Scaling of the elastic contribution to the surface free energy of a nematic liquid crystal on a sawtoothed substrate

Scaling of the elastic contribution to the surface free energy of a nematic liquid crystal on a sawtoothed substrate Jose Manuel Romero-Enrique,1Chi-Tuong Pham,2and Pedro Patrício3,4 1Departamento de Física Atómica, Molecular y Nuclear, Area de Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain 2Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, CNRS-UPR 3251, Université Paris-Sud 11, BP 133, F-91403 Orsay Cedex, France 3Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro 1, P-1949-014 Lisboa, Portugal 4Centro de Física Teórica e Computacional, Universidade de Lisboa, Avenida Professor Gama Pinto 2, P-1649-003 Lisboa Codex, Portugal 共Received 7 October 2009; published 16 July 2010兲 We characterize the elastic contribution to the surface free energy of a nematic liquid crystal in the presence of a sawtooth substrate. Our findings are based on numerical minimization of the Landau-de Gennes model and analytical calculations on the Frank-Oseen theory. The nucleation of disclination lines 共characterized by nonhalf-integer winding numbers兲in the wedges and apexes of the substrate induces a leading order proportional to qln qto the elastic contribution to the surface free-energy density, with qbeing the wave number associated with the substrate periodicity. DOI: 10.1103/PhysRevE.82.011707 PACS number共s兲: 61.30.⫺v I. INTRODUCTION Topological defects are ubiquitous in many branches of physics, spanning from condensed-matter physics 关1,2兴to cosmology 关3兴. They may emerge in systems with broken continuous symmetry, and their presence can introduce essential singularities in the free energy which lead to infiniteorder phase transitions such as the Kosterlitz-Thouless transition 关4,5兴. Liquid-crystalline phases, such as nematics, are prototypical for showing topological defects, such as point hedgehogs and disclination lines 关6,7兴. Topological arguments show that within the bulk and in three dimensions the only stable disclination lines have ⫾1/2 winding number 关2兴. The presence of substrates may induce the nucleation of disclinations with other winding numbers. Theoretically, non-half-integer disclination lines are predicted to nucleate in the cusps 共wedges and/or apexes兲of substrates 关8–11兴,as well as in sharp boundaries between domains characterized by different anchoring conditions in flat surfaces 关12,13兴. Furthermore, there is some experimental evidence of the formation of these unusual non-half-integer disclination lines on surfaces 关14兴. It is well known that the nematic director field in the presence of structured substrates may be distorted, leading to an elastic contribution to the free energy. Since the seminal work by Berreman 关15,16兴, this problem has been extensively studied and generalized in the literature 关8–11,17–22兴. However, most of these studies focused on smooth substrates or on the effect of isolated cusps. In this paper we will consider the effect that the disclination lines nucleating on the cups of the substrates have on the elastic contribution to the surface free energy of the nematic on periodic and cusped structures. In particular, we will consider a sawtoothed substrate which favors homeotropic anchoring. Other geometries may also be considered with our formalism, such as steplike substrates, which have been studied numerically and experimentally in the