Structure and interfacial tension of a hard-rod fluid in planar ......Structure and interfacial tension of a hard-rod uid in planar con nement P. E. Brumby,1, a) H. H. Wensink,2, b)

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Structure and interfacial tension of a hard-rod fluid in planar confinement.Paul E Brumby, H H Wensink, Andrew John Haslam, and George Jackson

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Structure and interfacial tension of a hard-rod fluid in planar confinementP. E. Brumby,1, a) H. H. Wensink,2, b) A. J. Haslam,3, c) and G. Jackson3, d)1)Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522,Japan2)Laboratoire de Physique des Solides, Université Paris Sud & CNRS, 91405 Orsay Cedex,France3)Department of Chemical Engineering, and Qatar Carbonates and Carbon Storage Research Centre,Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom

(Dated: 29 August 2017)

The structural properties and interfacial tension of a fluid of hard-spherocylinder rod-like particles in contactwith hard structureless flat walls are studied by means of Monte Carlo simulation. The calculated surfacetension between the rod fluid and the substrate is characterized by a non-monotonic trend as a function of bulkconcentration (density) over the range of isotropic bulk concentrations. As suggested by earlier theoreticalstudies, a surface-ordering scenario can be confirmed from our simulations: the local orientational order closeto the wall changes from uniaxial to biaxial nematic when the bulk concentration reaches about 85% of thevalue at the onset of the isotropic-nematic phase transition. The surface ordering coincides with a wettingtransition whereby the hard wall is wetted by a nematic film. Accurate values of the fluid-solid surface tension,the adsorption, and the average particle-wall contact distance are reported (over a broad range of densitiesinto the dense nematic region for the first time), which may serve as a useful benchmark for future theoreticaland experimental studies on confined rod fluids. The simulation data are supplemented with predictions froma second-virial density functional theory, which are in good qualitative agreement with the simulation results.

PACS numbers: 61.30.Cz,64.70.M-,82.70.Dd

The interaction between a solid surface and a dense ne-matic liquid-crystalline fluid can be characterized by theinterfacial tension, a challenging property to quantify forfluids of anisotropic molecules. A detailed understand-ing of the interfacial properties would provide substan-tial insight into how these complex fluids are influencedby confinement1 – insight that would be invaluable, forexample, in the context of liquid-crystalline displays, atechnology that has facilitated a revolution in portabledisplay devices. As a result the surface thermodynam-ics of anisotropic molecules remains an important area ofongoing research.

Confinement – or the presence of coexisting phases –gives rise to inhomogeneities, increasing the richness ofthe structure and phase behaviour with an additionalsurface tension contribution to the thermodynamics ofthe system associated with the corresponding interfaces.This interfacial tension, or surface free energy, is respon-sible for many important and fascinating processes. How-ever, considering the challenges involved in its quantifi-cation, it is perhaps not surprising that there is still alack of understanding of the interfacial tension of liquid-crystalline systems in contact with solid surfaces, a de-ficiency that is particularly striking for higher-densitystates.

The purpose of our current work is to improve theoverall thermodynamic understanding of such systems

a)Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected]

and quantify the interfacial tension and surface struc-tural properties of a simple model of a confined liquid-crystalline system, across a broad range of concentra-tions. Hard spherocylinders represent a suitable sim-ple model of purely repulsive rod-like particles to studythe rich morphology of liquid-crystalline matter. Thehard-spherocylinder system is known to exhibit the pro-totypical nematic (orientationally ordered) and smectic(orientationally ordered with one degree of positionalorder in layers) liquid-crystalline states, as well as themore usual isotropic (orientationally disordered) fluidand solid (orientationally and positionally fully ordered)phases2–6. Despite the apparent simplicity of the model,our work will nevertheless provide a benchmark for morecomplicated effective molecular topologies7, interactionpotentials8, and/or confined structures9.

Prior simulation studies on confined liquid-crystallinesystems have provided a detailed appreciation of the in-fluence of surface-liquid interactions on the behaviour ofthese complex fluids. In addition to extensive work oncolloidal systems, Dijkstra and co-workers10–12 have sim-ulated hard-spherocylinder fluids confined between twohard planar walls. In agreement with subsequent experi-mental observations for rod-like particles at interfaces13,Dijkstra and co-workers found that the bulk isotropicphase will partially wet the hard planar wall with in-creased nematic ordering, biaxiality, and planar anchor-ing. The density profiles of these lower-density statesreveals a distinctive peak at a distance of half the lengthof the particle from the wall surface. In addition it isfound that the separation distance between the confiningwalls must exceed two particle lengths before a first-orderisotropic-nematic phase transition can be observed in the

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“bulk” region of the system14. In close proximity to theirsurfaces, hard confining walls may induce nematic order-ing, even if the molecules are not sufficiently elongatedto naturally form nematic phases in the completely ho-mogenous system.15

As well as simulations of fluids of hard spherocylin-ders confined between hard walls, hard-ellipsoid16,17 andhard Gaussian overlap18–20 particles have been studied incontact with soft walls. Depending on the nature and de-gree of the wall-particle interaction, one finds that eitherplanar or homeotropic anchoring of molecules relative tothe wall surfaces can be observed. Further modificationof the wall-particle interaction can even encourage sur-face anchoring with tilted orientations21. Recent experi-mental work confirms that similar variations in particle-interface contact angles are possible in real systems22.

Introducing surface roughness, as in the simulationstudies of Cheung and Schmid23 with soft ellipsoids, hasinteresting consequences. The degree of roughness actsto both inhibit the formation of nematic phases at thewall and to shift the bulk isotropic-nematic phase tran-sition towards higher pressures (densities). The pres-ence of particle-particle and/or particle-surface attrac-tive interactions, can also have a substantial effect onthe phase transitions and wetting behaviour of the con-fined fluid24–26. Other comprehensive studies of surfaceadsorption and density profiles of confined fluid from thewall surface have involved polymers27,28, flexible rods29,Gay-Berne attractive particles8,30–32, platelets33,34, hardrectangles35, and Lennard-Jones molecules in contactwith grooved surfaces36,37 to name but a few representa-tive examples.

Our detailed understanding of the effect of confine-ment on the structure, phase behaviour, and wettingtransitions of fluids is, however, in stark contrast to thecurrent knowledge of the solid-fluid interfacial tension ofanisotropic particles. In addition to some unresolved fun-damental issues and challenges in the analysis of the sur-face thermodynamics of non-spherical particles, the factthat the interfacial tension is a thermodynamic deriva-tive property leads to practical difficulties; obtaining thesolid-fluid tension to an acceptable level of accuracy re-quires sampling over a very large number of configura-tions. Unless efficient techniques are employed, the com-putational expense can, depending on the type and size ofsystem, be insurmountable. Notwithstanding, a varietyof different methods are now available for the computa-tion of the interfacial tension in molecular simulations.

The most common methodology for the computationof the interfacial tension is based on the mechanical def-inition of the restoring force due to a deformation in thearea of the system38,39: the interfacial tension is calcu-lated from the direct evaluation of the components ofthe pressure tensor (the negative of the stress tensor) interms of the virial – specifically, the forces acting in direc-tions normal and tangential to the interface. This is themethod of choice in molecular-dynamics simulations, andthere are many such examples to be found in the litera-

ture40–44. Although novel approaches based on the me-chanical route are still being developed45, the technique isunsuitable when the particle-wall and/or particle-particleinteraction potentials are discontinuities or when the in-terface is not planar. As a consequence of these short-comings a number of alternatives to the mechanical/virialroute have been developed.

The thermodynamic approach of Bennett46 is amongstthe earliest alternative methodologies for the determina-tion of the interfacial tension (surface free energy) wherea multi-stage method is employed to sample the free-energy difference of selected systems. While the tech-nique is well suited for high-density states, the imple-mentation is not straightforward, requiring multiple sim-ulations to obtain the value of the interfacial tension ateach state. Salomons and Mareschal47 have developedthe method of Bennett to arrive at an expression that canbe readily employed in molecular-dynamics simulation.As with the test-area approach39, which will be discussedin more detail later in this section, the implementation ofthe Bennett method is typically based on the assumptionthat a decrease in the interfacial area leads an equiv-alent free-energy change to that brought about by theequivalent increase in surface area. While this holds truefor molecules interacting through continuous sphericallysymmetrical potentials, it is, unfortunately, generally in-valid for systems comprising non-spherical molecules.

The finite-size scaling method of Binder48 and its vari-ants offer an attractive route to the determination of theinterfacial tension. Examples of its application includefluid interfaces comprising Lennard-Jones49 and square-well50 molecules. The method relies on the computa-tion of the probability of states in terms of the numberof molecules in the system from simulations performedin the grand-canonical ensemble. Accurate sampling ofhigh-density states is therefore difficult to achieve, par-ticularly in the case of non-spherical hard-body systemsfor which the probabilities of successful particle insertionsare acutely small51,52. The use of the Binder finite-sizescaling method with grand-canonical transition-matrixMonte Carlo and histogram-reweighting can alleviate theproblem to a degree53. The treatment of dense fluid sys-tems characterized by hard-body interactions neverthe-less remains challenging with approaches involving par-ticle insertions. Recent work using finite-size scaling, aspart of the ensemble switching method, have further ex-tended the applicability of the Binder approach54–56. Indoing so the interfacial tension can be computed reliablyfor vapour-liquid systems and, with a certain degree ofsuccess, for solid-liquid interfaces.

The interfacial tension of a fluid phase of anisotropicparticles in contact with a surface can be obtained byintegrating the Gibbs adsorption equation as describedby Mao et al.57. This type of thermodynamic integra-tion approach is, however, again implemented withinthe grand-canonical ensemble and thus suffers from thesame drawbacks inherent with the method of Binder ofpoor insertion statistics when applied to higher-density

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states. The related Gibbs-Cahn thermodynamic integra-tion technique58,59 has also been used to determine thewall-fluid interfacial tension of confined systems compris-ing hard-sphere59–61 and Lennard-Jones62 particles, al-though the method requires careful parameterization.

