Observation of Surface Nematization at the Solid− Liquid Crystal

15998

J. Phys. Chem. C 2007, 111, 15998-16005

Observation of Surface Nematization at the Solid-Liquid Crystal Interface via Molecular Simulation† Luis F. Rull* and Jose´ M. Romero-Enrique* Departamento de Fı´sica Ato´ mica, Molecular y Nuclear, Area de Fı´sica Teo´ rica, UniVersidad de SeVilla, 41080 SeVilla, Spain

Erich A. Mu1 ller* Department of Chemical Engineering, Imperial College London, London SW7 2AZ, U.K. ReceiVed: May 18, 2007

We report molecular simulations of a model liquid crystal that evidences the existence of surface phase transitions when its vapor is placed in contact with an unstructured attractive wall. Monte Carlo simulations are performed in the grand-canonical ensemble on a wide pore at a range of temperatures corresponding to states below and above the bulk nematic-isotropic-vapor triple point of a Gay-Berne fluid. At bulk coexistence conditions, the system exhibits complete wetting of the vapor-wall interface either by nematic (low temperature) or isotropic liquid (high temperature). There are no hints of a thin-thick prewetting transition, but a surface nematization of the first fluid layer close to the substrate; that is, a surface phase transition between a disordered film and a nematic-like film is observed. The nature of the ordered film is studied and is shown to be analogous to the transition seen on a two-dimensional Gay-Berne fluid and persists even at the highest temperatures studied, corresponding to isotropic liquid bulk conditions.

I. Introduction Phase equilibria of confined liquid crystals (LC) is an area of ongoing interest because many industrial applications of LC crystals involve the direct interaction with surfaces. The effect of surfaces on the phase behavior is far from being totally understood. In this scenario, new surface phenomena come into play, including, but not limited to, the appearance of new phases and wetting phenomena. For a vapor at bulk conditions close to (but not at) the vaporliquid coexistence transition, the presence of a surface may induce the adsorption of a film on the surface, even if such a condensed phase is not expected in the bulk, that is, even though the chemical potential. µ, is below the value of the coexistence value µo(T). As the chemical potential is increased, the thickness of the layer may diverge as bulk coexistence is approached, leading to a complete wetting of the surface. Conversely, partial wetting corresponds to a situation where the adsorption on the surface remains finite at coexistence. The change from partial to complete wetting is known generically as wetting transition. Predicted originally by Cahn,1 it has been well documented for isotropic fluids and studied theoretically2,3 and experimentally (e.g., see the review by Bonn and Ross4). For first-order wetting transitions, the initial thin film formed on the wall may suddenly experience a first-order (discontinuous) transition onto a thick film before reaching the coexistence. This thin-thick transition is known as prewetting, and its existence has been confirmed by molecular simulations.5-12 For liquid crystals (LC), the scenario becomes more complicated. LCs are characterized by the presence of ordered phases, for example, nematic or smectic, which are unseen in †

Part of the “Keith E. Gubbins Festschrift”. * E-mail: [email protected]; [email protected]; [email protected].

fluids composed of spherical molecules. The transition between these phases is usually dictated by subtle energetic and entropic balances, which may be altered significantly when LCs come in contact with surfaces, be it either by confinement or by the termination of a bulk phase with a wall. Because of the expected ordered nature of the adsorbed films, speculation arises about the possibility of having a nematic thin film adsorbed unto the walls. Although theoretical studies13-19 have addressed this situation showing that there is an interplay between wetting and orientational transitions, molecular simulation is posed as the tool of choice to understand the details accompanying such phenomena. It is this issue that is addressed in this paper. A recent review by Wilson20 shows the state of the art of molecular simulation of LCs. However, only a few reported studies deal with inhomogeneous cases. Chalam et al.21 have performed molecular dynamics (MD) studies of the changes that ocurr in a fluid of Gay-Berne (GB) LCs when confined in pores with parallel homeotropic walls (which favor a perpendicular aligment to the wall plane), showing how the confinement shifts the phase transitions to higher temperatures and stabilizes the LC phases. Stelzer et al.,22 Zhang et al.,23 and Wall and Cleaver24 have performed MD studies on the effect of confinement of GB molecules in atractive smooth-walled pores. They have performed MD simulations on wide pores (such that it would be expected that no correlation exists between each wall). By varying the wall potential, various smectic and planar layers were obtained near the wall, in spite of the isotropic, nematic, or smectic nature of the coexisting bulk phase (i.e., the middle of the pore). Gruhn and Schoen25 have performed grandcanonical Monte Carlo (GCMC) simulations on confined GB fluids in pores formed by layers of spherical molecules. The bulk conditions corresponded to an isotropic fluid. However, upon confinement, the molecules near the walls developed either

