Density functional theory study of the nematic-isotropic transition in an hybrid cel

Density functional theory study of the nematic–isotropic transition in an hybrid cell I. Rodrıguez-Ponce, J. M. Romero-Enrique, and L. F. Rull Citation: The Journal of Chemical Physics 122, 014903 (2005); doi: 10.1063/1.1829041 View online: http://dx.doi.org/10.1063/1.1829041 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/122/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The isotropic-nematic and nematic-nematic phase transition of binary mixtures of tangent hard-sphere chain fluids: An analytical equation of state J. Chem. Phys. 140, 034504 (2014); 10.1063/1.4860980 Density-functional theory of the columnar phase of discotic Gay–Berne molecules J. Chem. Phys. 116, 7744 (2002); 10.1063/1.1467340 The nematic-isotropic phase transition in semiflexible fused hard-sphere chain fluids J. Chem. Phys. 114, 3314 (2001); 10.1063/1.1340606 Interfaces, wetting, and capillary nematization of a hard-rod fluid: Theory for the Zwanzig model J. Chem. Phys. 113, 7689 (2000); 10.1063/1.1288903 The nematic–isotropic phase transition in linear fused hard-sphere chain fluids J. Chem. Phys. 110, 11630 (1999); 10.1063/1.479102 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 Density functional theory study of the nematic–isotropic transition in an hybrid cell I. Rodrı ´guez-Ponce Physik Department, Technische Universita ¨tMu ¨nchen, James-Franck-Strasse, D-85747 Munich, Germany J. M. Romero-Enrique Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Area de Fı ´sica Teo ´rica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain and Department of Mathematics, Imperial College 180 Queen’s Gate, London SW7 2BZ, United Kingdom L. F. Rull Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Area de Fı ´sica Teo ´rica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain 共Received 2 August 2004; accepted 13 October 2004; published online 13 December 2004兲 We have employed the density functional theory formalism to investigate the nematic–isotropic capillary transitions of a nematogen confined by walls that favor antagonist orientations to the liquid crystal molecules 共hybrid cell兲. We analyze the behavior of the capillary transition as a function of the fluid–substrate interactions and the pore width. In addition to the usual capillary transition between isotropiclike to nematiclike states, we find that this transition can be suppressed when one substrate is wet by the isotropic phase and the other by the nematic phase. Under this condition the system presents interfacelike states which allow us to continuously transform the nematiclike phase to the isotropiclike phase without undergoing a sharp phase transition. Two different mechanisms for the disappearance of the capillary transition are identified. When the director of the nematiclike state is homogeneously planar-anchored with respect to the substrates, the capillary transition ends up in a critical point. This scenario is analogous to the observed in Ising models when confined in slit pores with opposing surface fields which have critical wetting transitions. When the nematiclike state has a linearly distorted director field, the capillary transition continuously transforms in a transition between two nematiclike states. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1829041兴 I. INTRODUCTION The effect of the confinement on simple fluids has been vastly investigated in the past. Most of studies are related to the effect of the confinement on the vapor–liquid transition, the so-called capillary transition. In the case of symmetric walls, where both walls attract or repel the molecules with the same strength, the capillary transition in large pores is governed by the well-known Kelvin law.1In addition, it may appear surface phenomenology that can affect the global phase diagram. Often this surface phenomenology is related to wetting states found in the single wall cases. The latter phenomenon is related to the spreading of drops of vapor 共liquid兲on the surface in coexistence with the bulk liquid 共vapor兲. The presence of these surface states can be considered in the description of the phase behavior in the confined system. The generalized Kelvin law for small pores 共where surface phenomenology is more dominant兲takes into account these effects so it is still possible to determine the vapor–liquid transition for small pores.2However, the predictions of the generalized Kelvin law reveal that the wetting phenomena does not affect essentially the phase diagram of the confined problem so they are often discarded. An interesting system is the hybrid cell 共also known as asymmetric cell兲where one substrate or wall is repulsive and the other attractive 共competing walls兲. In this case, the wetting properties present in the single wall cases play a crucial role in the determination of the critical point in the confined problem. Here, given a pore width, the coexistence of two confined phases characterized by distinct order parameter profiles can only occur for temperatures below TW, which corresponds to the total wetting temperature of the single wall cases,3,4 provided the substrate–fluid interaction strengths are equal in absolute value. A very good insight in the confined phase diagram topology for the general case in which the substrate–fluid interaction strengths are neither equal nor opposing has been reported for a polymer model.5 These studies have