Surface-induced alignment at model nematic interfaces

PHYSICAL REVIEW EVOLUME 52, NUMBER 5NOVEMBER 1995 Surface-induced alignment at model nematic interfaces E. Martin del Rio,*M. M. Telo da Gama, and E. de Miguel* Departamento de Ftsica da Faculdade de Ciencias eCentro de Fssica da Materia Condensada, Universidade de Lisboa, Avenida Prof Ga. ma Pinto 2, 46gg Lisboa Codex, Portugal L. F. Rull Departamento de Fisica Atornica, Molecular yNuclear, Universidad de Sevilla, Apartado 1085, Sevilla $1080, Spain (Received 30 June 1995) We have applied the generalized van der Waals theory to amodel liquid crystal that includes, explicitly, all of the second-order terms in the spherical harmonic expansion of the anisotropic intermolecular potential. We have investigated the orientational order induced by each one of these terms, as well as the order resulting from the competition of various terms included in the potential. It was shown that, for appropriate choices of the relative strengths of the spherical harmonic coeKcients, the theory is capable of accounting qualitatively for all the orientational e8'ects observed at nematic interfaces, including tilted orientations. In particular, different molecular alignments at the nematic-vapor and nematic-isotropic interfaces of a given nematogen were described as the result of competing terms in the anisotropic interactions. Additionally, we have shown that temperaturedriven orientational transitions may occur in systems characterized by this type of interaction. For agiven choice of parameters, we have also found an orientational transition, which is related to awetting transition at the nematic-vapor interface. Finally, it was shown that complete wetting of the isotropic liquid-vapor interface by the nematic phase may be destroyed as aresult of the competition between di8'erent terms in the potential. Similarly, areentrant wetting transition at the nematic-vapor interface by the isotropic phase was found, as aresult of this competition, at the nematic-vapor interface. PACS number(s): 61.30.Cz, 61.30.Gd, 68.10.Cr, 68.45.Gd I. INTRODUCTION The understanding of the interfacial properties of molecular liquids has long been recognized as afundamental problem in the statistical physics of inhomogeneous systems. In the simplest case ofcoexisting isotropic phases, aplanar interface breaks the translational symmetry of the system and induces, in general, alower rotational symmetry at the interface. Several years ago, Gubbins [1] predicted apreferred molecular orientation at the liquid-vapor interface of fl.uids of nearly spherical molecules, such as N2 and C12, based on the results ofperturbation theories and molecular dynamics simulations. These efFects are, however, too small to be observed experimentally and of little practical relevance. By contrast, surface-induced molecular alignment is expected to be important when liquid crystalline phases are involved and it is known to have awide range of practical applications. The theoretical understanding of the anchoring mechanisms at liquid crystal surfaces and interfaces is still rather poor. This is due in part to the *Permanent address: Departamento de Fisica Atomica, Molecular yNuclear, Universidad de Sevilla, Apartado 1065, Sevilla 41080, Spain. complexity of realistic liquid crystal interactions, which renders atheoretical description of nonuniform systems almost unfeasible. Additionally, the tensorial nature of the orientational order parameter is often exhibited by non-uniform nematogens (i.e.,broken biaxial sym- ~rnetry may spontaneously occur at interfaces) and thus even phenomenological theories are considerably harder to solve than similar Landau theories for systems with scalar order parameters. For simplicity, we shall study translationally invariant phases only, i.e.,isotropic liquid, vapor, and nematic phases. In the absence of surfaces or external fields, the average direction of molecular alignment in the bulk nematic is arbitrary. This is no longer the situation at the interface between the nematic and the vapor (or isotropic liquid). The translational symmetry of the system is now broken and the nematic director will be determined by the direction that minimizes the free energy of the inhomogeneous system. This e8'ect is not expected to be universal, but seems to depend sensitively on the details of the intermolecular interactions [2]. This is indeed what has been observed experimentally. The preferential molecular orientation at the nematicisotropic (NI) and nematic-vapor (KV) interfaces was reported to be either perpendicular [3— 5], parallel [6— 8], or tilted [9— 11],depending on the specific nematogen. In some cases, atemperature-dependent oblique tilt angle was also found [4,6,12,13]. 1063-651X/95/52(5)/5028(12)/$06. 