Table Of ContentEffective triplet interactions in nematic colloids
M. Tasinkevych1,2 and D. Andrienko3
1Max-Planck-Institut fu¨r Metallforschung, Heisenbergstrasse 3, 70569 Stuttgart, Germany
2Institut fu¨r Theoretische und Angewandte Physik,
Universit¨at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
3Max-Planck-Institut fu¨r Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
(Dated: February 6, 2008)
7
0 Three-body effective interactions emerging between parallel cylindrical rods immersed in a ne-
0 maticliquidcrystalsarecalculatedwithintheLandau-deGennesfreeenergydescription. Collinear,
2 equilateral and midplane configurations of the three colloidal particles are considered. In the last
n two cases theeffectivetriplet interaction is of the same magnitude and range as thepair one.
a
J PACSnumbers: 61.30.Dk,82.70.Dd,61.30.Jf
9
2
I. INTRODUCTION asuperpositionofone-particlesolutionsofacorrespond-
] inglinearizedtheory(so-calledNicolsonapproximation),
ft In colloidal suspensions knowledge of particle interac- which yields vanishing many-body potentials. In addi-
o tion,a linearsuperpositionofone-particlesolutions does
s tions is the key to understanding their thermodynamic
not satisfy boundary conditions at all particle surfaces.
. properties. Typically it is assumedthat the main contri-
t
a bution comes fromthe pairwiseinteractions betweenthe In this paper we would like to address the role of
m three-body interactions for long cylinders immersed in
particles. This simplifies the corresponding equations of
a nematic mesophase. This system can serve as a two-
- motion which can then be solved using efficient numeri-
d dimensionalrepresentationfornanoparticlesinanematic
calalgorithms,suchasmoleculardynamics. However,in
n host. It is also a realistic model for carbon nanotubes-
many systems the governing physical equations are non-
o
nematic liquidcrystalcomposites(discotic carbonaceous
c linear and this, in principle, can result in a modification
mesophases), which have recently been studied theoreti-
[ of the interaction between two particles by the presence
cally [26] and experimentally [33].
of other ones, meaning that the total interaction energy
1
isnolongerpairwiseadditive,andadditionalmany-body
v
6 potentials appear. The importance of many-body inter-
1 actionshasbeenrecognizedinanumberofsystems,from II. MANY-BODY POTENTIALS AND
7 electron screening in metals [1], catalysis and island for- NEMATIC FREE ENERGY
1 mation on surfaces [2], to charged colloids [3, 4, 5, 6, 7],
0
binary mixtures [8], colloidal crystals [9, 10] and simple
7 Duetotheanisotropicnatureofthesolvent,ananalyt-
liquids [11, 12].
0 ical treatment of particle interactions is practically im-
/ Attempts to characterize the solvent-mediated inter- possible already at a pairwise level. We therefore choose
t
a actions have been made in case of nematic colloids [13, here a numerical approach: equations for the nematic
m 14, 15], where colloidal particles experience long-ranged order tensor are solved by minimizing the free energy
- interactions mediated by the distortions of the nematic numerically using finite elements with adaptive meshes
d director around the particles [16, 17, 18]. These effec- [34]. For a system of N colloids at positions r we in-
n i
tive interactions result in particle clustering and self- troduce an excess free energy F (defined below) over
o N
c organization [17, 19]. Due to the presence of topological the free energyofauniaxialnematic. FN is thendecom-
: defects [20, 21, 22, 23, 24] the interparticle interaction posedinto effective n-body potentials F(n), with n≤N,
v
is inherently anisotropic: depending on the symmetry of as following [3]
i
X thedirectorfieldaroundtheparticleseitherdipole-dipole
r orquadrupolar-typeinteractionscanbeobserved[17,25]. N N
a The long-ranged nature of interactions mediated by F =NF(1)+ F(2)(i,j)+ F(3)(i,j,k)+..., (1)
N
the nematic director can result in a substantial many- Xi<j i<Xj<k
body contribution to the interaction energy already for
small concentrations of colloids. Indeed, recent theoret- where we introduce the short-hand notation for the cen-
ical studies of N-mers [26] and trimers [27] dispersed in ter of mass positions (r ,...) ≡ (i,...). The n-body po-
i
a nematic host have shown that the three-body interac- tentialisdefinedin the systemwith ncolloidalparticles.
