Direct Nanomechanical Measurement of an Anchoring Transition in a

Article pubs.acs.org/Langmuir

Direct Nanomechanical Measurement of an Anchoring Transition in a Nematic Liquid Crystal Subject to Hybrid Anchoring Conditions Marina Ruths*,† and Bruno Zappone*,‡ †

Department of Chemistry, University of Massachusetts Lowell, Lowell, Massachusetts 01854, United States CNR-IPCF, Liquid Crystal Laboratory c/o University of Calabria, Rende (CS) 87036, Italy

‡

ABSTRACT: We have used a surface forces apparatus to measure the normal force between two solid curved surfaces confining a film of nematic liquid crystal (5CB, 4′-n-pentyl-4cyanobiphenyl) under hybrid planar−homeotropic anchoring conditions. Upon reduction of the surface separation D, we measured an increasingly repulsive force in the range D = 35−80 nm, reaching a plateau in the range D = 10−35 nm, followed by a short-range oscillatory force at D < 5 nm. The oscillation period was comparable to the cross-sectional diameter of the liquid crystal molecule and characteristic of a configuration with the molecules parallel to the surfaces. These results show that the director field underwent a confinement-induced transition from a splay−bend distorted configuration at large D, which produces elastic repulsive forces, to a uniform planar nondegenerate configuration with broken homeotropic anchoring, which does not produce additional elastic forces as D is decreased. These findings, supported by measurements of the birefringence of the confined film at different film thicknesses, provide the first direct observation of an anchoring transition on the nanometer scale.

I. INTRODUCTION Nematic liquid crystals (NLCs) are increasingly used as host materials for the guided assembly of colloidal and nanoscopic particles into novel micro- and nanostructured ordered systems such as photonic crystals and metamaterials.1−5 A key property of the NLCs in these applications is their ability to generate structural forces between particles, which arise from the tendency of the anisotropic NLC molecules to align parallel to one another along the common director n. When the director field is deformed (e.g., in the presence of boundaries that are curved and/or impose conflicting directions of alignment (anchoring) on n), the free energy E increases and structural forces appear. However, the mechanisms of force generation on the nanometer scale, where applications are most anticipated, are complex and not clearly understood. Even in the relatively simple geometry of a hybrid NLC film (Figure 1), where two flat parallel interfaces at a distance h apart impose incompatible homeotropic (normal) and planar (parallel) anchoring directions (easy axes), the director field can adopt different configurations. Theoretical studies6−8 and simulations9−12 have shown that three equilibrium configurations can be expected: splay−bend uniaxial (Figure 1a,b), uniform uniaxial (Figure 1c), and biaxial with eigenvalue exchange (Figure 1d). The first two configurations can be analyzed within the classical framework of nematic elasticity:13 the director n is specified by a (zenithal) tilt angle θ from the interface inducing planar anchoring and rotates by a total angle Δθ across the film thickness to comply with the anchoring conditions. In the approximation of a single (average) elastic © 2012 American Chemical Society

constant K for the splay and bend NLC distortion modes, the energy E of director distortion per unit area of the film is KΔθ2/2h, whereas a surface energy of W/2 (where W is the anchoring strength) is required to deviate n by 90° from the anchoring direction. When the film thickness is large compared to the anchoring extrapolation length L = K/W of both anchorings, i.e., h ≫ L, n rotates by Δθ = ±90°, and a disclination defect is expected at the boundary between domains with opposite signs of rotation (Figure 1a).10,14 When h becomes comparable to one or both extrapolation lengths, h ≈ L, n deviates from the anchoring direction(s) by an angle δ (Figure 1b). As h is decreased, δ increases and Δθ decreases until a critical distance hc is reached, where an “anchoring transition” occurs:6 n becomes uniform across the film thickness and oriented along the easy axis of the strongest anchoring (i.e., rotated by 90° from the weakest anchoring direction (Figure 1c)). Such a transition is expected to have a distinct signature in the structural forces because as n no longer changes as h is further decreased, the free energy E remains equal to the value W/2 required to break the weakest anchoring completely and the force per unit area f = −dE/dh drops to zero. The eigenvalue exchange configuration (Figure 1d) is expected to be an alternative to the anchoring transition when both anchorings are strong with extrapolation lengths L comparable to the correlation length of the biaxial order, ξb ∼ 10 nm.8−12 In this case, the NLC remains uniaxial close to the plates, with n parallel to the Received: December 1, 2011 Revised: February 14, 2012 Published: May 23, 2012 8371

dx.doi.org/10.1021/la204746d | Langmuir 2012, 28, 8371−8383

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we describe the results obtained under symmetric anchoring conditions (planar−planar and homeotropic−homeotropic). Then, similar experiments under hybrid anchoring conditions are described, and the results are contrasted with the symmetrical cases. This is followed by a discussion of the anchoring transition and anchoring strengths in the hybrid system.

II. MATERIALS AND METHODS A. Materials and Sample Preparation. The NLC, 4′-n-pentyl-4cyanobiphenyl (5CB), was purchased from BDH (EM Industries, purity >99%), and the surfactant hexadecyltrimethylammonium bromide (CTAB) was from Fluka (≥99.0%). Both were used as received. The ordinary and extraordinary optical indices of 5CB were no = 1.54 and ne = 1.72, respectively, in the temperature and optical spectral ranges considered.37 Surfaces for the SFA measurements were prepared from a flexible sheet of muscovite mica with uniform thickness in the range 2−6 μm, cleaved from a thicker block (grade no. 2, V-2 ASTM, S&J Trading, Glen Oaks, NY) and coated on its backside with a semireflective silver layer deposited by thermal evaporation (thickness 53 nm). Square pieces of such a sheet were glued with EPON 1004F (Miller−Stephenson, Danbury, CT) onto half-cylindrical quartz glass supports with a radius of curvature of R = 2 cm. The surfaces were then mounted in the SFA facing one another in a crossed-cylinder configuration (Figure 2). The separation

Figure 1. Possible configurations of an NLC confined under hybrid planar−homeotropic anchoring conditions between two parallel plates separated by a distance h. The anchoring strength is given by the extrapolation length L. (a) Splay−bend uniaxial configuration for strong anchoring conditions, h ≫ L. The director n rotates by a total tilt angle Δθ = ±90° from one surface to the other. A disclination defect (central dot) is expected at the boundary (dashed vertical line) between domains with opposite signs for Δθ. (b) Splay−bend configuration for finite anchoring conditions, h ≈ L. The director rotates by Δθ < 90° and deviates by the angles δi from the easy axes at the surfaces. (c) Uniform uniaxial configuration expected upon confinement when the planar anchoring is stronger than the homeotropic one. (d) Eigenvalue exchange biaxial configuration. Ellipses represent the tensor order ellipsoid in the vertical plane defined by the easy axes. The ellipse axes are proportional to the eigenvalues and become equal between the plates (central dot).