00 52 5028 1995 The American Physical Society 52 SURFACE-INDUCED ALIGNMENT AT MODEL NEMATIC INTERFACES 5029 Although various theoretical approaches have been used to date to describe orientational efFects at nematic interfaces, afull understanding of the underlying mechanisms is still lacking. Moreover, some of these treatments yield somewhat contradictory results and thus the status of the models and/or of the approximations used in these studies is far from clear. The first attempt to account for the existence of a preferred molecular orientation at the NI interface was made by de Gennes [14]. By using aLandau — de Gennes expansion of the &ee energy i~~powers of asingle order parameter, both homeotr6pic (perpendicular to the interface) and homogeneous planar alignments were obtained, depending on the sign of the square-gradient term (which is related to the elastic constants of the isotropic phase). Although several theories were developed following de Gennes's pioneering work, their phenomenological nature restricts their predictions to rather general qualitative trends. Furthermore, the connection between the expansion coefFicients and the microscopic details of the molecular interactions is far from trivial, making it difFicult to interpret the physical mechanisms that are responsible for the orientational order of aparticular nematogen. Adi8'erent approach was followed by the so-called microscopic theories, which start by considering specific models for the intermolecular interactions (molecular shape, anisotropic attractive interactions, etc.)and then, based on agiven theory, deduce the mechanisms that are ultimately responsible for the preferred interfacial alignment of aspecific (model) nematogen. In line with bulk studies, two diferent models have been investigated in detail: an "Onsager-type" model, which takes into account solely the efFects of anisotropic hard-core repulsions, and a"Maier-Saupe-type" model, which considers explicitly anisotropic interactions that are, on average, attractive. Some of these studies, however, may be considered as acombination of the two approaches, as will be discussed in the following paragraphs. Most studies of the interfacial properties of liquid crystals based on hard-body models consider the molecular units to be spherocylinders. As far as the &ee nematic surface is concerned, it is ageneral Gnding that the hardcore repulsions favor perpendicular alignment at the interface [15,16]. Discrepancies, however, appear regarding the orientation at the NI interface. While most studies report planar alignment [17— 20], Holyst and Poniewersky [16] predict an oblique tilt angle of 60', which was found to be independent of the molecular elongation, in agreement with experimental results for the 4-cyano-4'- (n-alkyl)biphenyl (nCB) series [2]. This result was seriously questioned by Moore and McMullen [21], who found, in general, planar alignment, with the exception of systems with very small elongations where an oblique tilt angle was also obtained. More recently, however, Chen and Noolandi [20] suggested that the oblique tilt angle obtained in Refs. [16,21] is an artifact of the approximations used in the solution of the Onsager &ee-energy functional, in the limit of short length to breath ratios. Indeed, Chen and Noolandi [20] claimed that, within the Onsager approximation, asystem of hard spherocylinders exhibits planar alignment at the NI interface for all molecular elongations. Arather difFerent approach includes anumber of theories that emphasize the interfacial ordering efFects arising from the longer-ranged anisotropic attractive interactions. Many of these theories are based on highly idealized models and predict the same preferred orientation at the NI and NV interfaces, by contrast with the hardbody models and the experimental observations. Additionally, planar or perpendicular alignment is generally reported and thus no account of an oblique tilt angle seems to be given. An exception is the model studied by Kimura and Nakano, which is much more elaborate [15,17]. In these references the authors accounted for diBerent orientations at the NI and NV interfaces of agiven nematogen and predicted a(temperaturedependent) oblique tilt angle as aresult of the competition between the repulsive and attractive interactions, for arange of model parameters. More than adecade ago, one of us [22] developed a generalized van der Waals theory for nonuniform molecular Auids. This theory was applied to model nematic liquid crystals with anisotropic attractive interactions truncated at the Maier-Saupe level. No preferred orientation at the interface of this model was obtained, since the orientational and spatial dependences of the intermolecular interactions are decoupled in the Maier-Saupe potential. This theory was subsequently applied to amore realistic (although still highly idealized) model by Thurtell et aL [23] and Tjipto-Margo et al. [24] by including an extra term in the intermolecular potential, which couples the translational and rotational degrees of freedom. This extended model yields perpendicular alignment at the free nematic surface for prolate molecules, while planar alignment is predicted for oblate molecules. These results disagree with those obtained by Parsons [25] and by Harrowell and Oxtoby [26], who reported the opposite trend for systems of perfectly alligned ellipsoidal molecules. In this paper we apply the generalized van der Waals theory to aliquid crystal model that includes, explicitly, all of the second-order terms in the spherical harmonic expansion of the anisotropic intermolecular potential. This model includes two terms that were not considered in closely related work [22— 24]. We have investigated the orientational order induced by each one of these terms as well as the order resulting from the competition of the various terms included in the