Studies with other approaches which are specific toconfined solid-fluid systems have been reported includ-ing: the phantom-wall method of Leroy and Müller-Plathe63; the thermodynamic-integration approaches ofHamada et al.64 and of Das and Binder65; the interfacepotential analysis method developed by Errington andco-workers66–68; and the ensemble mixing method of Debet al.69, wherein one gradually inserts a wall potential ina system without walls. All of these methods provide aroute to the interfacial tension – and in some cases thecontact angle – of confined fluids.

In a similar manner to the multi-stage simulationscheme of Bennett, obtaining the interfacial tension viathe expanded ensemble method70–73 requires the simula-tion of multiple connected sub-ensembles. Ideally, theseconnected systems should have identical properties (i.e.,number of particles N , volume V , and temperature T )excepting an incremental difference in interfacial area be-tween one system and the next. The expanded ensem-ble thermodynamic method proceeds according to thestandard canonical (NV T ) ensemble Monte Carlo algo-rithm74, but with extra trial moves where a jump fromone system to another is attempted. The difference infree energy between the systems with different interfa-cial area is then used to compute the interfacial tension.The related wandering-interface method of MacDowelland Bryk75 involves a slightly different procedure. In thiscase, the domain lengths of the system are allowed to fluc-tuate freely, while maintaining a constant system volume.An analysis of histograms of the probability distributionof the domain lengths allows one to extract the surfacetension as the logarithm of the distribution, as it tendstowards zero. This thermodynamic approach has the im-portant advantage of not being constrained to the grand-canonical ensemble. In addition, it is equally applicableto both vapour-liquid and solid-liquid systems compris-ing non-spherical particles. The perturbative wandering-interface method has been used by Blas et al.76 to deter-mine the interfacial tension for systems of freely rotatingchain molecules, including subsequent work on the effectof long-range interactions on the interfacial tension77–79.

In our current work we employ a free-energy pertur-bation methodology to determine the solid-fluid inter-facial tension of hard spherocylinders in contact with ahard wall. The approach is related to the aforementionedperturbative test-area method39 which is not restrictedto the grand-canonical ensemble precluding problems as-sociated with high-density states. As its name impliesthe test-area method is a thermodynamic approach in-volving the computation of the change in free energy ac-companying a vanishingly small perturbation in the sur-face area of the system. The versatility of the test-areamethod for the determination of the interfacial tension is

apparent from its broad implementation to systems in-volving diatomic molecules80, chain molecules76,81, liquiddrops82,83, mixtures84,85, and fluids confined within slit-pore86,87 and cylindrical88 geometries. It is now a pop-ular alternative for the simulation of the vapour-liquidinterfacial tension of fluid systems comprising moleculescharacterized by continuous interactions. There are anumber of inadequacies associated with the direct im-plementation of the test-area method for systems involv-ing discontinuous potentials or hard non-spherical parti-cles; in this case it is more appropriate to determine theinterfacial tension from the components of the pressuretensor obtained from a thermodynamic route involvingtest-volume perturbations.

The test-volume perturbation technique was firstadopted by Eppenga and Frenkel89 for the determina-tion of the bulk (macroscopic) pressure of the isotropicand nematic phases of systems of purely repulsive harddiscs; owing to the discontinuous nature of the poten-tial the pressure can be obtained directly by examiningthe probability of configurations with overlapping par-ticles resulting from vanishingly small (isotropic) vol-ume perturbations (compressions). Extensions of themethodology to systems with attractive interactions suchas Lennard-Jones90, square-well91, and anisotropic Gay-Berne92 fluids have been reported. The test-volume ap-proach has also been used to determine the bulk pressureof other systems comprising anisotropic particles such ashard-Gaussian overlap molecules93 and hard spherocylin-ders94, and has become the method of choice for purelyrepulsive particles of arbitrary shape95,96.

As mentioned earlier, the interfacial tension can be de-termined from knowledge of the normal and tangentialcomponents of the pressure tensor. Test-volume pertur-bation methods for the calculation of the macroscopiccomponents of the pressure tensor are well suited forthe simulation of the interfacial tension of systems withplanar interfaces. This type of perturbative approach isnon-invasive and is applicable over a wide range of den-sities, as it is not anchored to a specific simulation en-semble. Test-volume perturbations have been applied forthe determination of the pressure tensor of hard spher-ical93 and non-spherical94 particles; it is important toemphasize that there are important subtleties to con-sider in the implementation of the approach to non-spherical particles owing to the lack of equivalence be-tween the compressive and expansive contributions tothe pressure tensor associated with anisotropic volumechanges94. Jiménez-Serratos et al.97 further extendedits utility and successfully applied the method to gen-eral non-spherical systems with discontinuous potentials,such as chain molecules and spherocylinders with square-well and square-shoulder interactions. For an excellentdiscussion and comparison of the various techniques forthe simulation of the pressure tensor of confined systemscharacterized by discontinuous interactions the reader isdirected to the review by Deb et al.69.

It may now have become apparent that volume-

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perturbation methods offer great promise for the sim-ulation of the solid-fluid interfacial tension, especially inthe case of high-density states of molecules interactingthrough discontinuous potentials – states which are inac-cessible with the majority of other approaches. The keygoal of our paper is to determine, for the first time, the in-terfacial tension of confined systems of hard spherocylin-ders confined between structureless hard walls, for statesranging from low-density isotropic phases, through the(bulk) isotropic-nematic phase transition, and deep intothe high-density nematic region of the phase diagram.In addition, a detailed analysis of the effect of confine-ment on the thermodynamic, structural and orientationalproperties of hard-spherocylinder fluid is made.

The remainder of this paper is organized as follows:the simulation methodology is detailed in Sec. I; we ex-amine the phase diagram of the unconfined bulk systemin Sec. II; profiles of the density and nematic-order pa-rameter for the confined systems are presented in Sec. III;the test-volume perturbation method is employed to de-termine normal and tangential components of the pres-sure tensor and thus the solid-fluid interfacial tension inSec. IV; an Onsager density functional theory for hardspherocylinders confined in slit-pore geometry and itspredictions are compared with the simulation results inSec. V; finally, some concluding remarks are made inSec. VI. Details of the perturbation method used in ourcurrent work are assigned to the appendices, wherein oursimulated interfacial-tension data are also tabulated.

I. SIMULATION METHODOLOGY

In all of the simulations described in our work, a collec-tion of N hard spherocylinders (each formed from a hardcylinder of length L capped by two hard hemispheres ofdiameter D where the aspect ratio is defined as L/D) areplaced in a rectangular box of volume V and dimensions`x, `y and `z. Standard periodic boundary conditions areused in all three directions in the case of the bulk systems.Planar confinement is considered by placing structurelesshard walls at positions z = 0 and z = `z of the simula-tion box while maintaining the periodicity of the othersystem boundaries; the closest distance of approach ofa particle from the hard wall is therefore D/2. We ex-plore the phase behaviour of these systems at various con-centrations (densities) of the rod particles. Sampling isperformed using Wood’s adaption98,99 of the MetropolisMonte Carlo method74 for the isobaric-isothermal (con-stantNpT ) ensemble in the case of bulk (unconfined) sys-tems, and using the standard Metropolis method74 in thecanonical (constant NV T ) ensemble for the confined sys-tems (see Refs. 5 and 94 for further details). Trial statesare created by performing a series of cycles: N attemptsto randomly translate or rotate one of the hard sphe-rocylinders (which are also selected at random for eachattempt) are made within each cycle, maintaining thecondition of detailed balance; where appropriate a single

system-volume expansion or compression is attempted atthe end of each cycle in the isobaric-isothermal ensem-ble. This process is carried out until equilibrium statesare attained, and the density and nematic-order param-eter are then calculated as configurational averages. Theamplitudes of the trial moves are chosen on an empiricalbasis so as to achieve an overall acceptance ratio of ∼30% for the translational and rotational displacements,and system volume changes.

The system density ρ = N/V can be convenientlyquantified in terms of the volume (packing) fraction as

η = vmN

V=(π

6D3 +

π

4D2L

) NV, (1)

where for our model vm = πD3/6 + πD2L/4 is the vol-

ume of the hard-spherocylinder particle. A dimensionlessconcentration is also often employed in theoretical stud-ies to characterize the density of the system, defined interms of the so-called Onsager limit for the second-virialcoefficient of infinitely long rod-like particles:

c =(π

4DL2

) NV. (2)

Both measures of density are employed interchangeablyin the ensuing discussion, the dimensionless concentra-tion being the more appropriate to facilitate compari-son with previous theoretical studies. With our choice ofGibbs dividing surface at z = 0 and z = `z, two inac-cessible layers of thickness D/2 in the proximity of thewalls has to be accounted for; this has implications inthe determination of the overall system volume, density,wall-fluid surface tension and surface adsorption (cf. Sec-tion IV). As a consequence of the external potential dueto the presence of the confining walls, the density is nothomogenous throughout the system but will depend onthe distance z from the wall. Local density profiles canbe obtained by dividing the space between the walls intoa number of equal-sized bins. The concentration c(z) ofhard spherocylinders within each bin, at a distance z, isdetermined from Eq. 2, where V and N now representthe volume of the bin and the number of particles withtheir centre of mass within the bin, respectively. In thecase of a sufficiently large separation distance betweenthe two walls, the density in the central (bulk) part ofthe system ceases to be a function of z; the correspond-ing values of the packing fraction and concentration ofthe bulk phase are obtained as averages of the densityprofiles in the central region of the simulation box andare denoted by ηb and cb, respectively.

