Van der Waals interaction in uniaxial anisotropic media

0.25 MB·English

by  Pavel Kornilovitch

Van der Waals Interaction in Uniaxial Anisotropic Media
Pavel Kornilovitch1,a)
Hewlett-Packard Company, Imaging and Printing Division, Corvallis, Oregon 97330 USA
(Dated: 10 January 2012)
Van der Waals interactions between flat surfaces in uniaxial anisotropic media are investigated in the nonre-
tarded limit. The main focus is the effect of nonzero tilt between the optical axis and the surface normal on
the strength of van der Waals attraction. General expressions for the van der Waals free energy are derived
using the surface mode method and the transfer-matrix formalism. To facilitate numerical calculations a
temperature-dependent three-band parameterization of the dielectric tensor of the liquid crystal 5CB is de-
veloped. A solid slab immersed in a liquid crystalexperiences a van der Waals torque that aligns the surface
2 normalrelativetotheopticalaxisofthemedium. Thepreferredorientationisdifferentfordifferentmaterials.
1 Two solid slabs in close proximity experience a van der Waals attraction that is strongest for homeotropic
0
alignmentof the intervening liquid crystalfor all the materials studied. The results have implications for the
2
stability of colloids in liquid crystal hosts.

n
a
J
I. INTRODUCTION anisotropic solid particles separated by an isotropic liq-

8 uid. Kats16 generalized the temperature Green’s func-

One component of the interaction between colloidal tion formalism to anisotropic media. He considered
]
t particles suspended in a fluid is van der Waals (vdW), two anisotropic particles separated by an isotropic liq-
f
o ordispersion,forces.1Inthenonretardedlimit(distances uid and calculated a vdW torque that rotates the two
s belowapproximately1µm)the dispersionforcesareme- particles relative to each other. Kats also considered
t. diatedbylongitudinalmatter-likemodeslocalizedonthe a cholesteric liquid crystal mediating vdW interaction
a between two isotropic bodies. Parsegian and Weiss17
m interfaces, while in the retarded limit (large distances)
studied two anisotropic bodies interacting across an
theyaremediatedbytransverselight-likemodesstanding
d- between the interfaces.2 Quantum-mechanicaltreatment anisotropic medium but with all three regions sharing
n of the nonretarded regime was pioneered by London3 one common optical axis. Smith and Ninham18 investi-
o and of the retarded regime by Casimir and Polder.4 Lif- gated the vdW forces applied by two anisotropic bodies
c shitz5introducedamacroscopictreatmentoftheproblem onafilmoftwistednematicsqueezedbetweenthem. The
[ torque exertedby the bodies was balancedby the elastic
where the electromagnetic properties of the bodies and
energy of the nematic, which determined the structure
1 the medium were characterized through dielectric per-
v mittivity functions. Dzyaloshinskii and Pitaevskii gen- of the twist. Sˇarlah and Zˇumer19 considered two semi-
5 infinite optically uniaxial bodies separated by a uniaxial
eralized the Lifshitz theory to non-uniform bodies using
1 theformalismoftemperatureGreen’sfunstions.6–8Later, medium with all three optical axes parallel. Those au-
6 thors derived an analytical expression for the Hamaker
van Kampen, Nijboer, and Schram developed a method
1
constant but did not consider vdW torques.
. of calculating the vdW interaction based on the knowl-
1
edge of electromagnetic surface states, which simplified
0
the treatment of complex geometries.9 Those pioneering
2
works started theoretical and experimental investigation
1
: of vdW forces, a research field that has remained active
v ever since.10–13
i Themaingoalofthepresentpaperistoinvestigatethe
X The present article concerns with vdW forces between
effectsofarbitraryorientationofthe opticalaxisrelative
r nanosize particles in anisotropic media. It is motivated
a byexperimentsoncolloidssuspendedinliquidcrystals.14 toparticlesurfaces. Duetothecomplexityofthegeneral
problem, treatment will be confined to the parallel-plate
In the cited paper, clustering of plate-like clay particles
geometryinthenonretardedlimit. Inthiscase,thevdW
in liquid crystal 5CB was observed by small-angle X-ray
energy and forces can be derived from the spectrum of
scattering. Apartfromfundamentalinterest,suspensions
longitudinalsurfacemodes.9,12,13Theelectrostaticmodel
ofcolloidsinnonaqueoussolutionsareofincreasingtech-
is formulated in Section II and the general solution is
nologicalimportance, most notably as the work medium
constructed in Section IV. For numerical evaluation of
in the electrophoretic reflective displays.
forces dielectric functions at imaginary frequencies are
The dielectric anisotropy was introduced into the
needed. Such functions for several materials are listed
vdW problem, apparently for the first time, by Ki-
in Section III. In addition, a temperature-dependent di-
hara and Honda.15 Those authors consideredelectrically
electric model of 5CB is defined for both polarizations.