local easy axis, but the order becomes biaxial between the plates. The force f is expected to decrease suddenly at the transition, occurring at h ∼ ξb, and increase again as h is further reduced.15 Experimentally, the uniaxial anchoring transition has been observed at a critical thickness hc ranging from 450 nm to 2.5 μm in hybrid NLC films confined between two flat, solid plates16−19 or free films at the interface between air and a solid substrate.20−22 For smaller film thicknesses, the eigenvalue exchange biaxial configuration has been observed for NLC confined between curved solid surfaces using force-measuring techniques,23,24 but the presence of an anchoring transition in free films deposited on solid or liquid substrates is under debate.25−31 In this article, we describe the first evidence of a hybridto-planar anchoring transition in nanoscale films of NLC that are thinner than 100 nm, induced by directly applying an external mechanical constraint. The film was confined between a curved solid surface of muscovite mica inducing planar nondegenerate anchoring and a curved mica surface coated with a monolayer of surfactant inducing homeotropic anchoring.32 For comparison, we also considered symmetric planar−planar or homeotropic− homeotropic anchoring conditions. We used a surface forces apparatus (SFA), a technique used previously to study structural forces in NLCs on the nanoscale.23,32−36 The forces measured under hybrid anchoring conditions are distinctly different from the cases of symmetric anchoring conditions: a structural transition occurs as the surface separation D is decreased below 35 nm, which is observed as a plateau in the force versus distance curve. At smaller distances (film thicknesses) of D < 5 nm, oscillatory forces reveal a layering of molecules parallel to the substrate. These observations are consistent with an anchoring transition induced by nanoscale confinement for a planar anchoring stronger than the homeotropic anchoring: as D is decreased below a critical value, the NLC becomes uniform and planar, with molecules parallel to the substrate. The article is organized as follows: After a description of the sample preparation and the techniques for measuring interaction forces and optical properties of the confined films,

Figure 2. Crossed-cylinder confinement geometry in the SFA. The contact position is marked with a black dot. The anchoring is planar at the bottom cylinder, whose axis is parallel to x, and homeotropic on the top cylinder, whose axis is parallel to y. r = (x, y) is the vector distance from the contact position. Because of curvature, the projections of the homeotropic and planar easy axes, nh and np, on the xy plane form a twist angle Δϕ. distance is D ≪ R at the point of closest approach, which eventually becomes the contact point as the surfaces are brought into contact. When a droplet of liquid crystal (with a typical volume of 50−100 μL) is confined between the surfaces, its lateral dimension is smaller than R and D ≪ R. Under these conditions, the separation h between the surfaces at a lateral distance r from the contact position is h(x, y) ≈ D + (r2/2R), which is equivalent to two spheres of equal radius 2R (or a sphere of radius R and a plane) at a distance D apart.38 The mica sheets are optically transparent and birefringent under normal illumination, with the extraordinary optical axis γ lying in the plane of the sheet. The easy axis of 5CB on such a sheet is specified by the “pretilt” (zenithal) angle from the mica substrate plane and the inplane (azimuthal) angle formed with γ. In dry air or N2 gas, the anchoring of 5CB on a clean mica surface is planar (zero pretilt).39 If the layered structure of the mica crystal could be cleaved off layer by layer, then the easy axis would be found at alternating sides of γ at azimuthal angles of ±30°. The actual cleaving process separates an unknown number of layers, leaving the azimuthal angle on a given sheet undetermined. When the NLC was confined between two such sheets with their γ axes crossed at an angle ψ, the anchoring conditions induced a twisted planar configuration at the contact position. We show data from one such experiment where the mica sheets were mounted with ψ ≈ 90° and the twist angle imposed by the anchoring conditions was Δϕ ≈ 30° (Figure 3). 8372


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Figure 3. FECO with wavelengths λ in the optical range 540−570 nm, obtained for symmetrical planar anchoring conditions (mica|5CB|mica). q is the chromatic order and r is the lateral distance from the contact position. The vertical line on the left in each image is a spectral line from a Hg lamp used to calibrate λ. (a) Bare mica surfaces in contact in dry N2 gas (D = 0). Strong adhesion flattens the surfaces at the contact position over an area of radius a surrounding the contact position because of the elastic deformation of glue layers outside the interferometer. The FECO appeared as singlets (δλ = 0) because the angle between the γ optical axes was ψ ≈ 90°. (Inset) Vertical cross section of the crossed-cylinder geometry. (b) Mica surfaces separated by a small distance D. The FECO have a parabolic shape around the contact position because of the curvature of the surfaces. (Inset) Confinement geometry with the nematic director n parallel to the surfaces. (c) At larger distances, the FECO show a finite splitting δλ. (d) Crossing between a “fast” fringe of order q and a “slow” fringe of lower order, occurring at large D. imposed tilt rotation decreases as Δθ0 = (π/2) − {(|x| + |y| sin ψ)/R} when moving away from the contact point so that the twist Δϕ becomes increasingly important in determining the local director configuration. Moreover, the sign of Δθ0 (the sense of tilt rotation) is uniquely defined at any point xy different from the contact position. The coupling between splay−bend and twist director distortions could not be avoided in our experiments. Even after determining the orientation of γ relative to x or y, the orientation of np relative to γ (i.e., either +30 or −30°) remained a priori unknown, and so was Δϕ. The anchoring of NLC of the nCB series on mica is known to change rapidly in the presence of volatile molecules, particularly water vapor.34,39,44 For this reason, the SFA chamber containing the surfaces was dried by purging with dry N2 gas before bringing the surfaces into contact with one another or introducing the NLC, and P2O5 was kept in the chamber as a desiccant throughout the measurements. All measurements were made in a temperature range of 23−25 °C, corresponding to the nematic phase of 5CB,45 with a stability of better than 0.5 °C/h. B. Force Measurements. The force F acting normally between the surfaces was measured as a function of the separation distance D with an SFA Mark II, described in detail in ref 46. The upper surface was attached to a fixed rigid mount whereas the lower surface was supported on a double-cantilever leaf spring with a known spring constant, k = 170 or 520 N/m, whose deflection was measured and equated to F/k. The fixed end of this spring was moved vertically using motor-driven mechanical stages. In our experiments, the sensitivity in F was about 10−7 N, the rate of approach and separation was ca. 0.5 nm/s, and the NLC was allowed to equilibrate at each film thickness before measuring D and F. Note that the spring-based force detection system is in a stable equilibrium only for dF/dD < k and ‘jumps’ across unstable regions of the F versus D curve.38,46 Typically, for each set of surfaces, the forces at two to three different contact positions were investigated. A ‘fresh’ contact position, where the surfaces had never been brought close to or in contact before, was found by separating the surfaces and translating them laterally at right angles with respect to one another. At each contact position, the force was measured in 2−10 cycles of approach and retraction of the surfaces, waiting 15 min to 5 h between consecutive cycles. No dependence of the forces on the number of force measurements or on the waiting time was