elucidated which mechanisms are involved in the surface transitions and how the properties of the substrate can determine the phase behavior within the pore. With respect to complex fluids like liquid crystal 共LC兲 there is an increasing interest in the understanding of the surface phenomena due to its crucial role in the advance of the LC technology. For the case of a single wall 共semiinfinite problem兲and symmetric pores favoring an specific orientation on the molecules, theoretical approaches6–12 as well as computer simulations13–17 reveal that the interaction wall–fluid particle can strongly affect the ordering not only close to the surface but also far from it. Additionally to finite size effects, novel phenomena often appear due to the combined presence of anchoring and orientational wetting tran- THE JOURNAL OF CHEMICAL PHYSICS 122, 014903 共2005兲 122, 014903-10021-9606/2005/122(1)/014903/10/$22.50 © 2005 American Institute of Physics Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 sitions. The interplay between both surface phenomena and its dependence on wall properties have been recently studied by us employing a microscopic density functional theory 共DFT兲approach for a simple model of liquid crystal.18 The effect of confinement by symmetric walls on these surface transitions was also studied.19 Our findings revealed that as in the case of simple fluids, the finite-size effects for not very narrow pores on the phase diagram are not of qualitative significant importance and only the location of the capillary transitions are affected by the confinement. However, recent theoretical20,21 and simulation results20 for hard-sphero- cylinder models show that for very narrow symmetrical pores the NI capillary transition may end up in a critical point. In a recent paper,22 we extended the study to the case with an hybrid cell where the surfaces favor antagonist orientations. This geometry has in LCs a particular interest due to the possible technological applications. This fact has stimulated an increasing interest by experimentalists and theoreticians.20,23,24 In particular, different nematiclike states with homogeneous and linearly distorted director fields were found to coexist under confinement. In the present paper, our goal is to complete this study with the investigation of the nematic–isotropic 共NI兲transition within the hybrid cell. Quintana et al. investigated a similar problem employing a Landau-type theory.25 They claimed that the NI transition either appears unaffected and its temperature TNI remains unchanged or the transition disappears. Basically what determined the existence of the transition was the value of fluid– wall interaction strength with respect to the value for the first-order wetting transition for the semi-infinite systems. For strength values below this threshold the transition occurs for all wall separations. In contrast, above the threshold, the transition disappears for large pore widths, where interfaciallike states appear. The disappearance of the phase transition must be understood as the gradual transformation from isotropiclike states to nematiclike states when increasing the chemical potential or decreasing the temperature, with no singularities in the free energy associated to the transition. However, for large pore widths it would be extremely difficult to distinguish this rounded transition from a true thermodynamic phase transition. The interfacial-like states are related to the wetting states by isotropic and nematic phase that each wall present in the semi-infinite geometry, respectively. In this paper we address the same problem of the behavior of the NI capillary transition and its possible disappearance from a more microscopic point of view by using a DFT approach. Our paper is organized as follows: First, we briefly present our liquid crystal model in Sec. II. In Sec. III we revisit and complete the study of a substrate which favors either planar or homeotropic anchoring in contact with a nematogen fluid at its isotropic or nematic phase. In particular, the different wetting regimes are found. This phenomenology will be a crucial guide to understand the behavior of the fluid under confinement. Section IV is devoted to the study of the fluid adsorbed in a hybrid cell, paying special attention to the disappearance of the NI capillary transitions and the mechanisms involved in it. Finally we end up with our conclusions. II. THE MODEL The theoretical model is a standard generalized van der Waals theory based on a perturbative expansion, using a hard-sphere 共HS兲fluid as reference system.8Details on the physical basis of the model and how to obtain its solutions numerically can be found elsewhere.26,27 Our starting point is the grand potential functional per unit system area A, ⍀关 ␳ 兴/A, whose functional minimum with respect to the oneparticle distribution function, ␳ 共r,⍀ ˆ兲, which depends on both molecular positions rand orientations ⍀ ˆ, gives the equilibrium structure of the interface. This function, ␳ (r,⍀ ˆ) ⬅ ␳ (z)f(z,⍀ ˆ), contains a mass distribution ␳ (z) and an angular distribution f(z,⍀ ˆ). These quantities vary locally with the distance from z⫽0toz⫽H, where His the pore width. The expression for ⍀关 ␳ 兴, in a mean field approximation is, ⍀关 ␳ 兴⫽Fr关 ␳ 兴⫹1 2 冕冕冕 drdr⬘d⍀ ˆd⍀ ˆ⬘ ␳ 共r,⍀ ˆ兲 ⫻ ␳ 共r⬘,⍀ ˆ⬘兲⫻v共r⫺r⬘,⍀ ˆ,⍀ ˆ⬘兲 ⫺ 冕冕 drd⍀ ˆ ␳ 共r,⍀ ˆ兲关 ␮ ⫺vW共r,⍀ ˆ兲兴,共1兲 where ␮ is the chemical potential and Fr关 ␳ 兴⫽ 冕 drfhs共 ␳ 共r兲兲⫹kBT 冕 dr ␳ 