tions can be of the same importance as two-body ones. F(1) =F is the excess free-energy coming from one iso-
1
In spite of this fact, most researchon nematic colloids lated colloid. The effective pair interaction between two
still relies on pairwise interactionpotentials between the colloids follows from Eq. (1) when N =2
particles [28, 29, 30, 31, 32]. Moreover, often the direc-
tor distribution around N particles is approximated by F(2)(1,2)=F (1,2)−2F . (2)
2 1
2
ThetripletpotentialF(3)(1,2,3)isdefinedinthesystem
of three colloids, N =3
3
F(3)(1,2,3)=F (1,2,3)− F(2)(i,j)−3F . (3)
3 1
X
i<j
By construction F(3)(1,2,3) tends to zero whenever one
of the r ≡ |r −r | → ∞. F(3)(1,2,3) is the quantity
ij i j
we are interested here.
The nematic host is modelled by the Landau-de
Gennes free energy density [35]
FIG. 1: (Color on-line) Order parameter and director dis-
tribution around colloidal particles with R/ξ = 10. Left:
f = a(T −T∗)Q Q −bQ Q Q +c(Q Q )2
ij ji ij jk ki ij ji r23 = 3R, r12 = 3.5R; right: r23 = 3R, r12 = 2.5R. Di-
L L rectororientation far from theparticles is parallel tothe line
1 2
+ Q Q + Q Q , (4)
ij,k ij,k ij,j ik,k connectingparticle centers. Notethatonly asmall part next
2 2
to theparticles is shown, the actual box size is 40R×40R.
whereQ istheorderparametertensor,summationover
ij
repeated indices is implied and the comma indicates the
spatial derivative. The positive constants a,b,c are as- The functional F is minimized numerically, using finite
∗
sumed to be temperature independent, and T is the elements with adaptive meshing. The two-dimensional
supercooling temperature of the isotropic phase. The domain of integration of the free energy density f is tri-
constants L and L are related to the Frank-Oseen angulatedandthefunctionsQ arelinearlyinterpolated
1 2 ij
elastic constants K = K = 9Q2(L + L /2)/2, within each triangle. Then the excess free energy F is
11 33 b 1 2 N
K = 9Q2L /2, and Q is the bulk nematic order pa- calculated as
22 b 1 b
rameter. We introduce the dimensionless temperature
τ = 24ca(T −T∗)/b2. The bulk nematic phase is stable
F =L (f −f )dxdy, (5)
forτ <τ =1withadegreeoforientationalordergiven N zZ b
NI
by Q =b(1+ 1−8τ/9)/8c; Q (τ >1)=0.
b b
where f = f(Q ) and L is the length of colloidal par-
For the conpstants entering the free energy density b b z
ticles. Here we will only talk about effective interactions
(4) we use typical values for a nematic compound 5CB
[36, 37]: a = 0.044×106J/m3K, b = 0.816×106J/m3, per length, F¯N = FN/Lz, F¯(n) = F(n)/Lz. It will be
c = 0.45×106J/m3, L = 6×10−12J/m, L = 12× understood that such scaling has been carried out, and
1 2
10−12J/m, T∗ = 307K. The nematic-isotropic transi- we shall omit the overbars in the text below.
tion temperature for 5CB is T = 308.5K. We choose
NI
T =308.3K,correspondingtoQ ≈0.31,closetotheNI
b
transition point, at which flocculation of colloidal parti- III. RESULTS
cles, driven by attractive interactions, normally occurs.
The nematic coherence length is ξ = 24L c/b2 1/2 ≈ A. Collinear geometry
1
10nm; it sets the smallest length-scale i(cid:0)n our syst(cid:1)em.
