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Monte Carlo simulations of liquid crystals near rough walls David L. Cheung ∗1 and Friederike Schmid1 1Theoretische Physik, Universität Bielefeld, 33615 Bielefeld, Germany The effect of surface roughness on the structure of liquid crystalline fluids near solid substrates is studied by Monte Carlo simulations. The liquid crystal is modelled as a fluid of soft ellipsoidal molecules and the substrate is modelled as a hard wall that excludes the centres of mass of the fluidmolecules. Surfaceroughnessisintroducedbyembeddinganumberofmoleculeswithrandom 5 positions and orientations within the wall. It is found that the density and order near the wall 0 are reduced as the wall becomes rougher (i.e. the number of embedded molecules is increased). 0 Anchoring coefficients are determined from fluctuations in the reciprocal space order tensor. It is 2 found that the anchoring strength decreases with increasing surface roughness. n a J I. INTRODUCTION surface anchoring to the free energy is often taken to be 5 of the Rapini-Popoular form11 ] The interaction between liquid crystalline (LC) fluids ft and solid surfaces has attracted much interest1. The Fsurf =W sin2(θ−θ0) (1) o presence of the surface breaks the symmetry of the LC s . phase. As well as being intrinsically interesting this is where θ − θ0 is the angle between the director at the t a technologically important - many applications of liquid surface and the surface’s ’easy-axis’. W is the surface m crystals depend on the interactionbetween the fluid and anchoring coefficient. This depends on both the prop- - an external field, strongly influenced by coupling with erties of the bulk liquid crystal and on the interaction d external surfaces. between the liquid crystal and the surface, so may be n Most previous studies of LC surface anchoring have expected to vary with surface roughness. As this is a o c assumed that the surface is homogenous. Two models key property in applications of liquid crystals it would [ are commonly used. In the first the wall is modelled be interesting to see how this is affected by changes in by a perfect crystalline array2. The second, more coarse the surface morphology. 1 grained model, uses an external potential function that This paper is organised as follows. Details of the sim- v 7 depends onlyonthe distance fromthe wall3,4. While at- ulation, including the method used for calculating the 7 tractive from a theoretical standpoint, it has long been anchoring coefficient, are given in the next section. The 0 recognised that deviations from these ideal surfaces can structure of the fluid confined between rough walls is 1 affect the properties of the surface5. One notable ex- given in Sec. III while results for the anchoring coeffi- 0 ample of this is the reduction of the order parameter cient are presented in Sec. IV. Finally some concluding 5 of nematic liquid crystals at SiO surfaces6,7. This con- remarks are given in Sec. V. 0 trasts with measurements made on other surfaces1 (e.g. / at rubbed polyimide) and with most simulation and theo- m retical studies that give a higher order parameter at the LC-solid interface. Electron micrographs show that SiO II. SIMULATION - d surfaces are extremely rough8, which gives rise to the n disordering effect of the surface. A. Simulated Systems o In this paper the structures of nematic and isotropic c : fluids near rough walls are studied. The effect of rough- v Inordertosimulatelargesystems,asimpleintermolec- nessisincorporatedbyembeddinganumberofmolecules i ular potential is used. This models the fluid as a system X in an otherwise smooth wall. These are placed and ori- ofsoftellipsoidalmoleculesinteractingthroughasimpli- ar esinmtaptleedflruaidnsd9o,1m0lya.ndSiitmiislahrompeoddetlhsathatvheisbseimenpluesemdodfoerl fied version of the popular Gay-Berne (GB) potential12. In particular this has two major simplifications. First may give insights into the behaviour of molecular fluids the orientation