共r兲 具 ln共4 ␲ f共z,⍀ ˆ兲兲 典 共2兲 is the reference system free energy. In the above expression fhs( ␳ (r)) is the hard-sphere free energy density of a uniform fluid with a density equal to the local density at rand 具¯典is an angular average. The attractive potential vcontains anisotropic 共dispersion兲forces driving the liquid-crystalline behavior of the model material: v共r,⍀ ˆ,⍀ ˆ⬘兲⫽vA共r兲⫹vB共r兲P2共⍀ ˆ•⍀ ˆ⬘兲⫹vC共r兲 ⫻关P2共⍀ ˆ•r ˆ兲⫹P2共⍀ ˆ⬘"r ˆ兲兴,共3兲 where r ˆ⫽r/r, and ris the center-of-mass distance. The functions vA(r), vB(r), vC(r) are the radial contribution to the potential as a result of considering dispersion-type forces. In this work we choose them to have a simple Yukawa form, i.e., vi(r)⫽⫺ ⑀ iexp(⫺␭i(r⫺d))/rfor r⬎d, and vi(r)⫽0 otherwise, where dis the diameter of a hard sphere. The vB and vCterms are the responsible of the liquid-crystalline behavior of our fluid model. So, vBdetermines the orientational order in bulk. The effect of vCon the orientational ordering is more subtle, since by symmetry its contribution to the bulk nematic free energy is zero at mean-field level. However, it plays an important role in determining the orientation of the particles close to interfaces and walls. We note that a positive value of ⑀ Cfavors the molecules to orientate parallel to the nematic–vapor 共NV兲interface and a negative value favors the perpendicular orientation to NV interface.28 As this behavior is caused by the absence of orientational ordering in the vapor phase, it is also observed for 014903-2 Rodrı ´guez-Ponce, Romero-Enrique, and Rull J. Chem. Phys. 122, 014903 (2005) Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 a NI interface and a nematic fluid in contact with a hard wall. This anisotropy in the orientational configuration of the nematic molecules is essential to understand the phenomenology in the problem with the walls. The walls are modelled via the following potentials: vW 共1兲共z, ␪ 兲⫽ 再 ⫹⬁z⬍0 ⫺ ⑀ W 共1兲e⫺␭W共z⫺d兲P2共cos ␪ 兲z⬎0, 共4兲 where ␪ is the angle between the molecule axis and the z-axis normal to the substrate. For ⑀ W (1)⬎0, this potential favors the molecules to align perpendicularly to the surface 共homeotropic anchoring兲. The potential corresponding to the other substrate is: vW 共2兲共z, ␪ 兲⫽ 再 ⫹⬁z⬎H ⑀ W 共2兲e⫺␭W共⫺共H⫺z兲⫺d兲P2共cos ␪ 兲z⬍H,共5兲 which favors the molecules to align in a plane parallel to the surface for ⑀ W (2)⬎0. However, this preferred alignment corresponds to a random planar anchoring, since there is not a preferred direction at the xy orientation plane. The parameters ⑀ W (1) and ⑀ W (2) measure the substrate–fluid interaction strength of each substrate. In our study, we will assume that the nematic director orientation is embedded in the xz plane. The formalism permits to define the orientation distribution of the molecules referred to a laboratory reference system, described by three extrinsic order parameters, ␩ 共z兲⫽ 冕 d ␾ sin ␪ d ␪ f共z,⍀ ˆ兲P2共cos ␪ 兲,共6兲 ␯ 共z兲⫽ 冕 d ␾ sin ␪ d ␪ f共z,⍀ ˆ兲sin2 ␪ cos ␾ ,共7兲 ␴ 共z兲⫽ 冕 d ␾ sin ␪ d ␪ f共z,⍀ ˆ兲sin2 ␪ cos2 ␾ .共8兲 The order also can be referred to an intrinsic reference system by three intrinsic order parameters: ␺ the tilt angle defined like the angle formed by the director with the zaxis, U 共uniaxial order parameter兲the amount of order along the director and B共biaxial order parameter兲which measures the amount of order along the perpendicular directions of the director. The nematic ordering is essentially uniaxial (B ⫽0) far from the substrate. The biaxiality induced by the surfaces was already discussed in a previous paper.22 These intrinsic order parameters can be obtained in terms of the set 兵 ␩ , ␴ , ␯ 其by the following expressions: tan ␺ 共z兲⫽ ␯ 共z兲 ␩ 共z兲⫺ ␴ 共z兲 2⫹ 冑 冋 ␩ 共z兲⫺ ␴ 共z兲 2 册 2 ⫹ ␯ 共z兲2 , 共9兲 U共z兲⫽ ␩ 共z兲 4⫹3 8 ␴ 共z兲⫹3 4 冑 冋 ␩ 共z兲⫺ ␴ 共z兲 2 册 2 ⫹ ␯ 共z兲2, 共10兲 B共z兲⫽ ␩ 共z兲 2⫹3 4 ␴ 共z兲⫺1 2 冑 冋 ␩ 共z兲⫺ ␴ 共z兲 2 册 2 ⫹ ␯ 共z兲2. 共11兲 It is convenient to define the average density and order parameter for the confined system as ␳ ¯ ⫽1 H 冕 0 H ␳ 共z兲dz,共12兲 U ¯ ⫽1 H 冕 0 HU共z兲dz.共13兲 As we will also make reference to the semi-infinite problems, the adsorption in the order parameter is defined as ⌫⫽ 冕 0 ⬁ 共U⫺Ub兲dz,共14兲 where Ubis the value of the intrinsic nematic order parameter of the bulk phase. Numerical values for the potential parameters were taken as ⑀ A⫽1共which sets the temperature scale兲, ⑀ B/ ⑀ A ⫽0.847 and ⑀ C/ ⑀ A⫽0.75. The range parameters ␭iare set, in units of d共throughout we choose this unit to set the length scale兲,to␭i⫽2, 4, 1.75, i⫽A,B,Crespectively, and ␭W ⫽1. In our calculations the temperature is fixed at the value T*⬅kBT/ ⑀ A⫽0.57, that corresponds to a typical situation in which vapor, isotropic and nematic phases can be observed. At this temperature, the isotropic–nematic coexistence occurs for a reduced chemical potential ␤␮ ⬅ ␮ /kBT ⫽⫺3.918. III. THE SEMI-INFINITE CASE In order to better understanding the behavior of the fluid inside the pore, we have first considered the semi-infinite cases of the fluid in presence of a single substrate where the substrate–fluid interaction potentials given by Eq. 