Table Of ContentMolecular Dynamics Study of the Nematic-Isotropic Interface
Nobuhiko Akino1, Friederike Schmid2, and Michael P. Allen3
1 Max-Planck-Institut fu¨r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany
2 Fakult¨at fu¨r Physik, Universit¨at Bielefeld, Universit¨atsstrasse 25, D-33615 Bielefeld, Germany
1 3 H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1TL, U.K.
0 (November7, 2000)
0
Wepresentlarge-scalemoleculardynamicssimulationsofanematic-isotropicinterfaceinasystem
2
of repulsive ellipsoidal molecules, focusing in particular on the capillary wave fluctuations of the
n interfacial position. The interface anchors the nematic phase in a planar way, i.e., the director
a
aligns parallel to the interface. Capillary waves in the direction parallel and perpendicular to the
J
director are considered separately. We find that the spectrum is anisotropic, the amplitudes of
1
capillary waves being larger in the direction perpendicular to the director. In the long wavelength
1
limit, however, thespectrum becomes isotropic and compares well with the predictions of a simple
capillary wave theory.
]
h
c PACS:68.10, 68.35, 83.70
e
m
I. INTRODUCTION position (capillary waves) are not present.
-
t In general, however, one would expect such fluctua-
a
t Surfacesandinterfacesinliquidcrystalshavebeenthe tionsinfluid-fluidinterfaces. Theusualsimpleargument
s
. subject of much interest both from a fundamental point readsasfollows: Onverylargelengthscales,the freeen-
at of view and because of their practical importance in the ergy of a system containing an interface should be gov-
m context of liquid crystal devices [1,2]. erned by the interfacial tension γ. Let us neglect bub-
- Liquidcrystalsareformedbyanisotropicparticles. De- bles and overhangsand parametrizethe localposition of
d pending on temperature and density, they exhibit vari- a fluctuating planarinterface by a single-valuedfunction
n ous liquid phases. Here, we shall be concerned with the h(x,y). Fluctuationsenlargetheinterfacialareaandthus
o isotropic phase (I), which is an ordinary fully symmet- cost the free energy [7,8]
c
[ ric fluid phase, and the nematic phase (N), where the
fluid retains the translational symmetry in all directions γ ∂h 2 ∂h 2
1 CW h = dxdy + , (1)
v (no positional order), but exhibits long range orienta- F { } 2 Z "(cid:18)∂x(cid:19) (cid:18)∂y(cid:19) #
tional order. According to a very general symmetry ar-
9
gument, the transitionbetweenthe isotropicand the ne-
6 (assuming small distortions with ∂h/∂x, ∂h/∂y 1).
1 matic phase is first order [1]. Therefore there exist re- Since this free energy functional|is qua|dr|atic, th|e≪par-
1 gions in phase space where nematic and isotropic fluids tition function can be evaluated exactly. h is eas-
0 coexist and are separated by interfaces. ily diagonalized by means of a two dimensFion{al}Fourier
1 The nematic state is characterized by a so-called ne-
transform. Onefinds that the squaredamplitude offluc-
0
/ matic director, which specifies the preferred direction of tuations with wavevector q is on averagegiven by
t alignment of the particles. Surfaces and interfaces break
a
m the isotropy of space and usually favour a certain di- 2 kBT
h(q) = , (2)
rector orientation. This effect, called surface anchoring, h| | i γq2
-
d is commonly characterized by the tilt angle θ between
n the preferredorientationand the surface (interface) nor- i.e.,diverges inthelong-wavelengthlimitq 0. Thear-
→
o mal, and the anchoring strength or anchoring energy. In gument thus not only predicts on fairly general grounds
c particular, the cases θ = 0◦ (perpendicular alignment) the existence of capillary waves at all temperatures, but
v: and θ = 90◦ (parallel alignment) are often referred to also that they are actually quite large.
i as homeotropic anchoring and planar anchoring, respec- On the other hand, it has been argued that the On-
X
tively. sager approach should be exact in the limit of infinitely
r
Theoretically, particular attention has been given to elongatedparticles[9]. TheOnsagertheoryassumesthat
a
NI-interfaces in systems of hardparticles. Studies of NI- the structure of the fluid is entirely determined by the
interfaces in hard rod systems based on Onsager’s ap- secondvirialcoefficient, i.e., by the statisticalproperties
proach [3] have indicated that the surface free energy of clusters of two particles. In systems of very long,very
has a minimum when the rods lie parallel to the inter- thin particles, the probability that one particle interacts
face [4,5], suggesting that the anchoring in this system with more than one other particle approaches zero, and
is planar. Similar results have been obtained in systems the Onsager assumption becomes exact. One concludes
of hard ellipsoids [6]. According to the Onsager theory, that capillary waves should vanish in this limit, in con-
the interfaces are flat, i.e., fluctuations of the interface tradiction to the argument presented just above.