intermolecular potential. In Sec. II we summarize the theory used in this work applied to our model liquid crystal. In Sec. III we present and discuss the results for the orientational order at the NVand NI interfaces. It is shown that, for appropriate choices of the relative strengths of the spherical harmonic coeKcients of the intermolecular potential, the theory is capable of accounting qualitatively for all the orientational efFects at nematic interfaces found in experiments. The wetting behavior of these systems, at the triple point, is also investigated for particular combinations of the intermolecular parameters. Finally, we summarize our results in Sec. IV and compare them with other theoretical works and available experimental data. 5030 MARTIN del RIO, TELO da GAMA, de MIGUEL, AND RULL II. THEORY f.r(r) =fHS(r) +po(r)kaT(l 4mnf(r, (u)), (2) We start by summarizing the mean-field theory for a nonuniform molecular fluid, which was proposed in the context of liquid crystal interfaces more than 10 years ago [22]. Applications of this theory to fluid and solidfluid interfaces of model nematogens have been reviewed recently [27]. For convenience we will consider an open system at temperature Tand chemical potential p, in a volume V. In the absence of external fields, the grand potential 0of the inhomogeneous system is the minimum of the functional 42(p(Pw )] =fdPf,.r(r) — pfdr dw p(P w) dr dr 'd~d(u'u„(r, (u; r', ~') 2 xp(r, (u)p(r ', ~'), where p(r, ~) is the density-orientational profile, which is afunction of the molecular center of mass r, and of the molecular orientation ufor rigid molecules. The molecular interactions, which are supposed to be pairwise additive, are split into ashort-ranged (generally repulsive) reference potential and alonger-ranged (usually attractive) perturbation. The &ee energy of the reference system is written within alocal density approximation; following previous work [22] we assume the reference system to be afluid of hard particles, i.e.,f„r(r)is the Helmholtz free-energy density of auniform hard-core fluid with density p()(r) =fd(up(r, u) and orientational distribution function f(r, (4)) =p(r, w)/p()(r). The contribution to the free energy from the long-ranged (attractive) interactions is given by amean-field average of the corresponding pair potential u„. In line with previous work [22— 24), we consider the simplest possible model characterized by spherical hard cores and linear molecules. In this case, the free-energy density of the reference fluid is greatly simplified and can be written as [22] tern of hard spheres and the effective potential u,rr(r, w) acting on the reference system is given by u. u(P, w) =jdP'dw'ur(P, w;P', w')Po(P')f(r', w'), (4) Equations (3a) and (3b) are solved numerically once the intermolecular potential u„ is specified. The (longranged) perturbation potential studied in this work is ageneralization of the potential used previously [22— 24] and it includes all the invariants in the spherical harmonic expansion of an arbitrary anisotropic potential, with inversion symmetry, up to second-order. Thus two different terms, neglected in Refs. [23,24], are included explicitly in our model. We note that asimilar potential was analyzed by Sullivan and Tjipto-Margo [28] within a simplified version of the theory described in this paper. We will return to this point in Sec. III, where we discuss our results for the interfaces of liquid crystals described by this model. The general expansion for an arbitrary anisotropic intermolecular potential is written as u~~2(r4 ~» ~2) =)) my, mg, m xC(lil2l; mim2m) xYi,~,(~1)Yi2m2 (~2)Y$~(~12) 4 O6 u(000;r) =— (4u) A(— )(6a) where r=[r2 —ri[is the intermolecular distance, C(lil2l; mim2m) is aClebsch-Gordan coeKcient, and Yi (u) is aspherical harmonic, all in the notation of Gray and Gubbins [29]. Our model for the anisotropic potential amounts to keeping terms in the expansion Eq. (5), with li — —0, 2and l2 =0, 2only. In addition, we will assume an rdependence for the expansion coeKcients u(lil2l; r), which is appropriate to dispersion interactions, but we will consider the signs and magnitudes of the various terms as free parameters. These can then be written as where fHS is the free-energy density of asystem of hard spheres (given for consistency with earlier results by the Percus-Yevick approximation), kH is Boltzmann's constant, and (A(r, u)) =fd~ f(r, ~)A(r, ~) is aweighted angular average over the orientational distribution function. The last term in (2) accounts for the loss of orientational entropy of the reference system, within arandom mixing (mean-field) approximation. The Euler-I agrange equations for po (r) and f(r, u) are obtained by minimization of Eq. (1) and are written in terms of the effective potential as (6b) (42r) /o6 u(202;r) =u(022;r) =C— 5r(6c) (6d) g6 u(222;r) =(4u)'~* 12 (— )— 10 r IA — )(AHs(r) +kHT(in4vrf (r, (42)) +(u,H(r, (4f)), (3a) exp [— u,s(r, w) /kryo T] 4 dm exp [— u,s(r, ~)/k~T] where pHS(r) is the chemical potential of auniform sys- (4~)'f 2 u(224; r) =3(6e) if r)o' and u(lil2l;r) =0if r(o, where ois the diameter of the spherical hard core. A, B, C, D, and Eare arbitrary constants. In what follows, each term of 52 SURFACE-INDUCED ALIGNMENT AT MODEL NEMATIC INTERFACES 5031 the anisotropic potential is denoted by the corresponding set of