Three identical colloidal particles, each of which we In this case the three colloids are positioned on a line
taketobealongcylinderofradiusRwiththesymmetry paralleltothenematicdirectorn,alongthey axis. Typ-
axis parallel to the z axis, are immersed into a nematic ical director orientation and order parameter maps are
liquidcrystal. Theorderparameteratthesystembound- shown in Fig. (1). Each particle is accompanied by two
aries is fixed to the bulk order parameter of the nematic defectlinesofstrength−1/2,andthe orientationofeach
phase; the director orientation at the box boundaries is defect pair is tilted with respect to the (fixed) director
fixedalongthey axis. Inthisgeometrythecolloidalpar- orientation far from the particles, Fig. (1a). Note that
ticles are orthogonal to the director field far from them if a single colloidal particle is immersed in the nematic
(the other geometry is when the rods are aligned with host, the accompanying defects will be positioned on a
the nematic). The problem of equilibrium orientation is line perpendicular to the director orientation. The tilt
discussed in Ref. [38]. angle (which breaks the mirrorsymmetry of the system)
Theorderparameterattheparticlesurfacesisfixedto is a pure two-body effect and is alreadypresent in a sys-
thebulkorderparameter. Weconsiderfixedhomeotropic tem of two colloids [39]. At small separations, the three
boundary conditions at the particle surfaces, i. e. the particles in a collinear geometry can serve as a model
director orientation is fixed along the surface normals. for a rodforwhich symmetry breakinghas alsobeen ob-
Assuming translational symmetry of the system along served[40]. When the distance between the particles (1)
the z axis, the equilibrium distribution of the order pa- and(2)isdecreased(seeFig.(1b)),theorientationofthe
rameter tensor Q minimizes the free energy functional defects next to these particles tilts even more. However,
ij
F = fdxdy, where f is the free energy density (4). this does notaffect the orderparameterandthe director
R
3
13
/ K31112 (a) rrr222333 === 345 RRR F / K31112 (a) rrr111222 === 456...469260 RRR
F 11
(b) F(2)(1,2)
(b)
(2)F / K110.51 FFF(((222)))(((112,,,233))) r23 = 3 R ((21)) n (2)F / K11-00..055 r12 = 4.42 R FF((22))((12,,33)) (1)(2 )θ n
(3)
0 (3) (c) r23 = 3 R
0 (c) r = const K11 0
(3)F / K11-0.3 (3)(2)| F/F|01..551 23 (3)F / --00..00480 20 40 θ° 60 80
0
-0.6 2 4 6
2 4 6 FIG. 3: (Color on-line) Collinear geometry, dependence on
r / R
12 the tilt angle with respect to the nematic director: (a) ex-
cess free energy F3; (b) two-body potentials; (c) three-body
FIG. 2: (Color on-line) Collinear geometry. Three colloids potential, solid, dashed and dash-dotted lines correspond to
are positioned on the line which is parallel to the director r12 = 4.42R,5.66R and 6.90R, respectively. Particle radius
n. Particle radius R/ξ = 10. (a) Excess free energy F3 as R/ξ=10.
a function of the separation between the particles (1) and
(2). The position of the particle (3) is fixed with respect to
the particle (2) at 3R, 4R, and 5R. (b) Pair potentials for
ofr ,affectingtheeffectiveinteractionbetweenthepair
23
r23 = 3R. (c) Three-body contribution to F3; solid, dashed and the third colloid [41].
and dash-dotted lines correspond to r23 = 3R,4R and 5R,
One can also see that in this geometry the three-body
respectively. Inset shows the ratio between the three- and
potentials always provide an attractive contribution to
two-body contributions and a power law fit of the decaying
tail for r23 =3R. the total interaction energy. To emphasize the role of
the three-body term, its ratio to the sum of two-body
potentials is shownin the insetofFig.(2c) forr =3R.
23
It clearly shows that at small separations, when rear-
distribution around the third particle, i. e. one might
rangement of the defects plays a major role, the three-
anticipate that in a collinear geometry the contribution
body term is dominating and therefore, cannot be ne-
of the three-body potential to the total interaction en-
glected. At larger distances the role of the three-body
ergy is rather small, in line with the results of Guzma´n
term becomes less important decaying as a power law
et. al. [27]. F(3)/F(2) ∼r−3.7.
Toassesthis contribution,the totalinteractionenergy 12
Wealsostudyhowthetwocontributionschangeunder
F as well as the pair F(2) and the triplet F(3) contribu-
3 the director rotation. Figure (3) shows the tilt angle de-
tionstoitareshowninFig.(2). Herewefixthedistance
pendenceoftheexcessfreeenergyF ,aswellastwo-and
between the particles (2) and (3) at 3R, 4R, and 5R 3
three-body contributions to it. It is interesting that our
and study the interaction of these two particles with the
calculations predict an equilibrium configuration with a
particle (1). The dependence of F3 on separation r12 is tiltangleabout70◦ withrespecttothenematicdirector,
similar for all three curves: particle (1) is repelled from
i. e. there is also a torque acting on the line of colloids.
the particles (2) and(3) atlargedistances and there is a
The tilt is due to the pairwise interactions: the three-
barrier which does not allow particle (1) to form a sta- body contribution, in fact, has a maximum at about 60◦
ble aggregate. The height of this barrier decreases with
and favors almost collinear with the director alignment
thedecreaseofthedistancebetweentheparticles(2)and ◦
(minimum at about 10 ).