To induce homeotropic orientation of 5CB, clean mica surfaces were coated with a monolayer of CTAB.32 The surfaces were immersed for 30 min at 25 °C in a 5 × 10−4 M solution of CTAB in purified water (Milli-Q purifying system, water resistivity 18 MΩ cm), prepared a day in advance and stored at 25 °C to ensure the complete dissolution of the surfactant.40 At this concentration, corresponding to about half of the CTAB critical micelle concentration, the surfaces became covered with a bilayer.41−43 The outermost layer was removed by withdrawing the surfaces from the CTAB solution and dipping them for a few seconds in purified water.43 A monolayer was left on the mica surface, which was dried thoroughly in a stream of N2 gas to remove traces of water. Whenever mica substrates were transferred in and out of the CTAB solution or pure water, the surface of the liquid was aspirated with a pipet to remove possible contamination accumulating at the interface. The advancing contact angle of water on these CTAB monolayers was 92°, in good agreement with values found in the literature.42,43 The anchoring of 5CB was homeotropic, as checked between crossed polarizers for closed cells made with CTABcoated mica plates and filled with 5CB. The adsorption of the CTAB monolayer is complex and of importance for the anchoring conditions, which will be further elaborated on in the Discussion. Because of curvature, the crossed cylindrical surfaces of the SFA (Figure 2) produced nonuniform anchoring conditions that were slightly different from those obtained between flat parallel surfaces (Figure 1). Consider a reference frame (x, y, z) such that x and y are the axes of the cylinders inducing planar and homeotropic anchoring, respectively, and z is the vertical direction. Let θ be the tilt angle between the director n(x, y, z) and the xy plane, and let ϕ be the azimuthal angle between the projection of n on the xy plane and an arbitrary direction in the plane. At the contact position (x = y = 0), the anchoring conditions are the same as for flat parallel surfaces: the homeotropic anchoring direction nh is parallel to the vertical axis z and the planar anchoring direction np lies in the horizontal plane xy. The tilt rotation imposed by the anchoring conditions going from one anchoring direction to the other is Δθ0 = ±90°. At any other point r = (x, y) on the xy plane (Figure 2), nh deviates from z and np deviates from the xy plane. In particular, nh acquires a nonzero projection along the x axis, which increases when moving away from the contact position along the x axis and forms a twist angle (azimuthal direction rotation) Δϕ with the projection of np (Figure 2). As a consequence, the 8373


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Figure 4. Analysis of FECO obtained for symmetric planar anchoring conditions (mica|5CB|mica). Triangles and circles indicate values measured at mica−mica contact (D = 0) and at D > 0, respectively. Solid curves were calculated using the eigenvalue method for the geometry shown in the inset in a: two mica sheets of thickness T = 37.790 μm, with γ optical axes crossed by ψ ≈ 90°, confining an NLC with an effective refractive index n = 1.63 ± 0.01 and birefringence δn = 0.16 ± 0.01. The planar easy axis formed an angle of σ = 30° with γ. The average azimuthal orientation of the director n is along the ψ bisector, with an expected twist Δϕ ≈ 30° (shaded area in the inset). (a) Mica−mica separation distance D as a function of the wavelength λ for different chromatic orders q, each appearing as a doublet. (b) Fringe splitting δλ as a function of the median wavelength ⟨λ⟩. The measurement errors in λ and δλ (vertical bar) were 0.2 and 0.5 nm, respectively. average orientation of the director n across the thickness of the confined film. In fact, n is generally tilted by a certain angle θ from the xy plane (Figure 1a) and deviates by an azimuthal angle ϕ from any of the two γ axes. The measured wavelengths were fitted with calculated values obtained for a uniform director field oriented at angles θ and ϕ and having the full (intrinsic) birefringence of 5CB, ne − no = 0.18.37 Such a configuration was considered to be optically equivalent to the actual director field, which may be distorted (with θ(z) and ϕ(z) varying along the vertical axis z) as a result of incompatible anchoring conditions and/or the curvature of the surfaces (Figure 2). In the latter case, the fit provided the angles z-averaged across the NLC film thickness: θ = ⟨θ(z)⟩ and ϕ = ⟨ϕ(z)⟩. Note that a uniform director oriented at angles (θ, ϕ) is also optically equivalent to a uniform planar director (θ = 0) with the same azimuthal angle ϕ but with an effective refractive index of n = (n* + no)/2 and a birefringence of δn = n* − no given by

found within this range. The forces F were normalized by the radius R to allow for comparison between experiments with different sets of surfaces and radii and for comparison with theoretical calculations using the Derjaguin approximation: F(D)/R = 2πE(D) where E is the free energy of interaction between two flat parallel surfaces.38 C. FECO Interferometry. The two semireflective Ag coatings, the mica sheets (bare or coated with CTAB), and the NLC film constitute a layered multiple-beam interferometer of the Fabry−Perot type that was used to determine the surface separation, D.47 When illuminated normally with white light, the interferometer produces a discrete set of transmitted wavelengths λq due to constructively interfering waves of different chromatic order q. The wavelength λq(h) increases as q decreases and decreases when the surface separation h decreases. Because h ≥ D increases as the lateral distance r from the contact position increases, the fringes of equal chromatic order (FECO) λq(h) measured as function of r appear curved in the spectrometer, with a minimum wavelength λq(D) corresponding to the contact position (Figure 3). The geometric mean radius of curvature R can be determined from the curvature of the FECO from the undeformed (separated) surfaces at the contact position (inset in Figure 3b).46 FECO are affected by the birefringence of the NLC and carry important information about the configuration of the director field n. When the interferometer comprises one or more birefringent layers, the FECO of order q are split into a doublet (two fringes) λq,1 and λq,2 corresponding to the two eigenmodes of polarization, generally elliptical, resonating within the interferometer (Figure 3c). The eigenmodes travel at different velocities through the interferometer, and their wavelengths vary at different rates in response to thickness variations of the birefringent layers. In particular, we can identify a slow and a fast mode as the thickness h of the NLC film is varied. To determine the thicknesses of the birefringent layers and the orientation of their optical axes, we used the method of eigenvalue analysis proposed by Rabinowitz48 and further developed in ref 23. The wavelengths of two or more FECO doublets were measured in the optical spectral range 540−570 nm with a resolution of a few angstroms using an imaging spectrometer coupled to a CCD camera (Figure 3). A resolution of better than 0.1 Å in the wavelength could be obtained by measuring a single fringe using an eyepiece attached to the spectrometer. For each surface separation D, the measured wavelengths were fitted with values calculated as a function of a set of parameters that describe the interferometer. First, the wavelengths obtained at direct mica−mica contact (D = 0, Figure 3a) were analyzed to determine their chromatic order q, the thickness T of the mica sheets, and the angle ψ between their γ optical axes. The doublet splitting δλq = |λq,1 − λq,2| was found to be directly proportional to cos ψ, in agreement with previous work.36,48,49 When an NLC layer was confined between the surfaces (D > 0, with ψ and T fixed), the fitting procedure was used to determine D and the

2 ⎛ sin θ ⎞2 ⎛ 1 ⎞2 ⎛ cos θ ⎞ ⎜ ⎟ = ⎜ ⎟ +⎜ ⎟ ⎝ n* ⎠ ⎝ ne ⎠ ⎝ no ⎠

(1)

For short-range force measurements (D < 400 nm), we measured the wavelengths of a single FECO of known order q to achieve subnanometer resolution in D using standard analytical formulas derived for isotropic liquids (setting the refractive index n to the value determined experimentally).47 When one or both of the surfaces was coated with CTAB, we used the same method to measure the thickness of the CTAB layer at direct mica−CTAB or CTAB−CTAB contact in dry N2 using nCTAB = 1.45 and δnCTAB = 0. As the surface separation D is increased from mica−mica contact, the slow wavelength of the qth FECO measured at contact is continuously passed by fast components of FECO with higher orders p. For example, when δn > 0, the fast eigenmode of order p (e.g., λp,2) eventually crosses the slow mode of the next FECO with lower order p − 1 (e.g., λp−1,1) (Figure 3d). The difference p − q at the first crossing varies between 8 and 12 for a planar uniform director (maximum δn = 0.18, θ = 0, Δθ = 0 in eq 1, Figure 1c) and between 15 and 25 for a hybrid splay−bend configuration (δn = 0.09, θ = 45°, Δθ = 90°, Figure 1a), whereas no crossing occurs for a homeotropic film (δn = 0, θ = 90°, Δθ = 0), providing a quick estimate of the director orientation at large D.