10.1021/jp0738560 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/09/2007

Observation of Surface Nematization

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planar (ordered layers with parallel aligment to the wall) or homeotropic alignments. For a wide enough pore, the middle of the pore maintained an isotropic-like behavior. Palermo et al.26 performed canonical (NVT) MC simulations of adsorption of GB molecules onto graphite-like walls. The opposing wall of the simulation cell was filled with a layer of either planar or homeotropic GB molecules fixed in space. These papers highlight the structure of liquid films near a single wall or within confinement. In all of them, it is evident that the effect of the wall does not extend more than a few layers into the bulk (or middle of the pore) and that the film formed on the wall is more directly influenced by the nature of the substrate than on the thermodynamic conditions. None of the above-mentioned papers address explicitly the case of wetting of a surface (from a vapor) by a nematic LC. Experiments show that such a situation is possible; for example, Sigel and Strobl27 have observed the wetting of a solid surface by a homeotropically oriented nematic layer in the isotropic phase of a cyanobiphenyl CB8. It is expected that by varying the surface properties the nematic wetting might change its orientation, as is the case for the wetting at the isotropic liquidvapor interface by a nematic phase, (see for example theoretical,28,29 computer simulation,30 and experimental31-33 evidence of this). Moreover, in a recent paper Wall and Cleaver34 showed, via canonical MD simulations, how a change in the substrate properties, particularly in the attraction potential, produced drastic changes in the properties of films adsorbed from a vapor phase. In a previous paper,35 we have studied the adsorption of a GB vapor on unstructured attractive walls. Although we had found no evidence of first-order pre-wetting transition, the question was left open to whether a suitable wall potential could induce such a transition. This is in part due to the early predictions of Nakanishi and Fisher36 who have shown that such a prewetting of a highly ordered film with finite thickness on the surface may occur and that it continues to first-order wetting transition. Alternatively, layering transitions may be observed associated to a continuous wetting transition. Only recently, however, experimental evidence37 points to such pre-wetting scenarios and shows the occurrence of isotropic-nematic surface phase transitions. By varying strength of the anchoring of LCs onto the surfaces, the latter authors found the transition to be of first order (weak anchoring) or continuous (strong anchoring). In this paper, we build on this idea. We thus used an attractive model for the wall, which, as opposed to our previous paper,35 allows us to favor the parallel orientation of the first layers of adsorbed molecules. We searched for the telltale signs of a firstorder transition, that is, abrupt changes in the structure and density of the thin films. II. Potential Model And Simulation Details The fluid-fluid intermolecular potential energy, Uff, is modeled using an anisotropic Gay-Berne38 potential. Uijff(uˆ i,uˆ j,rˆ ij) ) 4(uˆ i,uˆ j,rˆ ij)

{[

σ0

]

rij - σ(uˆ i,uˆ j,rˆ ij) + σ0 σ0

[

12

-

]}

rij - σ(uˆ i,uˆ j,rˆ ij) + σ0

6

(1)

where uˆ i and rˆ ij are unit vectors that define the direction of the main symmetry axis of the molecule and intermolecular distance, respectively. Details on the particular functional form of the potential are given in elsewhere (e.g., refs 38 and 39). In particular, (uˆ i,uˆ j,rˆij) and σ(uˆ i,uˆ j,rˆij) are defined as

[ (

σ(uˆ i,uˆ j,rˆ ij) ) σ0 1 -

2 χ (uˆ ij‚uˆ i + rˆ ij‚uˆ j) + 2 1 + χ(uˆ i‚uˆ j)