共4兲either favor homeotropic or planar alignment, i.e., ⑀ W (1)⬅ ⑀ W⬎0or ⑀ W⬍0, respectively. We are interested in the wetting behavior, so the system will be at bulk nematic–isotropic coexistence, unless stated otherwise. We have considered both situations in which the bulk fluid is either isotropic or nematic. First we consider the case of a nematic bulk phase in presence of the substrate. This situation was studied for substrates that favor homeotropic anchoring in Ref. 18. We extend this analysis to substrates which favor planar alignment. For both negative and small positive values of ⑀ Wthe fluid far from the substrate orientates parallel to the substrate. For the case in which ⑀ W⬍0, biaxiality is developed in a microscopic layer close to the substrate, leading to the decay of the orientational order in that area. If the substrate favors a homeotropic alignment, we observe a microscopic isotropiclike layer close to the substrate, corresponding to a partial wetting situation of the substrate–nematic interface by the isotropic phase. As ⑀ Wis increased, the isotropic layer close to the substrate becomes macroscopic as the isotropic phase completely wets the nematic–substrate interface at ⑀ W ⫽ ⑀ W w1. Our calculations showed that the wetting transition is continuous. The orientation far from the substrate remains 014903-3 Nematic-isotropic transition in an hybrid cell J. Chem. Phys. 122, 014903 (2005) Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 planar in the complete wetting case. By further increasing ⑀ W, an anchoring transition occurs for ⑀ W⫽ ⑀ W d, as the nematic orientation changes from planar to homeotropic far from the substrate, and the wetting becomes partial again since the isotropic–nematic interface favors a planar anchoring in our model. The anchoring transition is first order and continues in the single nematic phase region. However, the complete wetting with planar orientation state remains metastable up to ⑀ W m. This fact will be important for the discussion of the confined cases. For the model parameters considered in this paper, ⑀ W w1/ ⑀ A⫽0.470, ⑀ W d/ ⑀ A⫽0.527, and ⑀ W m/ ⑀ A⫽0.72. We turn to the case in which there is an isotropic phase in bulk. For small values of 兩 ⑀ W 兩 , the inhomogeneities of the density and order parameter profiles are restricted to a microscopic layer close to the substrate. Although the fluid is isotropic far from the substrate, the anisotropic character of the substrate–fluid potential induces a random planar ordering near the substrate, i.e., the particles orientate parallel to the substrate but without any preferred direction in that orientation plane 共 ␩ ⬍0, ␴ ⫽ ␯ ⫽0兲. Such a condition can be seen as an extreme biaxial case, where B⫽2U, i.e., the highest eigenvalue of the orientational order parameter tensor is twofold degenerate. The layer width remains microscopically finite as the NI coexistence is approached, corresponding to a partial wetting situation. For large enough values of 兩 ⑀ W 兩 ,we observe complete wetting situations. For negative values of ⑀ Wand at the NI coexistence, there is a sudden change at ⑀ W⫽ ⑀ W w2from the previous surface state to a complete wetting state of the isotropic–substrate interface by the nematic phase. This corresponds to a first order wetting transition. For our model parameters, ⑀ W w2/ ⑀ A⫽⫺0.359. Due to its first order character, there is a prewetting first-order transition for ⑀ W⬍ ⑀ W w2共see Fig. 1兲. At this transition the particles at the first layer orientates along some direction parallel to the substrate, breaking the rotational symmetry of the state 共see insets of Fig. 1 for the order parameter profiles兲. Consequently, the prewetting transition can be seen as a two-dimensional isotropic to nematic transition at the fluid layer close to the substrate. We will denote Iand INto the surface states with the random planar and nematic layer close to the wall, respectively. For the INstate, there is also biaxiality near the substrate, as occurred when there was a nematic fluid in bulk. A similar behavior was found for the Zwanzig model and hard spherocylinder fluids,20 although their ‘‘prewetting’’ transition is continuous 共second order for the Zwanzig model and it is expected to be Kosterlitz–Thoulesslike for the spherocylinder system兲. The prewetting surface nematization transition shifts towards lower chemical potentials as ⑀ Wis decreased, and eventually disappears at a critical point. For large positive values of ⑀ W, i.e., ⑀ W/ ⑀ A⬎ ⑀ W w3/ ⑀ A ⫽0.691, we also found that the nematic phase completely wets the isotropic–substrate interface. We have checked indirectly this by studying the numerically obtained profiles that minimize our free energy functional. The order parameter profiles that we obtain in our numerical minimization are inhomogeneous along the wetting layer 共see Fig. 2兲. This fact is due to the mismatch of anchoring conditions at the interfaces. We first note that the intrinsic order parameters are homogeneous along the wetting layer except close to the interfaces 共see inset in Fig. 2兲. So the origin of the inhomogeneity of the extrinsic order parameters is the change of the orientation of the nematic director field along the wetting layer. As previously mentioned, the molecules align parallel to the interface between a nematic and isotropic phase. On the other hand, the molecules anchor homeotropically close to the substrate due to the strong anchoring coupling between the substrate and the fluid. As the nematic layer width diverges, it is possible to fulfill both anchoring conditions by slowly varying the nematic director from homeotropic to planar anchoring. However, any numerical calculation requires minimization in a finite range Rfor the zcoordinate, imposing bulk isotropic conditions for z⬎R. In this situation, the nematic director profile