The pressure p of the system is expressed in dimension-less form throughout our work. A convenient measure forthe athermal hard-spherocylinder system is

p∗ =pvmkBT

, (3)

where kB is the Boltzmann constant. Equivalent dimen-sionless expressions for averages of the normal pN and

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tangential pT components of the pressure tensor pαα areemployed in our subsequent analysis of the interfacialtension. In the case of the inhomogenous system confinedbetween two parallel planar walls (in the xy plane) at me-chanical equilibrium, the normal component pN = pzz ofthe pressure tensor is constant and corresponds to thebulk scalar pressure p; for this planar interfacial geome-try the tangential component pT (z) = (pxx(z)+pyy(z))/2of the pressure tensor is cylindrically symmetrical and isa function of the position z from the surface of the wall(only the macroscopic average of pT over the box dimen-sion `z needs to be evaluated in order to determine theinterfacial tension of the system).38

The average local orientational order of the sphero-cylinders can be quantified from the second-rank trace-less, symmetric tensor5,89:

Q =1

N

(N∑i

3

2〈ûi ⊗ ûi〉 −

1

2I

), (4)

where û represents the orientational unit vector alongthe main axis of the hard spherocylinder, ⊗ denotes thedyadic product, I the unit second-rank tensor, and 〈·〉the canonical ensemble average.

The nematic-order parameter S, defined as the largesteigenvalue S = λ+ of the tensor Q, is a key parameter todistinguish between nematic (S > 0) and isotropic order(S = 0). The corresponding eigenvector n̂ corresponds tothe nematic director, which for the confined systems in-dicates the principal direction of alignment with respectto the wall normal. In the case of uniaxial order, the tworemaining eigenvalues (λ0 and λ−) are equal and there isno preferential orientational order in the plane perpen-dicular to the nematic director. The presence of a hardwall, however, will break the uniaxial nematic symme-try of the fluid and induce local biaxial nematic order: anon-zero difference ∆ = λ0 − λ− between the two small-est eigenvalues of Q can be used as a measure to quantifythe degree of biaxial order across the range of distancesz. As for the concentration profile c(z), the spatial reso-lution of the orientational-order parameters between thewalls can be obtained in a straightforward manner by dis-cretizing `z into a finite number of slabs of equal volume(see Ref. 100).

II. BULK PROPERTIES OF THEHARD-SPHEROCYLINDER FLUID

Prior to studying the system under confinement, wefirst characterize the unconfined fluid of hard sphero-cylinders, paying particular attention to the degree oforientational order over a range of concentrations. Thisallows for a better appreciation of the (sometimes sub-tle) effects of confinement on the liquid-crystalline statsformed by the system, as described later in our paper.For the bulk systems, we study N = 3080 hard sphero-cylinders, each with an aspect ratio of L/D = 10. Sim-ulations are performed in the isobaric-isothermal (NpT )

ensemble, starting from an initially perfect face-centred-cubic crystal-lattice arrangement. The pressure of thesystem is reduced until an equilibrium low-density dis-ordered isotropic phase is achieved. The system is thencompressed from this state by increasing the pressure inincremental steps. A minimum of 2 × 106 Monte Carlocycles are required to equilibrate these systems at eachpressure. The average values of the equilibrium densityand order parameters are then determined over a further2 × 106 cycles. The values obtained for the compres-sion runs are shown (as the red circles) in Fig. 1. Ina similar manner, expansion runs (involving the incre-mental reduction in the pressure) are performed, start-ing with a stable nematic state at a density well abovethe isotropic-nematic phase transition pressure. Fromthe equation of state depicted in Fig. 1 one can estimatethe isotropic-nematic coexistence pressure as p∗ ∼ 1.88corresponding to the sharp discontinuity in the nematic-order parameter. For the isotropic branch this pressurecorresponds to a concentration of cI ∼ 2.29 (ηI ∼ 0.24)and nematic-order parameter of SI ∼ 0.04, while for thenematic branch the corresponding values are cN ∼ 2.50(ηN ∼ 0.27) and SN ∼ 0.71.

III. STRUCTURE AND ORIENTATION OF THEHARD-SPHEROCYLINDER FLUID CONFINED BYHARD WALLS

Having simulated systems of hard spherocylinders withperiodic boundaries along all three dimensions, we nowfocus on the systems confined between two hard walls,highlighting the influence of confinement on the isotropic-nematic phase transitions (relative to that of the bulkunconfined system). We examine the local properties(density and orientational order) in the regions wherethe solid walls interact with the hard spherocylinders.As mentioned earlier, the confinement implemented inour current work involve a pair of flat, featureless, andimpenetrable walls positioned parallel to each other adistance `z apart at the boundaries of the z Cartesianaxis; this arrangement is often referred to as a slit-poregeometry.

In our case, we require `z to be large enough for aclearly distinct bulk phase to form in the centre of thesystem. In order to maintain a fixed inter-wall separa-tion distance, these systems are simulated in the canoni-cal (NV T ) ensemble. The initial configuration is createdfrom a high-density state of N = 3000 non-overlappinghard spherocylinders arranged in a perfect face-centred-cubic crystal lattice. The specific box length dimensionsof the system are `x = 39.8926 D, `y = 39.2271 D, and`z = 43 D. Particles are removed from the high-densitystate to create a series of new lower-density states rangingfrom N = 20 to 2900 hard spherocylinders in the simula-tion box. Monte Carlo simulations are then performed toestablish equilibrium states and obtain the appropriateaverage properties.

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(a)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

p*

c

η

1.7

1.8

1.9

2.0

2.2 2.3 2.4 2.5 2.6

(b)

0.00

0.25

0.50

0.75

1.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

S

c

η

FIG. 1. Isotropic-nematic equation of state for bulk systemsof hard spherocylinders characterized by an aspect ratio ofL/D = 10 obtained from isothermal-isobaric Monte Carlo(MC-NpT ) simulations of N = 3080 particles. (a) The depen-dence of the dimensionless pressure p∗ on the concentrationc (lower horizontal scale) and volume fraction η (upper hor-izontal scale). The red circles denote the equilibrium statesobtained from compression runs, which originate from an ini-tially isotropic system, and the blue diamonds denote theequilibrium states obtained from expansion runs, which origi-nate from a state deep in the nematic region. The red dashedline indicates an estimate of the density of the isotropic phaseat the isotropic-nematic transition, and the blue dashed linethe corresponding density of the coexisting nematic phase.(b) The dependence of the nematic-order parameter S on theconcentration and packing fraction of the system.

For the calculation of density profiles in terms of thedistance from the surface of one of the walls, the sys-tem volume is divided into nbin = 200 bins, positionedadjacent to one another along the z axis. A minimumof 1 × 106 Monte Carlo cycles are performed for eachstate, and averages of the density for each bin are deter-mined. Due to the sensitivity of the calculation of thelocal nematic-order parameter to the number of particlessampled in each bin, it is necessary to increase the systemsize. In order to simulate a larger number of particles,the systems are replicated in the x and y axes, such thatthere are at least N = 25, 000 hard spherocylinders in

the simulation cell. The extra periodicity created by thisreplication process is removed by performing an addi-tional series of equilibration cycles until new equilibriumstates are generated. The considerably larger systems arethen simulated for a further 1× 105 cycles to determineequilibrium profiles of the nematic-order and biaxialityparameters. To further increase the average number ofparticles per bin, the overall system volume is dividedinto a smaller number nbin = 20 of bins along the z axis(reducing uncertainties relating to finite-size effects of thesample, at the cost of a lower resolution).

Isotropic Nematic(a)

0

1

2

3

4

5

0.0 0.5 1.0 1.5 2.0

c(z)

z*

cb = 2.195cb = 2.105cb = 2.003cb = 1.897cb = 1.674cb = 1.215cb = 0.746

01234567

0.0 0.5 1.0 1.5 2.0

c(z)

z*

cb = 3.279cb = 3.047cb = 2.811cb = 2.571cb = 2.448

(b)

-0.5

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0

S(z)

z*

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0

S(z)

z*

(c)

0.0

0.1

0.2

0.3

0.4

0.0 0.5 1.0 1.5 2.0

∆(z)

z*

0.00

0.05

0.10

0.15

0.0 0.5 1.0 1.5 2.0

∆(z)

z*

FIG. 2. Profiles of the (a) local concentration c(z), (b) uniax-ial nematic order S(z), and (c) biaxial nematic order ∆(z)) asfunctions of the dimensionless distance z∗ = z/(L+D) fromthe wall for systems of hard spherocylinders with an aspectratio of L/D = 10 in planar slit-pore confinement obtainedfrom canonical Monte Carlo (MC-NV T ) simulation. The sep-aration distance between the two parallel hard walls is fixed at`z = 43 D. The plots on the left correspond to bulk isotropicphases, and those on the right to bulk nematic phases.

A detailed analysis is first made of the local densityc(z), uniaxial nematic order S(z), and biaxial order ∆(z)for equilibrium states with stable bulk isotropic phases.The profiles of these properties as functions of the dis-tance from one of the walls are presented in Fig. 2 (left),and representative snapshots of configurations of thesesystems are displayed in Fig. 3 (a) and (b). At low bulkconcentration, a particle dewetting of the wall surface isobserved as the rods are depleted from the impenetra-

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ble wall. The nematic orientational order near the wallis anti-nematic limz→0 S(z) = Sw ∼ −1/2 and uniaxiallimz→0 ∆(z) = ∆w ∼ 0 indicating planar anchoring withrods oriented perpendicular to the wall normal. Finite-size effects are clearly observable for the biaxiality at thewall, as a result of the relatively low average number ofhard spherocylinders per bin; this is particularly notice-able for the systems with lower concentrations.

As the bulk density is increased there is a markedchange in the surface structure. A sharp surface phasetransition is observed at cb ∼ 2 where the orientationalorder at the surface changes abruptly from uniaxial tobiaxial. This is more-clearly reflected in the behaviour ofthe profiles of the nematic order S(z) in close proximityto the wall surface of Fig. 4 (left). The steep increasein the biaxiality ∆w of the particles in contact with thewall at concentrations of cb ∼ 2 coincides with a markedincrease in the uniaxial nematic-order parameter Sw atthe wall. This is a direct consequence of a change of ne-matic director which defines the reference frame for theorientational-order parameters.