1 - χ(uˆ i‚uˆ j) (uˆ i,uˆ j,rˆ ij) 1 ) 0 (1 - χ2(uˆ ‚uˆ )2)ν/2

[ (

i

)]

(rˆ ij‚uˆ i - rˆ ij‚uˆ j)2

-1/2

(1b)

)]

j

2 (rˆ ij‚uˆ i - rˆ ij‚uˆ j)2 χ′ (rˆ ij‚uˆ i + rˆ ij‚uˆ j) 1+ 2 1 + χ′(uˆ i‚uˆ j) 1 - χ′(uˆ i‚uˆ j)

µ

(1c)

where χ ) (κ2 - 1)/(κ2 + 1) and χ′ ) [(κ′)1/µ - 1]/[(κ′)1/µ + 1]. The potential requires four parameters: κ, the measure of the length-to-breadth ratio of the molecule; k′, the measure of the well-depth anisotropies; and µ and ν, which adjust the relative strengths of the intermolecular interactions. Additionally, two parameters σ0 and 0 correspond to the characteristic length and energy. We have set the values µ ) 2, ν ) 2, κ ) 3, and κ′ ) 1.25, which are the same as those used in the bulk-phase simulations of de Miguel et al.40 and correspond to the BatesLuckhurst39 notation of GB(κ, κ′, µ, ν) ) GB(3, 1.25, 2, 2). With this choice of parameters, the bulk liquid exhibits a vaporisotropic-nematic triple point approximately located in a range of reduced temperature T* ) kT/0 ) 0.53-0.55, where T is the temperature and k is Boltzmann’s constant. Above this temperature, the isotropic liquid is in equilibrium with the vapor phase; below it, nematic-vapor equilibria may be found. The wall fluid intermolecular potential, Uwf, takes the shape of

Uwf i ) aw w(z)(1 + bw/awP2(cos θi(z)))

(2)

where aw ) 1, bw ) -1, and w(z) is given by

w(z) )

[ ( ) ( )]

σw 2 σw 9 2π Fwσ3ww 3 15 z z

3

(3)

where z is the normal distance to the wall. P2 is the secondorder Legendre polynomial:

3 1 P2(cos θi(z)) ) cos2 θi(z) 2 2

(4)

where θi is the angle formed by uˆ i, the molecular axis and, nˆ , the unit vector normal to the attractive wall. A value of P2 ) -0.5 corresponds to molecules completely parallel to the wall (although their orientations may be at random in this plane), and values of P2 ) 1 correspond to molecules perfectly perpendicular to the wall. The wall potential used favors the parallel orientation of the adsorbed GB molecules on the wall. Because it is expected that due to the constraints imposed by the interfacial tension, the free (liquid-vapor) interface will also show the same parallel orientation, the director should not experience a change in direction.34 The potential is similar to that suggested by Finn and Monson7 in their simulations of prewetting in Lennard-Jones fluids. As in that reference, we choose Fwσ3w ) 0.988, σw ) 1.09457σ0, and w ) 1.2770. The ratio of the well depth (potential minima) between the side-side (i.e., parallel configuration) fluid-fluid interaction and the fluid-solid interaction is chosen to be roughly equivalent to the ratio found between bulk fluid argon and the argon-graphite interactions. This latter system, modeled as a Lennard-Jones fluid adsorbing onto a 9:3 wall, serves as a

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Rull et al.