that corresponds to the minimum of FIG. 1. The adsorption ⌫plotted vs the chemical potential for a semiinfinite geometry where the fluid is isotropic in bulk and the substrate favors homeotropic anchoring. The dashed line corresponds to ⑀ W/ ⑀ A⫽0.3 and the continuous line to ⑀ W/ ⑀ A⫽0.66. The I–INtransition for the latter case is represented by the dotted line. The insets represent the reduced density ␳ * ⬅ ␳ d3and extrinsic order parameter profiles ␩ , ␴ , and ␯ for different chemical potentials and ⑀ W/ ⑀ A⫽0.66. FIG. 2. Plot of the reduced density ␳ d3and extrinsic order parameter profiles ␩ , ␴ , and ␯ for ⑀ W/ ⑀ A⫽0.70 and ␤␮ ⫽⫺3.918, obtained by numerical minimization for R⫽435d. Inset: the intrinsic order parameters Uand ␺ corresponding to the same state. The intrinsic order parameter Bis zero except close to z⫽R共not shown for clarity兲. 014903-4 Rodrı ´guez-Ponce, Romero-Enrique, and Rull J. Chem. Phys. 122, 014903 (2005) Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 the grand-canonical free energy is expected to be linear, as it is indeed observed by numerical minimization 共see inset of Fig. 2兲. This fact induces a 1/Rdependence of the apparent equilibrium excess grand-canonical free energy 共or surface tension ⌺兲 due to the elastic deformation. Such decay is much slower than the usual exponential correction when elastic deformations are not involved, and we have to consider much larger values of Rto obtain reliable values of the true equilibrium free energies. As Rincreases, the apparent surface tension decreases and it is always larger than the complete wetting state excess free energy, that it is the sum of the surface tension ⌺wN corresponding to the interface between substrate and a nematic phase oriented homeotropically far away, and the surface tension ⌺NI between corresponding to the nematic-isotropic interface where the nematic phase is planar-anchored with respect to the interface. The latter surface tension is independent of the fluidsubstrate interactions, and its value is ␤ ⌺NId2⬇0.0259 for the NI coexistence at T*⫽0.57. Figure 3 summarizes the results of this section. We represent the cosine of the contact angle ␪ of a droplet of isotropic phase in coexistence with bulk the nematic phase on a planar substrate as a function of ⑀ W. The Young equation states that cos ␪ ⫽⌺wN⫺⌺wI ⌺NI ,共15兲 where ⌺wN ,⌺wI , and ⌺NI are the surface tensions corresponding to the substrate–nematic, substrate–isotropic, and nematic–isotropic interfaces, respectively. For ⑀ W⬍ ⑀ W d共including negative values兲, the nematic director is parallel to the substrate far from the substrate. Otherwise, the nematic director is homeotropically anchored. The complete wetting region corresponds to cos ␪ ⫽1. The curve meets the cos ␪ ⫽1 line at ⑀ W⫽ ⑀ W w1with no change in slope. This fact is a signature of the second-order character of the wetting transition. On the contrary, the curve departs from the cos ␪ ⫽1 line at ⑀ W⫽ ⑀ W dwith nonzero slope, indicating the first-order character of the dewetting transition. Figure 3 also gives information about the wetting properties when there is an isotropic phase in bulk. By reversing the roles of the nematic and isotropic phases in Eq. 共15兲,itis straightforward to see that the contact angle ␪ ⬘of a nematic droplet on a planar substrate is related to ␪ via ␪ ⬘⫽ ␲ ⫺ ␪ . Consequently, when cos ␪ ⫽⫺1 we observe complete wetting by nematic phase of the isotropic–substrate interface. This fact is clear for ⑀ W⬍ ⑀ W w2⬍0. However, for ⑀ ⬎ ⑀ W w3the apparent values of cos ␪ are less than ⫺1. As previously discussed, this is a finite-size effect of our numerical calculations and the predicted value is cos ␪ ⫽⫺1. Nevertheless, since the nematic layer remains finite as ⑀ W→ ⑀ W w3from below, the localization of the wetting transition is quite accurate. For both wetting transitions at ⑀ W w2and ⑀ W w3there is a change of slope of the cos ␪ curve, that it is in agreement with their first-order character. IV. THE CONFINED CASE We turn to the case of the fluid confined in a hybrid slit pore, where each substrate favors an antagonist anchoring, i.e., homeotropic and planar, respectively. Under confinement, it is well known that the bulk transitions have lowerdimensional counterparts that are known as capillary transitions. The thermodynamic conditions 共temperature, pressure, chemical potential兲for such transitions are usually shifted with respect to the bulk coexistence values. Before considering our DFT results, we consider the predictions of the Kelvin equation, which gives the leading order for the shift on chemical potential ⌬ ␮ ⫽ ␮ NI c⫺ ␮ NI bof the capillary nematic–isotropic transition ␮ NI cwith respect to the bulk value ␮ NI bat large pore widths H. Thermodynamic considerations2,21,29 show that the chemical potential shift ⌬ ␮ of a capillary transition with respect to the bulk one between the ␣ and ␤ phases at a given temperature obeys asymptotically ⌬ ␮ ⬇⌺ ␣␤ 共cos ␪ 1⫹cos ␪ 2兲 共 ␳ ␤ ⫺ ␳ ␣ 兲H.共16兲 In this expression, ⌺ ␣␤ is the surface tension for an ␣ – ␤ interface, and ␳ ␣ and ␳ ␤ are the bulk densities of the ␣ and ␤ phases, respectively. Finally, ␪ 1and ␪ 2are the contact angles corresponding to the droplets of ␣ phase that nucleate on each substrate when the fluid is in bulk in the ␤ phase, respectively. In our case, we can identify ␣ as the isotropic phase and ␤ as the nematic phase. It is implicitely assumed in Eq. 