1
12 6 1 6 1
FirHsto,wcacpainllatrhyiswbaev?esTvawnoisehxipflathneatiinotnesrfaarceiaclotnencesiiovanbdlei-. V(ui,uj,rij)= 04ǫ(X −X + 4) oXthe>rw2ise (3)
(cid:26)
verges. However, this is unlikely here, since the coex-
istence density of the nematic and the isotropic phase with
moves towards zero as the particle elongation is in- σ
s
X = (4)
creased. Second, capillary waves might be suppressed r σ(u ,u ,ˆr )+σ
i j ij s
by long-range interactions, which are disregarded in the −
functional(1). Indeed,elasticlong-rangeinteractionsare Here,u andu arethe orientationsofellipsoidsiandj,
i j
present in the nematic phase. We recall that the inter- andr isthecenter-centervectorbetweenthe ellipsoids,
ij
face orients the director of the nematic liquid in a cer- of magnitude r and direction ˆr . The distance function
ij
tain direction. Long-range interface fluctuations there- σ [13] approximates the contact distance between two
fore lead to long-range elastic distortions of the director ellipsoids and is given by
field, which are penalized.
In order to shed more light on these issues, we have σ(u ,u ,ˆr)=σ 1 χ (ui·ˆr+uj ·ˆr)2 (5)
carriedoutextensive MolecularDynamics simulationsof i j s − 2 1+χui uj
an N-I interface in a system of over 100000 particles of n (uh ˆr u ˆr)·2 −1/2
i j
elongation15,andanalyzedthe capillarywavespectrum + · − · ,
1 χu u
in detail. − i· j io
Toourknowledge,thereexistonlyfewnumericalstud- where the parameter χ is related to the elongation κ
ies of N-I interfaces [10,6,11]. In one of them [10], a σ /σ through ≡
l s
system of molecules with length-to-width ratio κ = 3
interacting via a Gay-Berne potential [12] was studied κ2 1
χ= − . (6)
in a simulation box with a temperature inhomogeneity, κ2+1
whichservedtomakeandmaintaintheNIinterface. The
molecules in the nematic phase were found to align par- Sincetheattractivetailofthepotentialeqn. (3)hasbeen
allel to the interface (planar anchoring). However, due cut off, its range is much shorter than that of the Gay-
to the use ofthe temperature inhomogeneity,the system Berne potential. This enables us to simulate systems
is not at thermodynamic equilibrium. Capillary waves with a large number of particles, which is essential to
of the interface position fluctuations are obviously sup- study the capillary wave effects of interest to us. For
pressed. In the other studies [6,11], repulsive ellipsoids convenience,σs =1 defines a unit oflength, ǫ=1 a unit
of revolution of axis ratio κ = 15 were used in a sys- of energy, and we take particle mass m=1; the particle
tem confined between two hard walls. The interactions moment of inertia is set at I =50mσ2.
s
of the walls with molecules were systematically varied As a preliminary run, we performed a molecular dy-
to study the interplay between surface and interface an- namics(MD)simulationofasystemwith7200molecules
choring. Planaralignmentatthe N-I interface wasagain at constant volume in a box geometry (Lx : Ly : Lz) =
observed. The system sizes considered were too small to (1:1:8) with periodic boundary conditions in all direc-
allow for an analysis of capillary waves. tions. The density was chosen in the coexistence region,
The present work builds on Reference [11]. We study ρ = 0.017/σs3, and the temperature kBT/ǫ = 1. A ne-
NI-interfacesinaverylargesystemofrepulsiveellipsoids matic slab bounded by two interfaces parallel to the xy
by means of extensive computer simulations, focusing in plane assembled as a result. More than 5.5 106 MD
×
particular on the capillary wave fluctuations of the in- stepswererequiredtoequilibratethissystem,whereeach
terfaces. The paper is organized as follows: In the next step covers 0.002 time units. The last configuration was
section, we describe the model and some simulation de- then reproduced 4 times in the x and y direction, which
tails. SectionIIIexplainsourwaytodeterminethelocal generated a system with N =115200 molecules and box
position of the interface. The results are described and size Lx = Ly = 150.1958σs L and Lz = 300.3916σs.