III. EXPERIMENTAL RESULTS A. Planar (Twisted) and Homeotropic Symmetric Anchoring Conditions. Figure 3a shows FECO with different q, obtained in adhesive mica−mica contact in dry N2 gas. In this case, the γ optical axes of the mica sheets were crossed at an angle ψ close to 90° and the fringe splitting δλ ∝ cos ψ was close to zero. After the NLC was introduced, the FECO 8374


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splitting increased as the distance D was increased (Figure 3b,c), eventually leading to a crossing between fast and slow fringes of different orders (Figure 3d), indicating a highly birefringent configuration of the director n. Figure 4 is a fit of FECO data with curves calculated using the eigenvalue method.48 The effective refractive index n = 1.63 ± 0.01 and birefringence δn = 0.16 ± 0.01 were close to the values n = (no + ne)/2 = 1.63 and δn = ne − no = 0.18 expected for a uniform planar configuration (θ = 0, Δθ = 0). The z-averaged orientation of n was along a bisector of the crossed γ axes, as expected (inset in Figure 4a). The presence of a twist angle, expected to be close to Δϕ = 30° for ψ ≈ 90° and strong planar anchoring conditions (inset of Figure 4a), was not apparent in our data. It is possible that the twist produced a value of δn that is slightly lower than that expected for a uniform planar alignment, as observed in ref 35. Figure 5 shows force curves (F vs D) measured by us and data found in the literature32,36 for two symmetrical surfaces (i.e., mica−mica or CTAB−CTAB), imposing planar (twisted) and homeotropic configurations, respectively. In Figure 5a, the data for the planar orientation at ψ = 0, 0) and increased as D was decreased down to a value of about 5 nm in the planar case and 15 nm in the homeotropic case. In the planar (twisted) orientation, the long-range force increases with increasing ψ up to ψ ≈ 40°, after which no further increase is observed,33 and the curves shown therefore represent the lower and upper ranges of the values found experimentally. These results can be explained qualitatively by the elastic theory of NLC, predicting a monotonic power law dependence of the force of F on the distance D under planar (twisted)32 and homeotropic orientations.50 Note that no plateau was observed or expected at small separations. At smaller D, the force oscillated periodically between repulsive maxima (F > 0) and attractive minima (F < 0), with increasing oscillation amplitude as D decreased (Figure 5b,c). In this region, the F versus D curve showed inaccessible regions where the stability condition, dF/dD < k, was not satisfied. The oscillatory force is due to a confinement-induced semiordered layered arrangement of molecules near the surface. This phenomenon has been extensively studied theoretically51,52 and experimentally for several NLCs including 5CB.32,33,36,53 The period of oscillation is equal to the layer thickness, and each oscillation (e.g., going from one maximum to the next upon approaching the surfaces) corresponds to the progressive straining and ‘melting’ of the central layer located midway between the surfaces. 5CB molecules have an average crosssectional diameter of 0.43−0.5 nm and a length of 1.8 nm and associate into dimers with a length of 2.57 nm in the bulk.54 In the mica|5CB|mica case (Figure 5b), the data at ψ = (2 ± 2)° and ψ < 10° are from refs 36 and 32, respectively, and the data at ψ ≈ 90° were measured in this study. The period of oscillation, (0.5 ± 0.1) nm, was independent of the angle ψ between the γ axes36 and close to the cross-sectional diameter of a molecule, showing that the layered structure is composed of molecules parallel to the substrate. We also observed that the minima in the force curve were not as deep at larger ψ, in agreement with previous work.36 At sufficiently strong forces (Figure 5b), the NLC can be entirely removed and the mica surfaces can be brought into direct contact.

Figure 5. Normalized structural forces F/R as a function of the distance D between two symmetrical bare (mica|5CB|mica) or two CTAB-covered (CTAB|5CB|CTAB) mica surfaces inducing planar and homeotropic anchoring, respectively. (a) Long-range forces on compression (data points omitted for clarity). (b) Oscillatory short-range forces for mica| 5CB|mica at small D, with a period of (0.5 ± 0.1) nm comparable to the cross-sectional diameter of a 5CB molecule. D = 0 at mica−mica contact. Open symbols indicate approach and filled symbols indicate retraction of the surfaces, except for the data at ψ < 10° (open circles) where a distinction between approach and retraction was not made in ref 32. Diamond symbols indicate points from where the surfaces spontaneously jumped apart upon separation. (c) Oscillatory short-range forces measured for CTAB|5CB|CTAB with a period of (2.5 ± 0.1) nm, comparable to the length of a 5CB dimer. D = 0 at CTAB−CTAB contact. Squares and circles show data from ref 32 for CTAB monolayers with and without suspected roughness, respectively, and triangles show data from the present study. The pull-off forces in each case are indicated in the lower left corner. In b and c, the lines are intended as guides only. Data from ref 32 (details in Experimental Results) are reproduced with the permission of EDP Sciences. Data from ref 36 are reproduced with permission. Copyright 2000 American Chemical Society. 8375


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Figure 6. FECO for hybrid anchoring conditions (mica|5CB|CTAB). (a) Adhesive contact between one bare mica and one CTAB-covered mica surface in dry N2 gas at a mica−mica separation D equal to the thickness TCTAB of the CTAB monolayer. The splitting δλ is caused by an angle ψ ≈ 30° between the γ axes of the mica sheets. (b−d) Progressive approach of the surfaces from a large distance after the introduction of 5CB. Note the large distortions, discontinuities, and fluctuations of the fast fringes. (e) D = 0.6 nm, corresponding to a single layer of planarly aligned 5CB molecules trapped between mica and CTAB at relatively high compression. (f) Distortion and fluctuations reappear upon separation from the contact in e.

B. FECO for Hybrid Anchoring Conditions. Figure 6 shows examples of FECO obtained between a CTAB-coated and an uncoated mica surface. The γ optical axes of the mica sheets were crossed at an angle of ψ ≈ 30°, producing a measurable splitting δλ at mica−CTAB contact in dry N2 (Figure 6a). After the 5CB was introduced and the surfaces were approached for the first time at separations D less than 80 μm, a sudden change in the shape of the fast fringes was observed, whereas the slow fringes remained unaffected. At a lateral distance r = 10−300 μm from the contact position, the fast fringes were perturbed and distorted over a small domain δr and continued in the outer regions of the confinement (large r) following a λ(r) curve with different position, slope, and curvature from those of the contact region (small r). In some cases, the distortion was concentrated in a very narrow domain δr, appearing almost discontinuous (Figure 6b,c). This indicates that the director field at large r switched to an alternative configuration with a different birefringence than the one at small r. Both the separation D and the lateral distance at which the transition occurred were difficult to reproduce from one experiment to another. The distortion generally became less prominent as D was decreased, eventually disappearing for D < 2 μm, but reappeared as D was increased again. Occasionally, it was possible to approach the surfaces on a fresh contact position without observing the transition, but the distortion always appeared upon subsequent separation. Once formed, the distortion could disappear after waiting for a few hours while keeping the surfaces still at a separation of tens of micrometers but then immediately reappeared as the surfaces