test case for simple adsorption studies and has been shown12 to have a first-order prewetting transition. All distances and energies considered are reduced with respect to the characteristic length, σ0, and well depth, 0, of the potential, for example, z* ) z/σ0, T* ) kT/0, F* ) Fσo3, µ* ) µ/0. The fluid is confined in a rectangular box of length L*x ) L*y in the x and y directions and L*z ) 50 in the z direction. The box has periodic boundary conditions in the x and y directions and walls in the z* ) 0 and z* ) 50 plane. To avoid capillary condensation, which may preempt wetting phenomena, we placed a purely hard wall at z* ) 50. The full potential is thus a superposition of the fluid-fluid interactions, the interactions of the molecules with the attractive wall at z* ) 0 and a hard wall at z* ) 50. We have run GCMC molecular simulations, where the temperature, T, chemical potential, µ, and volume, V, are fixed. Each MC cycle consists of a total of N translation and rotation steps on a randomly chosen particle (where N is the number of molecules in the system) and between 10 and 100 insertion/ deletion steps. The reader is referred to standard textbooks41,42 for a full description of the GCMC procedure. The volume of the box is fixed and initially is set up with 10 randomly placed molecules. The potential is cut off and shifted at a distance of 4σ0, and no long-range corrections are applied. At the end of the simulations, the simulation boxes exhibited a maximum mean occupation of O(103) molecules. For averaging purposes, the box has been divided in the z direction into slices of width ∆z ) 0.1σ0 with the exception of the calculation of the orientational order where each slice corresponds to a density layer close to the substrate, and we use a width ∆z ) 0.65σ0 in the rest of the profile. This choice is made in order to guarantee a sufficient number of molecules in each slice in order to obtain reasonable statistics. The area, A, of a slice is A ) (L*x).2 The systems were run for up to 4.5 × 106 cycles and statistics collected at least during the last 106 cycles. In each slice, the adsorption, the local density, and the nematic order parameters are calculated. We define the adsorption, Γ, as

Γ)

∫0L

/ z

[F*(z*) - F/b]dz*

(5)

where L/z is the simulation box length in the z direction, F*b is the reduced bulk number density, and F*(z*) is the density profile in the z direction

〈N〉σ03 F (z ) ) A∆z *

*

(6)

and 〈N〉 is the average number of molecules in a given slice. The bulk values are consistent with the data reported by de Miguel et al.40 The order tensor QRβ(z) is defined as

QRβ(z) )

1

N(z)

∑ N(z) i)1

(

D

1 uiRuiβ - δRβ 2 2

)

(7)

where R and β are combinations of the D-dimensional Cartesian axis, N(z) is the instantaneous number of molecules in given slice centered at z, and δRβ is the Kronecker delta. We will consider in this paper the cases of dimensionalities D ) 2 and 3. In both cases, the nematic order parameter, S, is defined as the ensemble average of the largest eigenvalue of the order tensor. Additionally for D ) 3, the biaxial order parameter, B, is defined as the ensemble average of the difference of the

remaining two eigenvalues. Values of S close to unity correspond to a perfect orientationally ordered phase, and B measures the ordering of the molecular axis projections to the plane perpendicular to the nematic director. There are two distinct ways to evaluate these parameters; one may evaluate the tensor along the course of the simulation and at the end, calculate the eigenvectors and through them the S and B order parameters. However, one could alternatively find block averages of the order tensor and “intrinsic” values for S and B for each of these blocks. The rolling averages of the resulting values should, in principle, give the same result as the formerly described (global) mentioned ones for dense fluids. However, differences are observed if the nematic director does not remain fixed in the simulation run, which is an indication that there is no true long-range ordering but algebraically decaying quasilong-range order. III. Results 3.1. Surface Nematization. We have studied 12 isotherms. The lowest reduced temperatures, T* ) 0.50, 0.51, and 0.52, are below the bulk nematic-isotropic-vapor triple point.40 Hence, one expects to encounter a nematic liquid at bulk saturation. The intermediate ones, T* ) 0.53, 0.54, and 0.55, are close to the triple point. Finally, the highest temperatures, T* ) 0.56, 0.57, 0.58, 0.59, 0.60, and 0.65, correspond to situations where isotropic liquid is expected in bulk. We have performed a complete adsorption isotherm in all cases, for example, made a series of constant-temperature simulations in which the chemical potential was varied. In all cases, we have considered that there is a continuous buildup of a condensed phase film close to the wall as bulk coexistence is approached. Complete wetting is observed in each isotherm as the layer width diverges continuously close to saturation conditions. The wetting layer shows a planar nematic ordering for T* < 0.52 and isotropic liquid characteristics for higher temperatures. Some representative results are given in Table 1 for the lateral size L*x ) 15. For comparison purposes, the chemical potentials corresponding to capillary condensation within the system (which we expect to be similar to the corresponding bulk coexistence values) are also listed in Table 1. For the lowest temperature, T* ) 0.50, the density profiles are shown in Figure 1. No hint of a thin-thick transition is observed. However, monitoring the ordering at the first layer close to the substrate for a lateral size L*x ) 15, we observe a qualitative change for chemical potentials between µ* ) -4.2 and µ* ) -4.1. For lower chemical potentials than µ* ) -4.1, the biaxial order parameter B takes higher values than the nematic order parameter S, which is an indication of a random planar ordering (ideally, B ) 3S). So the GB molecules at the first layer are oriented parallel to the wall, but there is no preferred orientation in the x-y plane. As the chemical potential is increased to higher values than µ* ) -4.1, there is an increase in the value of S, which takes values higher than 0.4, and B takes values lower than 0.4. This change in the orientational order parameters indicates that the first layer experiences a symmetry-breaking transition where the molecules orientate preferentially along a director on the x-y plane. In this sense, it resembles a 2D nematic-isotropic transition, so we call it surface nematization. This transition is analogous to the uniaxial-biaxial surface transition, which has been observed previously43 on simulations of hard-rod fluids for the wetting of the isotropic liquid-hard wall interface by the nematic phase. Figure 2 shows snapshots of equilibrium configurations at T* ) 0.50, as seen from the perspective of the wall, that is, the