共16兲that there is a partial wetting situation for both substrates in semi-infinite geometries. However, if either there is complete wetting only at one substrate or in both but by the same phase ( ␪ 1⫽ ␪ 2⫽0or ␲ 兲, the effective pore width in Eq. 共16兲is reduced to H⫺ ␬ l, where ␬ is a constant that depends on the range of the wall–fluid interactions and l is the sum of the widths of the adsorbed wetting layers. Nevertheless, this modification does not change qualitatively the FIG. 3. Plot of the cosine of the contact angle ␪ of an isotropic phase droplet on the substrate–nematic interface as a function of the fluid–substrate interaction strength ⑀ W共thick continuous line兲. The thick dashed line represents the complete wetting metastable states corresponding to a planar anchoring far from the substrate. The values of ⑀ W w1, ⑀ W w2, ⑀ W w3, ⑀ W d, and ⑀ W mare also marked. The filled circles correspond to the numerical estimates for ⑀ W ⬎ ⑀ W w3共see text for explanation兲. 014903-5 Nematic-isotropic transition in an hybrid cell J. Chem. Phys. 122, 014903 (2005) Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 results. Its only effect is to observe the behavior predicted by the Kelvin equation at larger values of the pore width Hthan for a partial wetting situation. A different situation arises when both substrates are completely wet by different phases at coexistence, i.e., cos ␪ 1cos ␪ 2⫽⫺1. Studies for symmetrically opposing surface fields Ising systems3,4 and Landau–de Gennes models of liquid crystals25 show that in this situation there is no capillary transition. Instead, an interfacelike state appears, where half the pore is filled with an ␣ -like layer, and the rest by ␤ -like phase. The position of the interface between both layers is controlled by the deviation of chemical potential with respect to the coexistence value. A rounded transition from the confined ␣ -like phase to the ␤ -like phase 共marked by the disappearance of the free energy singularities associated to a sharp capillary transition兲is building up, becoming a sharp first-order transition only for H⫽⬁. When the opposing surface fields are not symmetrical 共at least for nearsymmetrical兲, it is expected to observe a similar behavior.3 Mean-field calculations for a confined polymer model with asymmetrical opposing fields also support this picture.5 This analysis allows us to envisage different scenarios for the confined case and large H. For simplicity, we will only consider the case ⑀ W (1)⫽ ⑀ W (2)⬅ ⑀ Win Eqs. 共4兲and 共5兲.We must note that this choice does not correspond to the symmetrical opposing fields, since the effect of each substrate is qualitatively different. For large Hand ⑀ W/ ⑀ A⬍0.48 and ⑀ W/ ⑀ A⬎0.53 we expect to have a nematic–isotropic capillary transition. The transition point will shifted towards lower or higher chemical potentials than the corresponding to the bulk transition if cos ␪ 1⫹cos ␪ 2is negative or positive, respectively. This dependence of the sign of the capillary transition shift on the thermodynamic conditions of the fluid 共temperature, etc.兲, which does not occur for simple fluids, has been observed in our model under confinement between symmetrical substrates that favor homeotropic anchoring,19 and also in confined spherocylinder model with the Onsager approximation for a symmetrical slit pore.21 On the other hand, if 0.48⬍ ⑀ W/ ⑀ A⬍0.53, no capillary transition is expected to be obtained for any pore width. We present now our DFT calculations. For the sake of clarity, we separate the results obtained for ⑀ W⬍ ⑀ W w1and ⑀ W⬎ ⑀ W d. A. The case ⑀ W Ë ⑀ W w 1 We performed our first DFT calculations for ⑀ W/ ⑀ A ⫽0.30, which correspond to partial wetting situation in the semi-infinite geometry for both substrates. Figure 4 shows the behavior of the averaged intrinsic order parameter U ¯ as the chemical potential is varied. We see that there is a transition from an isotropiclike state 共lower branch兲to a nematiclike state 共upper branch兲. The order of the transition is firstorder, and it is found as the coincidence point of the grandcanonical free energies of the nematiclike and isotropiclike states. We have shown also the metastable continuations of each branch: nematiclike states on the left of the capillary NI transition and isotropiclike states on the right of the transition. Examination of the order parameter profiles show that the director field in the nematiclike state is oriented parallel throughout the pore to the substrate with an intrinsic order parameter similar to the bulk value, except in thin layers around the substrates, where the density and order parameter profiles are similar to the ones corresponding to the semiinfinite cases. In particular a very thin layer close to the z ⫽0 substrate is homeotropically anchored. This state corresponds to the Sstate in Ref. 22 and we will denote it by NS. On the other hand, the isotropiclike state 共which we will denote as I兲develops microscopic homeotropically and random-planar layers close to the z⫽0 and z⫽Hsubstrates, respectively, and the orientational ordering decays to zero in the middle region of the pore. We have also compared our DFT calculations with the Kelvin law prediction Eq. 