context of zenithal bistable switching in nematic devices 关17,23,24兴. We will show that the disclination lines induce a contribution to the elastic contribution to the free-energy density 共per unit projected area兲which depends only on the geometric characteristics of the substrate, and that scales with the wave number qassociated with the periodicity of the substrate as −qln q The paper is organized as follows. The different models for the nematic in the presence of a sawtooth substrate are presented in Sec. II. We focus in particular on substrates which favor homeotropic anchoring and their consequences. In Sec. III we report the numerical and analytical results obtained from these approaches. In Sec. IV we present our conclusions. II. MODEL We consider a nematic phase in contact with a sawtooth substrate characterized by an angle ␣ and a side length L共see Fig. 1兲. At the substrate the nematic molecules preferentially align homeotropically, i.e., parallel to the local normal to the substrate. The system is translationally invariant along the Nematic p h ase α L Lz z x y ∞ FIG. 1. Schematic picture of the geometry of the system, characterized by the side length Land the angle ␣ . PHYSICAL REVIEW E 82, 011707 共2010兲 1539-3755/2010/82共1兲/011707共7兲©2010 The American Physical Society011707-1 out-of-plane axis zand periodic along the xaxis. We will only consider azimuthal distortions for the nematic director n, which can be parametrized in terms of the angle ␪ between the director and the ydirection as n=共−sin ␪ ,cos ␪ ,0兲. Out-of-plane or twist deformations may also be important under other conditions, as a twist instability may occur 关25兴, but we checked numerically that they are not relevant for our choice of parameters as free-energy minimization always leads to azimuthal distorted textures. Finally, we impose that, far from the substrate, the bulk nematic phase orients homogeneously along either the x共the N 储 texture兲or the y共the N⬜texture兲direction, which are the only relevant situations allowed by symmetry considerations. We study the orientational ordering of this system within two different models: the Landau-de Gennes 共LdG兲model and the Frank-Oseen 共FO兲model. A. Landau-de Gennes model In the LdG model, both isotropic and nematic phases can be locally represented by a traceless symmetric order parameter tensor with components Qij, which can be represented as Qij=3 2S关ninj−1 3 ␦ ij兴+1 2B关lilj−mimj兴, where niare the Cartesian components of the director field n,Sis the nematic order parameter which measures the orientational ordering along the nematic director, and Bis the biaxiality parameter, which measures the ordering of the molecules on the orientations perpendicular to n, characterized by the eigenvectors land m. The LdG free energy can be written as FLdG=兰V共 ␾ bulk+ ␾ el兲dV+兰A ␾ surfds, where ␾ bulk is the bulk free-energy density, ␾ el is the elastic free-energy density, and ␾ surf is the surface free energy defined as 关16兴 ␾ bulk =aTr Q2−bTr Q3+c关Tr Q2兴2,共1兲 ␾ el =L1 2 ⳵ kQij ⳵ kQij +L2 2 ⳵ jQij ⳵ kQik,共2兲 ␾ surf =−2 3wTr关Q·Qsurf兴,共3兲 where adepends linearly on the temperature, band care positive constants, and L1and L2are positive parameters related to the elastic constants. If we will rescale all the variables as follows 关26兴:Q ˜ =6cQ/b, the positions r ˜ =r/ ␰ , where the correlation length ␰ is defined as ␰ 2=8c共3L1+2L2兲/b2, and F ˜ LdG=242c3FLdG/ ␰ 3b4,weget F ˜ LdG=兰V ˜ 共 ␾ ˜ bulk+ ␾ ˜ el兲dV ˜ +兰A ˜ ␾ ˜ surfds ˜ , with rescaled freeenergy densities ␾ ˜ bulk =2 3 ␶ Tr Q ˜ 2−8 3Tr Q ˜ 3+4 9关Tr Q ˜ 2兴2,共4兲 ␾ ˜ el =1 3+2 ␬ 关 ⳵ ˜ kQ ˜ ij ⳵ ˜ kQ ˜ ij + ␬ ⳵ ˜ jQ ˜ ij ⳵ ˜ kQ ˜ ik兴,共5兲 ␾ ˜ surf =−2 3w ˜ Tr关Q ˜ ·Q ˜ surf兴.