Observation of Surface Nematization

J. Phys. Chem. C, Vol. 111, No. 43, 2007 16001

TABLE 1: Adsorption (Γ) and First-Layer Nematic Order Parameter (S), Biaxial Order Parameter (B), and Density (G*) as a Function of the Reduced Chemical Potential, µ*, for Three Selected Temperatures below (T* ) 0.50), Close to (T* ) 0.53), and above (T* ) 0.56), the Bulk Nematic-Isotropic-Vapor Triple Pointa T* ) 0.50 µ*

Γ

S

B

F*

-4.4 -4.3 -4.2 -4.15 -4.1 -4.05 -4.0 -3.9 -3.8 -3.7a

0.162(43) 0.241(38) 0.293(26) 0.416(71) 0.550(45) 0.620(62) 0.717(33) 1.172(99) 1.907(29) 7.177(68)

0.348 0.342 0.274 0.388 0.430 0.505 0.693 0.775 0.805 0.806

0.478 0.477 0.542 0.420 0.380 0.306 0.130 0.066 0.040 0.040

0.803 1.053 1.193 1.345 1.430 1.492 1.540 1.604 1.641 1.644

µ*

Γ

S

B

F*

-4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6b

0.191(18) 0.219(27) 0.323(32) 0.429(28) 0.741(31) 0.924(31) 1.605(45) 3.298(83)

0.283 0.371 0.314 0.335 0.668 0.703 0.748 0.727

0.519 0.451 0.483 0.462 0.129 0.114 0.075 0.101

0.855 0.931 1.161 1.272 1.475 1.528 1.539 1.550

T ) 0.53

T ) 0.56 µ*

Γ

S

B

F

-4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5b

0.104(30) 0.162(50) 0.217(39) 0.278(35) 0.341(48) 0.457(54) 0.592(48) 0.779(41) 1.259(63) 3.280(36)

0.352 0.312 0.297 0.333 0.329 0.344 0.362 0.524 0.627 0.691

0.445 0.474 0.485 0.451 0.451 0.424 0.407 0.274 0.164 0.100

0.459 0.673 0.853 1.004 1.094 1.216 1.305 1.376 1.423 1.474

Figure 2. Snapshots of surface layers at T* ) 0.50. The fluid is seen form the perspective of the wall, i.e., the attractive wall (z* ) 0) is behind the reader, the first layer is seen in forefront. Top figure (a) corresponds to µ* ) -4.15, bottom (b) is µ* ) -4.1.

a An error of 1.23(4) refers to 1.23 ( 0.04. b Corresponds to capillary condensation within the system.

Figure 3. First-layer (thin lines) and second-layer (thick lines) particlenumber histograms at T* ) 0.50 and reduced chemical potentials µ* ) -4.20 (dotted lines), -4.15 (continuous lines) and -4.10 (dashed lines). The lateral size for these simulations is L*x ) 15.