In the case of CTAB|5CB|CTAB (Figure 5c), the period was (2.5 ± 0.1) nm, corresponding to the length of a dimer, showing that these were oriented normal to the layers and substrates. In our experiments, homeotropically oriented 5CB could be expelled from between the two CTAB-covered surfaces at an F/R value of a few mN/m. Qualitatively similar forces have been reported previously for 5CB confined between monolayers of CTAB32 and for 8CB between close-packed monolayers of other surfactants.33 In our data and in ref 32, the thickness of the last two 5CB layers in direct contact with the surfaces was 2.5 nm (Figure 5c), the same as for bulk layers interacting with other 5CB layers, showing that the penetration of perpendicularly oriented molecules into the CTAB layers was very limited. However, the “expulsion” force in our experiment (triangles in Figure 5c) was much lower than the one required to remove the last homeotropic layer of 5CB trapped between the CTAB monolayers in ref 32 (circles and squares in Figure 5c) and also lower than the force required to remove the last planar layer between uncoated mica surfaces (Figure 5b). We will return to this observation in the Discussion. The CTAB could not be expelled from confinement even under the highest compressive forces (F/R) of tens of mN/m. The thickness of a CTAB layer in adhesive contact with an opposing bare mica sheet was 1.2−1.5 nm as measured both before and after introducing the NLC and 1.5−1.7 nm per monolayer in CTAB−CTAB contact, in good agreement with previous results.32,41 For symmetric planar and homeotropic anchoring conditions, the behavior of the FECO (Figures 3 and 4) and the force curves (Figure 5) was reproducible as the surfaces were repeatedly approached and retracted. 8376


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Figure 7. Analysis of FECO of Figure 6 (mica|5CB|CTAB). The solid curves were calculated using the eigenvalue method for the geometry shown in the inset in a: two mica sheets of thickness T = 27.453 μm, with γ optical axes crossed by ψ = 35°, confining an NLC with an effective refractive index n = 1.56 ± 0.01 and birefringence δn = 0.05 ± 0.01. The planar easy axis formed an angle σ = 30° with one of the γ axes, lay outside ψ, and was coplanar with the splay−bend director n (no twist). Triangles, circles, and squares correspond to mica−mica distances D = 0, 468 nm, and 2.581 μm, respectively. (a, b) D as a function of the wavelength λ for different chromatic orders q. (c) Splitting δλ as a function of median wavelength ⟨λ⟩. The measurement errors in λ and δλ (vertical bar) were 0.3 and 0.6 nm, respectively.

free of defects before, during, and after contact, but defect lines were present at a distance r of several hundred micrometers away from the contact position. The defects formed a random pattern of lines and loops that changed, in response to both surface separation and induced flow, as the surfaces were moved over large distances on the order of several tens of micrometers. The defect pattern and the distortion of the FECO are likely to be connected, although the relation is still unclear. In the small range of distances 0 < D < 200 nm where structural forces could be measured, the defect lines remained practically fixed far from the contact during the measurements. C. Structural Forces for Hybrid Anchoring Conditions. Figure 8 shows the normal force F measured as a function of the surface separation D upon approach of one bare and one CTAB-coated mica surface in 5CB. The position D = 0 corresponds to the mica−CTAB contact in dry N2. The data in Figure 8 come from three separate experiments with surfaces of different radii of curvature (R = 0.8−1.2 cm), angle ψ (in the range 20−40°), and twist Δϕ and various contact positions. The force F was repulsive and increased monotonically as D was decreased from D ≈ 80 to 35 nm (Figure 8a). At smaller distances, a distinct plateau was observed in the F versus D curve, extending down to D ≈ 10 nm. In this region, we observed the slight increase in the fringe splitting δλ mentioned above, indicative of an increase in birefringence. The forces in the region 15 nm < D < 80 nm were reversible (i.e., they showed no hysteresis upon reversing the direction of movement of the surfaces). At D ≈ 5 nm (Figure 8b), the force became oscillatory with repulsive maxima and attractive minima similar to those observed for symmetric anchoring conditions (cf. Figure 5b,c). In the range 1 nm < D < 5 nm, the period of the oscillation was 0.5 ± 0.1 nm, equal to the value obtained for symmetric planar anchoring conditions (Figure 5b), clearly due to a layering of 5CB molecules aligned parallel to the surfaces. Layers were strained and sequentially removed upon progressive confinement, eventually reaching direct mica−CTAB contact when F/R exceeded 9 mN/m. However, in contrast to the systems in Figure 5b,c, removing a bulk layer that interacted with neighboring layers of 5CB molecules was different from removing the surface layers of 5CB in direct contact with mica or CTAB.24 After the last bulk layer was removed (going

were moved again. These observations suggest that the transition was triggered mainly by the flow of NLC created in the outer regions by the motion of the surfaces. Such behavior may be linked to the curvature-induced twist and nonuniformity of the anchoring conditions and will be further discussed in the next sections. The director configuration in the contact region remained stable and was not affected by the transition. The crossing between fast and slow modes of different chromatic orders continued as the distance D was further decreased. The last crossing occurred for a value of p − q (Materials and Methods), indicating a birefringent nonplanar director configuration (0 < δn < ne − no with 90° > θ > 0°). The fast fringes showed random fluctuations in wavelength (Figure 6d and also at all larger D), which were not observed for symmetric anchoring conditions. According to eq 1, thermal fluctuations of the tilt angle δθ produce maximum (first order) fluctuations of the birefringence δn and therefore of the FECO splitting when the tilt angle is θ = 45°, whereas fluctuations around θ = 0 and 90° produce negligible (second order) effects. These observations indicate that the z-averaged director tilt was close to the value expected for a hybrid configuration (Figure 1a,b). This was further confirmed by the eigenvalue analysis of the wavelengths at the contact position, shown in Figure 7 for the FECO in Figure 6. The effective refractive index n = 1.56 ± 0.01 and the birefringence δn = 0.05 ± 0.01 were both close to but slightly lower than the values n = (3no + ne)/4 = 1.58 and δn = (ne − no)/2 = 0.09 expected for an average tilt angle of θ = 45°. Note that the NLC remained birefringent nonplanar, and no anchoring transition to a uniform homeotropic or planar director configuration was detected as D was decreased down to about 400 nm. For smaller distances, we measured a single FECO to increase the precision in D because the precision in δn was too small for a quantitative analysis of the orientation. However, we observed a slight increase in the fringe splitting δλ for distances D < 35 nm, suggesting an increase in birefringence and an evolution of the director configuration toward a planar configuration in this distance regime. For hybrid anchoring conditions, a defect is expected at the contact position where the surfaces can be approximated as flat parallel plates (Figure 1a). However, the contact region was 8377


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homeotropically oriented 5CB layers). The final distance in Figure 8b corresponds to the direct mica−CTAB contact measured before introducing the 5CB, showing that the 5CB was entirely removed. The force needed for this was slightly lower than that reported for the symmetric mica−mica system (Figure 5b). The length of the inward jump upon removal (D → E in Figure 8b) was 0.4 ± 0.1 nm, close to the value measured for bulk layers (i.e., the cross-sectional diameter of the molecule). Therefore, the compression produced a last confined layer with 5CB molecules oriented parallel to the surface. The forces measured under hybrid anchoring conditions appeared to be independent of the location, width, and magnitude of the distortion observed on the fast FECO at the boundary of the contact region (Figure 6b−d,f and other images not shown). Over the time of an experiment (a couple of days), the anchoring conditions eventually degraded to symmetric homeotropic, in agreement with previous observations,23,34,35 producing zero birefringence and force curves similar to the ones observed for two CTAB-coated mica surfaces.