共6兲 Here, ␶ =24ac/b2is a dimensionless temperature, ␬ =L2/L1 is an elastic dimensionless parameter 共 ␬ ⬎−3/2兲and w ˜ =16wc/b2 ␰ is the dimensionless anchoring strength. Hereafter we will consider these rescaled expressions, so we will drop the overtilde notation. For ␶ =1, the bulk free-energy density has two minima corresponding to ␾ bulk=0 for rescaled scalar order parameters SI=0 共isotropic phase兲and SN=1 共nematic phase兲, so both phases are at coexistence. It is important to note that the order parameter Sin the coexisting nematic phase is rescaled, so its value in real units is b/6c, which must be smaller than 1 共typically ⬇0.4兲.Ifthe elastic parameter ␬ is positive 共negative兲, the nematic prefers to align parallel 共perpendicular兲to a possible nematicisotropic 共NI兲interface. Finally, Qsurf defines the favored tensor at the substrate. We will favor a homeotropic alignment of the nematic by setting Qsurf=共3 ␯ 丢 ␯ −1兲/2, with ␯ as the normal vector to the substrate, establishing a direct connection to previous papers 关27–30兴. Hereafter we will restrict ourselves to the nematic phase at the NI transition 共 ␶ =1兲, with ␬ =2. B. Frank-Oseen model The FO model can be considered as an approximation to the LdG model, in which we assume that the variations of the nematic order parameter are restricted to the neighborhood of the substrate 共with a width on the order of ␰ 兲and inside defect cores, and takes the bulk value Selsewhere. Assuming that there are no disclination lines in the bulk, substitution of this Ansatz into the LdG free-energy functional leads to the following FO approach for large L共in units of ␰ 兲: F⬇FFO ⬅ ␾ bulk共S兲V+ 冕 A ds⌺„ ␪ 共s兲…+K 2 冕 V dV兩ⵜ ␪ 兩2, 共7兲 where Vis the total volume of the nematic, Ais the substrate area in contact with the nematic, the elastic constant K=共9/2兲S2共2+ ␬ 兲/共3+2 ␬ 兲and ⌺共 ␪ 兲is an effective anchoring potential to be determined in the following that is due to the nematic order parameter distortions close to the substrates. In order to make a quantitative comparison with the results within the LdG model, instead of considering an ad hoc expression for the anchoring potential, we will derive it from the LdG model. As it was mentioned above, the nematic order parameter distortions are assumed to be confined to a layer of width ␩ ⬃ ␰ close to the substrates. On the other hand, the variations of the nematic director field close to the walls are rather small in directions parallel to the surfaces for large L共except close to the wedges and apexes兲. So, we may estimate ⌺共 ␪ 兲as the LdG excess free energy per unit area of a slab of width ␩ , for which, at the boundary y=0,weimpose a surface field ␾ surf, and for yⱖ ␩ we consider a bulk nematic phase with an uniform nematic director character- ROMERO-ENRIQUE, PHAM, AND PATRÍCIO PHYSICAL REVIEW E 82, 011707 共2010兲 011707-2 ized by a tilt angle ␪ with respect to the yaxis. Figure 2 shows ⌺共 ␪ 兲obtained by numerical minimization with a conjugate-gradient method for ␩ =1.5 ␰ 共similar results are obtained for other values of ␩ 兲. For each value of w, the minimum value of ⌺共 ␪ 兲corresponds to the homeotropic alignment ␪ =0, where it takes the 共true兲nematic-wall surface tension value ␴ NW共w兲. For not too large