dependence of the energy parameter is near rough or porous surfaces. Two aspects of the ef- suppressed. Secondly the potential is cut off and shifted fectofthe surfaceroughnessonthe LCfluidarestudied. at the potential minima. These changes lead to a much Firstly the change in the structure of the fluid was ex- simplier phase diagram than the GB potential, showing amined. Secondly the effect of surface roughness on the onlynematicandisotropicphases,closertothephasebe- anchoring properties of the LC. The contribution of this haviourofthehardellipsoid13 orhardgaussianoverlap14 potentials. This potential is also more computationally efficient than the full GB potential. ∗Presentaddress: Dept. ofPhysics,UniversityofWarwick,Coven- The interaction between two molecules i and j, with try,CV47AL,UK positions ri and rj, and orientations ui and uj is given 2 (a) by 4ǫ ρ−12−ρ−6 +1, ρ≤21/6 V(r ,u ,u )= 0 ij i j 0 , otherwise (cid:26) (cid:2) (cid:3) (2) where ǫ is the energy unit, r =r −r , and 0 ij i j r −σ(ˆr ,u ,u )+σ ρ(r ,u ,u )= ij ij i j 0. (3) ij i j σ 0 r =|r |,ˆr =r /r ,andσ istheσ(ˆr ,u ,u )isthe ij ij ij ij ij 0 ij i j shape function given by15 (b) χ (ˆr .u +ˆr .u )2 σ(ˆr ,u ,u ) = σ 1− ij i ij j ij i j 0 2 1+χu .u (cid:26) (cid:20) i j (ˆr .u −ˆr .u )2 −1/2 ij i ij j + . (4) 1−χu .u i j (cid:21)(cid:27) This approximates the contact distance between two el- lipsoids. InEq. 4χ=(κ2−1)/(κ2+1)is theanisotropy parameter, where κ is the elongation (for the molecules studied here κ=3). The wall is represented by a hard core potential act- FIG.1: (Coloronline)Exampleroughwallconfigurationsfor ing upon the centres of mass of the molecules. Previous (a) Σ=0.2 and (b) Σ=0.4. studies have shown that this gives rise to homeotropic alignmentatthe wall16. Roughnessisintroducedby em- bedding a number of molecules, N in the wall. These w weregivenrandompositionsandorientationswhichwere where u is the orientation of the ith molecule and δ i αβ keptfixedduringthesimulations. Whilegeneratingthese is the Kronecker delta function. It may also be infor- surface configurations interactions between the surface mative to consider the order parameter in the cell bulk molecules were ignored, thus these molecules may over- and near the surface, Sbulk and Ssurf. Sbulk is calcu- lap. It should be noted that these molecules do not cor- lated for molecules within the region l /4 ≤ z ≤ 3l /4, z z respond to real molecules, rather they are used as a con- while Ssurf is calculatedformoleculeswithin1σ ofthe 0 vient way of introducing inhomogenity into the wall.The surface. roughness of the wall was characterised by the surface Thedistributionofmoleculesinthesimulationcellcan density of these embedded molecules Σ = N /A. Some w be described by the density profile ρ(z). To describe example wall structures are shown in Fig. 1. To ensure the ordering through the cell, the ordering tensor Eq. some sampling of surface configurations three different 5 can be calculated throughout the cell. Diagonalising surfaces were studied for each pair of ρ and Σ. this givesthe order parameterprofiles (q (z), q (z), and Simulations were performed at two average densities, + 0 ρ = 0.314 and ρ = 0.30. For the higher density the q−(z)). These canbe expressedas S(z), S(z)+ 21Sxy(z), and S(z) − 1S (z), where S(z) is the nematic order fluid confined between smooth walls was nematic, while 2 xy parameter and S (z) is the biaxiality parameter. it is isotropic for the lower density. The simulated sys- xy tems were composed of 1200 fluid molecules and up to The nematic director n(z) can be identified with the 63 molecules embedded in each wall. Throughout this eigenvector of the ordering tensor corresponding to the reducedunits defined by the molecularwidth σ andthe largest eigenvalue. 