IV. DISCUSSION A. Evidence of a Nanoscale Confinement-Induced Anchoring Transition. The observations of a plateau in the force curve at separations of 10 nm < D < 35 nm and a layered planar structure at smaller separations indicate that the NLC underwent a confinement-induced anchoring transition of the type predicted by Barbero and Barberi6 within the framework of the uniaxial description of NLC order. In contrast, when the confinement induces a transition to the biaxial eigenvalue exchange configuration (Figure 1d), the force curve is expected to show nonmonotonic behavior as the separation D decreases, reaching a maximum force and showing an inward jump at a separation near the transition.23,24 Moreover, a plateau like that in Figure 8a was never observed for symmetric (twisted) planar or homeotropic anchoring conditions, where the force follows a monotonic32−34 dependence on D. The appearance of the plateau can be explained by considering the bulk elastic energy and the anchoring energy of the distorted director field n. In the absence of twist deformations, the energy per unit area due to the elastic distortion of a 5CB film confined between two parallel plates at a distance h (Figure 1a,b) can be written as13

Figure 8. (a) Normalized force F/R as a function of the mica−CTAB distance D, measured upon approach of one bare mica and one CTABcoated surface in 5CB. Note the plateau in the forces at 10 nm < D < 35 nm. D = 0 at mica−CTAB contact. (b) Enlarged view of the oscillatory force at small D. Open and filled symbols indicate forces measured during the approach and retraction of the surfaces, respectively. Diamond symbols indicate points from which the surfaces jumped out to regions of zero force (D > 15 nm). Some of the jumps outward and inward are also indicated by arrows next to the symbols. The distances (and number of 5CB molecular layers) at points A−E are respectively 1.6 nm (three layers), 1.1 nm (two layers), ∼0.7 nm (two layers, after reorientation of the 5CB layer at the CTAB surface), 0.4 nm (one layer), and zero.

E(h , Δθ0) =

2 W W K (Δθ0 − δ1 − δ2) + 1 sin 2 δ1 + 2 sin 2 δ2 2 h 2 2 (2)

where K = 6 × 10−12 N is the average splay−bend elastic constant of 5CB,55 Δθ0 is the director tilt rotation imposed by the anchorings, and δ is the angle of deviation of n from the easy axis at the surface. The anchoring energy is written in the Rapini−Papoular form, (W/2)sin2 δ, where W is the anchoring strength.13 Deviations δ are negligible as long as h is much larger than the extrapolation lengths Li = K/Wi of both anchorings so that n actually rotates by a tilt angle Δθ0 going from one surface to the other (Figures 1a and 9a). In this case, the energy E increases as h−1 as h is decreased (Figure 9b). When h is progressively decreased, becoming comparable to the extrapolation length L2 of the weaker anchoring (i.e., L2 > L1), the deviation δ2 is no longer negligible compared to Δθ0, and the actual tilt rotation becomes Δθ < Δθ0 (Figure 1b). In this case, the energy E increases less rapidly than h−1 as h is decreased (Figure 9b). When h reaches the critical distance hc = L2 − L1, the energy saturates to the value Ec = W2/2 = K/2L2, the deviation δ2 reaches the maximum value Δθ0, and δ1 drops to

from A to B in Figure 8b), the two surface layers came into contact. Further compression first produced a compaction of one or both of the surface layers by a total of 0.45 nm (from B to C in Figure 8b) and then the complete removal of one layer (from C to D). The compression cannot be attributed to the compaction of the CTAB monolayer because of its high thickness compression modulus,41 K = pressure/(ΔD/D)) ≈ 0.5 × 108 N/m2. The value of F/R ≈ 5 mN/m at point B in Figure 8b corresponds to a much lower compressive pressure than that at the midpoint (r = 0) in adhesive mica−CTAB or CTAB−CTAB contact and implies a negligible change in the thickness of the CTAB (smaller than 0.1 nm). Also, no such compaction at an almost constant force was seen in homeotropic (CTAB|5CB|CTAB) systems at similar and higher F/R values (Figure 5c, where the gradual changes seen in the layer thicknesses are consistent with the compressibility of 8378


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Figure 9. Anchoring transition between two parallel plates that favor hybrid planar−homeotropic alignment, inducing a splay−bend distortion. The plates are at a distance h apart, and hc = L2−L1 is the critical distance. The two possible cases are shown: L1 < hc < L2 (solid curves, L1 = hc/3, L2 = 4hc/3) and hc < L1 < L2 (dashed curves, L1 = 40hc, L2 = 41hc). (a) Deviation δi of the surface director from the easy axes (cf. Figure 1b) as a function of h/hc. (b) Normalized energy per unit area E/(W2/2) as a function of h/hc.

film underwent an anchoring transition toward a uniform planar configuration. B. Confinement to Single-Molecule Film Thickness. The oscillatory forces measured for D ≤ 5 nm (Figure 8b) in such a surface-frustrated configuration, with broken homeotropic anchoring, were different from those observed in uniform or twisted planar films obtained under symmetric planar anchoring conditions (Figure 5b).32,33,36 As D was decreased, bulk layers were sequentially removed from the center of the confinement, where molecules interacted with neighboring layers via 5CB−5CB interactions (going from large film thicknesses toward point A in Figure 8b). The situation is similar to the case of symmetric planar anchoring conditions (mica|5CB|mica) and produces similar forces in this distance regime.32,33,36 However, after the removal of the last bulk layers, surface layers of 5CB directly interacting with the bare mica and CTAB-coated mica came into contact (going from A to B in Figure 8b). A further reduction of D first produced a compression, rather than the removal, of one or both of the surface layers (from B to C in Figure 8b). Most likely, this corresponded to a realignment of 5CB molecules in contact with CTAB. These molecules are expected to be aligned normal to the CTAB surface when D is large and form a thicker layer than the one in contact with the mica surface (which is oriented planarly). Indeed, the thickness of the 5CB layer at the CTAB surface before compression (approximately equal to the B−D distance in Figure 8b) was larger than the 0.5 nm diameter of a single 5CB molecule (Figure 5b) but smaller than the 2.5 nm thickness of bulk homeotropic layers found for symmetric CTAB|5CB|CTAB (Figure 5c). Such a hybrid layer, midway between planar and homeotropic, is due to a complex arrangement of 5CB molecules trapped between two surfacesmica and CTABwith very different structures and physical−chemical properties. The layer structure may be similar to the “trilayer” arrangement commonly observed for 5CB free films spreading under hybrid anchoring conditions on solid and liquid substrates.25−27,29,31 The rearrangement produced a planar layer that was expelled upon further compression (going from C to D in Figure 8b), leaving a single planar layer in contact with both mica and CTAB. This layer was expelled last (going from D to E in Figure 8b), indicating a stronger adsorption of 5CB on mica than on CTAB, in agreement with a stronger anchoring. This was