values of ␪ ,⌺共 ␪ 兲takes a Rapini-Papoular form 关31兴 ⌺共 ␪ 兲⬃ ␴ NW+⌺0 ⬙共w兲sin2 ␪ . There are deviations for ␪ around ␲ /2, but in any case the involved energies are much larger. In the range of values of wwe will consider in this paper 共0⬍w⬍2兲, both ␴ NW and ⌺0 ⬙are on the order of w. Indeed ␴ NW can be obtained analytically from minimization of the LdG functional in the presence of a flat wall and for homeotropic anchoring 关27,28兴. In this situation the equilibrium nematic director field does not show any deformation and there is no biaxiality, so the resulting free energy at NI coexistence can be expressed in terms of the nematic order parameter profile S=S共y兲 as FLdG=A兰0 ⬁关S2−2S3+S4+共S⬘兲2/2兴dy−wS共0兲. where S⬘共y兲=dS/dy. By introducing a magnetizationlike field m共y兲⬅2S共y兲−1, the LdG functional reduces to the Landau- Ginzburg free-energy functional of an Ising model for zero applied magnetic field in the presence of a flat wall. This problem has been extensively studied in the literature 关32,33兴, leading to a nematic order parameter profile S共y兲=关1+gexp共−冑2y兲兴−1, where gis obtained from the boundary condition S⬘共0兲=−w. The resulting expression for ␴ NW is ␴ NW =冑3共g+3兲g2 6共1+g兲3−w 1+g,共8兲 where g=−共1+1/冑2w兲+冑共1+1/冑2w兲2−1. In order to obtain the azimuthal angle ␪ field, we have to minimize the energy functional given by Eq. 共7兲. In the bulk, we thus have to solve the Laplace equation ⵜ2 ␪ =0 with appropriate anchoring conditions at the boundaries. Let ␯ =共−sin ␪ 0,cos ␪ 0,0兲now be the local normal to the substrate. Using the approximation ⌺共 ␪ 兲 ⬃ ␴ NW+⌺0 ⬙共w兲sin2共 ␪ − ␪ 0兲found previously, the nematic director field then satisfies approximately that K ␯ ·ⵜ ␪ +⌺0 ⬙sin关2共 ␪ − ␪ 0兲兴=0. Note that we may assume strong homeotropic anchoring conditions when Lis large compared to the extrapolation length K/2⌺0 ⬙⬃K/wor, equivalently, wLⰇ1. We can justify this assumption by considering the following rescaling: rⴱ=r/L, ␪ ⴱ共rⴱ兲= ␪ 共r兲, and FFO ⴱ=FFO/L. In this rescaled description, we must minimize the FO free energy 共7兲in the rescaled domain subject to an effective anchoring potential ⌺ⴱ共 ␪ ⴱ兲=L⌺共 ␪ 兲. This means that the dependence on the size of the system can be absorbed into the coefficients of the anchoring potential: ␴ NW 共which does not affect the anchoring conditions兲and ⌺0 ⬙共which is rescaled by a factor of L兲. Strong anchoring condition is then satisfied when the rescaled extrapolation length K/2⌺0 ⬙LⰆ1, in agreement with our previous estimate. III. RESULTS A. Numerical results within the LdG model We now turn to the evaluation of the distortion contribution in the LdG model We consider values of ␣ between 0 and ␲ /2, 8ⱕLⱕ96, and 0ⱕwⱕ2. In order to get the orientational ordering, we numerically minimize the LdG freeenergy functional by using a conjugate-gradient method. The numerical discretization of the continuum problem is performed with a finite element method combined with adaptive meshing in order to resolve the different length scales that may emerge in the problem 关34兴. The numerical procedure is similar to that used to study the wetting transition by the nematic phase in this geometry 关29兴. The numerical minimization of the LdG functional FLdG shows that the N⬜texture has lower free energy if ␣ ⬍ ␲ /4 owing to lesser distortion. Conversely, the N 储 texture has lower free energy for ␣ ⬎ ␲ /4. These results are in agreement with previous studies in the literature 关8–11兴. As a consequence, we observe