0 energy unit ǫ are used. A reduced temperature of 0.5 0 While the presence of layersmay be deduced from the was used for both densities. density profiles it may be useful to quantify the degree oftranslationalorder. Thesmectic orderparametermay be introduced for this purpose17,18. This is given by B. Simulation observables The orientational order may be characterised by the 1 N 2πiz j usual nematic order parameter. This is given by the ρ1 = (cid:12)N exp d (cid:12) , (6) largest eigenvalue of the ordering tensor, defined as *(cid:12) j=1 (cid:18) (cid:19)(cid:12)+ (cid:12) X (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1 N 3 1 (cid:12) (cid:12) Q = u u − δ , α,β =x,y,z (5) wheredisthelayerperiodicity. Thisisinitiallyunknown αβ iα iβ αβ N 2 2 and is take to be the value that maximises ρ 18. i=1(cid:18) (cid:19) 1 X 3 C. Director fluctuations and surface anchoring III. FLUID STRUCTURE The surface anchoring coefficient is determined by the A. High Density Fluid director fluctuation method19,20. This method relates thermal director fluctuations in a confined geometry to Thedensityprofilesforthehighdensityfluidareshown the zenithalanchoringcoefficient, ina similarmanner as in Fig. 2a. The effect of the wall roughness is most ap- the fluctuations in a bulk LC can be related to the bulk parent near the wall. Here the density near the surface elasticconstants21,22. Thetheoryforthishasbeenexten- decreases with increasing Σ. This is caused by the de- sively developed elsewhere and this section will contain crease in available volume near the wall due to the em- only the briefest of outlines. bedded molecules. Values of the density near the wall As for the bulk elastic constants the zenithal anchor- are presented in Tab. 1. The surface density falls from ing coefficient may be determined by fitting elastic the- 0.72 for the smooth wall to 0.34 for the rough wall with ory predictions of fluctuations in the ordering tensor to Σ=0.4. thosedeterminedfromsimulations. Thereciprocalspace Another noticeable difference is that the second peak ordering tensor is given by (at z = 2.8 for the plain wall) becomes broader. This arisesfromthesurfacedisorderdisturbingthelayerstruc- V ture and has been observed in simulations of Lennard- Qαβ(k)= N Qjαβexp(ik.ri). (7) Jones fluids10. This can more clearly be seen in the in- j set,whichshowsthe detailofthe densityprofilesaround X the minima. The disruption of the translational order- Fluctuations can be calculated from simulation ing caused by the embedded molecules can be seen by considering the smectic order parameter (Eq. 6). Values 2 for these are presented in Tab. 1. As can be seen ρ 1 V2 |Q (k )|2 = Qj cos(k z ) markedly decreases with increasing grafting density, as αβ z N2  αβ z j  would be expected for increasing translational disorder. D E  Xj Far from the wall the profiles all tend to a constant  2 values, indicating a layer of bulk fluid. The density of + Qj sin(k z ) , (8) this layer increases slightly with increasing grafting den-  αβ z j   sity. Thisarisesastheembeddedmoleculesexcludefluid j X  moleculesfromregionsnearthewall,increasingthenum-    ber of molecules in the cell bulk. This is a consequence Thecorrespondingelastictheorypredictsthattherefluc- of having a fixed cell size and may be avoided by using tuations are given by19 NpT simulations. Quantitively this can be seen by ex- aminingthe densities inthe cellbulk. Values forthis are 9 S2V χ2+ζ2 presented in Tab. 1. The density in the bulk of the cell |Q (k )|2 = k T × αβ z 8 B K q2(2ζ+ζ2+χ2) goesfrom0.29forthesmoothwallto0.31forthehighest D E 33 Xqz z grafting densities. ei(κ+χ)−1 iχ−ζ ei(κ−χ)−1 2 Figure 2(b) shows the orderparameter profiles for dif- + κ+χ iχ+ζ κ−χ ferent values of Σ. As can be seen the value of the or- (cid:12) (cid:18) (cid:19) (cid:12) der parameter at the wall is lower for higher grafting (cid:12) ((cid:12)9) (cid:12) (cid:12) densities. This is caused by the disorientating effect of (cid:12) (cid:12) the embedded molecules. This disorientating effect also where K is the bend elastic constant. q is a wave 33 z leads to a deeper minima. For Σ ≥ 0.2 this leads to vector with a discrete spectrum19, χ = q L , and κ = z e a small layer (approximately 1 molecular width thick) k L . ζ is the anchoring strength parameter z e of almost isotropic fluid. The position of this minima moves closer to the wall with increasing surface rough- WLe Le ness. For the smooth wall this minima is at approxi- ζ = = (10) K λ mately z = 2, while for the highest grafting densities it 33 appears at about z = 1.1. Again this is attributable to whereW isthezenithalanchoringcoefficientandλisthe the disruption in the surface induced layering. As for extrapolation length. L is the cell thickness appearing ρ(z) the second peak becomes broader with increasing e in the elastic theory; this is not necessarily equal to the Σ. Finally, as can be seen from Tab. 1 the bulk order simulationcellthickness,L . Infittingthe elastictheory parameter Sbulk increases with increasing Σ. This is a z tosimulationprofilesL andζ arethefittingparameters. consequenceoftheincreasingdensityinthecentreofthe e K has been determined from simulation for a nearby cell due to the excluded volume effect of the embedded 33 state point23 (ρ = 0.30). While this value (K = 1.48) molecules. ItisnoticeablethatforΣ≥0.2Sbulk becomes 33 is likely to be too large for some of the systems studied larger than Ssurf. here, this should be sufficient for a qualitative study. The biaxiality profiles are shown in Fig. 2(c). For the 4 4 1 ρ Σ ρ ρ S Ssurf Sbulk ρ surf bulk 1 1.0 0.8 0.314 0 0.72(1) 0.286(3) 0.60(3) 0.84(1) 0.53(6) 0.14(2) 3 ρ(z)0.5 0.314 0.1 0.61(1) 0.294(2) 0.64(2) 0.76(2) 0.65(3) 0.12(2) 0.6 ρ(z)2 0.00 1 2z 3 4 S(z)0.4 00..331144 00..23 00..4483((35)) 00..330048((34)) 00..6679((22)) 00..6655((39)) 00..7725((22)) 00..1009((22)) 1 0.314 0.4 0.33(2) 0.307(9) 0.66(8) 0.58(7) 0.73(7) 0.07(2) 0.2 0.300 0 0.70(1) 0.273(2) 0.28(3) 0.81(1) 0.10(4) 0.14(2) 0 0 0.300 0.1 0.59(1) 0.281(2) 0.34(7) 0.72(3) 0.27(9) 0.12(2) 0 2 4 6 8 0 2 4 6 8 z z 0.300 0.2 0.46(2) 0.291(3) 0.52(6) 0.61(3) 0.59(5) 0.10(2) 1 0.300 0.3 0.41(2) 0.293(3) 0.51(4) 0.53(9) 0.60(4) 0.09(2) 0.3 0.8 0.300 0.4 0.38(1) 0.300(3) 0.59(6) 0.54(5) 0.69(3) 0.07(2) 0.2 0.6 S (z)xy n (z)z 0.4 0.1 Figure 2(d) shows the z component of the director for 0.2 eachΣ. Inthecellbulkthedirectorisessentiallyparallel 0 0 to the z axis. For Σ ≥ 0.2 there is a tilt away from the 0 2 4 6 8 0 2 4 6 8 z axis at about the position of the order minima. As z z maybeexpectedthisismostpronouncedfortheΣ=0.4 FIG. 2: (a) Density profiles for the high density fluid near wall. InFig. 2dthe profilesforeachofthe Σ=0.4walls rough walls. The density profile for grafting density Σ = are shown separately. It can be seen that the size of this 0 is shown by the solid line, for Σ = 0.1 dotted line, Σ = tilt differs strongly for different wall configurations (for 0.2 dashed line, Σ = 0.3 long dashed line, and Σ = 0.4 the ◦ the largest the tile angle is approximately 79 ). For the dashed dotted line. Inset shows the density profiles around largertiltanglesthispropagatesintothebulkofthefluid theminima. Symbols as in main figure. ◦ leading to a director tilted up to 16 from the z-axis. It (b)Orderparameterprofilesforhighdensityfluidnearrough is not clear how a randomly generated wall gives rise to walls. Symbols as in (a). (c) Biaxiality (Sxy) profiles for high density fluid near rough a titled configurationin the bulk. Similar behaviour has wall. Symbolsas in (a). been seen in a recent study of a LC near a planar wall (d) z component of the director for the high density fluid. with perpendicularly grafted rods25. In that case the Symbolsas in (a). 