≡
summarized in section IV. This was used as the starting point for a MD simulation
6
of 4.2 10 MD steps.
×
II. MODEL AND SIMULATION DETAILS
III. BLOCK ANALYSIS AND THE DIVIDING
SURFACE
Our system consists of idealized ellipsoidal particles
with elongation κ=σ /σ =15 where σ and σ are the
l s l s
lengthandwidthoftheparticles,respectively. Thelarge In order to study the interfacial position fluctuations
value of the elongation κ = 15 ensures that the order as a function of the system size, one can either perform
of the transition is fairly strong, which makes it easier simulations of various system sizes or simulate a single
to generate and maintain the interface. The interaction (large) system and analyze subsystems of it. In this
between two ellipsoids i and j is given by study, we took the latter approach. We split our sys-
tem of size L L L into columns of block size B B
z
× × ×
2
1 where u is the α component of the unit vector which
iα
order parameter profile points along the axis of the i-th molecule (α = x,y,z),
15 bins andnisthenumberofmoleculesinthebin. Thenematic
0.8 30 bins orderparameterS(z)inthe bincenteredatz is givenby
45 bins
thelargesteigenvalueoftheorderingtensor. Thefurther
procedureisillustratedinFig.1. Once the orderparam-
0.6 Dividing
eter profile S(z) is obtained (solid line), we compute at
) surface
(z least two averagedprofiles over Nave bins, where Nave is
S
chosen such that an averagedprofile still reflects the ex-
0.4
istence of the interface, but short range fluctuations are
averagedout(dottedanddashedlinesinFig.1). Finally,
the position of the interface is estimated as the intersec-
0.2
tion of two averagedprofiles.
The method is motivated by the following considera-
tion: If the order parameterS(z) did not fluctuate atall
0
−120 −80 −40 0 40 80 120 inthebulk,theintersectionwouldlocatea“dividingsur-
z/σ face”, where the negative order parameter excess on the
s
nematic side just balances the positive order parameter
FIG.1. Illustration ofourmethodofestimating thedivid-
excess on the isotropic side. It should then be indepen-
ing surface. Solid line shows an example of an order param-
eter profile for one of 16 columns with B = L/4. Dotted, dent of Nave, as long as LNave/Nbins is larger than the
dashed and long dashed lines show averages of this profile interfacialwidth and smaller than the total width of the
overNave =15, 30, and45 bins,respectively. Wedefinedthe nematic slab. This is not exactly true in an actual con-
position of theinterface zint tobe theintersection of thetwo figuration due to the bulk fluctuations of S(z), but the
profiles averaged with Nave =15 and Nave =30. methods still works well even for small block sizes (see
Fig. 2).
1
IV. RESULTS
0.8 The fluctuations of the total director (the direction of
the eigenvalue corresponding to the largest eigenvalue of
the total ordering tensor) turned out to be so slow that
0.6
the director hardly changed throughout the run. It al-
)
z wayspointedinthey direction. This madeitconvenient
(
S
to resolve wave-vectorcomponents along and perpendic-
0.4
ular to the director, without the need to apply any kind
of director constraint.
We used the procedure sketched in section III to de-
0.2
termine the local deviations h(x,y) = zint(x,y) z¯int of
−
the local interface position from its average for various
block sizes B. The landscape obtained for the block size
0
−120 −80 −40 0 40 80 120 B =L/8 was further analyzed and Fourier transformed.
z/σ Fourier modes h(q) are labeled by n = (n ,n ), where
s x y
n and n are positive integers with q= 2πn.
FIG.2. SameasFig.1butforB=L/10. Thefluctuations x y L
First, we inspect the relaxationtimes and the correla-
in theorder parameter profile are much larger.
tion times of the Fourier modes. The time evolution of
themodesn=(0,1),(1,0),(0,4),and(4,0),isshownas
a function of time (in MD steps) in Fig. 3. From there
and height L . The columns are further divided into
z we estimate the number of MD steps needed to equili-
Nbins =200bins inthez direction. The localpositionof brate the system. For the slowest mode n = (0,1), in
the interface zint(x,y) [14] in each column is estimated the direction parallel to the director, the equilibration
as follows. process seems to require roughly 1.0 106 MD steps.