The single-slab problem is solved in Section V and the
two-slabprobleminSectionVI.Someimplicationsofthe
obtained results and future directions are discussed in
a)Electronicmail: [email protected] Section VII.
2
5 4 3 2 1 electrictensorisdefinedasfollows. (i)Inthestartingori-
entation the optical axis coincides with the z-axis. This
Liquid crystal Solid body Liquid crystal Solid body Liquid crystal
implies εzz = εk, εxx = εyy = ε⊥, and all off-diagonal
termsarezero. (ii)Themediumisrotatedaroundaxisy
d4 L d2 by anangleθ. (iii) The medium is rotatedaroundaxis z
byanangleψ. Asaresult,the dielectrictensorinregion
r assumes the form
ψ εxx εxy εxz
3 r r r
X εˆjrk(θr,ψr,ω,T)= εyrx εyry εyrz , (1)
εzx εzy εzz
r r r
θ  
3
Z where
Y
εxx =ε⊥(cos2θ cos2ψ +sin2ψ )+εk(sin2θ cos2ψ )
r r r r r r r r
L54 L43 L32 L21 εxry =εyrx =(εkr −ε⊥r)sin2θrsinψrcosψr
εxrz =εzrx =(ε⊥r −εkr)sinθrcosθrcosψr
FIG. 1. The model geometry. Two finite thickness slabs 2 εyry =ε⊥r(cos2θrsin2ψr+cos2ψr)+εkr(sin2θrsin2ψr)
ainndun4iamxiaadlelioqfudidieclercytsrtiaclasll1y,is3o,taronpdic5m. Tatheeriasllasbasrearime minefirnsietde εyrz =εzry =(ε⊥r −εkr)sinθrcosθrsinψr
in the x and y directions and parallel to each other. The εzrz =ε⊥r sin2θr+εkrcos2θr. (2)
optical axes of the liquid crystals are tilted away from the
All five regions are assumed to be spatially uniform,
surfacenormalbyrespectiveanglesθ1,3,5 androtatedaround
i.e. the dielectric functions are independent of x, y, and
thez-axis by respective angles ψ1,3,5.
z. This implies no distortion of the director field and
hence a zero bulk elastic energy. Physically, this cor-
responds to either very weak or very strong anchoring.
II. MODEL AND METHOD
In the first case, the surface energy is zero, and the ori-
entation is an arbitrary parameter. In the second case,
TheoverallgeometryisshowninFig.1. Thesystemof the orientation is fixed and the vdW energy is a small
interest consists of two parallel slabs 2 and 4 a distance correction to the dominant surface energy. In any case,
L ≡ L32−L43 apart. All slabs’ surfaces are parallel to the goal of the present calculation is to characterize the
the xy-plane of the coordinate system, and the z-axis is vdW energy for a given orientation which is considered
perpendicular to the interfaces. The slabs are infinite in a model parameter.
thexandy directionsandhavefinitethicknessesinthez In the surface mode method,9,11–13 the vdW energy
direction: d2 andd4,respectively. Thematerialsof2and is derived from the spectrum of electromagnetic surface
4areassumedtobeopticallyanddielectricallyisotropic. modes. If W(q,ω) = 0 is a spectral equation, then the
They are characterized by the scalar dielectric functions vdW free energy is a sum over all quantum numbers q
ε2(ω) and ε4(ω). [The formalism developed below could and bosonic imaginary frequencies ξn =(2πkT/~)n
beeasilygeneralizedtoanisotropicsolidmaterials. How-
∞
ever,itwouldhaveobscuredthemainfocusofthepaper, ′
F =kT ln[W(q,iξ )] . (3)
which are the effects derived from the anisotropy of the n
medium. Suchageneralizationisleftforfutureresearch.] Xq nX=0
Thespacebetweentheslabsaswellasoutsidetheslabs Theprimeatthesumsignindicatesthen=0termmust
isfilledwithdielectricallyanisotropicmedia. Therespec- betakenwithweight1/2. Differentiatingwithrespectto
tive regions are labeled r = 1, 3, and 5. The media are the distance between the bodies yields the interaction
assumedtobeopticallyuniaxialsuchasmostliquidcrys- force.
