic and Columnar Liquid Crystals[END_REF] and is beyond the scope of this article. However, the formation of 1d patterns at the transition into the intermediate state of both LC and superconductor films suggests a deep and general connection between frustration of the low-temperature (smectic or superconductor) phase and spontaneous breaking of symmetry in the film plane. It is noteworthy that the period d of the P-texture increases when T and/or B decrease in 8CB (Figs. 5 and6), whereas the opposite behavior (pattern coarsening) is observed in the intermediate state of a metal [START_REF] Poole | Superconductivity[END_REF][START_REF] Prozorov | Equilibrium topology of the intermediate state in type-I superconductors of different shapes[END_REF][START_REF] Prozorov | Suprafroth in type-I superconductors[END_REF]. This suggests that d increases to accommodate periodically spaced domains of the S phase that grow to the expense of undercooled nematic regions. Fig. 7B shows the likely internal structure of the P-texture based on available experimental data [START_REF] Michel | Optical gratings formed in thin smectic films frustrated on a single crystalline substrate[END_REF][START_REF] Coursault | Linear self-assembly of nanoparticles within liquid crystal defect arrays[END_REF][START_REF] Gryn | Controlling the self-assembly of periodic defect patterns in smectic liquid crystal films with electric fields[END_REF][START_REF] Parodi | A possible magnetic transition in smectics-A[END_REF][START_REF] Coursault | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF][START_REF] Zappone | Self-ordered arrays of linear defects and virtual singularities in thin smectic-A films[END_REF][START_REF] Coursault | Modeling the optical properties of self-organized arrays of liquid crystal defects[END_REF]. To satisfy both surface anchoring conditions and keep a constant layer spacing, the smectic layers tend to be flat and parallel near the surface inducing homeotropic anchoring, whereas they tend to form hemicylindrical domains near the surface inducing planar anchoring. Although previous models have tried to resolve the complex balance of bulk elastic forces and surface anchoring in the P-texture mainly by introducing various types of topological defects at the boundary between, the structure should also comprise domains of bent undercooled N phase at the boundary between curved but bend-free smectic domains. Further theoretical effort is needed to build a predictive model starting from Eq. 1.

Conclusions

Our study shows that fundamental aspects of the N-S phase transition can be studied by deforming the director field in thin LC films using external aligning field (i.e., surface interactions) [START_REF] Dozov | Quantized grain boundaries in bent smectic-A liquid crystals[END_REF][START_REF] Wang | Bend-induced melting of the smectic-A phase: Analogy to a type-I superconductor[END_REF][START_REF] Ruan | Coexisting nematic and smectic-A phases in a twisted liquid-crystal cell[END_REF], as opposed to observing thermodynamic fluctuations in uniform LC samples [START_REF] Davidov | High-resolution x-ray and light-scattering study of critical behaviour associated with the nematic-smectic-A transition in 4-cyano-4′-octylbiphenyl[END_REF][START_REF] Litster | High resolution x-ray and light scattering studies of bilayer smectic A compounds[END_REF][START_REF] Sprunt | Equality of X-ray and light scattering measurements of coherence lengths at the nematic-smectic-a phase transition[END_REF][START_REF] Primak | Critical behavior at the nematic-to-smectic-A transition in a strong magnetic field[END_REF]. In particular, our findings confirm De Gennes' prediction of a depressed N-to-S transition leading to the "intermediate" frustrated smectic phase. In view of the bottom-up production of micropatterns with a large response to external stimuli, we anticipate that the 1d P-texture of the intermediate LC state can be conveniently produced by applying a strong electric field to a uniform LC film with planar anchoring conditions [START_REF] Gryn | Controlling the self-assembly of periodic defect patterns in smectic liquid crystal films with electric fields[END_REF]. Our findings are particularly relevant to the design of self-assembled LC templates that are used for guiding the ordering and assembly of functional nanoparticles into anisotropic structures with interesting plasmonic and charge-transport properties [START_REF] Coursault | Linear self-assembly of nanoparticles within liquid crystal defect arrays[END_REF][START_REF] Coursault | Tailoring anisotropic interactions between soft nanospheres using dense arrays of smectic liquid crystal edge dislocations[END_REF][START_REF] Honglawan | Synergistic assembly of nanoparticles in smectic liquid crystals[END_REF][START_REF] Gryn | Electric-field-controlled alignment of rod-shaped fluorescent nanocrystals in smectic liquid crystal defect arrays[END_REF][START_REF] Ro | Oriented gold nanorods and gold nanorod chains within smectic liquid crystal topological defects[END_REF].

Materials and Methods

The LC compound 8CB was purchased from Sigma-Aldrich. Homeotropic anchoring was obtained by dip-coating a glass substrate with a silane monolayer (octadecyl-trimethoxy-silane from Sigma-Aldrich), whereas unidirectional parallel anchoring was obtained by spin-coating the glass substrate with a thin polymer film (poly-imide PI2555 from Hitachi Chemical DuPont MicroSystems) that was gently rubbed with a velvet cloth along the anchoring direction.