zero.6 For h ≤ hc, the director is uniform between the plates and aligned along the direction of strongest anchoring, whereas E = Ec no longer depends on D. In this regime, the E versus h curve shows a plateau (Figure 9b) whereas the force per unit area of flat plates, f = −dE/dh, is zero. Such a theoretical description must be adapted to the curved confinement geometry of our SFA experiments (Figure 2). When D ≫ L2 and/or both anchorings are infinitely strong, the normal elastic force acting between the crossed cylindrical SFA surfaces is expected to be F/R ≈ πKΔθ02/D.32 This force satisfies the Derjaguin relation F/R = 2πE(D, Δθ0),38 where the total force F is related to the energy per unit area E at the contact position, given by eq 2 with L1 = L2 = 0 and Δθ0 = 90°. When D ≈ L2, the energy E in the contact region increases less rapidly as D is decreased and the total force F is therefore smaller. In fact, when D is decreased below the critical value hc, the director becomes uniformly aligned over an increasingly large circular area surrounding the contact position, which no longer contributes to the force F. The Derjaguin approximation ceases to be valid as F continues increasing while E at the contact position remains equal to the constant value Ec. As shown in the Appendix, the expansion of the uniformly aligned circular area counterbalances the increase in force due to the compression of the outer regions, and F tends to saturate to a value close to 2πREc, likely producing a plateau in the F versus D curve at small separations. Our results agree with this scenario of a confinement-induced anchoring transition. The FECO show that the director remains hybrid-aligned at the contact position for film thicknesses D as small as 468 nm (Figure 7c). In the range 35 nm < D < 80 nm, the force F/R increased from zero to 0.5 mN/m (Figure 8a), which is of the same order of magnitude as (but smaller than) the force F/R = 2πE(D, π/2) = 0.6−1.3 mN/m expected for infinite anchoring strengths (i.e., δ = 0) at the same distance. Therefore, in this regime a significant tilt rotation Δθ of the director was still present in the contact region, although the surface director was already in the process of being deviated from one or both of the easy axes, producing a smaller value of F/R than expected for infinitely strong anchorings. Indeed, a further reduction of D to below 35 nm produced a plateau in the F versus D curve (Figure 8a) with a slight increase in fringe splitting δλ and finally a layered structure with 5CB parallel to the substrate (Figure 8b), all of which confirm that the 5CB 8379


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adsorbed monolayer is very loosely packed62,63 or the interaction with the underlying substrate is very strong.66 The adsorption of CTAB on mica from aqueous solution is complex, and no full monolayer without at least a partial bilayer is found at any concentration.43 In the present experiments, the CTAB monolayer was adsorbed following a procedure (including dipping and rinsing in pure water, see Materials and Methods) that is known to produce very few second-layer molecules and patches on top of it.40,42 The thickness of the last 5CB layer trapped between two CTAB monolayers in our experiment was the same as the bulk layers at larger D (Figure 5c), indicating no detectable penetration of 5CB into the CTAB monolayers. In ref 23, CTAB layers were formed on mica from a solution that was about 4 times more concentrated than the one we used here, with no rinsing step, which likely produces a rougher interface with large bilayer patches. In ref 15, homeotropic alignment of 5CB was obtained on a monolayer of DMOAP chemisorbed on glass. The molecules of DMOAP have a larger polar headgroup than CTAB and adsorb on the surface leaving spaces where 5CB can penetrate and reach the substrate.67 Such a monolayer is therefore more similar to a rough or patchy CTAB monolayer than to a well-formed one. The likely explanation for the differences between our results and previous SFA and AFM observations is that the anchoring of 5CB is stronger on nonrinsed CTAB and DMOAP layers than on a rinsed CTAB monolayer and comparable to the strength of the planar anchoring on mica. We hypothesize that the greater roughness and density of patches or holes on nonrinsed CTAB and DMOAP layers may strengthen the homeotropic anchoring by allowing 5CB molecules to penetrate deeper into the layer and remain ‘locked in’ with a homeotropic (normal) orientation.68 However, the relation between the physical−chemical properties of the surfactant monolayer (polarity, density, roughness, thickness, etc.) and the orientation and strength of the NLC anchoring is subtle and still not completely understood.56,57,62,64,65,67−71 Further experiments, including a direct detailed characterization of the surfactant layer, are required to clarify the causes of the different anchoring strengths observed in various SFA and AFM experiments. D. Structural Defects. Disclination lines passing at the contact position are expected to play a major role in determining the force between two surfaces inducing hybrid anchoring conditions.15 It is therefore noteworthy that, in our experiments, defect lines were located far from the contact position, as mentioned in section III.B. A possible explanation is that a small pretilt τ ≪ Δθ was present at the homeotropic surface. When D ≫ Lh, a pretilt as small as τ = 3° is sufficient to displace the defect position (i.e., the point where the director tilt rotation is Δθ = ± 90°) by a distance as large as R sin τ ≈ 1 mm from the contact position without significantly changing the force F/R = πK(Δθ)2/D.23 Also note that the anchoring transition produces a uniform planar director field when D < hc ≈ Lh over a circular area surrounding the contact region, thereby expelling the defects from this area. Predicting the type and location of defect lines in the curved geometry of the SFA is outside the scope of this article because it requires a consideration of the full 3D director field including both zenithal and azimuthal deviations from the anchoring directions. The presence of a curvature-induced twist in the anchoring conditions (Figure 2) may favor the creation of mixed splay−twist−bend configurations and complex defect structures. Indeed, a twist deformation may spontaneously

further confirmed by the smaller force required for its removal (F/R = 9 mN/m, Figure 8b) compared to the force necessary to remove the last layer trapped between two bare mica surfaces (F/R ≈ 14 mN/m, Figure 5b). C. Anchoring Strengths. Our SFA measurements enable us to estimate the anchoring strengths Lh and Lp for homeotropic and planar anchoring, respectively. The plateau in the force curves upon compression appears at the critical thickness hc = Lh − Lp ≈ 30 nm, which gives the difference between the extrapolation lengths. Lh and Lp cannot be much larger than hc because then the director would strongly deviate from the easy axes for D = 35−80 nm and the force would be much smaller than the one measured in this range (Figure 9). However, the planar anchoring of 5CB on mica, a crystalline mineral, is due to different molecular interactions and is expected to be very different from the homeotropic anchoring on a CTAB monolayeran amorphous, soft organic surface. Therefore, we expect Lp ≪ Lh ≈ 30 nm, which corresponds to anchoring strengths of Wp ≫ Wh ≈ 0.2 mJ/m2. These values are much higher than those measured for mica56 (Wp ≤ 0.1 mJ/m2) and for CTAB and other surfactant monolayers57 (Wh ≤ 0.01 mJ/m2) using electro-optical methods on thick NLC samples. The discrepancy is common to other experiments where the NLC was confined to a nanoscale thickness (smaller than about 100 nm) under hybrid and strong anchoring conditions using mechanical constraints23,24 or the spontaneous spreading of a free NLC film on a substrate.25,26 The origin of such a discrepancy is not yet clear28−30 and may be attributed to different factors: an inadequacy of the Rapini−Papoular expression in describing large deviations δ of the director from the easy axes; gradients of nematic order near the surfaces that may affect the elastic constants and the transition temperature;58 biaxial order;8,59,60 and the presence of unaccounted for spontaneous ‘escaped’ configurations under strong confinement (such as a twist component in the geometry of Figure 1).27,31,60,61 Recent molecular dynamics simulations60 confirm that characterizing the complex NLC− surface interaction on the nanoscale in terms of simple anchoring may lead to anchoring strengths of Wh,p > 10 mJ/m2, which are much higher than the values commonly accepted on large scales. It is noteworthy that previous force measurements conducted with SFA23 and with atomic force microscopy15 (AFM) on 5CB films confined to less than 100 nm between curved surfaces of mica and surfactant monolayers (CTAB23 or N,Ndimethyl-N-octadecyl-3-aminopropyltrimethoxysilyl chloride, DMOAP15) did not show an anchoring transition but a transition to the biaxial eigenvalue exchange configuration (Figure 1d), revealed by a sudden decrease in the force F when the separation D was decreased to about 10 nm. Such a difference from our SFA force measurements is to be attributed to a different structure of the surfactant monolayer. Generally, on a surfactant monolayer, the interactions between the polar part of the 5CB molecules and the underlying mica substrate may be lowered sufficiently by the surfactant to allow a homeotropic orientation to be favored either because of dispersion interactions62 or because of steric interactions between the 5CB and the surfactant monolayer (small-scale roughness, penetration of 5CB molecules into loosely packed monolayers,63,64 “holes” in the monolayer, or “patches” of bilayer65). All of these factors contribute to the strength and pretilt of the homeotropic anchoring of 5CB on the surfactant monolayer, which may be even planar in cases where the 8380