tals. (The terms “uniaxial media” and “liquid crystals” The method is particularlysimple in the non-retarded
willbeusedinterchangeablythroughoutthepaper.) The limit when surface modes are obtained from solutions of
dielectric andoptical properties are characterizedby the the anisotropic Laplace equation rather than of the full
parallel and perpendicular dielectric functions εkr(ω,T) set of Maxwell’s equations:
and ε⊥(ω,T). In uniaxial liquid crystals, both dielectric
r ∂ ∂φ
functions are strong functions of temperature T. The εjk =0. (4)
∂x (cid:18) ∂x (cid:19)
theoretical formalism allows the three materials to be j k
different. However in all the examples considered below Themodesarefixedbyappropriateboundaryconditions
the material will be the same in all three regions. atthephaseboundaries. Thisiscompletelyanalogousto
Ineachregionr, the directionofthe opticalaxisis de- aprobleminoptics. Mathematically,the solutioncanbe
finedbyapolarangleθ andazimuthangleψ ,asshown constructed by using the formalism of transfer matrices.
r r
in Fig. 1. Within Cartesian coordinate system, the di- This will be done in Section IV.
3
T, (K) C1e C2e C1o C2o 3
εll(T)
298.2 0.10253 0.06161 0.10748 0.03737 2.8
ε⊥(T)
300.3 0.09724 0.06115 0.10840 0.03808 2.6
2.4
303.0 0.09205 0.05973 0.11008 0.03962
2.2
305.7 0.08470 0.05842 0.11241 0.04193
307.9 0.07587 0.05030 0.11590 0.04631 εξ(i) 2
1.8
TABLEI.Temperature-dependentcoefficientsC ofthethree- 1.6
band dielectric model, Eqs. (5)-(6), of 4-cyano-4-n-pentylbi- 1.4
phenyl(5CB).20 Other model parameters are: ω =9.19 eV,
0
1.2
ω1 =5.91 eV, ω2=4.40 eV, Tc =308.3 K,and β=0.142.
1
10−2 10−1 100 101 102 103
ξ (eV)
III. MATERIAL PROPERTIES
FIG. 2. Dynamic dielectric functions of liquid crystal com-
pound5CB, Eqs. (5)-(6) for thethreetemperatures 298.2 K,
Numerical evaluation of the vdW energy and forces
305.7 K, and 307.9 K. εk decreases with temperature, while
requires dielectric functions of the interacting materi-
ε⊥ increases with temperature.
als. Accurate knowledgeofthe entire tensorεˆ(ω,T)isof
paramount importance. Before solving the electrostatic
surface mode problem, material properties are discussed
are temperature-dependent. In particular, they define
in this Section.
thetemperaturevariationofbirefringence. Convertingto
For a growing number of substances, the dielectric imaginary frequency and using εk,⊥(iξ,T) = n2 (iξ,T),
e,o
functions on the imaginary axis are obtained through the model reads
a Kramers-Kronig transformation of the absorption or
2
reflection data followed by a fit to a multiple Lorentz
0.460 C (T) C (T)
oscillator model.12 In this paper, the parameters given εk (iξ,T)=1+ + 1e + 2e ,(5)
by Parsegian13 and van Zwol and Palasantzas21 are 5CB  1+ ωξ202 1+ ωξ212 1+ ωξ222 
used to construct dielectric functions of the following 2
isotropic materials: silica21 (set 1); polytetrafluoroethy- 0.414 C (T) C (T)
lene21 (Teflon or PTFE); polystyrene21 (set 2), mica13 ε⊥5CB(iξ,T)=1+ 1+ ξ2 + 11+o ξ2 + 12+o ξ2 .(6)
(Table L2.7, set b), gold13 (Table L2.4, set 3), silver13  ω02 ω12 ω22 
(Table L2.5, set 1), and copper13 (Table L2.6).