Figure 1. Density profiles F*(z*) as a function of µ* for a temperature of T* ) 0.50. Bottom curve corresponds to µ* ) -4.4. Subsequent curves correspond to increments of ∆µ* ) 0.1. Top curve corresponds to µ* ) -3.8

observer is placed at z* < 0 and looks into the simulation box. At low chemical potential, only a sparse monolayer is formed and there is no preferential ordering. As the density of the first layer is increased, the layer suffers a nematization and at

chemical potentials above the transition the layer shows a clear nematic-like ordering. Although we mentioned that there is no sharp discontinuity in the adsorption isotherm, the surface nematization is accompanied by the emergence of a second peak in the density profiles that rises to a value above the bulk value, as seen in Figure 1. Furthermore, simulations performed at µ* ) -4.15 show an increase of the adsorption fluctuations that correspond to the broadening of the number particle histograms associated to the second layer (cf. Figure 3) with respect to the cases µ* ) -4.20 and µ* ) -4.10. However, statistic uncertainties do not allow us to see if the histogram may exhibit a bimodal shape,

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Figure 4. Density F* (solid line), nematic order parameter S (dashed line), and biaxial order parameter B (dotted line) profiles as a function of the distance to the wall z* of the fluid for a temperature of T* ) 0.56 and a chemical potential µ* ) -3.5. The orientational order parameters have been calculated by averaging the instantaneous order tensor eigenvalues.

signaling a weak first-order transition. If this were not the case, then the nematization transition would be of continuous nature, and because of its quasi-two-dimensional (2D) character should be a Kosterlitz-Thouless transition.44 We will come back to this point in the next section. For higher temperatures, similar transitions are seen between the values of µ* ) -4.1 and µ* ) -4.0 for T* ) 0.53 and between the values of µ* ) -3.9 and µ* ) -3.8 for T* ) 0.56. However, the adsorption fluctuations close to the nematization transition become less important as the temperature increases. It is important to point out that the nematic ordering of the first layer is observed even when the bulk equilibrium phase is an isotropic one; that is, the surface transition is observed even when the temperature is above the isotropicnematic-vapor triple point and no nematic would be observed in a bulk. Figure 4 shows the average layer order parameters S and B for a state point above the triple point, (T* ) 0.56, µ* ) -3.5). Although there is a definitive signature of ordering in the adsorbed walls, the coexisting fluid in the thick film is isotropic. We note that we are calculating the order parameters from averaging the eigenvalues of the instantaneous order tensor. This is the most-appropriate method for studying the nematic phase because it eliminates the fluctuations of the director, but it has the disadvantage of reporting spurious values in the isotropic phases. In an ideal isotropic phase, the probability distribution is at its maximum in S ) 0, but the value will have a spread consisting of entirely positive quantities. Because we are performing an average, the mean value ends up being a nonzero value, due to the fact that it is not an infinitely narrow distribution. In our case, we are sampling small numbers of molecules in the slices, so the discrepancy between the expected null value is enhanced. (In the limiting case of one molecule, S ) 1). This “size effect” has been discussed in detail by Eppenga and Frenkel.45 In these types of systems, low values of the order parameter S < 0.3 can be considered isotropic. Figure 5 shows a sequence of snapshots for T* ) 0.53. At lower chemical potentials (points a and b), there is a sparse filling of the surface and no observable order. After the full development of the thick film (point d), a clear order is seen in the first surface layers. It is seen how, in spite of the nematic ordering at the solid surface, the “bulk” of the thick film is isotropic. Finally, we have to note that the order shown in this figure beyond the vapor-isotropic layer interface is spurious

Figure 5. Snapshots of a lateral view of the adsorbed layers at T* ) 0.53. (a) and (b) correspond to chemical potentials µ* ) -4.3 and -4.1, respectively, (lower than that required for the thin-thick film transition); c and d correspond to chemical potentials µ* ) -4.0 and -3.6, (higher than the transition value).