At low density, the nematic director is oriented ran-domly with respect to the wall normal while, at the sur-face phase transition, the director rotates perpendicularto the wall normal. It is evident that this sudden reori-entation signals the formation of a nematic wetting layerclose to the wall. The extent of this layer can be gleanedfrom the peaks in the density profiles c(z) in Fig. 2. Thetransition from uniaxial to biaxial nematic surface orderfor a rod fluid in contact with a hard wall has been pre-dicted from Onsager theory. A careful stability analysisreveals101 that one would expect a wetting transition atcb ∼ 2.79, well before the bulk isotropic-nematic transi-tion which occurs at cb ∼ 3.29. From this one can inferthat a surface ordering transition will occur universallywhen the bulk concentration reaches about 85–87 % ofthe coexistence value at the isotropic-nematic transition.The precise value will depend only weakly on the aspectratio provided the rods are sufficiently anisometric, i.e.,L/D � 1.

From purely thermodynamic arguments one would ex-pect that hard-core mesogens confined between parallelhard walls in slit-pore geometry would exhibit a tran-sition to a bulk nematic phase at densities below theisotropic-nematic transition of the unconfined system.Indeed the occurrence of a first-order transition froman isotropic phase (with a biaxial nematic film at eachwall) to a condensed nematic phase that fills the slit porehas already been reported for confined hard rods10,11,14.The slit-pore geometry essentially stabilizes the nematicphase relative to the isotropic shifting the isotropic-nematic transition to lower densities (and higher tem-peratures in the case of mesogens with attractive interac-tions such as the Gay-Berne fluid24). This stabilizationof the nematic phase due to confinement is now com-monly referred to as capillary nematization10,11,14. Wealso observe a capillary nematization transition in oursystems of confined hard spherocylinders at a density of

cb ∼ 2 which is below the onset of the isotropic-nematictransition of the unconfined bulk system, cb ∼ 2.29.

(a)

(b)

(c)

(d)

FIG. 3. Representative configurations of hard spherocylin-ders with an aspect ratio of L/D = 10 in planar slit-pore con-finement obtained from canonical Monte Carlo (MC-NV T )simulation. The separation distance between the two parallelhard walls is fixed at `z = 43 D. Different colours are usedto highlight the different relative orientation of each particle.The snapshots on the left are taken from a position normal toone of the wall surfaces (made invisible for clarity), and thesnapshots on the right are taken from a position between thetwo wall surfaces at `z/2. Snapshots with concentration of (a)cb = 0.7459 and (b) cb = 2.1950 correspond to bulk isotropicstates, while (c) cb = 2.4480 and (d) cb = 3.2792 correspondto bulk nematic states.

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(a)

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Sw

cb

ηb

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

∆w

cb

ηb

FIG. 4. The dependence of the (a) uniaxial nematic orderSw and (b) biaxial nematic order ∆w for particles at the wallsurface on the bulk concentration cb and packing fraction ηbfor hard spherocylinders with an aspect ratio of L/D = 10 inplanar slit-pore confinement obtained from canonical MonteCarlo (MC-NV T ) simulation. The separation distance be-tween the two parallel hard walls is fixed at `z = 43 D. Thefilled circles represent the data for systems with bulk isotropicstates, while the hollow circles the data for systems with bulknematic phases.

When the bulk phase of the confined systems is ne-matic, the trends in the local density and particle or-dering differ from those for systems with bulk isotropicphases, but smoothly follow the characteristics of the ne-matic wetting layer formed at concentrations below thebulk isotropic-nematic phase transition. The concentra-tion c(z), nematic-order S(z), and biaxial-order ∆(z)profiles for these systems are shown in Fig. 2 (right);representative configurations of two systems with differ-ent bulk densities corresponding to the nematic state areshown in Fig. 3 ((c) and (d). As the density is increased,the biaxial nature of the nematic phase gradually de-creases since the rods progressively align perpendicularto the wall normal, thereby reducing the orientationalsymmetry breaking imposed by the hard wall; the bi-axiality of the high-density states is dominated by thatof the bulk nematic region. The overall scenario for the

surface ordering of rod-like fluids borne out by our sim-ulations is in line with the findings of earlier simulationsand Zwanzig-type density functional theory for hard rodsof finite anisotropy10,14,102,103 as well as the predictionsfrom Onsager density functional theory for infinitely slen-der rods101,104.

IV. INTERFACIAL TENSION, SURFACE ADSORPTION,AND AVERAGE ROD-WALL CONTACT DISTANCE

In this section, we provide a detailed analysis of theinterfacial thermodynamic properties pertaining to theevolution of the structure of the rod fluid at the wallsurface upon increasing the bulk concentration. Of par-ticular interest is the effect of the onset of local biaxial ne-matic order and the formation of a nematic wetting layeron the fluid-wall surface tension, a quantity of great prac-tical interest in understanding the behaviour of liquidcrystals at surfaces. A test-volume free-energy pertur-bation method94 (outlined in Appendix A) is applied todetermine the fluid-wall surface tension of our system ofhard spherocylinders in planar slit-pore confinement fromcanonical Monte Carlo (MC-NV T ) simulation. This ap-proach requires the evaluation of averages of the normalpN = pzz and tangential pT = (pxx + pyy) /2 componentsof the pressure tensor of the system using the test-volumemethod with appropriate anisotropic volume perturba-tions. The fluid-wall surface tension can be expressed indimensionless form as

γ∗ =γvm

kBTηbL=

1

2

`zηbL

(p∗N − p∗T ) . (5)

where the average components of the pressure tensor areexpressed in dimensionless form, p∗αα = pααvm/(kBT ),and the factor of a half accounts for the presence of twosurfaces in our system. We should note that the bulkpacking fraction is included in the denominator in thisdefinition of the surface tension to aid direct compari-son with exiting theoretical estimates. The normal andtangential components of the pressure tensor can be ex-pressed as a sum of the ideal contribution (which in ourcase simply corresponds to the overall packing fractionη of the system) and the corresponding excess contri-butions determined from expansive p∗+αα and compressivep∗+αα perturbations:

p∗N = η + p∗+N + p

∗−N ; (6)

p∗T = η + p∗+T + p

∗−T . (7)

For the slit-pore geometry considered here,p∗+N = p

∗+zz , p

∗−N = p

∗−zz , p

∗+T =

(p∗+xx + p

∗+yy

)/2 and

p∗−T =(p∗−xx + p

∗−yy

)/2. The excess contributions are

determined by performing expansive and compressivetest-volume perturbations using the relations

p∗+αα = lim[∆V αi→j+→0]

V

∆V αi→j+

lnP+novN

, (8)

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and

p∗−αα = lim[∆V αi→j−→0]

V

∆V αi→j−

lnP−novN

, (9)

where ∆V αi→j+ and ∆Vαi→j− represent changes in sys-

tem volume brought about by the test (non-permanent)anisotropic affine deformations, achieved by increasingand reducing the length of the α axis, respectively. Themethodology essentially requires the calculation of theprobabilities P+nov and P

−nov that the aforementioned per-

turbations produce configurations without overlaps be-tween particles and/or between particles and either ofthe two walls.

Following the methodology described in previous stud-ies93,94, 20 evenly spaced values of the expansive ∆V αi→j+and compressive ∆V αi→j− perturbations are consideredfor each state, selected so that the resulting values ofP+nov and P

−nov are evenly spread between zero and unity.

The values of P+nov and P−nov in the limit of infinitesimal

volume changes are obtained using a linear least-squaresfit. The anisotropic test-volume perturbations are un-dertaken every 20 Monte Carlo cycles in order to achieveefficient convergence (see Ref.105). The averages are eval-uated over 4.4 × 107 cycles for the low-density systems(0 < cb ≤ 2.1046), while the averages for the higher-density (cb > 2.1046) systems – which require additionalsampling due to slow re-orientation of the nematic direc-tor – are taken over 3.6× 108 cycles.

The wall-fluid interfacial tension of the hard sphero-cylinders confined it slit-pore geometry calculated withthe test-volume approach across a range of bulk con-centrations, from the low-density isotropic to the high-density nematic state, is displayed in Fig. 5 (a). Thevariation of the surface tension with concentration in theisotropic bulk phase is non-monotonic. The maximumin the isotropic fluid-wall tension is reached at a bulkconcentration cb ∼ 2 which corresponds to the onset ofthe wetting transition and surface biaxial order. Theinterfacial tension can then be seen to decrease as thedensity is further increased into the nematic region andappears to level off to a plateau value at high concentra-tions close to the bulk nematic-smectic transition, whichis estimated to occur at cb ∼ 5 for the aspect ratio ofL/D = 10 considered here6. The concentration depen-dence of a dimensionless interfacial tension defined asγ′ = γD2/(kBT ) = (1/2)

(`zD

2/vm)

(p∗N − p∗T ) in the di-lute isotropic phase phase is highlighted in the inset ofFig. 5 (a) to enable direct comparison with the corre-sponding values determined by Mao et al.57 by integrat-ing the Gibbs adsorption equation using grand canoni-cal simulation data. Good agreement between the test-volume and Gibbs adsorption wall-fluid interfacial ten-sion is found for the isotropic state; it becomes increas-ingly difficult to employ grand canonical simulations asthe density is increased limiting the applicability of thelatter approach.