indices (lil2l). We note that for particular combinations of the constants A— E, the potential given by Eqs. (5) and (6) corresponds to well defined microscopic models of the anisotropic interactions. We mention two of these that are relevant in studies of liquid crystals: anisotropic dispersion interactions [29] and aspherical harmonic expansion of the Gay-Berne potential [30]. We recall at this point that the thermodynamic properties of the uniform bulk phases, within the mean-Geld theory discussed in this section, depend only on the values of the parameters A(which sets the energy scale) and B(which determines the isotropic-nematic-vapor triple point) [23]. Stability of the isotropic liquid-vapor and nematic-isotropic liquid transitions requires that Aand. Bbe positive (respectively). The reader is referred to Refs. [22,31]where full details of the bulk phase diagrams, including adiagram characterized by anematic-nematic critical point [31],may be found. In this study we do not consider bulk phases with broken translational invariance (e.g.,smectic) and thus the interaction parameters that may be used in Eqs. (6) are somewhat restricted in order to guarantee the stability of the nematic phase. We consider aplanar interface in the x-y plane. Without loss of generality, the nematic director (average direction of alignment ofthe molecules) is assumed to lie in the x-z plane due to the rotational invariance of the system in the interfacial plane. The orientational structure is then characterized by three independent order-parameter profiles [22], defined as angular averages of linear combinations of second-order spherical harmonics: )7(z) =deaf(z, m)P2(cos8), (7a) v(z) =d~ f(z, ~) sin 20 cos P, (7b) o(z) =du f(z, cu) sin |)) cos 2P .(7c) g=1when there is full alignment of the molecules perpendicular to the interface, o=1when the molecules are fully aligned along the xaxis, and v=1when all the molecules are aligned parallel to the bisector of the x-z plane. In the following sections, we will take Aand o(the hard-core diameter) as units of energy and length, respectively. In addition we simplify the notation by taking Boltzmann's constant k~ — —1. In terms of the order parameters defined through Eqs. (7a)— (7c), the effective potential Eq. (4) reads u,ir(z, (u) =dz'po(z') I~o(lz —z'I) +cue(lz —z'I) Igjz') ypg(case)I +n(z') P.(cos 0) [— B«(lz — z'I) — Du2(lz — z'I) +3&u4(lz — z'I)] +-~(z )»n2«os& Buo(lz — zI) u2(lz — zI) 2@u4(lz — zI) /~/D// 2 +-~(z')»n' «os 24 — B«(lz — z'I) +Du2(lz — z'I) +— «(lz — z'I) 2 where the functions ui(z) are given by (10) these equations numerically, since it has proved much more accurate [24] than the simpler method used in the earlier calculations [22,23]. Our results, however, difFer slightly from those of Ref. [24] (typically 1part in 10 ) due to asmall error that has been found subsequently in the latter [32]. III. RESULTS — (35/8z4 — 5z +3/4) vr/(32z4) if IzI) 1. The solution of the Euler-Lagrange equations [Eqs. (3)]is now reduced to the solution of the one-dimensional integral equation [Eq. (3a)] for po(z) and the consistency relations [Eqs. (7)] for the orientational order parameters. We follow the method used in Ref. [24] to solve A. Orientational order at the interface As we have mentioned, the fluid (nematic-isotropicvapor) phase diagram is independent of the parameters C, D, and Ewithin the mean-Beld approximation discussed in Sec. II; however, these terms play asigniGcant role in spatially inhomogeneous situations (both at interfaces and in bulk modulated phases [33]). In our cal- 5032 MARTIN dcl RIO, TELO da GAMA, de MIGUEL, AND RULL 52 culations, we have set B=0.3, in line with earlier work [22— 24], and varied the other parameters in u„;(C, D, and E). For this value of B, the phase diagram exhibits three uniform Huid phases [22]: nematic (N), isotropic liquid (I), and vapor (V). These phases coexist simultaneously at the triple point, which occurs at atemperature Tt, — — 0.218156. At lower temperatures T(Tt, the vapor coexists with an ordered Nphase and at higher temperatures T)Tt, it coexists with adisordered Iliquid. Ifonly the l=0 terms are included in the intermolecular potential u„; [Eq. (5)], no preferred orientation at the interface is obtained [22]. This arises from the fact that if (220) is the only anisotropic interaction, the orientational and translational degrees of freedom are decoupled, with the result that the interface is rotationally invariant. We refer to this model as the Maier-Saupe model. The effect of including the term (202) in u„;, [Eq. (5)] (C g0and D=E=0) was analyzed by Thurtell et al. [23] and by Tjipto-Margo et al. [24]. Defining the tilt angle 4as the angle between the nematic director and the normal to the interface (z axis), Thurtell et al. [23] found ill =0' (homeotropic alignment) for C&0 (prolate molecules) and ill =90' (homogeneous planar alignment) for C(0(oblate molecules) at both the XV and NI interfaces. Although the tilt at the NV interface appears to be either 0' [3— 5] or 90 [6,7], in agreement with the results of this model, no oblique tilt angle, as observed at the NI interface for aseries of nematogens [10,11], is obtained. More terms in the potential seem to be required in order to account for the whole range of experimental results. In this paper we go one step in that direction