bistability in a range of values of ␣ around ␲ /4. Figure 3 displays some typical textures. We see that on the substrates the nematic orientation is preferentially homeotropic. Inspection of these textures shows that in general there are no disclinations in bulk, except for large wand ␣ around ␲ /4, but there is an important distortion on the nematic director field close to the wedges and apexes. On the other hand, when disclinations are observed in bulk 关see, for example, Fig. 3共b兲or Fig. 3共c兲兴, they are at a distance on the order of ␰ from a wedge or apex, which is almost independent of L. The analysis of the calculated equilibrium free energies confirms that the leading-order contribution to the equilibrium free energy per unit area is ␴ NW, which is a further confirmation that there is strong anchoring on the substrates. As a consequence, we may obtain the elastic part of the free energy per cell 共i.e., a slice parallel to the yz plane with width along the xaxis equal to the substrate period兲and unit length in the zdirection, felastic,as2L共FLdG兩eq/A− ␴ NW兲. Numerical evaluation of felastic shows a clear dependence on L 0 0.2 0.4 0.6 0.8 1 sin2 θ -3 -2 -1 0 Σ( θ ) w=0.5 w=1.0 w=1.5 w=2.0 FIG. 2. Plot of the effective anchoring potential ⌺共 ␪ 兲as a function of sin2 ␪ , for a layer width ␩ =1.5 ␰ and w=0.5 共circles兲,w=1 共squares兲,w=1.5 共diamonds兲, and w=2 共triangles兲. The dashed horizontal lines correspond to the values of ␴ NW obtained from Eq. 共8兲, and the continuous lines correspond to the fit to a Rapini- Papoular expression for small values of the tilt angle. The values of the slopes ⌺0 ⬙are 0.5655共w=0.5兲, 0.8729共w=1兲, 1.0678共w=1.5兲, and 1.17711 共w=2兲. SCALING OF THE ELASTIC CONTRIBUTION TO THE…PHYSICAL REVIEW E 82, 011707 共2010兲 011707-3 in a broad range of values of w共see Fig. 4for ␣ = ␲ /6 and ␣ = ␲ /3兲. It is interesting to compare this result to the prediction within Berreman’s approach for the elastic contribution to a smooth substrate-nematic surface free energy 关15兴. Although originally this approach was introduced for shallow sinusoidal substrates with strong tangential anchoring, it can be extended straightforwardly to the present case with homeotropic anchoring, either weak or strong. By using the electrostatic analogy, we may expand the azimuthal angle field ␪ as ␪ 共x,y兲=兺 n=0 ⬁ Mnsin共nqx兲exp共−nqy兲,共9兲 where qis the wave number associated with the substrate periodicity, and the coefficients Mnare chosen to satisfy the imposed boundary conditions. Substitution of this expression in the FO free energy leads to the elastic contribution to the surface free energy which depends only on Aq, with Aas the roughness amplitude. For sinusoidal substrates under moderate or strong anchoring conditions, Berreman’s approximation felastic⬇ ␲ K共Aq兲2is valid if qA is smaller than 1 关21兴.In any case, if Ascales like L, the surface free energy should be independent of L. In order to explain and characterize the anomalous scaling for the elastic free energy obtained within the LdG framework, we will now resort to the FO approximation. B. Results with the FO model Disclination lines are known to nucleate on wedges and apexes 关8–11兴. Their presence introduces a singular contribution to the orientation field ␪ which cannot be expanded as in Eq. 