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

γ*

cb

ηb

0.0

0.1

0.0 1.0 2.0

γ’

(b)

-0.02

-0.01

0.00

0.01

0.02

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Γ*

cb

ηb

(c)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Z*

cb

ηb

FIG. 5. The dependence of the dimensionless (a) wall-fluidinterfacial tension γ∗, (b) surface adsorption Γ∗, and (c) aver-age contact distance Z∗ (between the rod centre and the wall)on the bulk concentration cb and bulk packing fraction ηb forhard spherocylinders with an aspect ratio of L/D = 10 inplanar slit-pore confinement obtained from canonical MonteCarlo (MC-NV T ) simulation. The separation distance be-tween the two parallel hard walls is fixed at `z = 43 D. Thefilled circles represent the data for systems with bulk isotropicstates, while the hollow circles represent the data for systemswith bulk nematic phases. A comparison with the work ofMao et al.57 (crosses) for the bulk isotropic phase, with theinterfacial tension expressed as γ′ = γD2/(kBT ), is also in-cluded as an inset in (a).

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In order to broaden our insight of the interaction of therod fluid with the hard walls two additional propertiesare assessed. The surface adsorption Γ which can bedetermined (in dimensionless units) from106

Γ∗ = ΓD2 =D2

2

(N −Nb)`x`y

, (10)

where the number of particles Nb associated with thebulk phase is computed from the bulk packing fraction asNb = ηbV/vm (see Fig. 5 (b)); it is important to note thatin the case of our confined system V denotes the overallvolume of the system including the two regions inacces-sible to the centre-of-mass of the hard spherocylinders inthe layers of thickness D/2 close to the two walls. Theaverage of the shortest distance Z between the wall andthe centres-of-mass of the hard spherocylinders in con-tact with the wall is also determined (see Fig. 5 (c)).This quantity can be established by recording the subsetof overlapping hard spherocylinders {1 . . . Nov} generatedby each trial displacement (be it a change of rod position,orientation, or system volume), and performing a canoni-cal average 〈·〉 of the centre-of-mass distance with respectto the wall.

The contact distance Z can be expressed (in units ofL) as

Z∗ =Z

L=

D

2L+

〈1

Nov

Nov∑i

1

2|ûi · ẑ|

〉, (11)

with ẑ denoting the direction of the wall normal, andûi the orientation of the principal unit vector of the rodi overlapping with the wall. To ascertain whether oursimulation reproduces the correct zero-density limit werecall that for an ideal gas of hard spherocylinders thesurface tension and average rod-wall contact distance aregiven by104,107

limcb→0

γ∗ = limcb→0

Z∗ =D

2L+

1

4, (12)

which corresponds to γ∗ = Z∗ = 0.3 for hard sphero-cylinders with L/D = 10 in accordance with the limitingvalues apparent from Fig. 5 (a) and (c). In the limit ofa perfectly ordered nematic state in planar confinementûi · ẑ = 0 corresponding to Z∗ = 0.05 for our system ofhard spherocylinders.

V. DENSITY FUNCTIONAL THEORY FOR ACONFINED ROD FLUID

In this section, we compare the results obtained fromthe Monte Carlo simulations with predictions from a den-sity functional approach based on the seminal second-virial theory of Onsager108. Although the theory isknown to give quantitative results only for bulk systemsin the limit of infinitely elongated rods with L/D →∞,we expect the theory to provide a useful qualitative guide

for our confined systems of hard spherocylinders with afinite aspect ratio of L/D = 10.

Within the framework of density functionaltheory109,110 the grand potential Ω of a fluid of Nhard spherocylinders in the presence of an externalpotential Uext can be expressed formally in terms of the(unknown) single-particle density ρ(s):

Ω[ρ(s)] = kBT

∫ds ρ(s)[lnVρ(s)− 1] + Fex[ρ(s)]

+

∫ds ρ(s)[Uext(s)− µ], (13)

where s = {r, û} collectively denotes both the position rand orientation vector û characterizing the phase space ofeach rod. Furthermore, µ is the chemical potential, andV denotes the total thermal volume of the rods incorpo-rating the translational and orientational contributionsto the kinetic energy111; the value of V will prove to beimmaterial in the subsequent analysis.

The first term in Eq. (13) is exact and represents theideal translational and rotational entropy of an ensembleof rods, while the second term Fex is the excess Helmholtzfree energy describing the interactions between the rods.In the spirit of second-virial theory of Onsager one canwrite:

Fex[ρ(s)] = −kBT

2

∫ds ρ(s)

∫ds′ρ(s′)Φ(s, s′), (14)

where Φ = exp(−Ur/kBT ) − 1 is the Mayer function ofthe pair potential Ur(s, s

′) between the rod particles. Inthe case of hard spherocylinders, Φ adopts a simple stepfunction signalling overlap of the spherocylindrical hardcores:

Φ(s, s′) =

{−1 if the rods overlap

0 otherwise,(15)

The slit-pore confinement of parallel hard walls consid-ered in our simulation corresponds to an external poten-tial which encodes the impenetrability of the hard walls.For a hard spherocylinder of length L and diameter D,the particle-wall potential takes the following form:

Uext(r, û) =

{∞ r · ẑ < σ(û · ẑ)0 otherwise,

(16)

where ẑ denotes the normal direction to the wall, and thecontact distance σ between the centre-of-mass of the rodand the wall is

σ(û · ẑ) = D2

+L

2|û · ẑ|. (17)

Eq. (16) needs to be supplemented with its mirrored formUext(`z − r · ẑ, û) to reflect the second wall located at adistance `z from the first.

Since the systems considered in our current study cor-respond to either isotropic or nematic phases in the bulk,

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the imposition of a slit-pore geometry will break thetranslational invariance of the fluid along a direction nor-mal to the wall. The particle-wall distance is denoted byz = r · ẑ. The equilibrium profile for the single-particledensity ρeq(z, û) (with 0 < z < `z) then follows froma formal minimization of the grand potential Eq. (13)via δΩ[ρ(s)]/δρ(s)|eq = 0. The equilibrium density pro-file associated with the minimum of the grand potentialis unique for any given set {µ,L/D, `z} and can be ex-pressed in the following self-consistent form:

ρeq(z, û) =eβµ

Vexp[−βUext(z, û)] (18)

× exp[−∫∫

dz′dû′A(|∆z|, û, û′)ρ(z′, û′)],

with β = 1/(kBT ) and ∆z = z − z′. The key parameterhere is the overlap area A(|∆z|, û, û′) which correspondsto the two-dimensional excluded volume swept by twospherocylinders at fixed mutual orientation (û, û′) andrelative lateral distance ∆z. Noting that the Mayer func-tion Φ depends only on the interparticle centre-of-massdistance ∆r = r−r′ and orientations (û, û′), the overlaparea can be formally derived as

A(|∆z|, û, û′) = −∫

d∆rΦ(∆r, û, û′)δ(∆r · ẑ−∆z) ,

(19)where δ(∆r · ẑ−∆z) is the appropriate Dirac delta func-tion.

In the case of bulk homogeneous phases, there areno inhomogeneities along the z direction and the maininteraction kernel corresponds to the excluded volumevexcl(û, û

′) for a pair of particles with fixed mutual orien-tations. The excluded volume can therefore be obtainedfrom A via a simple integration over ∆z. For hard sphe-rocylinders this quantity is well known and can be ex-pressed as

vexcl(û, û′) = −

∫d∆zA(|∆z|, û, û′)

= 2L2D|û× û′|+ 2πLD2 + 43πD3 . (20)

Whilst the expression for the excluded volume is eas-ily obtained by geometric inspection, a similar analyticalderivation forA proves to be highly non-trivial and closedexpression are known only up to O(LD) 104. The full re-sult, which includes all contributions including those ofO(D2) arising from the overlap of the hemispherical endcaps, can be readily obtained numerically by means ofa simple two-dimensional integration scheme by employ-ing an efficient overlap routine for hard spherocylinders,similar to the one used in the simulations.

It is useful to locate the bulk phase transition from theisotropic to the nematic fluid before embarking on a cal-culation of the density profiles of the particles from thewalls. In the absence of an external field (Uext = 0) thesingle-particle density reduces to ρ(z, û) = ρf(û) reflect-ing a spatially homogeneous fluid with number density

ρ and orientational distribution function f(û); the latteris normalized according to

∫dû f(û) = 1, which may ei-

ther describe an isotropic phase with f = 1/(4π), or anematic phase with f(û · n̂) peaked around a commonnematic director n̂. The numerical procedure has beendescribed in detail elsewhere112. The coexistence den-sities obtained from Onsager’s second-virial theory forthe bulk isotropic-nematic transition of a fluid of hardspherocylinders with an aspect ratio of L/D = 10 areobtained as cI = 3.39 (ηI = 0.36) for the isotropic stateand cN = 3.82 (ηN = 0.41) for the nematic state witha nematic-order parameter of S = 0.70; the value of thechemical potential at coexistence is µ∗ = βµ = 10.89 andthe bulk pressure is p∗ = 5.853 (corresponding to the di-mensionless pressure of βp(πDL2/4) = 54.87 employedin the theoretical studies).

Once the equilibrium single-particle density is estab-lished using Eq. (19), the fluid-wall surface tension γ canbe obtained from the grand potential Ω = −pV + 2γA,where p is the bulk pressure, V is the total volume of theconfined system, and A is the surface area of the wall;the factor of 2 again accounts for the presence of twowalls. Note that, here, p is the scalar pressure deep inthe bulk where, locally, p = pN = pT . Inserting the equi-librium density profile from Eq. (19) into Eq. (13) onecan express the minimum of the grand potential as

βΩeqA

= −12

∫∫dz dû ρeq(z, û)

∫∫dz′dû′ρeq(z

′, û′)

×A(|∆z|, û, û′)−∫∫

dz dû ρeq(z, û) . (21)

In the particular case of the bulk homogeneous system,the minimum of the grand potential Ωb,eq can be ex-pressed in terms of the equilibrium single-particle ori-entational distribution function feq(û) as

βΩb,eqA

= −`z2ρ2b

∫dûfeq(û)

∫dû′feq(û

′)vexcl(û, û′)

−`zρb . (22)

The difference between the equilibrium grand potentialof the inhomogeneous system and the bulk homogeneoussystem provides a route to calculate the fluid-wall sur-face tension: γ = (Ωeq − Ωb,eq)/(2A), again expressed indimensionless form as γ∗ = γvm/(kBTηbL) as for Eq. (5).