(due to the method of evaluating the order parameter profiles, which is not valid for low densities). Even for the highest temperature (T* ) 0.65) we observe this nematization transition occurring at chemical potentials above the vapor-isotropic liquid coexistence value, which corresponds to a bulk isotropic liquid. In such conditions, it may be identified with the uniaxial-biaxial surface transition reported in ref 43. 3.2. Adsorbed First Layer and the Two-Dimensional GB Fluid. The results reported in the preceding section were obtained from simulations performed in a simulation box with a lateral size L*x ) 15. However, some of the quantitative results seem to be dependent on the size of the sample studied, that is, the size of the surface domain. Particularly, we have calculated the order parameters for some selected cases at different surface sizes, to evidence the finite size effects. Figure 6 exemplifies a typical case, at T* ) 0.51 and µ* ) -4.0. The histograms of the 3D order parameter S in the first layer are shown for different system sizes (corresponding to different areas of adsorption). Apparently, as the system size increases the order parameter decreases until in-plane orientational order disappears from the system (S ≈ 0.25, corresponding to perfect random-planar ordering). Inspection of the a snapshot of the larger systems shows that although the first layer does not show long-range correlations and a unique director there is a strong local ordering (Figure 7a). This lack of long-range order in the first layer is also indicated by the reduction in the mismatch between the nematic order parameter obtained by averaging the instantaneous eigenvalues of the order tensor and by diagonalizing the averaged order tensor for larger system sizes (see Table 2). This size dependence also depends on the thickness of the nematic layer. So, for T* ) 0.51 and µ* ) -3.8, where a thick wetting nematic layer has been developed, the size dependence is much smaller for the considered system sizes (although it also may be expected for

Observation of Surface Nematization

Figure 6. Normalized histograms of the nematic order parameter, S, at T* ) 0.51 and µ* ) -4.0. The different curves correspond to different system (wall) sizes. The area, A, of a surface corresponds to (15σ0)2, (25σ0)2, (50σ0)2, and (100σ0)2, respectively.

J. Phys. Chem. C, Vol. 111, No. 43, 2007 16003

Figure 8. Normalized histograms of the nematic order parameter, S, for a 2D fluid at T* ) 0.51 and P* ) 1.0. The different curves correspond to different system sizes, N ) 100, 400, 900, 1600, and 2500, respectively.

TABLE 2: Comparison between the First-Layer Nematic Order Parameter S and Biaxial Order Parameter B Obtained by Averaging the Eigenvalues of the Instantaneous Order Tensor, and the Corresponding Order Parameters S′ and B′ Obtained by Diagonalizing the Averaged Order Tensor, for the States µ* ) -4.0 and µ* ) -3.8 at the Reduced Temperature T* ) 0.51, and Different Lateral Sizes L*x

Figure 7. Snapshot of (a) the surface layer at T* ) 0.51 and µ* ) -4.0 corresponding to a system with surface area A ) (100σ0);2 (b) a 2D system at T* ) 0.51 and P* ) 1.0 with 2500 molecules.

very large systems) and there is almost no mismatch between the two methods to evaluate the orientational order parameters (see Table 2).

m*

L*x

S

B

S′

B′

-4.0 -4.0 -4.0 -4.0 -3.8 -3.8 -3.8

15 25 50 100 15 25 50

0.472 0.363 0.263 0.246 0.775 0.756 0.739

0.331 0.430 0.527 0.542 0.063 0.067 0.081

0.225 0.265 0.210 0.243 0.752 0.734 0.732

0.564 0.523 0.579 0.544 0.068 0.085 0.085

This behavior is consistent with that of a 2D GB fluid. In 2D, the long-range nematic order is destroyed by thermal fluctuations.46 However, the nematic phase may appear as a (globally) isotropic fluid that shows algebraically decaying quasi-long-ranged orientational correlations. The transition between the low-temperature nematic phase and the hightemperature isotropic liquid phase with short-ranged orientational correlations may be first-order or continuous. In the latter case, the transition occurs via an unbinding of vortex-antivortex pairs in the nematic director field, analogous to the phenomenology observed in the XY model: the Kosterlitz-Thouless transition.44 To test this hypothesis, we have performed isothermalisobaric Monte Carlo (NPT) simulations of a 2D GB fluid for a fixed temperature, T* ) 0.51, and reduced pressures P* ) Pσ02/0 ) 0.75, 1.00, 1.25, and 1.50. Figure 8 shows the results of this 2D GB fluid where the number of molecules is varied from 100 to 2500. The order parameter clearly shifts between the smallest to the largest system in a completely analogous fashion as occurred in the first layer of the adsorbed GB fluid (cf. Figure 6). A snapshot of this system (Figure 7b) shows clear resemblance to the adsorbed GB film (Figure 7a). For the molecules on the first layer close to the wall, and for the 2D systems, we have studied the 2D second rank orientational correlation function