Before depositing the silane and polymer, the surfaces were etched with a solution containing hydrofluoric acid to create channels with rectangular cross-sections, step-like edges narrower than 50 μm, and depths between 70 and 700 nm [START_REF] Zacheo | Fast and safe microwave-assisted glass channel-shaped microstructure fabrication[END_REF]. The step height Δh was measured using a stylus profilometer (Dektak 8 by Veeco) with a precision of about 20 nm. After assembling the surfaces and before filling the cell with the LC, the thickness h was determined with an accuracy of about 20% via analysis of the cell's optical transmittance, showing multiple-beam interference due to reflections at air-glass interfaces.

The dependence of transition temperatures on the LC film thickness h (e.g., the difference ΔT in Fig. 5) was studied in a narrow region across a step-like edge to avoid temperature gradients, i.e., both sides of the step were at the same temperature. The cells were sealed in a temperaturecontrolled enclosure with 0.01 °C accuracy and observed in transmission between crossed polarizers. Note that the reported temperature values were read from the temperature controller and may differ from the actual values reached in the region where texture transitions were observed.

The authors have shared as SI Appendix all datasets appearing in this article.

  Analogy between periodic patterns in thin smectic liquid crystal films and the intermediate state of superconductors Bruno Zappone a,1 , Atilla Eren Mamuk b , Iryna Gryn c , Valentina Arima d , Alessandra Zizzari d , Roberto Bartolino c , Emmanuelle Lacaze e , and Rolfe Petschek f a Consiglio Nazionale delle Ricerche, Istituto di Nanotecnologia, c/o Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy; b Laboratory of Liquid and Solid Crystals, Department of Physics, Mu gla Sitki Koçman University, TR48000, Mu gla, Turkey; c Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy; d Consiglio Nazionale delle Ricerche, Istituto di Nanotecnologia, c/o Campus Ecotekne, Università del Salento, 73100 Lecce, Italy; e Institut des Nano-Sciences de Paris, CNRS, Sorbonne Université, 75005 Paris, France; and f Department of Physics, Case Western Reserve University, Cleveland, OH 44106

  liquid crystals | superconductors | phase transitions | pattern formation | phase frustration

Fig. 1 .Fig. 2 .

 12 Fig. 1. Schematic phase diagram of a type I superconductor showing a frustrated intermediate state between the normal metal and superconductive states. The magnetic field B is completely expelled from the latter state, whereas it penetrates in the intermediate state via finite-sized domains of undercooled normal metal. T c and B c are the critical temperature and field of the transition from the normal state.

  C and D, Right; Fig. 6A; and SI Appendix, Fig. S2 in SM). Near T c , the slope of (dd c )/d c vs. temperature was 4.4 K -1 in average and did not correlate with d c . The slope decreased and d became almost independent from temperature at room conditions, where previous observations were conducted (10, 20). When different thicknesses h (and therefore different temperatures T c ) were considered, the period d c increased with h following an approximately linear relation: d c = d 0 + αh with d 0 = (0.6 ± 0.1) μm and α = (1.1 ± 0.1) (Fig.

Fig. 3 .

 3 Fig. 3. Textures observed across Q: 14 a step of thickness upon cooling an 8CB film with intermediate (1.5 μm < h < 2.0 μm) and small (h ≤ 1.5 μm) thickness. P is a texture with smooth and straight stripes perpendicular to the planar anchoring direction x.

Fig. 4 .

 4 Fig. 4. Transition from the L-texture to the P-texture observed upon cooling an 8CB film with intermediate thickness (1.5 μm < h < 2.0 μm). The circles highlight an isolated cluster of FCDs within the P-texture. The cross indicates the direction of the crossed polarizers. (A-D) The P-texture was completely extinguished when the analyzer was parallel to the planar anchoring direction x, whereas the L-texture could not be extinguished. (E-H) The same region observed with the polarizers oblique to x to highlight the P-texture. The rectangle at the bottom schematically shows the in-plane director modulation of the L-texture.

Fig. 5 .

 5 Fig.5. Variation ΔT c of the critical temperature for the appearance of the P-texture as a function of thickness variations Δ(1/h γ ). T c was obtained upon cooling. The dotted line is a linear fit to the theoretical equation ΔT = T 0 Δ[(h 0 /h) γ ] with h 0 = (0.60 ± 0.03) nm, T 0 = 306.5 K, and γ = 0.98. The circles and squares indicate measurements taken, respectively, across a step or at the crossing between two steps (each on opposite cell boundaries, SI Appendix, Fig.S2). Representative error bars are shown for three different measurements.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Variation of the P-texture period d as a function of the temperature T and film thickness h. (A) Increase of d as a function of decreasing T relative to the critical values T c and d c = d(T c ). Different colors indicate different values of d c , specified in the color bar. (B) Increase of d c with h. The dashed line has equation d c = d 0 + αh with d 0 = (0.6 ± 0.1) μm and α = (1.1 ± 0.1). The data and symbols correspond to those of Fig. 5.

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| www.pnas.org/cgi/doi/10.1073/pnas.2000849117 Zappone et al.

ACKNOWLEDGMENTS. We are grateful to Charles Rosenblatt at Case Western Reserve University for his comments and suggestions. We thank Roberto Termine of Consiglio Nazionale delle Ricerche, Istituto di Nanotecnologia, for his help with the stylus profilometer. A.E.M. also acknowledges the support of the Scientific and Technical Research Council of Turkey-International Doctoral Research Fellowship Programme 2214/A.

AUTHOR QUERIES

AUTHOR PLEASE ANSWER ALL QUERIES


Syou -P'heng Do

Amine Missaoui

Alessandro Coati

Delphine Coursault

Haifa Jeridi

Andrea Resta

Nicolas Goubet

Michal M Wojcik

Arnaud Choux

Sébastien Royer

Emrick Briand

Bertrand Donnio

Jean Louis Gallani

Brigitte Pansu

Emmanuel Lhuillier

Yves Garreau

David Babonneau

Michel Goldmann

Doru Constantin

Bruno Gallas

Bernard Croset

Emmanuelle Lacaze
email: emmanuelle.lacaze@insp.jussieu.fr

From Chains to Monolayers : Nanoparticle Assembly Driven by Smectic Topological Defects

Keywords: Smectic, Topological defects, Oily Streaks, Nanoparticles, gold, X-ray, dislocations, assembly

In this article, we show how advanced hierarchical structures of topological defects in the so-called smectic oily streaks can be used to sequentially transfer their geometrical features to gold nanospheres. We use two kinds of topological defects, 1D dislocations and 2D ribbon-like topological defects. The large trapping efficiency of the smectic dislocation cores not only surpasses that of the elastically distorted zones around the cores but also the one of the 2D ribbon-like topological defect. This enables the formation of a large number of aligned NP chains, within the dislocation cores that can be quasi-fully filled without any significant aggregation outside the cores. When the NP concentration is large enough to entirely fill the dislocation cores, the LC confinement varies from 1D to 2D. We demonstrate that the 2D topological defect cores induce a confinement that leads to planar hexagonal networks of NPs. We then draw the phase diagram driven by NP concentration, associated with the sequential confinements induced by these two kinds of topological defects. Owing to the excellent large-scale order of these defect cores, not only the NP chains but also the NP hexagonal networks can be oriented along the desired direction, suggesting a possible new route for the creation of either 1D or 2D highly anisotropic NP networks. In addition, these results open rich perspectives based on the possible creation of coexisting NP assemblies of different kinds, localized in different confining areas of a same smectic film that would thus interact thanks to their proximity but also would interact via the surrounding soft matter matrix.

Liquid crystal (LC) topological defects are fascinating objects, which are actively studied nowadays as model systems for a deep understanding of topological matter [START_REF] Isaac | Cosmology in the Laboratory: Defect Dynamics in Liquid Crystals[END_REF][START_REF] Tom | Phase-Transition Dynamics in the Lab and the Universe[END_REF][START_REF] Ball | Orientability and Energy Minimization in Liquid Crystal Models[END_REF][START_REF] Campo | Universality of phase transition dynamics: Topological defects from symmetry breaking[END_REF] .It has been also discovered that they can attract and trap inorganic nanoparticles (NPs). The trapping of NPs within topological defect cores allows for the release of the defect core energy and for the stabilization of the composite systems [START_REF] Yoshida | Nanoparticlestabilized cholesteric blue phases[END_REF][START_REF] Eva | Nanoparticle-induced widening of the temperature range of liquid-crystalline blue phases[END_REF][START_REF] Brigita | Theoretical and experimental study of the nanoparticle-driven blue phase stabilisation[END_REF][START_REF] Bohdan | Different modulated structures of topological defects stabilized by adaptive targeting nanoparticles[END_REF][9] . LC topological defects may thus be useful as a matrix for new kinds of NP organizations templated by the defect geometry [10][START_REF] Mohamed | Reversible Nanoparticle Cubic Lattices in Blue Phase Liquid Crystals[END_REF][START_REF] Haridas | Tuning and Switching a Plasmonic Quantum Dot Sandwich in a Nematic Line Defect[END_REF] . The research on LC topological defects is constantly expanding and many attractive systems are emerging, in particular characterized by the formation of different kinds of topological defects in a single system, like LC droplets or shells [START_REF] Lopez-Leon Teresa | Drops and shells of liquid crystal[END_REF][START_REF] Lopez-Leon | Frustrated nematic order in spherical geometries[END_REF][START_REF] David | Geometrical frustration of chiral ordering in cholesteric droplets[END_REF][START_REF] Alexandre | Waltzing route toward double-helix formation in cholesteric shells[END_REF] or even oriented thin smectic films [START_REF] Delphine | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF][START_REF] Ian | Observations of a streak texture in the hybrid-aligned smectic-C phase[END_REF] .