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force, which typically scales as R/h (eq 2) and therefore as (R/r)2. In the presence of an anchoring transition, the effect of curvature can be qualitatively understood by assuming that n is uniform in a circular region surrounding the contact position where D ≤ hc, whereas the surface deviations δ (Figure 9a) are negligible in the outer regions where D > hc. This amounts to overestimating the force contribution due to outer regions. The total energy is

appear even under pure splay−bend anchoring conditions in the flat geometry of Figure 1 when the film thickness is less that 100 nm.31,72 The escaped component of the n field, perpendicular to both anchoring directions, would help reduce the energy by relaxing the mechanical constraint with a twist deformation, associated with a lower elastic constant K2, or preferentially violate the azimuthal part of the planar anchoring, which typically requires a lower anchoring energy.57 Such escaped configurations typically produce complex disclination lines and point defects in NLC confined under competing anchoring conditions in thin cylindrical capillaries73 or between parallel plates.74 It is therefore likely that the sudden distortion of the FECO, often observed to occur at separations of D = 80−100 μm (cf. description of Figure 6b,c), was due to a transition to a twisted configuration in the outer regions of the confinement, promoted by the curvature-induced nonuniformity of the anchoring conditions that change from splay−bend to planar twisted when moving away from the contact position (Figure 2). Such an alternative configuration could not propagate to the contact region, where the alignment remained hybrid. Interestingly, the boundary between the two domains appeared to be located at or outside the area where the CTAB monolayer had been compressed in a flattened contact, and possibly, the pretilt thus increased because of the reorientation of the hydrocarbon chains (compare Figure 6b to 6f). Additional experiments are needed to establish the relationship between the location of the domain boundary and the edge of the compressed CTAB area and their connection with the network of disclination lines observed in the outer regions of the confinement.

⎛h ⎞ U (D) = EcAc(D) + πRK Δθ0 2 ln⎜ max ⎟ ⎝ hc ⎠

(A1)

where Ec = K/2L2 is the critical energy, Ac = 2πR(hc−D) is the area of the uniform region, and hmax is a large cutoff separation found in the outer regions, beyond which the contribution to the force is negligible. We have also neglected the variation of the director tilt rotation, set equal to the value Δθ0 imposed by the anchoring conditions at the contact. The total force is given by F(D) = −

⎛ 1 ⎞ dU = 2πRK ⎜ ⎟ = 2πREc dD ⎝ 2L 2 ⎠

(A2)

where we have used L2 ≪ hmax. Therefore, as D is decreased below hc, the force is expected to reach the critical value 2πREc with a slope dF/dD much more slowly than in the case of infinite anchoring, possibly reaching a plateau. As noted in Materials and Methods, the curvature of the SFA surfaces reduces the tilt rotation imposed by the anchoring conditions and induces a twist (azimuthal rotation in the xy plane) in the region surrounding the contact position (Figure 2). Assuming that the planar anchoring is stronger than the homeotropic anchoring, the presence of a twist does not affect the behavior of the NLC at the contact position in the limit D ≪ hc because the most stable director configuration in this regime is still uniform, planar, and oriented along the (local) planar easy axis. However, unlocking azimuthal (twist) distortion in the director field amounts to lowering the distortion energy for a given D and, therefore, the force required to confine the field to that value of D. As a consequence, azimuthal distortions are expected to increase the rate at which the 5CB film reaches the uniform planar configuration and F reaches the plateau as a function of D. Our SFA experiments were conducted at various values of the imposed twist angle Δϕ (Figure 2) but produced similar force curves. Therefore, the curvature-induced twist produced a contribution to the force that was not detectable over the distance range 0 < D < 300 nm of our measurements, either because it was too weak or its rate of variation with D was very slow so that it was indistinguishable from the linear background subtracted from the force curves for D > 150 nm. Most likely, this force had the same order of magnitude of the curvature effects calculated for pure splay−bend anchoring conditions (Δϕ = 0) in ref 32.

V. CONCLUSIONS The SFA measurements presented here provide the first direct experimental evidence of the confinement-induced anchoring transition from a splay−bend distorted configuration to a uniform planar orientation on the nanoscale, which has been predicted theoretically but thus far has remained elusive on this length scale. The unusually high strengths of the anchorings deduced from our measurements suggest that the classical picture of the uniaxial NLC order, with elastic constants determining the bulk director field and the anchorings determining the surface alignment, is inadequate to describe fully the structure of NLCs constrained in nanoscale films. A complete description should also take into account the complex molecular arrangement of NLC molecules near the surfaces, which may induce a lower symmetry (layering, polarity, and biaxiality) and spatial variations of the order parameters.

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APPENDIX Here, we discuss the effect of curvature on the anchoring conditions and normal forces in the crossed-cylinder geometry of the SFA (Figure 2). For the surface separations D and cylinder radius R ≫ D considered in our experiments, the gradients of the director field n in the xy plane of the cylinder axes can be neglected compared to the gradient along z over the circular area of the xy plane of radius much smaller than R occupied by the NLC droplet.32 The total free energy of elastic deformation can be calculated by integrating the energy E of eq 1 over the droplet area, approximating the surface separation around the contact position by h(x, y) ≈ D + (r2/2R), where r is the lateral distance from the contact position (r = 0) and h/R, r/R ≪ 1.38 This region produces the largest contribution to the

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AUTHOR INFORMATION

Corresponding Author

*(M.R.) Fax: 978-934-3013. E-mail: [email protected]. (B.Z.) Fax: +39 0984 494401. E-mail: bruno.zappone@fis. unical.it Notes

The authors declare no competing financial interest. 8381


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ACKNOWLEDGMENTS We thank R. Horn, T. Gruhn, J. Israelachvili, and P. Rabinowitz for many helpful discussions and the reviewers for their constructive comments. M.R. gratefully acknowledges access to SFA setups in the laboratories of W. Knoll (Max-PlanckInstitute for Polymer Research, Mainz, Germany) and J. Israelachvili (University of California, Santa Barbara, CA), financial support through a grant for bilateral researcher exchange between the Academy of Finland and Deutscher Akademischer Austauschdienst (DAAD) (grant no. 51588), a travel grant from the Magnus Ehrnrooth Foundation, and a CAREER award from the National Science Foundation (award no. 0645065). B.Z. thanks R. Bartolino and R. Barberi for their support at the Liquid Crystal Laboratory (LICRYL).

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