〈

g2(r*) ) cos2 θ12 -

〉

1 2

(8)

where here the angle θ12 corresponds to the angle between the principal molecular axis of two molecules that are separated

16004 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Figure 9. 2D orientational correlation function g2(z*) for the 2D GB model at T* ) 0.56 and P* ) 1.00 (isotropic case) and P* ) 1.25 (nematic case). For the latter, different system sizes N ) 900, 1600, and 2500 are plotted.

Rull et al. not to be related to the wetting by the nematic phase because it is observed above the triple point and even for isotropic liquid bulk conditions. Furthermore, when compared to the LennardJones prewetting transition, the nematization transition ocurrs at chemical potentials that are further from the coexistence values.12 Finally, the smoothness in the growth of the wetting layer precludes the possibility of observing the prewetting by fine-tuning the chemical potential close to the coexistence value. Because for long-ranged interactions (as the GB and the wallparticle potential) a first-order wetting is expected,3 it is possible that the prewetting line exists in a range of temperatures lower than those considered in this paper, and it may be preempted by the appearance of the smectic phase. Further work will be needed in this direction. The characterization of the nematization transition is beyond the possibilities of the molecular simulation techniques used in this work. Particularly, the slow decay of the orientational correlations at the nematized phase yields to a critical slowing down of the Metropolis sampling, analogous to the observed close to critical points. However, this problem may be bypassed by using a combination of cluster algorithms47,48 and histogram reweighing.49,50 In fact, this procedure has been applied successfully51 to characterize the order-disorder transition in the closely related bidimensional Maier-Saupe liquid. Acknowledgment. L.F.R. and J.M.R.-E. acknowledge financial support from the “Ministerio de Ciencia y Tecnologı´a” (Spain) through grants BQU2001-3615-C0-02 and CTQ200407730-C02-02, and Junta de Andalucı´a through “Plan Andaluz de Investigacio´n” (FQM-205). J.M.R.-E. also acknowledges a “Ramon y Cajal” Fellowship funded by the “Ministerio de Ciencia y Tecnologı´a” (Spain).

Figure 10. First-layer orientational correlation function g2(z*) at T* ) 0.51 and µ* ) -4.0 (random-planar layer) and µ* ) -3.8 (nematized layer). For the latter, different lateral sizes L*x ) 15, 25, and 50 are plotted.

by a distance r*. This function decays to zero in all cases. However, it decays exponentially in the first layer before the nematization and in the 2D isotropic liquid phase. Alternatively, an algebraic decay is observed above the nematization transition and for the 2D nematic phase. Figure 9 shows the decay of the correlation function in the 2D GB case for T* ) 0.56 and P* ) 1.00 (isotropic phase) and P* ) 1.25 (nematic phase). Clearly there is a crossover from an exponential to a power-law behavior (note the loglog scale). Figure 10 shows the corresponding correlation function for the first layer for before (µ* ) -4.0) and after (µ* ) -3.8) the nematization transition for different lateral sizes at T* ) 0.51. These results confirm our prediction of an algebraic decay of the orientational correlations in the nematized phase. IV. Discussion And Conclusions In this work, we have performed GCMC simulations of a system of GB molecules in contact with an attractive planar wall. We have not been able to detect a thin-thick film transition, but instead we report a continuous (or at most a weakly first-order transition) nematization of the first layer close to the wall. This transition is analogous to the 2D isotropicnematic transition and occurs even at temperatures above the 3D isotropic-nematic-vapor bulk triple point. The nematization transition presents characteristics that distinguish it from the prewetting transition. Its existence seems

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