Our aim is thus to go beyond the demonstration of the trapping phenomenon of NPs by LC topological defects and to understand the specificity of each topological defect regarding their NP templating property. In particular, can we use hierarchical structures of topological defects in LC materials to hierarchically build composites made of different kinds of NP assemblies ?

To answer this question, we have used smectic oily streaks made of an array of oriented topological defects, smectic dislocations (1D topological defect), together with ribbon-like grain boundaries (2D topological defect) [START_REF] Delphine | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF] . We have increased the concentration of gold NPs whose Localized Surface Plasmon (LSP) resonance in the smectic oily streaks is followed as a function of the local concentration. By combining spectrophotometry, Rutherford Back Scattering (RBS) measurements and X-ray diffraction at the Soleil synchrotron facility, we bring evidence for two different confinement features for the gold NPs. A 1D confinement starts first, where NP chains are formed until the dislocations are quasi-fully filled by the NPs, followed by a 2D confinement at higher concentration. We demonstrate that the 2D confinement induced by 2D topological defects leads to formation of planar oriented hexagonal networks of NPs ( 10 direction more favorably oriented parallel to the LC defect orientation). Our results also highlight how a hierarchical structure of topological defects can be used to sequentially build different kinds of NP assemblies, in relation with different kind of induced confinement by each topological defect. Ultimately this leads to the coexistence of 1D and 2D NP assemblies at different localizations in the same film, opening rich perspectives for future composites formed within LC systems made of different kinds of topological defects. We have used arrays of oriented smectic defects of 8CB (4'-octyl-4-biphenylcarbonitrile) thin films deposited on rubbed poly(vinyl alcohol) (PVA) surfaces, the so-called smectic oily streaks [START_REF] Michel | Structure of Smectic Defect Cores: X-Ray Study of 8CB Liquid Crystal Ultrathin Films[END_REF] . Due to hybrid anchoring at the two interfaces (air/8CB and 8CB/PVA, respectively), the smectic layers become curved in flattened hemicylinders perpendicular to the anchoring on the substrate, itself defined by the rubbing of the substrate (Figure 1). [START_REF] Michel | Structure of Smectic Defect Cores: X-Ray Study of 8CB Liquid Crystal Ultrathin Films[END_REF][START_REF] Michel | Structure of Smectic Defect Cores: X-Ray Study of 8CB Liquid Crystal Ultrathin Films[END_REF][START_REF] Bruno; Lacaze Emmanuelle | Surface-frustrated periodic textures of smectic-A liquid crystals on crystalline surfaces[END_REF] These flattened hemicylinders, with typical periodicity of several hundreds of nanometers, can be detected by polarized optical microscopy (POM) between crossed polarizers, leading to the observation of parallel stripes (Figure 1a). Their internal structure has been determined using combined X-ray diffraction and ellipsometry measurements (Figure 1b). [START_REF] Delphine | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF] It is associated with the presence of two rotating grain boundaries per hemicylinder (in red in Figure 1b). Along each rotating grain boundary, three edge dislocations are expected (in blue in Figure 1b), in relation with a different number of curved and flat smectic layers from each part of the rotating grain boundary. For one given hemicylinder, this leads to six edge dislo-cations, all being parallel to each other (i.e., oriented along the OX direction, perpendicular to the substrate rubbing, parallel to the hemicylinder axis and thus parallel to the stripes visible by POM -Figure 1) and separated from each other by a distance of approximately 50 nm. It has been shown that the rotating grain boundary profile does not depend on the thickness of the smectic film. [START_REF] Delphine | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF] This consequently leads to a constant number of 6 1D-linear topological defects per hemicylinder. A sublayer made of smectic layers perpendicular to the substrate has been also revealed, allowing to preserve the planar unidirectional anchoring on the PVA surface. A central defect is thus induced (green line in Figure 1b), directly on top of the sublayer. It possibly corresponds to a locally melted 2D area, thus forming a 2D ribbon-like topological defect per hemicylinder, of width around 400nm. [START_REF] Delphine | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF] Composite films (Oily streaks of 8CB containing gold nanoparticles) have been obtained (see section 'Materials and Methods'). Different diameters of nanoparticles were investigated : D = 5, 6, 7, and 8.3 nm, with a specific focus on the diameter D = 6 nm (Figure SI2).

The latter NPs are covered by dodecanethiol ligands. We tuned the average concentration of NPs, c avg , in the 8CB films between 700 and 2500 NPs µm -2 . These surface coverages were determined from Rutherford Back Scattering (RBS) measurements (see section 'Materials and Methods'). For c avg between 700 and 1300 NPs µm -2 , we have identified by POM two distinct structures: either the normal oily streaks, corresponding to well-dispersed NPs (Figure 2a); or locally more densely concentrated domains of NPs forming superstructures superimposed onto the normal oily streak pictures (Figure 2b). For both structures, the oily streak texture observed by POM between crossed polarizers appears only slightly modified with respect to pure 8CB (Figure SI3). The two different kinds of NP localization appeared to be difficult to control in relation with the solvent evaporation process. However, for average concentrations above c avg = 1300 NPs µm -2 , only the second kind of texture was observed (Figure SI3).