共9兲. We may write ␪ 共x,y兲as ␪ s共x,y兲+ ␪ ns共x,y兲, where ␪ sis the singular contribution due to the disclination lines and ␪ ns is the nonsingular part. Note that Berreman assumed that ␪ s⬅0. The singular ␪ field associated with one isolated disclination line located at the origin and characterized by a winding number Itakes the form −Iarctan共x/y兲=II关ln共i ␨ 兲兴, where ␨ =x+iy and I共 ␨ 兲is the imaginary part of ␨ .Inthe bulk, the values of the winding number Iare restricted to half-integer values to avoid discontinuities in the nematic director field. For a periodic linear array of disclination lines, we may obtain the singular nematic orientation field from a conformal transformation, ␨ 哫 ␨ ⬘= sin 冉 q ␨ 2 冊 = sin 冉 qx 2 冊 cosh 冉 qy 2 冊 +icos 冉 qx 2 冊 sinh 冉 qy 2 冊 , 共10兲 where the wave number which characterizes the substrate periodicity is q= ␲ /Lcos ␣ . The transformation maps a neighborhood of any point ␨ n=n共2Lcos ␣ 兲, with nas an integer, to a neighborhood of the origin in the ␨ ⬘-complex plane. If we denote ␨ = ␨ n+ ⑀ 共兩 ⑀ 兩small兲, then the transformation leads to a complex number ␨ ⬘⬇共−1兲nq ⑀ /2. Thus, the orientation field II关ln共i ␨ ⬘兲兴=−Iarctan关tan共qx/2兲coth共qy/2兲兴, which is a solution to the Laplace equation, periodic in x with period 2Lcos ␣ and singular at each ␨ n, reduces to the field corresponding to an isolated defect of winding number Iin the neighborhood of any ␨ n. Focusing on the range x苸关−Lcos ␣ ,Lcos ␣ 兴, the azimuthal angle generated by this function is ␪ =0 at x=0 and x=⫾Lcos ␣ . On the other hand, for 兩qy/2兩Ⰷ1, ␪ ⬇−sgn共y兲Iqx/2 for x⫽⫾Lcos ␣ , where sgn共y兲=y/兩y兩is the sign of y. Note that there is a jump from ␪ =−sgn共y兲I ␲ /2to ␪ =sgn共y兲I ␲ /2 when crossing x=Lcos ␣ . In the bulk, this solution is acceptable if Iis an FIG. 3. 共Color online兲Contour maps of the nematic order parameter S关blue 共dark gray兲for small S, white for the bulk value S=1, and orange 共light gray兲for higher S兴and the nematic director field n共gray segments兲for the equilibrium textures obtained from minimization of the LdG model for L=16 and w=1.0: 共a兲 ␣ = ␲ /6 共N⬜兲,共b兲 ␣ = ␲ /4共N⬜兲,共c兲 ␣ = ␲ /4共N 储 兲, and 共d兲 ␣ = ␲ /3共N 储 兲. The singular nematic director field from the FO approach 关Eqs. 共11兲and 共12兲兴 is also shown for comparison 共black segments兲. Note that they are indistinguishable almost everywhere. 00.5 11 .5 w 0 2 4 6 8 felastic L*=8 α=30ο L*=8 α=60ο L*=16 α=30ο L*=16 α=60ο L*=32 α=30ο L*=32 α=60ο L*=48 α=30ο L*=48 α=60ο L*=64 α=30ο L*=64 α=60ο L*=96 α=30ο L*=96 α=60ο FIG. 4. 共Color online兲Plot of felastic as a function of the anchoring strength wfor ␣ = ␲ /6共black symbols兲and ␣ = ␲ /3关orange 共light gray兲symbols兴. We considered different cell sizes: L=8 共circles兲,L=16 共squares兲,L=32 共diamonds兲,L=48 共up triangle兲, L=64 共left triangle兲, and L=96 共down triangle兲. ROMERO-ENRIQUE, PHAM, AND PATRÍCIO PHYSICAL REVIEW E 82, 011707 共2010兲 011707-4 integer 共note that the nematic state is invariant under inversion of the nematic director兲.IfIis a half-integer, we may add to this solution the orientation field Iarctan关tan共qx/2兲兴 due to an array of disclination lines with winding number −Ilocated at ␨ n−i⬁, which in the xinterval 共−Lcos ␣ ,Lcos ␣ 兲reduces to Iqx/2. The compound orientation field I兵−arctan关tan共qx/2兲coth共qy/2兲兴 +arctan关tan共qx/2兲兴其 has no discontinuity at x=⫾Lcos ␣ for y⬎0共in fact, it goes to zero as qy/2Ⰷ1兲, but it jumps from I ␲ to −I ␲ when crossing x=Lcos ␣ for y⬍0. This is again physically acceptable if Iis a half-integer. Finally, we must mention that there are different physically equivalent representations of the same singular field associated with a periodic array