by analyzing, in turn, the various anisotropic terms in the potential of Eqs. (5) and (6). We have studied the preferred interfacial orientation induced by the terms (with /g0) in u„;,at the nematicvapor interface, at afixed temperature T=0.175 (which is far from the triple point). In the absence of competition between terms of the intermolecular potential, the preferred molecular orientation (equilibrium tilt angle) is the same at the NI and NV interfaces. This is not true for real nematogens [2], which suggests that in these systems such acompetition may be relevant, resulting in different tilt angles at the NI and NV interfaces. We will return to this point in Sec. IIIC. In. order to solve Eqs. (3), the orientation of the bulk nematic director is required. Since the bulk phase is rotationally invariant, at equilibrium, the bulk director is determined by the interfacial tilt angle. Thus it may be difFicult to obtain convergent numerical solutions starting from an arbitrary direction of the bulk nematic director. In order to overcome this difhculty, we have used an ansatz proposed by Tarazona and Evans [34] in the context of wetting phenomena. We start by fixing the angle 4between the bulk nematic director and the normal to the interface and consider an initial guess for the densityorientational profile compatible with the given 4; then the integral equations (3) are iterated for afixed number of times M(M depends on the proximity to the triple point under wetting conditions, but it is otherwise arbitrary). The above process is repeated for different values of 4. The minimum of the excess grand potential as a function of 4yields an estimate of the equilibrium tilt angle. The equilibrium tilt is now calculated by solving Eqs. (3) for different values of 4' in the neighborhood of the minimum, until apredetermined convergence criterium is satisfied. Since the tilt angle may vary slowly at the interface, we use the bulk value of 4' as the equilibrium tilt angle. The value of the excess grand potential corresponding to the equilibrium tilt angle is the equilibrium surface tension. Let us consider first the NV interface. We start by choosing C=+0.3(D =E=0). In agreement with previous work [23,24], we find that the surface tension is lowest when @=0' (90 )for C&0((0) (see Fig. 1). A similar analysis of the (222) term was carried out by setting D=+0.3(with C=E=0). The results [Fig. 1(a)] indicate that the (222) term yields an equilibrium tilt an0.58 0.570.560.550.540.53C=-0.3 D= 0.3 E= 0.3 ~-~ ~ -~~"M'»: ~~-u "4' NV interface 0.520.51 0.500.490I m/4 'P z/2 0.009D=0.3 Nl interface 0.008 E= 0.3 LC=-0.3 E=-0.3C= 0.3 0.0070.006 0 D=-0.3 I vt/4 'P 7t/2 FIG. 1. Excess grand potential p(in units of Ao. )as a function of the orientation 4of the bulk nematic director for each of the terms in u„; [Eq. (5)]. The straight line is the surface tension of the Maier-Saupe nematogen. The other curves are the surface tensions when one of the other terms is included: (202) with t=+0.3, (222) with D=+0.3, and (224) with R=+0.3, respectively. (a) NV interface (T=0.175) and (b) NI interface (T=0.225). 52 SURFACE-INDUCED ALIGNMENT AT MODEL NEMATIC INTERFACES 5033 gle that is opposite the tilt due to the (202) term, i.e., when D&0the molecules are aligned parallel to the interface while for D(0the molecules are normal to the interface. The behavior of the surface tension (strictly speaking, we should reserve the term surface tension for the minimum of the excess grand potential, but following current practice we speak of asurface tension as afunction of 4') with @is monotonic in both cases [see Fig. 1(a)]. The change in the surface tension of the NV interface characterized by the Maier-Saupe potential (220) (straight line in Fig. 1) due to the inclusion of the (202) term is significantly larger than that due to the inclusion of a(222) term of similar magnitude. Finally, the effects of the (224) term (proportional to E) are particularly interesting. For E&0, the surface tension pis lowest when 4—49.5.This value is very close to that given by afirst-order perturbation expansion in terms of E(see the next paragraph) and to the value given by asharp-kink approximation of this mean-field theory [28]. 4=0' corresponds to the maximum of the surface tension while 4=90' is alocal maximum (Fig. 1). In contrast, when E&0, the equilibrium tilt angle is 4=0;in this case 4=49.5corresponds to the maximum ofthe surface tension and 4=90 corresponds to alocal minimum. The change in the surface tension of the Maier-Saupe nematogen [straight line in Fig. 1(a)]due to the inclusion of the (224) term alone is much smaller than the changes due to either the (202) or the (222) term discussed in the preceding paragraph. We note that, although the preferred orientations at the NI interface, induced by the various terms with t$0, are the same as those found at the NV interface, the relative change in the NI surface tension is largest when the (222) term is included [see Fig. 1(b)]. Although we do not expect realistic values of Eor D to be larger than A, our theory breaks down for D&1.2 or E&0.9, for asystem with B=0.3and C=O. This is signaled by the appearance of "structured" phases at the nematic &ee surface when Dor Ereaches those values. Whether these phases are real (for the given potentials) or are an artifact of