Firstly focusing on the well-dispersed NPs (Figure 2a), Figure 2c displaying the LSP resonance of the gold NPs with a diameter D = 6 nm embedded in the 8CB oily streaks. For a polarization of the incident light perpendicular to the oily streaks (black spectrum), the extinction wavelength is slightly smaller than the one of NPs dispersed in toluene (Figure SI2, λ o = 516 nm), with λ ⊥ = 500 nm. This confirms previous findings showing that the optical index for LSP resonance is dominated by the dodecanethiol ligands grafted around the NPs. [START_REF] Mulvaney | Surface Plasmon Spectroscopy of Nanosized Metal Particles[END_REF][START_REF] Delphine | Tailoring Anisotropic Interactions between Soft Nanospheres Using Dense Arrays of Smectic Liquid Crystal Edge Dislocations[END_REF] Otherwise a red-shift of the LSP resonance would have been observed. In contrast, a clear red-shift of the LSP resonance is observed for a parallel polarization, with λ = 550 nm. This phenomenon is not due to the 8CB birefringence (n o = 1.52, n e = 1.67) since the 8CB variation of optical index with polarization in relation with oily streak geometry would have induced a blue-shift of the LSP resonance. The observed red-shift indicates that a significant coupling between NPs occurs only in the direction parallel to the oily streak dislocations, in relation with the formation of chains all parallel from each other and parallel to the dislocations. [START_REF] Delphine | Tailoring Anisotropic Interactions between Soft Nanospheres Using Dense Arrays of Smectic Liquid Crystal Edge Dislocations[END_REF][START_REF] Delphine | Linear Self-Assembly of Nanoparticles Within Liquid Crystal Defect Arrays[END_REF] All the extinction measurements for the welldispersed NPs with D = 6 nm are represented with blue stars in the graph of λ as a function of λ ⊥ (Figure 3). It appears that while λ ⊥ varies only little from one area to another, i.e.

the NPs remain individual in this direction, λ changes substantially, with a maximum for λ equal to 562 nm. This shows that only chains parallel to the LC linear defects are formed, the longer chains being associated with λ = 562nm. Secondly focusing on the zones with locally dense domains (Figure 2b), the optical properties of the NPs differ from the zones with well-dispersed NPs (Figures 2c and2d). On Figure 2d, λ is still largely red-shifted, but now λ ⊥ also appears red-shifted with respect to the LSP band wavelength of NPs in toluene (Figure SI2). This indicates that a coupling between NPs also occurs in the direction perpendicular to the smectic oily streaks. However λ ⊥ is still smaller than λ . Figure 3 displays in yellow and purple the corresponding values of (λ ⊥ , λ ) for each measured zone.

Each extinction graph associated with the yellow and purple spots has been fitted by a single gaussian curve to obtain λ ⊥ and λ mean values . It shows that λ ⊥ regularly increases but remains always smaller than λ , finally reaching a maximum corresponding to λ ⊥ ≈ λ ≈ 580 nm (purple spot), associated with the densest and/or largest NP organizations.

In order to gain more insight on the second part of the diagram drawn in Figure 3, we have analyzed the sample corresponding to the purple dots displayed in Figure 3 by grazing incidence small-Angle X-ray scattering (GISAXS) and transmission small-Angle X-ray scattering (TSAXS) performed at Soleil synchrotron facility (SIXS beamline). The sample corresponds to c avg = 2500 NPs µm -2 (RBS measurements), with POM pictures displaying the presence of dense NP domains, similar to Figure 2b (Figure SI3). Figure 4b shows the GISAXS pattern collected with the X-ray beam parallel to the 8CB stripes (see 'Materials and methods') while GISAXS data collected with the same NPs without 8CB (ie. just deposited on rubbed PVA) are presented in Figure 4a. We will denote by q the scattering vector, q z its component normal to the substrate, q its in-plane component parallel to the oily streaks and q ⊥ its in-plane component perpendicular to the oily streaks. In Figure 4a, three symmetric pairs of rods are visible with their respective q ⊥ positions being in the ratio 1, √ 3, 2. This result indicates that the spherical NPs exhibit a planar hexagonal organization, as usually observed when NPs are directly deposited from the solution on a substrate. This is thus in agreement with observations carried out by scanning electron microscopy (Figure SI4). The presence of well-defined rods indicates a good quality of the NPs in terms For the sake of simplicity, the intensity map is the sum of two simulations corresponding to q 10 parallel to q ⊥ and to q 11 parallel to q ⊥ . The DWBA calculations were done using the FitGISAXS software package [START_REF] Babonneau David | FitGISAXS: software package for modelling and analysis of GISAXS data using IGOR Pro[END_REF] .