Here ω = 9.19 eV, ω = 5.91 eV, and ω = 4.40
0 1 2
There is much less information on εˆ(iξ,T) of liquid eV. The refractive indices of 5CB were measured by
crystals. Theusualdifficultyofknowingtheopticalspec- polarized UV spectroscopy22 and tabulated by Wu et
tra in a wide energy interval is multiplied here by the al.20Thetemperature-dependentcoefficientsC extracted
need to know them separately for two polarizations and from those data are given in Table I.
atdifferenttemperatures. Onlyafewliquidcrystalshave Rotational relaxation of 5CB molecules and other low
been studied experimentally well enough to enable a full frequency processes are neglected here based on the fa-
model. One of the most studied compounds is 4-cyano- miliarargument19thattheircharacteristicenergiesofless
4-n-pentylbiphenyl(5CB), which will be used here as an than0.01eVaremuchsmallerthanthe firstbosonicfre-
exemplary positive uniaxial material. quency at room temperature ξ ≈ 0.16 eV. The conclu-
1
The dielectric functions of 5CB used in this paper sion is Eqs. (5)-(6) represent the entire dynamical part
are based on the three-band dispersion model developed of the dielectric functions. The functions are plotted in
by Wu and co-workers.20,22–24 The model accurately de- Fig. 2.
scribes the experimentallymeasuredrefractiveindices in Static response requires a separate treatment. The
the (0.4-0.8) µm spectral interval for the entire temper- staticdielectricconstantcanbesplitintoatemperature-
ature intervalof the nematic phase 295.3K ≤T ≤ 308.3 independent isotropic part (≈ 10.7 for 5CB) and a
K. According to the model, the index dispersion in the temperature-dependent birefringent part. It is reason-
visible is governed by three electronic transitions: one able to assume that the temperature dependence comes
σ →σ∗ transition with λ ≈0.135 µm, and two π →π∗ from the order parameter. According to Li and Wu24
0
transitions with λ = 0.210 µm and λ = 0.282 µm. the order parameter of 5CB follows a universal relation
1 2
The oscillator strength of the σ → σ∗ transition is very ∝(1−T/T )β,whereT =308.3isthe nematic-isotropic
c c
weaklytemperaturedependent. Itcanbeextractedfrom transitiontemperatureandβ =0.142isauniversalexpo-
the dispersion of the isotropic part of the refractive in- nent. Adjusting the dielectric constants to the measured
dex. The oscillator strengths of the π → π∗ transitions experimentalvaluesarelowertemperatures,25onearrives
4
on either side of one interface, and the full amplitude is
20 found by matrix multiplications. This method is partic-
εll(T)
ularly suited to the geometry at hand with its four in-
18 ε⊥(T)
terfaces. Inthe restofthis Section,the twofundamental
εiso(T)
transfer matrices are derived.
16
The general solution in each region is a linear combi-
14 nation of the transmitted and reflected waves. Consider
ε interface(21). Thesolutioninmediumregion1issought
12 in the form
10 λ+1 λ−1
z z
φ1 =A1eεz1z +B1eεz1z ei(qxx+qyy). (9)
8
 
 
6  
295 300 305 310 315 320
T (K) Here A1 and B1 are unknown amplitudes, εz1z in the ex-
ponent is introduced for convenience, and the last ex-
FIG. 3. Static dielectric functions of liquid crystal 5CB, ponential factor reflects the uniformity of the Laplace
Eqs. (7)-(8). equation in the (xy) plane. The exponents λ± follow
1
from Eqs. (4) and (2):
at the following parameterization λ±1 =−i(qxεx1z+qyεy1z)±p1, (10)
β
T
εk5CB(0,T)=10.7+13.0(cid:18)1− T (cid:19) , (7) p21 ≡qx2 ε⊥1εk1 cos2θ1sin2ψ1+cos2ψ1
c (cid:16) (cid:17)(cid:0) (cid:1)
ε⊥ (0,T)=10.7−6.5 1− T β, (8) +qx2 ε⊥1 2 sin2θ1sin2ψ1
5CB (cid:18) Tc(cid:19) +2q(cid:0)q ε(cid:1)⊥(cid:0)εk −ε⊥ sin(cid:1)2θ cosψ sinψ
x y 1 1 1 1 1 1
with T = 308.3 K and β = 0.142. These functions are (cid:16) (cid:17)(cid:0) (cid:1)
c +q2 ε⊥εk cos2θ cos2ψ +sin2ψ
plotted in Fig. 3. y 1 1 1 1 1
(cid:16) (cid:17)(cid:0) (cid:1)
(i)TTohcelodsyentahmisicSaelcmtioodne,ls(e5v)e-r(a6l)cisomdemfiennetdsoanrleyianto5rddeisr-. +qy2 ε⊥1 2 sin2θ1cos2ψ1 . (11)
(cid:0) (cid:1) (cid:0) (cid:1)
crete temperatures listed in Table I. It seems reasonable Thequantitypisdefinedasthepositivesquarerootofp2.