the mean-field approximation requires further study. In this paper we have restricted our attention to values of Dand. Ethat are much smaller than those corresponding to the instability. The results of Fig. 1can be understood using aperturbation expansion of the surface tension about the MaierSaupe nematogen (C =D=E=0) similar to that used in Ref. [24]. Following Tijpto-Margo et al.,the surface tension difference between asystem with nonzero C(D =E=0) and areference Maier-Saupe system is written, to first order in C, as CJcPz (cos 4'), — where J~ is apositive number and P2(x) is the secondorder Iegendre polynomial. As pointed out in [24], Eq. (12) implies that pis ininimized by @=0when C&0 andby 4=90 when C(0. Similarly we take pto be the surface tension of asystem with nonzero D(C =E=0) and find, to first order in D, =— DJ~P2(cos 4), p1(i3) where JD is aconstant given by JD =dz pz7p zy dzl xdz2 p( z2 gl )z2 G2 z~ — z2 .14 dz2 In the above expression, p~ l(z) is the density profile of the reference system, gl~ l(z) is the "intrinsic" uniaxial order parameter defined in [24], and G2(z) is apositive, monotonically decaying function of z[see Eq. (A7) in Ref. [24]]. As both p~ l(z) and ql~ l(z) are also monotonic functions, JD is always positive. From Eq. (13) it follows that pis minimized when iII =90' (0') if D&0 (&0) Finally, the difFerence between the surface tension of a system with E$0 (C =D=0) and that of the reference Maier-Saupe system is written to first order in Eas EJ~ 3P— 2(cos iIJ) —— sin 2iII +— sin iII (pl 23~4 228 (15) where J@ is aconstant given by J~ — —dz] pz] r/I z] dzy d"" .(')""(')"G. "-" 16 dz2 and pool(z) and gl~ol(z) have the same meaning as before. The function G4(z) is the solution of the equation d2G4(z)/dzz =u4(z). Using u4(z) given by Eq. (11),it follows that G4 z(7z' — 20z' +i8z' — 4) if ~z~& i (i7) From the behavior of p~ol(z) and gl~ol(z) [24], it follows that &[pool(z)ill~ol(z)] is nonzero in the interfacial region only; this implies that the main contribution to J~ arises from small values of ~zi — z2 ~. In this region (z &0.5), the function G4(z) is negative and, as aconsequence, Eq. (15) implies that pis minimized by @=49' when E&0and by 4=0when E(0. As the contribution to the expansion coefficient u(224; r) arising from anisotropic dispersion interactions is always negative [29] (E &0), these results indicate, in agreement with those of Refs. [28,30], that this contribution cannot account for an oblique tilt angle at the interface. On the other hand, an electrostatic quadrupole-quadrupole contribution is indeed positive and, as pointed out by Sullivan and Tjipto-Margo [28,30], this interaction may be relevant in stabilizing an oblique tilt angle as observed at the NI interfaces of many nematogens [9— 11]. 5034 MARTIN del RIO, TELO da GAMA, de MIGUEL, AND RULL 52 B.Competing surface orientations and tilt angle transitions 0.5355 -- ~-..D=0.75 0.5345 0.5335 0.5325 0z/2 FIG. 2. Excess grand potential pat the IVV interface (T=0.175) as afunction of the orientation 4of the bulk nematic director for C=0.3, E=0, and difFerent values of D, close to the orientational transition from 4~—0to 90 . Since the (202) and (222) terms compete as far as the alignment at the interface is concerned, orientational transitions may occur as the relative magnitude of these terms is changed. We have looked for orientational transitions by taking Cconstant and varying D(with CD )0). We have set C=0.3and studied the orientation at the NV interface at T=0.175, as D(with E=0) is increased from 0. An orientational transition was found to occur for avalue of Din the range 0.70— 0.75. In systems with D(0.70 the tilt angle is 4=0 while for D&0.75 the equilibrium tilt angle is 4=90 . The transition is of first order, as indicated by the presence of an energy barrier in Fig. 2~We note that, close to the transition, the molecular orientation at the interface is neither perpendicular nor parallel. The tilt angle has avalue that is rather close to 0' or 90 (see Fig. 2). We have not attempted to calculate the exact value of Dat which the orientational transition takes place. The transition occurs in aregion where the linear terms of the perturbation expansion cancel out and thus aperturbation theory for the the excess grand potential requires the inclusion of higher-order terms. We stress that this competition does not yield equilibrium tilt angles of the order of 45, as observed experimentally; indeed, within our model, these values of the tilt can only be obtained by inclusion of (224) terms in the intermolecular potential. Similar transitions also occur at the NI interface. For asystem with C=0.3(and Z=O), at afixed temperature T=0.261, we have found that 4=0for D(0.01 and 4=90 for D&0.02. Note that the reverse transition takes place in asystem characterized by symmetrical values of Cand D. Sullivan and Tijpto-Margo [28] have studied the same model potential using amean-field theory. Rather than solving numerically the resulting integral equations„ they have obtained analytical results by resorting to asharpkink approximation for the density-orientational profile. Within this approximation afirst-order transition between 4=0and 4=90 takes place at the NV interface, at T=0.175, for asystem with C=0.3and D=0.7536, which is close to the range of Dobtained from the full