of polydispersity (size and shape). The three pairs of rods correspond to the ((10), (10))

, (( 11), (11))and (( 20), (20)) rods of the reciprocal hexagonal mesh, with the q ⊥ position of the (10) rod being q (10) = 0.92 nm -1 . At large q z , the different characteristic features of the different rods can be unambiguously attributed to the form factor of a dense sphere,

F (q) = 4π sin(qR) -qR cos(qR) q 3
. The (10) rod profile exhibits two detectable maxima, (10) and (10) , in contrast with the (11) rod, whose first maximum appears at a larger q z value than for the (10) and (20) rods. This form factor effect is clearly evidenced in Figure 4c, which shows the GISAXS intensity calculated in the DWBA using the FitGISAXS package [START_REF] Babonneau David | FitGISAXS: software package for modelling and analysis of GISAXS data using IGOR Pro[END_REF] for a planar hexagonal network of spheres (D = 6 nm) supported on a PVA/glass substrate.

The main difference between the experimental and simulated patterns lies in the shape of the (10) rod at q z ≈ 0.7 nm -1 that suggests the presence of some multilayers coexisting with monolayers in the sample without LC.

In Figure 4b, a half-circle with radius q LC ≈ 2 nm -1 appears. It corresponds to the scattering of the 8CB smectic layers, stacked on top of each other and forming hemicylindrical structures (Figure 1) [START_REF] Delphine | Self-organized arrays of dislocations in thin smectic liquid crystal films[END_REF] . Although this half-circle almost prevents the observation of the (20) and (20) rods, the ((10), (10)) and (( 11), (11)) rods are discernable at the same q z position as without 8CB, their q ⊥ positions being in the ratio 1, √ 3, with q (10) = 0.99 nm -1 . At large q z around 1.7nm -1 , the (10) signal is also clearly visible, with the same intensity variation along q z with respect to the (10) than without 8CB (see Figure SI5). The presence of vertical rods and the similarity of their profile above the horizon with the one of Figure 4c demonstrate that in 8CB also the NPs are arranged in a planar 2D hexagonal network of spheres parallel to the substrate. The (10) rod on Figure 4b has a shape very close to the one of Figure 4c, without any bump at q z ≈ 0.7 nm -1 , in contrast with Figure 4a. The presence of this bump at q z ≈ 0.7 nm -1 was observed in only 3 areas over 10, meaning that in 8CB, monolayers are preferentially formed rather than multilayers. The (10) rod has been analyzed in addition through TSAXS measurements with the sample perpendicular to the X-ray beam ('Materials and methods'). The obtained scattered circle (Figure 5a) shows that the NP domains are made of hexagonal meshes of NPs presenting a large distribution of possible crystallographic orientations (2D powder). However a careful analysis of the scattered intensity along this circle reveals the presence of an intensity modulation with a 60 o period and maxima of scattered intensity at 0 o , 60 o , 120 o and 180 o (Figure 5b). This confirms the hexagonal structure of the NP networks in 8CB. A preferred orientation of the hexagonal meshes in 8CB can be deduced in contrast with the NP network without LC (Figure SI6). The dense direction of the NP network, the 10 direction, is preferably parallel to the oily streaks, in other words parallel to the chain direction observed at smaller concentration and parallel to the LC dislocations. In agreement with the comparison between Figure 4b and Figure 4a, this favorable orientation observed in LC leads to a smaller intensity of the (11) rods in GISAXS measurements in LC (Figure 4b) with respect to the case without LC (Figure 4a). The q ⊥ position of the (10) rod in 8CB q (10) = 0.99 nm -1 corresponds to an average gap between NPs equal to s = 4 × π × √ 3/q (10) -D = 1.34 ± 0.05 nm, after averaging over 12 areas of the sample. For the NP hexagonal network on rubbed PVA without LC (q (10) = 0.92 nm -1 ), a gap of s = 1.93 nm ± 0.05 nm can be extracted as averaged over 5 zones. This value is very close to the one previously obtained for similar NP hexagonal networks with NPs of diameter D = 6 nm, also covered by dodecanethiol [START_REF] Nicolas | Which Forces Control Supracrystal Nucleation in Organic Media?[END_REF][START_REF] Melanie | Mechanical Properties of Au Supracrystals Tuned by Flexible Ligand Interactions[END_REF] . This demonstrates a LC-induced shortening of the inter-NP gap in the hexagonal network of NPs in LC with respect to hexagonal networks without 8CB. This is consistent with the largest LSP resonance wavelength for the purple dots on Figure 3 with respect to the green dot associated with hexagonal networks without LC on PVA substrates.