of disclination lines located at x= ␨ nas, for example, the field I兵arctan关cot共qx/2兲tanh共qy/2兲兴−arctan关cot共qx/2兲兴其. This orientation field shows a physically acceptable discontinuity at x=0 for y⬍0ifIis a half-integer. When disclination lines are located on surfaces, their winding numbers Iare not constrained to half-integer values 关8–11兴, as we may tune the winding numbers in order to match the boundary conditions close to any wedge or the apex in the strong anchoring regime. Alternatively, a Schwartz-Christoffel transformation may be used 关8兴. For the N⬜texture, we find that the winding numbers I1and I2for the disclination lines at the wedge bottom and at the apex top, respectively, are I1=− ␣ /共 ␲ /2− ␣ 兲and I2= ␣ /共 ␲ /2+ ␣ 兲. For the N 储 texture, the topological charges are I1=1 and I2=−共 ␲ /2− ␣ 兲/共 ␲ /2+ ␣ 兲. The resulting singular orientation fields, ␪ s ⬜and ␪ s 储 , can be written in terms of the wave number qand for x苸共−Lcos ␣ ,Lcos ␣ 兲as ␪ s ⬜=− ␣ ␲ 2− ␣ 冋 − arctan 冉 tanqx 2cothqy 2 冊 + arctan 冉 tanqx 2 冊 册 + ␣ ␲ 2+ ␣ 再 − arctan 冋 tanqx 2tanhq共y−Lsin ␣ 兲 2 册 + arctan 冋 tanqx 2 册 冎 ,共11兲 ␪ s 储 = ␲ 2+ 冋 − arctan 冉 tanqx 2cothqy 2 冊 + arctan 冉 tanqx 2 冊 册 − ␲ 2− ␣ ␲ 2+ ␣ 再 − arctan 冋 tanqx 2tanhq共y−Lsin ␣ 兲 2 册 + arctan 冋 tanqx 2 册 冎 .共12兲 These expressions do not show any discontinuity above the substrate. We checked that these solutions approximate quite well the boundary anchoring conditions 共but not exactly兲, so that ␪ ns may be neglected in most of the cases. Furthermore, the agreement between our Ansätze for the director orientation field and the numerical LdG textures is excellent except close to the defect cores, as can be seen in Fig. 3. The elastic energy can be obtained by standard contour integration techniques. By symmetry, felastic can be obtained as twice the contribution of the half-cell region R共see Fig. 5兲. By using the identity ⵜ·共 ␪ ⵜ ␪ 兲=兩ⵜ ␪ 兩2+ ␪ ⵜ2 ␪ , and using the divergence theorem and the fact that ⵜ2 ␪ =0 in R,we obtain that felastic can be expressed in terms of an integral over the contour Cof R, felastic =K 冖 C ␪ ␯ ·ⵜ ␪ ds.共13兲 Along the segments x=0, x=Lcos ␣ , and y→⬁, we impose that ␪ = ␪ sing= ␣ ⬁, where the far-field azimuthal angle is ␣ ⬁=0 for the N⬜texture and ␣ ⬁= ␲ /2 for the N 储 texture. We link C1to the x=0 and x=Lcos ␣ segments by circle arcs of small radii ⑀ . These circles take into account the presence of a defect core, and their radii are proportional to the nematic coherence length, which provides a microscopic cutoff for the continuum Frank-Oseen model. In these boundaries, ␪ approaches, up to a constant, the field of one isolated disclination line, which satisfies the following property ␯ ⬜ⵜ ␪ as ⑀ →0. So we can neglect the contribution of the circle arcs to the contour integral. Finally, ␪ = ␣ along the segment C1for strong anchoring conditions. Taking into account that 养C ␯ ·ⵜ ␪ ds=0, we may rewrite Eq. 共13兲as felastic =K 冖 C 共 ␪ − ␣ ⬁兲 ␯ ·ⵜ ␪ ds =K共 ␣ − ␣ ⬁兲 冖 C1 ␯ ·ⵜ ␪ ds. 共14兲 If ␪ is split into singular and nonsingular contributions, we see that the nonsingular term, which can be expressed by Eq. 