numerical solution. Similarly, within the sharp-kink approximation we estimate the transition at the NI interface, at T=0.261, to occur for asystem with C=0.3and D=0.0137. The agreement with the full numerical results is somewhat surprising, since the sharpkink approximation is expected to be less reliable for the NI interface. Note, however, that the relative range of D's estimated using the full mean-field theory is much larger for the transition at the NI interface. Qualitative discrepancies have also been found. The first is the fact that the full mean-field theory predicts atransition to occur between two tilted orientations rather than between perpendicular and planar alignment. The second concerns the order of the transition when temperaturedriven orientational transitions are considered for fixed values of the parameters. This point will be further discussed in Sec. IIID. C. Competing surface orientations and tilt angles at the NV and NI interfaces It has been found experimentally [2] that for most systems, the orientation at the NV interface difFers from that at the NI interface. This behavior may be described with the present theory, by including more than one term with l$0 in the anisotropic intermolecular potential. Following our previous arguments, an appropriate combination of the parameters Cand Dmay be chosen in order to obtain difFerent molecular orientations at the NV and NI interfaces. If we take CD )0, the (202) and (222) terms compete and may yield different orientations at the NV and NI interfaces; when D/C =1, the molecular alignment at the NI interface is dictated by the (222) term (more precisely by the sign of D) and the orientation at the NV interface is determined by the (202) term, i.e.,by the sign of C. In Fig. 3we have plotted the density and orderparameter profiles at the NI interface (T=0.225) and at the NV interface (T=0.218) for asystem with C=D=0.3and E=0. This choice of parameters may be used to describe the experimental observations of 4-methoxybenzilidene-4'-(n-butyl) aniline (MBBA) molecules close to the triple point: homeotropic alignment (4' =0') at the NV interface [12,13] and homogeneous planar alignment (4 =90 )at the NI interface [8]. Note that while the NV interface is uniaxial, the NI interface is biaxial and two order parameters are required to describe the interfacial order. We note that it is not necessary to take D/C 1in order to obtain difFerent orientations at the NI and NV interfaces. For example, by taking C=0.3(— 0.3) and 0.02 &D&.7(— 0.7&D&— 0.02) we find ill =0(90') 52 SURFACE-INDUCED ALIGNMENT AT MODEL NEMAATIC INTERFACES 5035 at the NV interface and 4=90 yO gtth NI In Sec. &aeinterface. nSec. IIIAit was pointed out that the (224) term has arelatively weak influence (compared with the other two terms with I$0) in determining the interfacial ordering. ique sang es, for posConsequently, we have found obl' t'lt l itive E, when Cand Dare such that the 'b aecontri utions oe( ) and (222) terms nearly cancel. The value of D/C required for this to occur at the NI interface is, in general, much smaller than at the NV interface (two or ers of magnitude, for asystem with C=0.3). Thus, within our model, it is not possible to find oblique tilted angles at both interfaces for agiven nematogen. scribe qualitatively the molecular orientation at the inserzes. In these sys-er aces of nematogens in the nCB '.I tems, the alignment is found to be perpendicular at the NV interface [3— 5] while it is oblique at the NI interface, wit atilt angle ranging from 48.5' (in 8CB) to 64.5' (in 6CB) [10,11].In Fig. 4we have plotted the density and or er-parameter profiles for this model nemato en at th (=.)and NI (T=0.225) interfaces .The NV interface is uniaxial with gthe onleon ynonzero or er paramietaxia caracterized eter, while the NI interface is tilt db1 ythree orientational order parameters. D. Temperature-driven tilt an lteransom sons Experiments on MBBA and 4-(n-ethoxy)benzilidene- '-(n-butyl) aniline [6,12,13] indicate that the molecular naacer ann temper-tilte to perpendicular orientation at t' aure 0slightly below the triple point. We have found 0.9 0.9 0.80.70.60.5 E td 0.4Ct. 0.3O 0.2Vinterface 0.80.70.6 0.50) E td 0.4O. L0.3O 0.2O.f NV interface 0.1-0.1-15 I -10 -5 t 010 -0.1-20 I -15 -10 I -5 Z010 0.9 0.80.8to E0.4Cd Cd CL 0.2O Nl interface 0.70.60.5td Cd CL 0.40) O0.30.2V Nl interface -0.2- ~t0NAP 0ahW q0 -20 -15 t -10 ~~ t -5 0 Z510 -0.4 -20 I -15 I -10 -5 0 Z I 10 15 FIG. 3. Density (m units of o)and order-parameter pro61es for asystem with C=D=0.3dE= interface (T=0.218). The equilibrium tilt angle is 4',~=0' (as for MBBA close to the triple point [12,13]);the interface is uniaxial. (b) NI interface (T=0.225). The equilibrium tilt angle is 4',~=90 (as for MBBA [8]);the interface is biaxial. zcoordinate is in units of cr. FIG. 4. ensity and order-parameter profiles for asysem with C=0.3, D=0.01, and E=0.5. aN The equilibrium tilt angle is 4=0' eq as or nematogens in the nCB series 3— 5theinterface is uniaxzs 4e =32.5~ ~ eequi I. rium tilt angle ial. (b) NI interface (T=0.225). Th 'l'b .5, i.e.,the director is oblique tilted with respect to the normal to the i interface (as for nematogens of the nCB series [10,11]). 