First, we compute the local ordering tensor S in each ×
Based on this information, we have discarded the initial
bin (of size B×B×(Lz/Nbins)), defined as 1.2 106 MDsteps,andcollectedresultsoverthefollow-
ing×2.96 106 MD steps only.
n
1 1 ×
S = (3u u δ ), (7) Onealsohastoensurethatthetotallengthofoursim-
αβ iα iβ αβ
n 2 − ulation run significantly exceeds the characteristic time
i=1
X
3
4 3 0.2
3 (0,1) (1,0)
B=L/3
2
2σs
2q)|/2
h( 1
|
1 B=L/5
)
0 0 (h 0.1 B=L/7
0 1 2 3 4 0 1 2 3 4 P
1.5 2
B=L/9
(0,4) 1.5 (4,0)
1
2σs
2/q)| 1
h( 0.5
|
0.5
0
−20 −10 0 10 20
00 1 2 3 4 00 1 2 3 4 h(x,y)/σ
106 MD steps 106 MD steps s
FIG. 5. Distribution of local interface positions for block
FIG.3. Fourier modes |h(q)|2 for n = (0,1), (1,0), (0,4),
sizes B =L/3, L/5, L/7 and L/9. The solid lines are Gaus-
and (4,0) vs. time in units of MD steps.
sian fits(eq.(9).
4
than the length of the total simulation run. Hence the
statistical error of our results is quite large.
We can now proceed to compare our results with the
3 predictions of the capillary wave theory.
First,weconsiderthedistributionofh(x,y). Fromthe
)] free energy functional (1), one derives the exact predic-
(t tion [15]
h
h 2
C
n[ 1 h2
−l P(h)= √2πs2 exp −2s2 , (9)
(cid:18) (cid:19)
1 (n,n)=(0,1)
x y where
(n,n)=(1,0)
x y
2 2
s = h (x,y)
h i
0 0 1 2 3 4 5 = 1 dq h(q)2 = 1 ln qmax . (10)
105 MD steps 4π2 h| | i 2πγ qmin
Z (cid:18) (cid:19)
FIG. 4. Autocorrelation function Chh (log scale) vs. MD The lower and upper cutoffs qmax = 2π/B and
steps for the two slowest modes n = (1,0) (normal to the qmin = 2π/L come into play, because the integral
director) and n = (0,1) (parallel to the director). Dashed dq h(q)2 dq/q diverges as q 0 and q .
lines are linear fits togive the correlation time. Theha|ctual|lioc∼al height distribution P→(h) obtained→fro∞m
R R
oursimulationsisplottedforseveralblocksizesinFig.5.
It can be fitted nicely by a Gaussian distribution (9).
scale of the slowest capillary wave mode. In order to Next, we study the width ω of the average order pa-
check this, we have computed the autocorrelation func- rameterprofileasafunctionoftheblocksize. According
tionsofthesquaredamplitudeoftheFouriermodesh(q) to the capillary wave theory, the capillary wave fluctua-
tions broaden it for large blocks B according to
2 2 2 2
h(t) h(0) h
C (t)= h| | | | i−h| | i e−t/τ (8)
hh h4 h2 2 ∝ 2 2 π 2 2 kBT qˆmax
h| | i−h| | i ω =ω0+ s =ω0 + ln . (11)
TheyareshowninFig.4fortheslowestFouriermodes 2 4γ qˆmin
(cid:18) (cid:19)
in the direction normal to the director n = (1,0) and
paralleltothedirectorn=(0,1). Thecorrelationtimeis Here the cutoff wavevectors are given by qˆmin = 2π/B
estimatedfromtheslopeofthefittedlineasτ 1.0 105 andqˆmax =2π/a0,wherea0isamicroscopiclengthwhich
5≈ × need not be specified here.