3 nz(z) profiles are shown for Σ=0.4. bulktiltwascausedbythecompetitionbetweenthewall (which promoted planar alignment) and the embedded molecules. As it appears only for a subset of the walls studiedhereitwouldbedesirabletoconsiderfurtherwall configurations. smooth wallthe this is essentially zero (the largestvalue The previousdiscussionmaybe illuminatedby exami- is 0.04) reflecting the cylindrical symmetry around the nation of simulation configurations. Figure 3 shows con- z axis. However, for the rough walls there is are sizable figurations of for Σ = 0.0 (smooth wall), Σ = 0.2, and peaks in the biaxiality profiles. These are stronger for Σ=0.4. The disordering effect of the rough wall can be larger values of Σ and are in the region of 0.5 ≤ z ≤ seen in the first and second layers (left and right most 1.3, corresponding to regions of low order. This surface pictures). However, the fluid between these two layers induced biaxiality has been seen for simulations of LCs showsthe mostnoticeablechangewithincreasingΣ. For near groovedsurfaces24. thesmoothwallthe moleculesinthis regionarestillwell ordered parallel to the z-axis. With increasing Σ the molecules in this region become increasingly disordered. Thisgivesrisetothedeeperminimaseenintheorderpa- rameter profile (Fig. 2(b)). Additionally it can be seen thatmanyofthemoleculeslieinthexy plane,givingrise to the biaxiality peak and the tilt of the director away Table 1. fromthe z axis. This behaviouris similarto thatseenin Densities and order parameters for the simulated simulations of smectic liquid crystals26 where molecules systems. ρ and ρ are the bulk and surface in the region between the layers are seen to align either bulk surf densities, S, Sbulk and Ssurf are the total, bulk and parallel or normal to the layers. These planar oriented surface order parameters, and ρ is the smectic order molecules possibly give rise to the bulk tilt seen in some 1 parameter. Errors in the last decimal place are in cases. Finally the number of molecules in this region parenthesises. visibly increases with Σ. 5 (a) 4 1 0.8 3 0.6 ρ(z)2 S(z) 0.4 1 0.2 (b) 0 0 0 2 4 6 8 0 2 4 6 8 z z FIG. 4: (a) Density profiles for the low density fluid near rough walls. The density profile for grafting density Σ = 0 is shown by the solid line, for Σ = 0.1 dotted line, Σ = 0.2 dashedline,Σ=0.3longdashedline,andΣ=0.4thedashed (c) dotted line. (b) Order parameter profiles for the low density fluid. Sym- bols as in (a). 100 80 200 FIG. 3: (Color online) Simulation configurations showing 2|Q (k)|/Vαz 4600 2|Q (k)|/Vαz100 molecules within 2.5 σ0 of the surface for (a) Σ = 0, (b) Σ = 0.2, and (c) Σ = 0.4. For each Σ the left most picture 20 shows molecules with 0≤ z ≤ 0.5, the centre pictures shows 0.5≤z≤1.5, and the right most shows 1.5≤z≤2.5. 00 0.1 0.2 0.3 0.4 0.5 00 0.1 0.2 0.3 0.4 0.5 k k FIG.5: Ordertensorfluctuations(normalisedbycellvolume) B. Low Density Fluid as a function of wavevector for (a) high density and (b) low densityfluids. Inboth graphsthesimulation datais denoted by symbols (circles Σ = 0, squares Σ = 0.1, diamonds Σ = Here the density and order parameter profiles for the 0.2, triangles Σ = 0.3, and crosses Σ = 0.4) and the elastic lowdensitysystemarediscussed. Forthesmoothwallthe theory data is shown by lines (continuous line Σ=0, dotted densityinthebulkofthecellis0.27(Tab. 1),justbelow line Σ = 0.1, dashed line Σ= 0.2, long dashed line Σ = 0.3, the isotropic-nematic transition density for this system and dot dashed line Σ = 0.4). The order tensor fluctuations (ρI−N =0.287). As the densityin the cellbulk increases for Σ=0.0 are shown only in (a). with Σ, for Σ ≥ 0.2 the fluid in the cell bulk is nematic rather than isotropic. Thedensityprofilesforthelowdensityfluidareshown and the local boundary conditions27. in Fig. 4(a). The changes in the density profile with in- creasingΣaresimilartothoseinthehighdensitysystem - the density at contactdecreaseswith Σ and the second IV. SURFACE ANCHORING peak becomes more diffuse. Again this can be gleaned from the decrease in the value of the smectic order pa- Shown in Fig. 5 are the order tensor fluctuations as a rameter with Σ (Tab. 1). It is interesting to note that function of wavevector. As can be there is good agree- the values of ρ1 obtained in this system are very simi- ment between the simulation and elastic theory curves, lar to those for the higher density system, indicating the especially for small k . z similarity in the structure of both systems. The fitted values for the anchoring coefficients are Shown in Fig. 4(b) are