共16兲 共see inset in Fig. 4兲. Although the transition value departs from this prediction for small pore widths, for H/d⬎20 the agreement between our numerical estimations and the Kelvin predicted value is quite reasonable. Note that the capillary NI transition is shifted to larger chemical potentials than the bulk NI transition value. However, as ⑀ Wincreases, this trend is reversed 共the Kelvin law predicts the turning point to occur at ⑀ W/ ⑀ A⫽0.329). As ⑀ Wexceeds ⑀ W w2, wetting in the z⫽Hwall begins to affect the pore adsorption. For small values of Ha single transition between an isotropiclike Istate to a nematiclike state occurs as for ⑀ W⬍ ⑀ W w2. However, as Hincreases the order of the ‘‘nematiclike’’ state decreases and the chemical potential for which this transition occurs converges to the I–INtransition value for the z⫽Hwall. The order-parameter profiles make clear that the ‘‘nematiclike’’ state corresponds to the formation of a nematic layer close to the z⫽Hsubstrate followed by an isotropic ordering along the pore. By analogy with the semi-infinite situation, we call INto the confined state after this transition, which we can see as a reminiscence of the I–INtransition in the semi-infinite phenomenology. As Hincreases, in addition to the I–INtransition, a new phase transition between the INto a proper nematiclike state NS共i.e., planar-anchored along the pore兲is observed. This FIG. 4. Plot of the averaged intrinsic order parameter throughout the pore U ¯ as a function of the reduced chemical potential ␤␮ for ⑀ W/ ⑀ A⫽0.3 and H/d⫽10 共continuous line兲and H/d⫽20 共dashed line兲. The dotted lines show the location of the capillary NI transition. Inset: the diamonds represent our DFT values of the chemical potential for ⑀ W/ ⑀ A⫽0.3 at the capillary NI transition for H/d⫽10, 20, and 35, and the continuous line is the Kelvin law prediction, Eq. 共16兲. 014903-6 Rodrı ´guez-Ponce, Romero-Enrique, and Rull J. Chem. Phys. 122, 014903 (2005) Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 transition can be regarded as the true capillary NI transition. Figure 5 shows our DFT calculations for H⫽20dand different values of ⑀ W. It becomes clear that the difference between the INand the NSstates at the transition decreases as ⑀ Wincreases, and eventually the NI capillary transition disappears in a critical point 共see inset of Fig. 5兲. The value of ⑀ Wat which the NI capillary transition disappears is lower than ⑀ W w1, i.e., the confinement destabilizes the two-phase region with respect to the one-phase region. We have also studied the effect of the pore width in the NI capillary transition. We set ⑀ W/ ⑀ A⫽0.465, slightly below ⑀ W w1. Figure 6 shows that the capillary NI transition does not exist for the smallest values of Hand reappears for H⬎50d. The disappearance of the NI capillary transition shown by our model is not related to the critical points observed for small values of Hin other models.20,21 Actually, those cases correspond to symmetrical slit-pores and consequently their phenomenology is not related to wetting properties of the substrates. In our model the NI capillary transition occurs at least for H⬎20dwhen the substrates at z⫽0 and z⫽Hare the same.19 However we cannot rule out the possibility of critical points for very small H, but we will not consider them since the predictions of our model are not reliable in that regime. Our DFT results about the existence of the NI capillary transition show the opposite trend to the one predicted by a Landau–de Gennes model in Ref. 25, which predicted that the confinement stabilized the two-phase region with respect to the one-phase region. Moreover for large values of Hthe boundary for the nonexistence region of the capillary transition was a triple point rather than a critical point. Actually the scenario shown by our DFT calculations is consistent with the phenomenology that presents a confined Ising model with opposing surface fields when the wetting transition is critical.3This analogy can be explained heuristically. For large Hand ⑀ Wclose to ⑀ W w1, the typical configuration close to bulk coexistence corresponds to an interfacial state formed by an isotropic layer around the z⫽0 and a nematic layer around the z⫽Hwall. In the first approximation the NI interface position Iis given by the minimum binding potential which is the superposition of the contributions from the two semi-infinite systems.3As there is a complete wetting situation at the z⫽Hsubstrate, we can consider its binding potential as a exponentially decreasing function of the interfacial position with respect to the z⫽Hwall cexp关⫺(H ⫺l)/ ␰ b ⬘兴, where c⬎0 and ␰ b ⬘is the nematic phase correlation length at bulk NI coexistence. Note that the local minimum corresponding to the bounded state, if any, plays no role in our argument as soon as we are far enough from the firstorder wetting transition corresponding to the semi-infinite case with the z⫽Hwall. On the other hand, the binding potential corresponding to the z⫽0 substrate takes the usual form ⫺aexp(⫺l/ ␰ b)⫹bexp(⫺2l/ ␰ b), where a⬀ ⑀ W w1⫺ ⑀ W⬎0, b⬎0 and ␰ bis the isotropic phase correlation length at bulk NI coexistence. So, the complete binding potential for the NI interface, up to irrelevant constants, is W共l兲⫽h共H⫺l兲⫺ae⫺共l/ ␰ b兲⫹be⫺共2l/ ␰ b兲⫹ce关共H⫺l兲/ ␰ b ⬘兴,共17兲 where h⬀⌬ ␮ . A critical point must verify h⫽a ␰ be⫺共l/ ␰ b兲⫺2b ␰ be⫺共2l/ ␰ b兲⫹c ␰ b ⬘e⫺关共H⫺t兲/ ␰ b ⬘兴,共18兲 0⫽⫺ a ␰ b 2e⫺共l/ ␰ b兲⫹4b ␰ b 2e⫺共2l/ ␰ b兲⫹c 共 ␰ b ⬘兲2e⫺关共H⫺l兲/ ␰ b ⬘兴.共19兲 Instead of solving simultaneously Eqs. 共18兲and 共19兲, we will study the spinodal line, which is the solution of Eq. 共19兲.We consider solutions of the form l⬅ ␹ H⫹⌬l, where 0⬍ ␹ ⬍1 and ⌬l⬃ ␰ b, ␰ b ⬘. Substituting this ansatz in Eq. 