The surface adsorption Γ is readily established froma simple spatial average of the difference between thedensity of the inhomogeneous system and the bulk ho-mogeneous system over the range of the z direction inthe slit-pore geometry:

Γ∗ = ΓD2 =D2

2

∫∫dz dû (ρ(z, û)− ρb) , (23)

which is equivalent to Eq. (10) employed in the analysisof the simulation data.

The average rod-wall contact distance Z can be deter-mined from the equilibrium density profile ρeq(z, û) with

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(a)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

γ*

cb

0.30

0.32

0.34

0.36

0.0 0.5 1.0 1.5 2.0 2.5

(b)

-0.05

0.00

0.05

0.10

0.15

0.20

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Γ*

cb

-0.04

-0.02

0.00

0.0 0.5 1.0 1.5 2.0 2.5

(c)

0.10

0.15

0.20

0.25

0.30

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Z*

cbFIG. 6. The dependence of the dimensionless (a) wall-fluidinterfacial tension γ∗, (b) surface adsorption Γ∗, and (c) av-erage contact distance Z∗ (between the rod centre and thewall) on the bulk concentration cb for hard spherocylinderswith an aspect ratio of L/D = 10 in planar slit-pore con-finement predicted with the second-virial density functionaltheory of Onsager. The separation distance between the twoparallel hard walls is fixed at `z = 43 D. The gaps corre-spond to the bulk isotropic-nematic transition at cb = 3.39and cb = 3.82. The local maximum in the tension and mini-mum in the adsorption at cb ∼ 1.8 can be seen more clearlyin the inset graphs.

the use of Eq. (17):

Z∗ =1

L

∫dûσ(û · ẑ)ρ(σ(û · ẑ), û)∫

dû ρ(σ(û · ẑ), û), (24)

expressed in units of L as for Eq. (11).The self-consistency equation Eq. (19) for equilib-

rium single-particle density is solved by discretising the

Isotropic Nematic(a)

0

1

2

3

4

5

0.0 0.5 1.0 1.5 2.0

c(z)

z*

01234567

0.0 0.5 1.0 1.5 2.0

c(z)

z*

(b)

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.0 0.5 1.0 1.5 2.0

S(z)

z*

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0

S(z)

z*

(c)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

∆(z)

z*

0.00

0.04

0.08

0.12

0.0 0.5 1.0 1.5 2.0

∆(z)

z*

FIG. 7. Profiles of the (a) local concentration c(z), (b) uniax-ial nematic order S(z), and (c) biaxial nematic order ∆(z)) asfunctions of the dimensionless distance z∗ = z/(L+D) fromthe wall for systems of hard spherocylinders with an aspectratio of L/D = 10 in planar slit-pore confinement predictedwith the second-virial density functional theory of Onsager.The separation distance between the two parallel hard wallsis fixed at `z = 43 D. The plots on the left correspond to bulkisotropic phases (with cb = 0.5, 0.8, 1.4, 2.2, 3.0 and 3.4, toppanel from low to high bulk concentration), and those on theright to bulk nematic phases (with cb = 3.8, 4.0, 4.3, 4.7, 5.2,top panel from low to high bulk concentration).

three-dimensional space {z, û} on an evenly spaced grid.The two-dimensional orientational phase space

∫dû =∫∫

dθ sin θdϕ = 4π is routinely parameterized in terms ofa polar angle 0 < θ < π and an azimuthal angle 0 < ϕ <2π, each discretized into 60 evenly spaced grid points. Inorder to enable comparison with the simulation data thewall-to-wall distance is fixed at `z = 43 D. Noting thatthe profiles are characterized by planar symmetry withrespect to the walls, i.e., ρ(z, û) = ρ(`z−z, û), we need toconsider only half of the spatial interval and the relevantrange of values of z: 0 < z < `z/2 is compartmentalizedinto 30 equally spaced sections per spherocylinder lengthL. Finer grids are found to only marginally improve ac-curacy whilst greatly increasing the computational bur-den. The results for a non-interacting system, equivalentto the limit of zero bulk density, ρb ↓ 0, are readily re-covered from Eq. (19) by noting that the exact density

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-0.2

0.0

0.2

0.4

0.6

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0cb

∆(D/2)∆(zmid)S(zmid)

FIG. 8. The dependence of the uniaxial nematic order S andthe biaxial nematic order ∆ at the wall (z = D/2) and at thecentre of the slab (z = zmid) on the concentration cb for theisotropic bulk phase of hard spherocylinders with an aspectratio of L/D = 10 in planar slit-pore confinement predictedwith the second-virial density functional theory of Onsager.The separation distance between the two parallel hard wallsis fixed at `z = 43 D. A capillary-induced nematic wettingtransition occurs at cb ∼ 2.6.

profile of an ideal gas exposed to an external wall poten-tial, cf. Eq. (16), corresponds to the Boltzmann factorρ(z, û) ∝ exp[−βUext(z, û)]. Substituting this expressioninto Eq. (21), omitting the first contribution of O(ρ2),and performing the spatio-orientational integration wereadily obtain the expression for the fluid-wall surfacetension and average particle-wall contact distance for anideal gas in Eq. (12).

The theoretically determined fluid-wall surface tensionshown in Fig. 6 (a) reveals a non-monotonic trend withthe bulk density, similar to what is observed with thesimulation data (cf. Fig. 5 (a)). The density functionaltheory is seen to overestimate the jump in the surface ten-sion for the transition from the isotropic to the nematicbranch. This feature is intimately linked to the fact thatthe predicted density gap between the isotropic and ne-matic phases (and the transition pressure) obtained withthe second-virial theory of Onsager is generally too largein comparison to the essentially exact simulation data.Moreover, the actual values of the tension are somewhatlower than the corresponding simulated values obtainedby simulation. This discrepancy could be attributed tothe inadequacy of the second-virial coefficient in describ-ing correlations in confined high-density fluids of rod-likeparticles where the local density may become very high(cf. the peaks in the density profiles of Fig. 2). Despitethese differences, the overall qualitative trends for the de-pendence of the surface tension, adsorption, and averagerod-wall contact distance on the bulk density are in goodagreement with the findings from simulation. The sharpdrop in the fluid-wall surface tension and the concomi-tant increase of the adsorption indicate the formation of anematic wetting layer predicted theoretically at cb ∼ 2.6.Recalling that the onset of the bulk isotropic-nematictransition is at cb = 3.39, the wall-induced nematiza-

tion occurs when the concentration exceeds a thresholdof 77 % of the value at the bulk phase transition. Close tothe bulk isotropic-nematic transition the theory predictsthat the surface tension becomes negative, which impliesthat the free energy of the system actually decreases uponincreasing the surface area of the wall in contact with therod fluid. The negative values of the wall-fluid interfa-cial tension can be qualitatively understood as a conse-quence of the formation of a nematic wetting layer atthe wall – it becomes thermodynamically favorable tocreate a nematic-wall (and nematic-isotropic) interface.The tension determined with the second-virial theory inthe vicinity of the isotropic-nematic transition is that ofthe nematic wetting film in contact with the wall ratherthan that between an isotropic fluid and the wall. Thenegative values obtained for the nematic film-wall tension(cf. Fig. 6 (a)) indicate the propensity of the particlesto spread on the surface. The negative tension obtainedwith the theory may be due to a finite-size effect in thenumerical treatment; it is apparent from the density pro-files depicted in Fig. 7 (a) that the bulk isotropic den-sity is not fully reached at large particle-wall distances,particularly at wetting conditions where there is a fast-growing nematic layer forming at the wall. It is alsopossible that the effect is an artefact of the second-virialapproximation in the theory where higher-body interac-tions between the rods are neglected. It is importantto realize that the theory is highly approximate and thetreatment clearly becomes inadequate close to the wet-ting transition.

The location of the wetting transition can be estimatedmore precisely from the behaviour of the uniaxial ne-matic and biaxial nematic order parameters at the wallsurface and at the centre of the slab, as depicted in Fig. 8.This suggests that the onset of biaxial order in the vicin-ity of the wall coincides with a capillary nematization(the transition from a bulk isotropic phase, which wetsconfining surfaces with a biaxial film, to a condensed ne-matic10,11,14). This behaviour is somewhat different fromwhat is observed in the so-called Zwanzig model in whichrod orientations are constrained to point along one of thethree Cartesian axes10. In the case of the Zwanzig modelthe uniaxial-biaxial surface ordering transition is foundto pre-empt capillary nematization. In line with the sim-ulation results, marked optima are observed in both thefluid-wall surface tension and adsorption at bulk con-centrations below the wetting transition. The extremado not, however, coincide since the minimum in the ad-sorption occurs at cb ∼ 1.7 whereas the surface tensionexhibits a maximum at cb ∼ 1.9. We remark that aslong as `z � L the surface structure is expected to befairly insensitive to a variation of the wall-to-wall dis-tance. We have verified this by repeating the theoreticalcalculations for a wall-wall separation of `z = 80D, andindeed find that the profiles and surface thermodynamicproperties deviate negligibly from those for `z = 43D,indicating that the two walls influence the properties ofthe fluid virtually independently from each other at these

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slit widths.Although the overall trends of the surface ordering and

wetting scenario found with the simulation are appro-priately captured by the simple second-virial approachadopted here, some important discrepancies are appar-ent. The main issue pertains to the microstructure ofthe wetting layer. Whilst the simulated density profiles(cf. Fig. 2(a)) are seen to exhibit a sharp peak in den-sity close to the wall, hinting at the formation of a co-herently structured nematic monolayer, the theoreticallydetermined density profiles (cf. Fig. 7(a)) appear muchmore diffuse at these high isotropic bulk densities. Moreimportantly, the sharp drop in the surface tension whicheventually becomes negative close to the bulk isotropic-nematic transition is not in accordance with the findingsof the simulations. This could point to a fundamentaldeficiency of the simple second-virial theory in captur-ing the highly anisometric many-body correlations thatprevail in the high-density confined fluid of rod-like parti-cles. Clearly, more realiable approximations are requiredin the development of accurate density functionals113,114

for confined fluid of anisometric particles in order to re-solve this issue allowing for more quantitatively reliabletheoretical predictions of the fluid-wall surface tensionand related properties.