MDstepsforthe(1,0)-modeandτ 3.0 10 MDsteps
≈ ×
forthe (0,1)-mode. Thisisonlybyafactorof10smaller
4
1.1 logarithmicincreaseofω2 withblocksizeB predictedby
eqn. (11). From the slope of the line one can estimate
0.9 B=L the interfacial tension, γ =0.016±0.002kBT/σs2.
B=L/2 For comparison, we have also determined the interfa-
S]I 0.7 BB==LL//46 cial tension from the anisotropy of the pressure tensor,
− B=L/8 making use of the relation [16,11]
N
S
∞
S]/[I 0.5 γ = dz[PN(z) PT(z)], (13)
− −∞ −
) Z
z 0.3
S( wherePN andPT arethenormalandtransversepressure
[
tensorcomponents. Here,weconsiderparticleswithpair
0.1 interactions V , eqn (3) andsystems with two interfaces
ij
inthe (xy)-plane. Neglectingfinite size effectsandinter-
−0.1 actions between the interfaces, eqn. (13) thus reads
−20 −10 0 10 20
(z−z )/σ
int s 1 ∂V ∂V ∂V
γ = rx ij +ry ij 2rz ij . (14)
L/F2I,GL./64., LO/r6dearnpdarLa/m8.eteSrNparonfidleSsIfoarrebtlohcekasviezreasgeBv=aluLes, 4L2 Xi<j* ij∂rixj ij∂riyj − ij∂rizj+
of the order parameter in the nematic and isotropic phase,
respectively. The simulation data yield γ = 0.0093± 0.003σs2/kBT.
This value agreeswith earlierresults obtainedin smaller
systems [17]. It is of the same order, but smaller than
thevalueestimatedfromtheinterfacialbroadening. The
90
quantitativedifferencepossibly stemsfromattractivein-
teractionsbetweenthetwointerfacesofthenematicslab.
80 More simulations, in which the thickness of the slab is
varied systematically, would be needed to elucidate this
point.
70 Finally, we turn to the analysis of the capillary wave
2 s spectrum h(q)2 ,whichispredictedtobeinverselypro-
2σ/ portionaltho| q2a|cciordingtoeqn(2). Westudyseparately
ω
60 the q-direction parallel (q (0,1)) and perpendicular
(q (1,0)) to the directo∝r. The inverse of h(q)2
∝ h| | i
in these two directions is shown in Fig. 8 as a func-
50
tionof the squaredwavevectors. Inthe long wavelength
limit q 0, the spectrum appears to be isotropic, and
1/ h(q)→2 approacheszero in agreementwith the capil-
40 h| | i
10 20 50 100 200 larywavetheory. Theinitialslopeisconsistentwitheqn.
FIG.7. Squaredinterfabcilaolcwki dstihzeω 2Bv/sσ.sblocksizeB. Solid (p2a)tiibfleonief ounseesuγse=s γ0.0=160k.B0T09/3σks2B,Ta/nσds2s.tiAllsroquginhclyrecaosmes-,
the capillary wave spectrum becomes anisotropic. As
line is a fit to eq.(11). onemightexpectintuitively,theamplitudes h(q)2 are
h| | i
larger for capillary waves in the direction perpendicular
to the director.
The broadening effect is demonstrated for different Deviations of capillary wave spectra from the straight
block sizes B in Fig. 6. The interfacial width can be es- linepredictedbythesimplecapillarywavetheory(2)are
timated by fitting order parameter profiles such as those oftendiscussedintermsofhigherordertermsinthecap-
shown in Fig. 6 to tanh-profiles, illary wave Hamiltonian (1), i.e., terms proportional to
squares of higher order derivatives of h(x,y). For exam-
1 1 z zint
S(z)= (SN+SI)+ (SN SI)tanh − , ple, including the next-to-leadingtermleads to a predic-
2 2 − (cid:18) ω (cid:19) tion of the form 1/ h(q)2 γq2+δq4, where γ is the
h| | i ∝
(12) interfacialtensionandδ isabending rigidity. Onemight
intuitivelyexpectthatthe NI-interfacehasbendingstiff-
wheretheparametersSN andSI arethevaluesofthe or- ness,basedonthe argumentthat the elasticinteractions
der parameter in the bulk nematic and isotopic phases.
should penalize interfacial bending.
(Note that SI is nonzerodue to the finite number ofpar- However,Fig.8indicatesthatthesignofthe“bending
ticlesinabin). Fig.7plotsthesquaredinterfacialwidth
energy” is negative. The naive argument sketched above
ω2 asafunctionofblocksizeB. Oneclearlyobservesthe
5
ACKNOWLEDGMENTS
3
n=(0,n)
y The simulations have been performed on the CRAY
n=(n,0)
x T3E of the HLRZ in Ju¨lich. N.A. receivedfinancial sup-
portfrom the German Science Foundation(DFG). MPA
> acknowledges the support of the Alexander von Hum-
2 2
q)|y boldt Foundation and the Leverhulme Trust. The paral-
q,x lel MD program used in this work was originally devel-
h( oped by the EPSRC Complex Fluids Consortium.