共19兲,we obtain FIG. 5. Plot of the averaged intrinsic order parameter throughout the pore U ¯ as a function of the reduced chemical potential ␤␮ for H/d⫽20 and ⑀ W/ ⑀ A⫽0.35 共dashed line兲, ⑀ W/ ⑀ A⫽0.41 共continuous line兲, and ⑀ W/ ⑀ A ⫽0.45 共dotted–dashed line兲. The dotted lines show the location of the I–IN transition 共left兲and capillary NI transitions 共right兲. Inset: The symbols are the coexistence values of U ¯ at the I–INand capillary NI transitions for different values of ⑀ W. The dotted line corresponds to the I–IN–NStriple point at ⑀ W/ ⑀ A⬇0.39. The continuous lines are to guide the eye. FIG. 6. Plot of the averaged intrinsic order parameter throughout the pore U ¯ as a function of the reduced chemical potential ␤␮ for ⑀ W/ ⑀ A⫽0.465 and H/d⫽10, 20, 40, 50, and 60. 014903-7 Nematic-isotropic transition in an hybrid cell J. Chem. Phys. 122, 014903 (2005) Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34 0⫽⫺共ae⫺共 ␹ H/ ␰ b兲兲e⫺共⌬l/ ␰ b兲⫹共4be⫺共2 ␹ H/ ␰ b兲兲e⫺共2⌬l/ ␰ b兲 ⫹ 冉 c ␰ b 2 共 ␰ b ⬘兲2e⫺关共1⫺ ␹ 兲H/ ␰ b ⬘兴 冊 e⌬l/ ␰ b ⬘.共20兲 In order to have two solutions for Eq. 共20兲, we need all the factors in brackets in Eq. 共20兲to be of the same order. Consequently we obtain that ␹ ⫽ ␰ b/(2 ␰ b ⬘⫹ ␰ b) and a ⫽a⬘exp关⫺H/(2 ␰ b ⬘⫹ ␰ b)兴, where a⬘⬃b⬃c. The value of a⬘ for which the critical point occurs corresponds to the case where the rescaled equation 0⫽⫺a⬘e⫺共⌬l/ ␰ b兲⫹4be⫺共2⌬l/ ␰ b兲⫹c ␰ b 2 共 ␰ b ⬘兲2e共⌬l/ ␰ b兲共21兲 has an unique real solution. For simplicity, we assume that ␰ b⫽ ␰ b ⬘, but similar results are found when both correlation lengths are different. We refrain to give explicit solutions and we just mention that indeed a⬘/c⬃1 and the value of ⌬lat the critical point is ⌬lc⬃ ␰ bfor reasonable values of band c. Consequently, the interfacial position at the critical point lc⬃H/3 and the NI capillary transition shift ⑀ W w1⫺ ⑀ W ⬃exp(⫺H/3 ␰ b) and from Eq. 共18兲⌬ ␮ ⬃exp(⫺2H/3 ␰ b). A numerical comparison of these predictions with our DFT calculations, although it would be very interesting, is extremely difficult due to the limitations of our numerical procedure and it will not be carried out in this work. However, at least qualitatively these predictions have been confirmed by the DFT calculations even for not so large values of H. B. The case ⑀ W Ì ⑀ W d Now we turn to study the confined system for ⑀ W ⬎ ⑀ W d. We first considered the case ⑀ W/ ⑀ A⫽0.70 and H ⫽20d. Different confined states are obtained as the chemical potential increases 共see Fig. 7兲. The first cases Figs. 7共a兲and 7共b兲clearly correspond to a vaporlike and isotropiclike states, and we will denote them as Vand I, respectively. There is in both cases a homeotropically-anchored nematic layer close to the substrate at z⫽0 and a random planar layer close to the substrate at z⫽H. As the chemical potential increases, the random planar layer at z⫽Hundergoes a transition to a planar biaxial state. Again this first-order transition is a reminiscence of the I–INsurface transition corresponding to the substrate at z⫽H, and we will denote both confined states Iand INby analogy with the semi-infinite phenomenology. As the chemical potential is further increased, the nematic layer increases and a state in which an isotropic and a nematic layer coexist is obtained 关Fig. 7共c兲兴. There is no phase transition between the isotropiclike INand the nematiclike NSstates, but a smooth transformation similar to the expected one when both substrates are completely wet by different bulk phases at coexistence. Finally, there is a transition to a new nematiclike state 共the L state in Ref. 22, that we will denote as NL), characterized by an almost linear distorsion from homeotropic to planar configuration on the tilt angle through the pore, although the intrinsic order parameter takes essentially the bulk value except close to the substrates 关see Fig. 7共d兲兴. This transition between the NLand NSstate is first order and converges to the anchoring transition that is observed in the semi-infinite case with a substrate that favors a homeotropic anchoring as H→⬁.18 A more detailed discussion about this transition can be found in Ref. 22. For larger values of H, the capillary NI transition appears as a transition between an INstate and a NLstate for chemical potentials smaller than the bulk NI transition value. As ⑀ Wapproaches ⑀ W d, the range of values of Hfor which capillary transition does not exist is wider. For ⑀ W/ ⑀ A ⫽0.53 共close to the dewetting transition point兲the capillary NI transition does not appear until H/d⬎150. Figure 8 shows the continuous change of the nematic layer as the chemical potential increases for H⫽20d, which is analogous to the behavior observed for ⑀ W/ ⑀ A⫽0.70. Figure 9 shows how the curves of the averaged intrinsic order parameter as a FIG. 7. Plot of the density ( ␳ *⬅ ␳ d3) and intrinsic uniaxial order parameter 共U兲for the fluid confined in a pore of width H⫽20dand ⑀ W/ ⑀ A⫽0.70 and chemical potentials: 共a兲 ␤␮ ⫽⫺4.7, 共b兲 ␤␮ ⫽⫺4.45, 共c兲 ␤␮ ⫽⫺3.95 and ␤␮ ⫽⫺3.7. FIG. 8. Density ( ␳ *⬅ ␳ d3) and intrinsic uniaxial order parameter profiles 共U兲for H⫽20dand ⑀ W/ ⑀ A⫽0.53. The chemical potentials run from ␤␮ ⫽⫺4.03 to ␤␮ ⫽⫺3.64 with a step of ⌬共 ␤␮ 兲⫽0.01. The intrinsic order parameter profiles at the NL–NStransition 共left兲and I–INtransition 共right兲 are highlighted. 014903-8 Rodrı ´guez-Ponce, Romero-Enrique, and Rull J. Chem. Phys. 122, 014903 (2005) Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.116 On: Fri, 01 Apr 2016 10:24:34