VI. SUMMARY AND CONCLUDING REMARKS

We present a detailed study of the surface behaviourand interfacial thermodynamics of a fluid of hard sphe-rocylinders with aspect ratio of L/D = 10 confined ina slit-pore geometry by means of Monte Carlo computersimulation. Anisotropic volume perturbations are em-ployed to determine the normal and tangential compo-nents of the pressure tensor of the inhomogeneous fluid asa function of density; the full data is reported in Table I.Accurate knowledge of the components of the pressuretensor and density profiles allows for an efficient calcula-tion of the fluid-wall surface tension and related quanti-ties such as the surface adsorption and average rod-wallcontact distance, over a wide interval of bulk densitiesranging from the low-density isotropic fluid to dense ne-matic states (for the first time). The results unveil amarked non-monotonic trend in the surface tension as afunction of bulk density, with a pronounced maximumjust below cb ∼ 2. The latter density indicates a surfacephase transition corresponding to a change in the localorientational order of the rod particles in the vicinity ofthe wall from uniaxial nematic to biaxial nematic. Thisbehaviour also signals the onset of the formation of a ne-matic wetting film at the wall surface. The methodologycould also be employed to determine the interfacial ten-sion between the wall and bulk smectic phases at higherdensities.

The novel data for the fluid-wall surface tension, re-ported in Table II, serves as valuable benchmarks for test-ing more sophisticated theoretical predictions from den-

sity functional theories such as those based on modified-weighted density functionals115 or fundamental measuretheory116. The simplest non-trivial functional, based ona second-virial theory of Onsager (which is accurate atlow density) is assessed here and found to give a reason-able qualitative account of the evolution of the surfacemicrostructure such as the symmetry-breaking of uniax-ial nematic order at the wall surface, the capillary nema-tization, and the non-monotonic trend of the surface ten-sion with bulk density for the isotropic bulk system.

The negative (tensile) values of expansive contri-butions to the pressure tensor (p∗+N and p

∗+T ) of the

hard-rod fluid seen in Table I have a subtle, but po-tentially far reaching, implication: when the particlesare anisotropic, the system as a whole possesses anexpansive tensile strength, even in the complete absenceof attractive interactions. In turn, this implies thata unidirectional stretching deformation of a liquid-crystalline fluid (either along or perpendicular to thenematic-ordering director) gives rise to an asymmetricalelastic response. It appears that this tensile strengthcorrelates with the orientation of the nematic director.This provides a possible explanation for the significantlylarger error bars found for the tangential componentsof the pressure tensor (p∗+T and p

∗−T ) of the nematic

phase when compared to the corresponding values of thenormal components (p∗+T and p

∗−T ). To address this issue

and obtain an equivalent level of precision, the numberof configurations generated must be large enough forthe nematic-ordering director to rotate such that it isuniformly sampled in the plane parallel to the interfaces.We note that the errors associated with the overall com-ponents of the pressure tensor p∗N and p

∗T , in Table II)

are significantly smaller than those of their constituentcomponents (p∗+N , p

∗−N , p

∗+T and p

∗−T ). When summed

together for each specific state, the positive and negativevolume perturbations uniquely capture the total residualresistance to the volume deformation. The asymmetryin the expansive and compressive deformations will alsobe reflected in the elastic constants (twist, bend, andsplay) of the system pertaining to a deformation of thedirector field (at constant volume).

AcknowledgmentsWe are very pleased to dedicate this work to Profes-sor Keith E. Gubbins on the occasion of his 80th birth-day. PEB, GJ, and AJH acknowledge funding to theMolecular Systems Engineering Group from the Engi-neering and Physical Sciences Research Council (EP-SRC) of the UK [grants EP/E016340 and EP/J014958].This work was carried out in part as an activity of theQatar Carbonates and Carbon Storage Research Centre(QCCSRC). PEB is grateful for financial support fromJSPS KAKENHI [Grant-in-Aid for Scientific Research(A)] Grant No. 15H02222. AJH also gratefully acknowl-edges the funding of QCCSRC provided jointly by QatarPetroleum, Shell, and the Qatar Science and Technology

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Park.

APPENDIX A: PERTURBATION METHOD FOR THEDETERMINATION OF THE FLUID-WALL INTERFACIALTENSION

Inhomogeneities in confined fluids cause imbalancesbetween the diagonal elements of the pressure tensor,creating tension at the interfaces between unlike phases.For fluids at equilibrium (with no shear forces present)confined in slit-pore geometry pT = (pxx(z) + pyy(z))/2,where (pxx(z) + pyy(z), and pN = pzz is constant and in-dependent of the distance from the confining walls38,92.The expression originally presented by Kirkwood andBuff117 can be used to used to determine the surface ten-sion:

γ =∆F

∆A=

∫ ∞−∞

dz (pN (z)− pT (z)) , (25)

where F and A are the Helmholtz free energy and in-terfacial area, respectively. For systems with planar in-terfaces92

∫dz pαβ(z) = `zpαβ . Conveniently, this allows

one to express the wall-fluid surface tension of our sys-tems in terms of the anisotropy of the normal and tan-gential components of the pressure tensor:

γ = `z (pN − pT ) . (26)

The separate components of the pressure tensor are cal-culated using the test-volume perturbation method pre-sented in Ref.94. This involves undertaking a series ofanisotropic volume deformations during the course of astandard Monte Carlo simulation. At periodic intervals,non-permanent perturbations are made to the volume Viof configuration i by scaling one axis only; separate per-turbations are made to reduce and increase the systemvolume anisotropically. Evenly spaced values are chosenfor ∆V αi→j+ and ∆V

αi→j−, the expansive and compres-

sive volume changes, respectively. The average values ofpN and pT are each calculated during the course of thesimulation from [94]:

pαα =NkBT

V+ lim

[∆V αi→j+→0]

kBT

∆V αi→j+lnP+nov

+ lim[∆V αi→j−→0]

kBT

∆V αi→j−lnP−nov. (27)

Here P+nov is the probability that no particle-particleand/or particle-wall overlaps occur upon a volume ex-pansion of ∆V αi→j+, and likewise P

−nov is the probabil-

ity that no overlaps occur for a volume compression of∆V αi→j−. Evaluating the limits for infinitesimal volumechanges using a mean-squares correlation gives us the ex-cess contributions to the pressure tensor.

APPENDIX B: TABULATED INTERFACIAL-TENSIONDATA

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TABLE I. Results for systems of N hard spherocylinders with an aspect ratio of L/D = 10 confined between two hard wallswith a separation distance `z = 43D obtained from canonical Monte Carlo (MC-NV T ) simulation. The packing fraction ηb,concentration cb, and nematic-order parameter Sb in the bulk region of the system are reported. The dimensionless componentsof the pressure tensor are represented by p∗αα, where αα is the axis of the appropriate component. The excess contributions tothe normal and tangential components of the pressure tensor (denoted by the subscripts N and T , respectively) relative to thewall surface are calculated from anisotropic test-volume perturbations in the xx, yy, and zz axes. p∗+αα and p

∗−αα are the excess

contributions from expansive and compressive changes, respectively, where p∗+N = p∗+zz , p

∗−N = p

∗−zz , p

∗+T =

(p∗+xx + p

∗+yy

)/2 and

p∗−T =(p∗−xx + p

∗−yy

)/2.