the order parameter profiles. given in Tab. 2 along with values of the extrapolation As in the high density fluid the value of the order pa- length λ and the surface anchoring coefficient W. For rameter at the wall decreases as Σ increases. The order both bulk densities ζ tends to decrease with increasing parameter profile also shows a deeper minima with in- Σ. creasing surface roughness. It is noticeable that even in this lower density case there is not an appreciable layer Table 2. of isotropic fluid between the wall and bulk fluid. This Fitting data for the order tensor fluctuations (Fig. 5). ζ has been predicted to happen near roughwalls as a con- is the anchoring coefficient, and L is the elastic theory e sequence of the competition between the bulk director cell width, which appear in Eq. 9. λ=L /ζ is the e 6 extrapolation length and W =K /λ is the surface correspondstoanincreasingsurfaceroughness. Thesim- 33 anchoring strength. ulations were performed at two bulk densities. For the higher density the fluid in the bulk of the cell is nematic ρ Σ ζ Le λ W for the smoothwallcase,while for the lowerdensity it is 0.314 0.0 5.62 16.00 2.85 0.52 isotropic. At both densities studied the effect of increasing sur- 0.314 0.1 6.51 28.39 4.36 0.34 face roughness is similar. Both the density and order 0.314 0.2 5.64 29.36 5.21 0.28 parameter in the region near the wall decrease as the 0.314 0.3 4.94 26.80 5.43 0.27 number of embedded molecules increases. The decrease 0.314 0.4 4.55 27.78 6.11 0.24 in the density arises from the excluded volume of the 0.30 0.1 3.05 18.32 6.01 0.24 embedded molecules,while the decreasein the order can 0.30 0.2 2.88 24.60 8.54 0.17 beattributedtothedisorientatingeffectoftherandomly orientated molecules in the wall. The rough walls also 0.30 0.3 2.58 23.85 9.24 0.16 act to smear out the secondary peaks in the density and 0.30 0.4 2.70 25.44 9.42 0.16 orderparameter profilesas the embedded molecules give anchoring points at positions other than at the wall sur- Thebehaviouroftheelastictheorycellwidth,L ,with e face. One side effect of the wall roughness is an increase Σ deserves comment. For the high density fluid, L for e in the density and order parameter in the centre of the the smooth wall is 16.00 σ , a few molecular lengths 0 cell. smaller than the physical cell width (L = 24.66). This z Alsostudiedwasthe effectofsurfaceroughnessonthe is similar to behaviour seen in previous simulations19,20 surface anchoring strength. For both systems the an- and is due to the formation of highly ordered layers in choring was found to become progressively weaker with the vicinity of the surface. For the rough walls however, increasing surface roughness. L is larger than L . This increase may be attributable e z A number of possible avenues for future work are pos- to the rough surface breaking up the highly orderedsur- sible. Calculationoftheanchoringcoefficientviaalterna- face layer. Thus instead of the elastic theory boundary tive methods16,28 wouldbe useful. As the formationof a conditions being applied at this layer, they are applied highlyorderedlayeratthesurfaceiscommonlyheldtobe closer to the wall, leading to an increase in L . e importantforthegrowthoforderinconfinedliquidcrys- For both bulk densities the extrapolation length in- tals,itmaybeinterestingtoinvestigatetheeffectofwall creases and the anchoring coefficient W decreases with roughnessonthephasebehaviouroftheconfinedfluid17. Σ. Thus, as may be intuitively expected, anchoring on Integral equation29 or density function theories30 have rough surfaces is weaker than on smooth surfaces. been applied to similar systems of simple fluids and ap- propriate generalisations to molecular fluids should also prove useful. V. SUMMARY In this paper results of Monte Carlo simulations for Acknowledgments a confined fluid of ellipsoidal molecules have been performed. The effect of surface roughness on the structure of the fluid has been examined. 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