(3). Figure(2b)showsthatthemajorcontributiontothe
totalinteractionenergycomesfromthe pairwiseinterac-
tion of particles (1) and (2), especially at large separa-
B. Equilateral triangle geometry
tions of particles (2) and (3). However, for r = 3R we
23
havesignificantcontributionofthethree-bodypotential,
which is shown in Fig. (2c). In fact, this contribution is The order parameter and the effective interaction po-
responsible for the decrease of the repulsive barrier and tentials for the equilateral triangle geometry are pre-
will lead to more efficient clustering of the colloids. For sented in Fig. (4) and Fig. (5), respectively. Already
r .3Rthepairsofdefectsaroundparticles(2)and(3) the director and order parameter profiles, depicted in
23
start to tilt. The tilt angle increases with the decrease Fig.(4),showthatthethree-bodypotentialissignificant
4
11.7
K11
F / 311.6 r = 4.42 R
r = 5.66 R
r = 6.90 R
11.5
K11 0 (3)
(2)F / -0.1 FF((22))((11,,23)) (1) θ n
(2)
-0.2 F(2)(2,3)
r =r =r ≡ r
0.05 12 13 23
K11
FIG. 4: (Color on-line) Order parameter and director dis- (3)F / -0.005
tribution for the case of equilateral triangle geometry, r12 =
0 20 40 60
r13 =r23 =2.8R,R/ξ =10. (a)threeparticles; (b)thesame θ°
geometry,buttheparticle(3) isremoved;(c)theparticle(1)
is removed. The director far from the particles is parallel to FIG.6: (Coloron-line)Equilateraltrianglegeometry,tiltan-
the line connecting the centers of particles (1) and (2). Only gledependenceoftheeffectiveinteractionenergies. R/ξ=10.
a small area nextto theparticles is shown. Bottompanel: solid,dashedanddash-dottedlinescorrespond
to r=4.42R,5.66R and 6.90R, respectively.
(a)
K1112 mostalwayspositive,andhasaminimumaroundr =3R.
F / 3 Contrary to the collinear geometry, the ratio F(3)/F(2),
10 whichisshownintheinsetofFig.(5),doesnotvanishat
large r. Moreover, it converges to a constant (negative)
1 (b) F(2)(1,2)
value. Thismeansthatatlargedistancesthe three-body
(2)F / K110 F(2)(1,3) = F(2)(2,3) (1) n pbootdeyntoianlesd.ecTayhserwefiothre,thite csaanmneotpobweernelgalwectaesdtheveentwaot-
(3) large separations between the colloidal particles.
-1 0
(2) The effective interaction potentials reveal rather weak
0.4 (c) (2)F-0.2 r12=r13=r23≡ r dependence on the tilt angle, which is shown in Fig. 6.
(3)F / K110.2 (3)F / --00..462 4 6 8 10 mThenetpaearokusnadrotuhnedp3a0r◦tiacrleesr.elatedtothedefectrearrange-
0
2 3 4 5 6
r / R C. Midplane geometry
FIG.5: (Coloron-line)Equilateraltrianglegeometry,R/ξ =
10. (a) total interaction energy F3; (b) two-body contribu- Inthisgeometrythepositionoftheparticle(3)isfixed
tions; (c) three-body contribution. Inset shows the ratio be- atadistancex3fromthelineconnectingparticles(1)and
tweenthethree-andtwo-bodycontributionsandapowerlaw (2) (the geometry is depicted in Fig. (7)). The addition
fit (r−3.5) of thetail. ofthethirdparticlechangestherepulsiveinteractionbe-
tweenparticles(1)and(2)totheattractiveone,favoring
their flocculation. The F(3)/F(2) ratio shows the same
in this case: the position of the defects and the director tendency as in the equilateral triangle case: it does not
orientation around three colloids (see Fig. (4a)) cannot reach zero at large r12, i. e. the three-body potential
be obtainedasasuperpositionofthe correspondingtwo- is long-rangedand provides about 10% correctionto the
particles profiles (see Fig. (4b) and Fig. (4c)), even at sum of pair potentials at all distances.
large separations. Since in the equilateral triangle ge-
ometry defect structures are less separated than in the
collinearone,amuchstrongerthree-bodycontributionis D. Four-body interactions
expected in this case.