5036 MARTIN del RIO, TELO da GAMA, de MIGUEL, AND RULL 52 that, within our simple model, the parameter E' must be positive in order to predict atilted orientation at the interface. The orientational transition may then be described as the result of acompetition between orientations dictated by different terms in the expansion of u„;. Approximations based on the model described by Eqs. (5) and (6) that neglect the variation of the density and order parameter profiles in the interfacial region, such as the sharp-kink approximation, yield an expression for the surface tension that may be cast in the form p(4) =po +p2P2(cos 4) +p4P4(cos @) 0.33 0.32 0.31 0.30 0.29 0.28 0.270.260.250.175 0.180 0.185 0.190 0.195 0.200 0.205 where the coefIicients po, p2, and p4 depend on the interxnolecular potential parameters (A E), as— well as on the bulk values of pand g[30]. This expression is identical to the one obtained by the so-called phenomenological theories, where it is derived using symmetry arguments and the coefficients are often arbitrary [35]. It is well known that, in this case, the tilt angle transition form oblique to homeotropic alignment is always continuous, as seen in experiments [2]. We have studied atemperature-driven orientational transition at the NV interface. Since the interfacial effects of the (224) term are much smaller than those of the other terms in the intermolecular potential with 1$0, in order to observe tilt angles we have to choose values of C and Dwhose combined effect in determining the surface alignment nearly cancels. We could also have set one of them to zero and chosen the other very small [36]. This, however, seems somewhat less realistic than our particular choice. We have set C=0.3, D=0.7, and E=0.3. As discussed in Sec. IIIC, asystem with these values of C and Dand with E'=0, at T=0.175, is close to atilt angle transition 4=0+90 .Under these circumstances, the orientation at the interface far &om the triple point is determined by the (224) term, which is proportional to E. When E=0.3we find that for T&0.1945 atilted orientation at the interface is obtained (see Fig. 5). At T=0.195, the orientation changes discontinuously from tilted to perpendicular. The change in slope of p(T) at this transition is too small to be clearly visible on the scale of Fig. 5. As mentioned previously, the continuous nature of this transition is predicted by most theories, although afirst-order transition was previously reported by Kimura and Nakano [15,17]. These authors used the sharp-kink approximation, but they included in the intermolecular potential excluded volume effects, which result in additional terms for the surface tension (18) [15,17]. These are ultimately responsible for driving the transition to first order. In our case, the first-order nature of the transition results from the coupling of the density and orientational order-parameter profiles in the interfacial region. It seems likely that our theory may also predict continuous orientational transitions for different sets of parameters (C, D, and E) [15,17]. We have not investigated this possibility, but very recently Braun et aL [37] have indeed observed acontinuous transition using this theory for asystem with D=0(and |g0and F/0) The sharp-kink approximation used by Sullivan and 0.210 I 0.215 I 0.220 I 0.225 FIG. 5. Liquid-vapor surface tension pversus temperature T(in units of A) close to the triple point T't, For T(Tt„ the surface tension corresponds to the NV interface, while for T)Tt„ it corresponds to the IV interface. The discontinuity in the surface tension at T=Tt, is equal in magnitude to the surface tension of the NI interface, indicating that the NV interface is wet by the isotropic phase. The Grst-order transition from tilted to homeotropic alignment occurring at T0.195 is shown in the inset. The second tilt angle transition, which takes place at T0.214, is indicated by the dotted line. E. Wetting transitions Tijpto-Margo et al. [24] have studied the wetting behavior of this model with D=E=0. At the triple point Tjipto-Margo predicts (for these values of the parameters) acontinuous transition from tilted to perpendicular alignment, at atemperature above the triple point, i.e.,in contrast to the results of the full theory there is no orientational transition at the NV interface for asystem with =0.3, D=0.7, and E=0.3. When the temperature is increased towards the triple point, the NV interface of this systexn exhibits a(second) orientational transition from perpendicular to planar alignment. This transition is also of first order and occurs at T=0.214, which is very close to the triple point temperature (Tt, — — 0.2182), suggesting that it may be related to awetting transition at the NV interface. This observation is further supported by the fact that, for this system, the equilibrium orientation at the NI interface (close or at the triple point) is 1II =90 .An inspection of Fig. 5shows that the surface tension is no longer linear with temperature for T)0.214. This departure &om linear behavior is due to an increase of the interfacial disorder as expected in aregime of wetting by the isotropic phase. The discontinuity in the surface tension, at the triple point, is equal to pNI, confirming that the NV interface is wet by the Iphase at that temperature. In the next subsection we will discuss other wetting related phenomena at the NV and at the isotropic liquid-vapor (IV) interfaces.