<|
2σ/s1
4
0
1
[1] P. G. de Gennes and J. Prost, The Physics of Liquid
0
0 5 10 15 20 Crystals, Clarendon Press, Oxford, 2nd edn., 1993.
n2 [2] B. Jerome, Rep.Prog. Phys. 54, 391 (1991).
[3] L. Onsager, Ann.N.Y.Acad. Sci. 51, 627 (1949).
FIG.8. Inverseofthemean-squaredFouriercomponentsof [4] Z. Y. Chen and J. Noolandi, Phys. Rev. A45, 2389
theinterfaceposition1/h|h(q)|2ivssquareofthewave-vector, (1992).
n2 =(qL/2π)2 for q parallel (circle) and normal (square) to [5] D. L. Koch and O. G. Harlen, Macromolecules 32, 219
the director. Dashed curves are guides to eye. Solid lines (1999).
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theory (eqn (2)) with γ = 0.016σs2/kBT (upper line), and [7] F.P.Buff,R.A.Lovett,F.H.Stillinger,Phys.Rev.Lett.
γ =0.0093σs2/kBT (lower line). 15, 621 (1965).
[8] J.S.RowlinsonandB.Widom,MolecularTheoryofCap-
illarity (Clarendon, Oxford,1982).
[9] Y. Mao, M. E. Cates, H. N. W. Lekkerkerker, J. Chem.
is thus clearly wrong. On the contrary, our unexpected
Phys. 106, 3721 (1997).
result suggests that the elastic interactions influence the
[10] M.A.BatesandC.Zannoni,Chem.Phys.Lett.280,40
capillary waves on the largest length scale: The inter-
(1997).
face is rougher on short length scales than one would
[11] A. J. McDonald, M. P. Allen, and F. Schmid, cond-
expect from the long wavelength fluctuations. This ob-
matt/0008056, Phys.Rev.E (to appear).
servationisconsistentwithourspeculationsintheintro- [12] J. G. Gay and B. J. Berne, J. Chem. Phys. 74, 3316
duction, that the elastic interactions may be responsible (1981).
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NI-interface in the Onsager-limit of infinitely elongated (1975).
particles. [14] A. Werner, F. Schmid, M. Mu¨ller, and K. Binder, Phys.
We note that this is not the first observationof a neg- Rev. E59, 728 (1999).
ativebending rigidityatafluid-fluid interface. Asimilar [15] J.D.Weeks,J.Chem.Phys.67,3106(1977);D.Bedeaux
phenomenon has been predicted theoretically for liquid- and J. D. Weeks,J. Chem. Phys. 82, 972 (1985).
vapour interfaces [18,20] and verified experimentally by [16] P.SchofieldandJ.R.Henderson,Proc.Roy.Soc.London
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attributed in their case to long range van der Waals in- [17] Thesimulationsofref.[11]yieldedtheinterfacialtension
teractions[19]. Indicationsofanegativebendingrigidity γ = 0.011σs2/kBT However, the systems studied there
were so small that the actual value of γ is not very reli-
ataliquid-liquidinterfacehavealsobeenfoundinMolec-
able.
ulardynamicssimulationsofsimpleliquidsbySteckiand
[18] E. M. Blokhuis, D.Bedeaux, Mol. Phys. 80, 705 (1993).
Toxvaerd [22,23].
[19] For a more general discussion of bending rigidities in
To summarize, we have studied the NI-interface by
isotropic fluids with arbitrary pair interactions see also
large scale molecular dynamics simulation of a system
N. Napiorkowsky, S. Dietrich, Phys. Rev. E 47, 1836
of repulsive ellipsoidal molecules. We find that the stan-
(1993); Z. Phys.B 97, 511 (1995).
dard capillary wave theory explains most of our results
[20] K. R. Mecke and S. Dietrich, Phys. Rev. E 59, 6766
very well. Discrepancies are encountered when looking
(1999).
at the amplitudes of capillary wave modes at large wave
[21] C. Fradin, D. Luzet, D. Smilgies, A. Braslau, M. Alba,
vectors, i.e., on small length scales. We find that they N. Boudet, K. Mecke, and J. Daillant, Nature 403, 871
are smaller than expected and anisotropic. (2000).
[22] J.Stecki,S.Toxvaerd,J.Chem.Phys.103,9763(1995).
[23] E. M. Blokhuis, D.Bedeaux, Mol. Phys. 96, 397 (1999).
6