N η ηb cb Sb Bulk phase p∗+N p

∗−N p

∗+T p

∗−T

20 0.0025 0.0029 0.0270 0.0091 Isotropic 0.0000 ± 0.0000 0.0005 ± 0.0000 0.0000 ± 0.0000 0.0001 ± 0.000040 0.0050 0.0057 0.0537 0.0091 Isotropic −0.0001 ± 0.0000 0.0013 ± 0.0000 −0.0001 ± 0.0000 0.0005 ± 0.000060 0.0075 0.0085 0.0801 0.0089 Isotropic −0.0002 ± 0.0000 0.0023 ± 0.0000 −0.0002 ± 0.0000 0.0011 ± 0.000080 0.0100 0.0113 0.1064 0.0092 Isotropic −0.0004 ± 0.0000 0.0036 ± 0.0001 −0.0004 ± 0.0000 0.0019 ± 0.0000100 0.0125 0.0141 0.1324 0.0087 Isotropic −0.0007 ± 0.0000 0.0052 ± 0.0000 −0.0007 ± 0.0000 0.0030 ± 0.0000200 0.0249 0.0277 0.2599 0.0097 Isotropic −0.0029 ± 0.0000 0.0172 ± 0.0001 −0.0029 ± 0.0000 0.0126 ± 0.0001300 0.0374 0.0410 0.3843 0.0093 Isotropic −0.0068 ± 0.0000 0.0368 ± 0.0001 −0.0070 ± 0.0000 0.0296 ± 0.0001400 0.0498 0.0540 0.5063 0.0094 Isotropic −0.0127 ± 0.0001 0.0649 ± 0.0001 −0.0130 ± 0.0000 0.0548 ± 0.0001500 0.0623 0.0669 0.6268 0.0099 Isotropic −0.0208 ± 0.0001 0.1017 ± 0.0002 −0.0212 ± 0.0001 0.0892 ± 0.0003600 0.0747 0.0796 0.7459 0.0099 Isotropic −0.0312 ± 0.0002 0.1491 ± 0.0001 −0.0320 ± 0.0001 0.1335 ± 0.0003700 0.0872 0.0922 0.8643 0.0107 Isotropic −0.0443 ± 0.0001 0.2074 ± 0.0004 −0.0456 ± 0.0001 0.1891 ± 0.0001800 0.0996 0.1047 0.9815 0.0109 Isotropic −0.0603 ± 0.0003 0.2774 ± 0.0003 −0.0622 ± 0.0002 0.2568 ± 0.0004900 0.1121 0.1172 1.0985 0.0110 Isotropic −0.0790 ± 0.0003 0.3608 ± 0.0008 −0.0821 ± 0.0002 0.3367 ± 0.00071000 0.1245 0.1296 1.2147 0.0115 Isotropic −0.1016 ± 0.0005 0.4581 ± 0.0009 −0.1060 ± 0.0003 0.4320 ± 0.00111100 0.1370 0.1419 1.3301 0.0110 Isotropic −0.1271 ± 0.0003 0.5699 ± 0.0002 −0.1334 ± 0.0001 0.5415 ± 0.00041200 0.1494 0.1542 1.4455 0.0111 Isotropic −0.1571 ± 0.0005 0.6969 ± 0.0008 −0.1656 ± 0.0003 0.6688 ± 0.00091300 0.1619 0.1663 1.5592 0.0139 Isotropic −0.1905 ± 0.0009 0.8406 ± 0.0034 −0.2024 ± 0.0003 0.8117 ± 0.00201400 0.1743 0.1785 1.6736 0.0129 Isotropic −0.2274 ± 0.0004 1.0024 ± 0.0025 −0.2431 ± 0.0001 0.9733 ± 0.00191500 0.1868 0.1905 1.7855 0.0140 Isotropic −0.2675 ± 0.0010 1.1794 ± 0.0043 −0.2894 ± 0.0008 1.1519 ± 0.00121600 0.1992 0.2023 1.8968 0.0138 Isotropic −0.3094 ± 0.0014 1.3646 ± 0.0013 −0.3400 ± 0.0012 1.3462 ± 0.00231700 0.2117 0.2137 2.0034 0.0147 Isotropic −0.3485 ± 0.0010 1.5581 ± 0.0026 −0.3930 ± 0.0005 1.5501 ± 0.00251800 0.2241 0.2245 2.1046 0.0276 Isotropic −0.3778 ± 0.0009 1.7288 ± 0.0039 −0.4507 ± 0.0035 1.7504 ± 0.00221900 0.2366 0.2341 2.1950 0.0507 Isotropic −0.3852 ± 0.0015 1.8629 ± 0.0025 −0.5040 ± 0.0073 1.9287 ± 0.00772000 0.2490 0.2449 2.2955 0.2587 Isotropic −0.3356 ± 0.0032 1.8668 ± 0.0067 −0.5347 ± 0.0150 2.0195 ± 0.01502100 0.2615 0.2611 2.4480 0.6562 Nematic −0.2751 ± 0.0017 1.8532 ± 0.0038 −0.5497 ± 0.0271 2.0823 ± 0.02702200 0.2739 0.2742 2.5705 0.7519 Nematic −0.2584 ± 0.0020 1.9588 ± 0.0031 −0.5856 ± 0.0488 2.2373 ± 0.04972300 0.2864 0.2871 2.6917 0.7896 Nematic −0.2502 ± 0.0016 2.0929 ± 0.0036 −0.7240 ± 0.0398 2.5221 ± 0.03922400 0.2988 0.2998 2.8109 0.8228 Nematic −0.2426 ± 0.0018 2.2525 ± 0.0046 −0.6912 ± 0.0873 2.6542 ± 0.08692500 0.3113 0.3122 2.9271 0.8486 Nematic −0.2368 ± 0.0022 2.4265 ± 0.0034 −0.6492 ± 0.1452 2.7930 ± 0.14382600 0.3237 0.3250 3.0470 0.8614 Nematic −0.2308 ± 0.0016 2.6242 ± 0.0052 −0.5578 ± 0.0485 2.8994 ± 0.04962700 0.3362 0.3374 3.1632 0.8884 Nematic −0.2239 ± 0.0030 2.8342 ± 0.0069 −0.6241 ± 0.0969 3.1792 ± 0.09702800 0.3486 0.3498 3.2792 0.9011 Nematic −0.2151 ± 0.0017 3.0556 ± 0.0059 −0.5401 ± 0.0388 3.3268 ± 0.03952900 0.3611 0.3623 3.3963 0.9146 Nematic −0.2027 ± 0.0040 3.2900 ± 0.0050 −0.9024 ± 0.2063 3.9380 ± 0.20703000 0.3735 0.3747 3.5132 0.9271 Nematic −0.1899 ± 0.0021 3.5458 ± 0.0031 −1.3187 ± 0.1054 4.6203 ± 0.1085

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TABLE II. Results for systems of N hard spherocylinders with an aspect ratio of L/D = 10 confined between two hard wallswith a separation distance `z = 43D obtained from canonical Monte Carlo (MC-NV T ) simulation. The packing fraction ηb,the concentration cb, the nematic-order parameter Sb in the bulk region of the system, the surface adsorption Γ

∗, the averageparticle-wall contact distance Z∗, the normal p∗N and tangential p

∗T components of the pressure tensor, and the fluid-wall surface

tension γ∗ are reported for a range of densities from the low density isotropic state to the high density nematic.

N η ηb cb Sb Bulk phase Γ∗ Z∗ p∗N p

∗T γ

∗

20 0.0025 0.0029 0.0270 0.0091 Isotropic −0.0010 0.2948 0.0030 ± 0.0000 0.0026 ± 0.0000 0.3079 ± 0.007340 0.0050 0.0057 0.0537 0.0091 Isotropic −0.0019 0.2978 0.0062 ± 0.0000 0.0053 ± 0.0000 0.3174 ± 0.011560 0.0075 0.0085 0.0801 0.0089 Isotropic −0.0028 0.2951 0.0095 ± 0.0000 0.0083 ± 0.0000 0.3116 ± 0.018880 0.0100 0.0113 0.1064 0.0092 Isotropic −0.0036 0.2954 0.0131 ± 0.0001 0.0114 ± 0.0000 0.3217 ± 0.0286100 0.0125 0.0141 0.1324 0.0087 Isotropic −0.0043 0.2889 0.0169 ± 0.0000 0.0148 ± 0.0000 0.3306 ± 0.0059200 0.0249 0.0277 0.2599 0.0097 Isotropic −0.0072 0.2794 0.0391 ± 0.0000 0.0346 ± 0.0001 0.3565 ± 0.0038300 0.0374 0.0410 0.3843 0.0093 Isotropic −0.0093 0.2685 0.0673 ± 0.0001 0.0599 ± 0.0001 0.3865 ± 0.0063400 0.0498 0.0540 0.5063 0.0094 Isotropic −0.0108 0.2574 0.1020 ± 0.0002 0.0916 ± 0.0001 0.4112 ± 0.0089500 0.0623 0.0669 0.6268 0.0099 Isotropic −0.0118 0.2467 0.1432 ± 0.0002 0.1302 ± 0.0003 0.4166 ± 0.0108600 0.0747 0.0796 0.7459 0.0099 Isotropic −0.0125 0.2367 0.1926 ± 0.0002 0.1762 ± 0.0003 0.4418 ± 0.0120700 0.0872 0.0922 0.8643 0.0107 Isotropic −0.0129 0.2262 0.2503 ± 0.0004 0.2307 ± 0.0002 0.4562 ± 0.0123800 0.0996 0.1047 0.9815 0.0109 Isotropic −0.0130 0.2161 0.3167 ± 0.0003 0.2942 ± 0.0005 0.4611 ± 0.0146900 0.1121 0.1172 1.0985 0.0110 Isotropic −0.0131 0.2059 0.3939 ± 0.0006 0.3667 ± 0.0009 0.4990 ± 0.01191000 0.1245 0.1296 1.2147 0.0115 Isotropic −0.0130 0.1967 0.4810 ± 0.0006 0.4504 ± 0.0010 0.5066 ± 0.01221100 0.1370 0.1419 1.3301 0.0110 Isotropic −0.0126 0.1874 0.5797 ± 0.0004 0.5450 ± 0.0004 0.5261 ± 0.00451200 0.1494 0.1542 1.4455 0.0111 Isotropic −0.0123 0.1778 0.6892 ± 0.0008 0.6526 ± 0.0010 0.5103 ± 0.01611300 0.1619 0.1663 1.5592 0.0139 Isotropic −0.0115 0.1689 0.8120 ± 0.0031 0.7711 ± 0.0017 0.5281 ± 0.03081400 0.1743 0.1785 1.6736 0.0129 Isotropic −0.0108 0.1603 0.9493 ± 0.0025 0.9045 ± 0.0018 0.5393 ± 0.02731500 0.1868 0.1905 1.7855 0.0140 Isotropic −0.0095 0.1499 1.0987 ± 0.0038 1.0492 ± 0.0009 0.5588 ± 0.04321600 0.1992 0.2023 1.8968 0.0138 Isotropic −0.0080 0.1401 1.2545 ± 0.0008 1.2054 ± 0.0013 0.5212 ± 0.01581700 0.2117 0.2137 2.0034 0.0147 Isotropic −0.0052 0.1288 1.4212 ± 0.0034 1.3687 ± 0.0025 0.5281 ± 0.05591800 0.2241 0.2245 2.1046 0.0276 Isotropic −0.0010 0.1159 1.5751 ± 0.0034 1.5238 ± 0.0025 0.4908 ± 0.04331900 0.2366 0.2341 2.1

Documents

Structure and interfacial tension of a hard-rod fluid in planar ......Structure and interfacial tension of a hard-rod uid in planar con nement P. E. Brumby,1, a) H. H. Wensink,2, b)