Our analysis reveals that in oily streaks, for NPs well-dispersed in small enough amount, chains of NPs are formed with NPs of diameter 6 nm, but also with NPs of diameter 5 nm, 7 nm and 8.3 nm (see Figure SI7). While it had been demonstrated that non distorted smectic films could avoid any aggregation of similar gold NPs 28 , for distorted smectic films, the role of linear defect cores had been previously highlighted: Nanorods are aligned in oily streaks perpendicular to the LC director but parallel to the defect cores [START_REF] Laurent | Alignment of Rod-Shaped Single-Photon Emitters Driven by Line Defects in Liquid Crystals[END_REF]30 . This demonstrates show the presence of 2D hexagonal monolayers that could coexist with 1D linear NP chains.

This would explain the small remaining anisotropy of the optical absorption associated with yellow and purple spots. In order to further test this hypothesis, we have subtracted from the extinction shown in Figure 2d the extinction of long chains (λ ⊥ = 500 nm, λ = 562 nm) without normalization and with the use of a multiplying factor of 0.48. This factor takes into account the not full coverage of NPs on the surface already suggested by Figure 2b.

A quasi isotropic extinction is obtained at λ = 558 nm (Figure SI8). It is consistent with coexisting long chains and 2D monolayer patches with the assumption that the anisotropy of the surrounding 8CB matrix is of negligible influence [START_REF] Mulvaney | Surface Plasmon Spectroscopy of Nanosized Metal Particles[END_REF][START_REF] Delphine | Tailoring Anisotropic Interactions between Soft Nanospheres Using Dense Arrays of Smectic Liquid Crystal Edge Dislocations[END_REF][START_REF] Yong Parka | Splitting of surface plasmon frequencies of metal particles in a nematic liquid crystal[END_REF] . This fitting procedure can be performed for each blue or orange spot shown on Figure 3. The obtained values for the λ of the hexagonal monolayer patches are shown in Figure 3e. λ increases from 553 nm to 586 nm when the local concentration increases. This may correspond to decreasing inter-NP gaps or/and increasing sizes of the hexagonal networks when the local concentration increases.

The question to solve now concerns the origin of these 2D NP hexagonal networks, their shape and localization in the 8CB matrix.

It is known that the dodecanethiol ligands grafted on gold substrates mostly induce tilted or homeotropic anchoring of LCs [START_REF] Hao; Hegmann Torsten | Multiple Alignment Modes for Nematic Liquid Crystals Doped with Alkylthiol-Capped Gold Nanoparticles[END_REF][START_REF] Blanc | Ordering nano-and microparticles assemblies with liquid crystals[END_REF] . We thus expect that if the NPs would be directly deposited on top of the substrate, they would destroy the 8CB oily streaks, which is not observed. In contrast, in presence of the dense NP domains, we observed clear GISAXS rings associated with the smectic layer structure of the oily streaks (Figure 4b). Moreover POM pictures between cross polarizers show the oily streak stripes everywhere including on top of the domains (Figure SI3). These POM observations also demonstrate that the NP domains do not sit at the 8CB/air interface. Otherwise they would lead to a decrease of the POM contrast between crossed polarizer which is not observed. The fact that dense NP domains do not sit at the 8CB/air interface is confirmed by a careful analysis of the RBS signal of the Au atoms in the 8CB film. A signal of larger width for the samples with dense NP domains with respect to the ones where only NP chains are formed is observed (Figure SI9). This is linked with a large number of Au atoms deeply embedded in the LC film in presence of NP domains, that are not observed when only chains are formed in the LC film. The only interpretation for these experimental results is thus the following: The 2D NP networks are localized elsewhere than in and around the LC dislocations. They are more deeply embedded in the LC film and must be confined in the planar 2D ribbon-like defects of each oily streak hemicylinder (green line in Figure 1). The 2D ribbon-like shape of this central defect, parallel to the substrate, of width around 400 nm may induce a 2D confinement leading to the observed

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The Supporting Information is available free of charge on the at Details about nanoparticles synthesis and characterization, simple preparation, X-ray and Rutherford backscattering spectroscopy (PDF).

Figure 1 :

 1 Figure 1: (a) Polarized optical microscopy image of oily streaks observed from the top of a 150 nm-thick sample. Each stripe along OX corresponds to one hemicylinder and two hemicylinders in perspective are represented (b) Detail in side-view (in the (Y, Z) plane) of the smectic layers stacked in one flattened hemicylinder with two rotating grain boundaries (RGB) in red, including dispersed edge dislocations (blue spots) and a central grain boundary in green 17 . The smectic layers are represented for a 8CB thin film with typical thickness of 230 nm.

Figure 2 :

 2 Figure 2: (a) POM picture with parallel polarizers for an area of well-dispersed NPs of diameter D = 6 nm. (b) POM picture with parallel polarizers for an area where the NPs (diameter D = 6 nm) form locally denser domains. (c) Normalized extinction spectrum measured in the center of Figure 2a with superimposed gaussian fits for the determination of the resonance wavelength (see Materials and Methods). In black is the extinction for light polarization perpendicular to the 8CB oily streak stripes, in red for the parallel polarization; (d) Normalized extinction spectrum measured in the center of Figure 2b.