In Fig. (5a) F as a function of the triangle size r is To get the feeling of the next (four-body) term in the
3
shown,demonstratingattractionofthe particlesatlarge expansion (1) we considered a system of four colloidal
r. Pairwisecontributionsshowthattheparticles(1)and particles positioned at the vertices of a square. Two-,
(2)repeleachotheratlargedistances,whilepairs(2)-(3) three-, and four-body potentials are shown in Fig. (8).
and (1)-(3) attract. The three-body contribution is al- The inset shows the ratio of the total three- and four-
5
12.5 (a) x = 2.5 R lows
3
F / K311 xx33 == 35 .R5 R F(2) = 2hF(2)(1,2)+F(2)(2,3)+F(2)(1,3)i,
10.5 F(3) = 4F(3)(1,2,3).
1 0.8 (b) F(2)(1,2) Fitting the ratiosF(i)/F(2) (i=3,4)tothe powerlaw
K1 F(2)(1,3) = F(2)(2,3) (a /x)γi +b in the region r ∈[6R..10R] yields γ =5.5,
2) / 0 (1) n b i=0.18 anid γ =3.6, b =−0.01. Again, at sh3ort dis-
(F x = 2.5 R 3 4 4
-0.8 3 (3) tances the contribution of the three-body potentials to
(c) F4 is comparable(and at small distances evenbigger)to
K11 0.2 0 xr(1323 )==r c23onst tthheefictonshtroiwbusttihoantobfotthhecotnwtor-ibboudtiyonpsohtaenvetiathlse.saMmoereroavnegre,
(3) / F -0.20 (3)(2)F/ F-0.1 atwnod-bthoadtyttherreme-sbeovdeynpaottleanrgtiealssepaamraotuinotnsu.pTthoe1f0o%ur-obfotdhye
2 4 6 8 term is slightly smaller than the three-body one. How-
2 4 6 8 ever, the slow convergence of the expansion (1) does not
r / R
12 allow to make a quantitative estimate of a many-body
contribution that can be neglected.
FIG. 7: (Color on-line) Midplane geometry, R/ξ = 10: (a)
totalinteractionenergy;(b)pairwisecontributions;(c)three-
bodycontribution;solid, dashedanddash-dottedlinescorre-
IV. CONCLUSIONS
spond to x3 = 2.5R,3.5R and 5R, respectively. Inset shows
the ratio between the three-and two-body contributions for
x3=2.5R. Our calculations indicate that a collinear array of col-
loids (in our case modeled as infinitely long rods) is not
1 (a) F(2)(1,2) = F(2)(3,4) as stable as the equilateral triangle configuration. The
(2)F / K110 FF((22))((21,,33)) == FF((22))((42,,14)) morefocttshoters.tsaidbeleseoqfutihlaetetrraialntgrilaenigslepacoranlfileglutroattihoennisewmhaetnicodnie-
-1
(1) (4) Russ at. el. [3] have shown that the triplet interac-
(b) F(3)(1,2,3) = F(3)(2,3,4) = (2) (3) n tion is a function of the summed distance r12+r23+r31
K110.5 F(3)(1,2,4) = F(3)(1,3,4) between the three particles only. However, this empiri-
(3)F / 1 rr3142==rr4213≡= r creaalsoobnsseravraet:ioannidsooetsronpoict hnoaltdurienoofutrhceasine.teTrahcetiopnosssaibnlde
0
0 presence of topologicaldefects, both of which are absent
0 in a system of charged colloids.
K11 -1 F (3)/F (2) The main message of this work is that in nematic
(4)F / -1 F (4)/F (2) cshololrotiddsistmaannceys-broedaryraenffgeecmtsenctasnonfoatccboempnaegnlyeicntgedd.efecAtst
(c) -24 6 8
stronglycontributetothethree-andfour-bodytermsin
2 4 6 8 10
r / R thefreeenergyexpansion. Insomegeometries(e.g. equi-
lateraltriangle)the exponents of the power law decay of
FIG. 8: (Color on-line) Effective interaction potentials for thetwo-andthree-bodycontributionswithinterparticle
four colloidal particles in a square geometry, with R/ξ =10. distancearethesame,i.e. thehigherordercontributions
(a) pairwise contributions; (b) three-body contributions; (c) cannot be neglected even at large separations.
four-body contribution. Inset shows the ratio between the
three-, four- and two-body contributions with the power-law
fits.
Acknowledgments
body contributions to the two-body one. The total two- Comments,remarks,anddiscussionswithMarkusDe-
andthree-body contributions to F arecalculatedas fol- serno are greatly acknowledged.
4
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