共9兲, leads to a contribution to felastic independent of L.Onthe other hand, the leading-order contribution to felastic comes from the singular orientation field, namely, the contribution close to the disclination lines and apex. If sis the distance of a point of the contour C1close to the wedge or apex, ␯ ·ⵜ ␪ sing⬇−I1/sor −I2/s, respectively. The leading-order contribution to felastic comes from integration on C1as ν 1 C 1 I 2 I C α R FIG. 5. Plot of the integration contour C共dashed line兲to get the elastic contribution to the free energy in the region Rinside the contour. The contour is deformed to avoid the cores of the disclination lines with wandering numbers I1and I2at the wedge and apex 共filled circles兲. We highlight the contour C1with an associated outward normal ␯ . SCALING OF THE ELASTIC CONTRIBUTION TO THE…PHYSICAL REVIEW E 82, 011707 共2010兲 011707-5 felastic ⬃K共 ␣ 兲ln L/ ⑀ ,共15兲 where Kis defined as K共 ␣ 兲=K共 ␣ − ␣ ⬁兲共I2−I1兲= 冦 K ␲ ␣ 2 共 ␲ 2 兲 2− ␣ 2, ␣ ⬍ ␲ 4 K ␲ ␲ 2− ␣ ␲ 2+ ␣ , ␣ ⬎ ␲ 4. 冧 共16兲 Expressions 共15兲and 共16兲show that the equilibrium texture is N⬜for ␣ ⬍ ␲ /4, and N 储 for larger values of ␣ , in agreement with our LdG calculations and previous results reported in the literature 关8兴. C. Analysis of the results within the LdG model Finally, in order to check the accuracy of our approximation, we analyze the results obtained with the LdG model by fitting the values of felastic obtained from minimization to an expression felastic =K共 ␣ 兲ln共L/ ␰ 兲+B共 ␣ ,w兲,共17兲 where K共 ␣ 兲is given by expression 共16兲and B共 ␣ ,w兲is expected not to depend explicitly on Lif wL is large enough 共i.e., under strong anchoring conditions兲. The resulting curves for different values of Lof B共 ␣ ,w兲as a function of w for a fixed value of ␣ collapse into a master curve 共see Fig. 6兲. Deviations only appear for small values of wⱗL−1,at which the strong anchoring condition breaks down. This result shows clearly that there is a contribution to the elastic part of the surface free energy which scales logarithmically with the periodicity of the substrate. For large L 共small q兲and wⰇL−1, the surface free-energy density 共i.e., per unit projected area on the xz plane兲of a nematic in the presence of a sawtooth substrate has the asymptotic behavior ␴ NW/cos ␣ −关K共 ␣ 兲/2 ␲ 兴qln q+O共q兲. The nonanalytical contribution −qln qintroduces a slow decay of the surface free energy of the nematic with increasing L. This may help to explain the large deviations with respect to the Wenzel law observed for the wetting transition by nematic of a sawtoothed substrate in contact with the isotropic phase 关29兴. IV. CONCLUSIONS In this paper we have analyzed the size dependence of the elastic contribution to the interfacial free-energy density of a nematic in the presence of a sawtoothed substrate. The nucleation of non-half-integer disclination lines of the nematic director field on the apexes and wedges of the substrate, predicted in the FO model, induces a nonanalytical contribution which scales with the substrate periodicity wave number qas −qln qfor small q. This has been confirmed by the numerical results in the full LdG model. The periodicity scaling of the elastic contribution to the surface free energy is different from that obtained for smooth surfaces, which scales linearly with q. Our arguments are not specific to this kind of substrate and can be extended straightforwardly to any surface shape which shows ridges, cusps, and similar singularities. On the other hand, the size scaling of the elastic free energy has consequences for surface transitions such as wetting. Finally, the recent impressive advances in microfluidic technology and surface patterning open the possibility of an experimental verification of our predictions. 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