Figure 3 :

 3 Figure 3: (a) Summary of the extinction measurements represented by the (λ ⊥ , λ ) values, obtained for NPs with diameter D = 6 nm. (b) In blue are shown the measurements for chains, corresponding to areas with well-dispersed NPs as seen by POM. (c) In red is presented the measurement for isolated nanoparticles in solution. (d) In yellow and purple are shown the measurements, corresponding only to areas where dense domains of NPs are identified by POM. Specifically the measurements performed on the sample with the highest NP concentration are shown in purple. In green is the result obtained for a monolayer of NPs without LC (λ ⊥ = λ = 558 nm). The dotted line is of slope 1, corresponding to isotropic optical properties (λ ⊥ = λ ). (e) the same data differently analyzed : the extinction of the "long chains" (blue spot of largest λ ) has been subtracted from the extinction of the yellow and purple spots to lead to isotropic extinction, of λ values again shown in yellow and purple.

Figure 4 :

 4 Figure 4: (a) GISAXS pattern of a sample with NPs deposited on a rubbed polymer substrate without LC, with the various rods of the hexagonal NP assembly being indicated. (b) GIXAXS pattern for a sample of average NP concentration of c avg =2500 NPs µm -2, collected with the X-ray beam parallel to the stripes, where the semi-circle corresponds to the hemicylindrical stacking of the smectic layers and the various scattering rods correspond to the signal of the hexagonal NP assembly. (c) Simulated GISAXS pattern for a planar hexagonal network of spheres (D = 6 nm) supported on a PVA/glass substrate. For the sake of simplicity, the intensity map is the sum of two simulations corresponding to q 10 parallel to q ⊥ and to q 11 parallel to q ⊥ . The DWBA calculations were done using the FitGISAXS software package[START_REF] Babonneau David | FitGISAXS: software package for modelling and analysis of GISAXS data using IGOR Pro[END_REF] .

Figure 5 :

 5 Figure 5: (a) TSAXS pattern of the same sample as in Figure 4b, observed in transmission with the X-ray beam perpendicular to the sample. The circle corresponds to the (10) rod visible in Figure 4b and demonstrates the possible orientations of the hexagonal crystals in the NP domains (b) Modulation of the intensity along the circle extracted after careful removal of the background. The angle α corresponds to the various positions along the circle shown in (a), the 0 and 180 o values corresponding to the horizontal, defined as parallel to the 8CB dislocations.

Figure 1 )Figure 5 .

 15 Figure1), which is parallel to the oily streaks. The 2D ribbon-like topological defect however only partially orients the trapped hexagonal network as shown by the structured circle of Figure5. When the NP concentration increases some monolayer patches could grow far from the central defect border and would become disoriented. This suggests an increase of disorder when the concentration increases, indeed observed in the hexagonal networks of small NPs of diameter 4 nm (FigureSI11). It would be now interesting to establish the critical concentration able to induce a large majority of well-oriented 2D NP networks. The enlargement of this feature to oriented 2D networks of nanorods could lead to formation of highly anisotropic 2D networks of nanorods. With gold nanorods, highly anisotropic LSP resonance would be expected, firstly due to nanorods all parallel from each other[START_REF] Liu Qingkun | Electrically and Optically Tunable Plasmonic Guest Host Liquid Crystals with Long-Range Ordered Nanoparticles[END_REF] , secondly due to highly anisotropic electromagnetic coupling between nanorods.In conclusion, we have shown how two different kinds of confinement induced by different topological defects lead to two different kinds of gold nano-sphere organizations. In the smectic oily streaks made of oriented arrays of 1D dislocations and 2D ribbon-like grain boundaries, we show that the dislocations induce formation of oriented NP chains, whereas the ribbon-like grain boundaries induce formation of planar 2D hexagonal networks of NPs.The efficiency of the 1D dislocations for the trapping of NPs is the highest. They can be filled at 85% by NPs without any significant aggregation outside the defect cores, allowing for the formation of a large number of long NP chains, all oriented parallel from each other, along a single direction defined by the rubbed polymer substrate. The 2D confinement also allows for the transmission of the anisotropy of the overall structure through a favorable orientation of the hexagonal network, the 10 direction being more favorably oriented parallel to the defect orientation, possibly due to the orienting influence of the border of the ribbon-like grain boundaries. Owing to the excellent large-scale order of these defect cores, not only the

Acknowledgement We thank D. Limagne, C. Rafaillac and B. Voisin for technical supports ; D. Demaille and L. Becerra for the electronic Microscopy ; Y. Vickridge for the analysis of the RBS data. We thank CNES (Centre National d'Etudes Spatiales) for supporting this project. Concerning the financial funding, S.P. Do, A. Missaoui and E. Lacaze thank the French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE. B. Donnio and J-L. Gallani thank the CNRS and the University of Strasbourg for support. We also thank Synchrotron SOLEIL for providing the beam on SixS beamline.