toextendthemodeltoanytemperaturebyfittingtheco-
pisafunctionoftheinterfacemomentumcomponentsq
x
efficientstothe sameuniversalfactora+b(1−T/T )β.24
c and q , optical axis angles θ and ψ, imaginary frequency
y
Thisisnotdoneinthepresentwork. (ii)Itispossibleto
ξ, and temperature T.
convertEqs.(5)-(6)toamorefamiliarformbyexpanding In slab region 2 the material is isotropic and the
thesquareandrefittingthefunctiontoalinearcombina-
Laplace equation simply yields
tion of oscillators. Such an additional fitting procedure
mightintroduceundesirableerrors,andthereforeitisnot φ2 = A2eqz+B2e−qz ei(qxx+qyy), (12)
employed here. (iii) Wu et al20 provided a data set for
(cid:0) (cid:1)
another liquid crystal compound 5PCH, thus enabling a
similar three-band dielectric model. q ≡+ q2+q2. (13)
x y
q
The matching conditions at z = L include the equal-
21
IV. TRANSFER MATRICES ity of the transverse components of the electric field
Ex =−∂φ/∂x and Ey =−∂φ/∂y (which lead to identi-
Inthis Section,the anisotropicLaplaceequation(4) is calrelationships),andtheequalityofthenormalcompo-
solved. Inanalogywiththeopticsofmultilayeredmedia, nents of Dz = −εzk(∂φ/∂x ). The resulting two equa-
k
it is convenient to construct solutions out of individual tions can be rearranged to express the wave amplitudes
transfer matrices. Each matrix links wave amplitudes of region 2 via the wave amplitudes of region 1:
5
λ+ λ−
1 1
 qε2+p1 e−qL21 eεz1zL21 qε2−p1 e−qL21 eεz1zL21 
(cid:18)BA22 (cid:19)= 2qε2 λ+1 2qε2 λ−1 (cid:18)AB11 (cid:19)≡Mˆ21(cid:18)AB11 (cid:19). (14)
 qε2−p1eqL21 eεz1zL21 qε2+p1 eqL21 eεz1zL21 
 2qε2 2qε2 
The last equality defines the transfer matrix Mˆ21 between the medium region1 and slab region2. A similar transfer
matrix defines scattering at the (34) interface after index substitution 1→3 and 2→4. On the other hand, at the
interface (23), the waves are incident from slab region 2. From the matching conditions one expresses the medium
amplitudes A3 and B3 via the slab amplitudes A2 and B2. After some algebra one obtains
λ+ λ+
 p3+qε2 eqL32 e−εz33zL32 p3−qε2 e−qL32 e−εz33zL32 
(cid:18)AB33 (cid:19)= p32−p3qε2eqL32 e−ελz3−3zL32 p32+p3qε2 e−qL32e−ελz3−3zL32 (cid:18)BA22 (cid:19)≡Mˆ32(cid:18)BA22 (cid:19). (15)
 2p3 2p3 
A similar matrix describes scattering at the (54) inter- SubstitutingthematrixelementsfromEqs.(14)and(15)
face. it becomes
Wt =1− (qε2−p3)(qε2−p1) ·e−qd2 =0. (18)
V. ONE SLAB (qε2+p3)(qε2+p1)
Thefreeenergyisthenobtainedasfollows: (i)Thespec-
In an isotropic liquid, a parallel-plate slab does not trum equation (18) is substituted in Eq. (3); (ii) Polar
experience any macroscopic forces or torques. In an coordinates q = qcosχ, q = qsinχ, are employed in
x y
anisotropicliquid,thedependenceofthedispersionforces the integral over the surface vector; (iii) A new function
on the inclination angle will result in a torque acting on ur ≡ pr/q is introduced. It is a function of the momen-
the slab. For planar and other non-homeotropic surface tumangleχbutnotofthemomentumamplitudeq. The
alignments,theorientationoftheopticalaxesonthetwo
sidesoftheslabmayinprinciplebe differentandthisef-
fectalsowarrantsanalysis. If,inaddition,thesolidmate-
1
rialitselfisanisotropic,therewillbeanothertorquethat
Silica
0.8
will rotate the plate aroundits normal. The latter effect Teflon
is not considered in this paper. (Note that all the cases 0.6 Polystyrene
Mica
mentioned are different from the mutual torque between 0.4
two anisotropic bodies studied by Kats16 and Parsegian zJ)
and Weiss.17) 0), ( 0.2
To determine the vdW energy of a single slab in an R( 0
−
anisotropic host, only three regions of Fig. 1 need to be θ) −0.2
taken into account, for instance 1, 2, and 3. The scat- R(
−0.4
tering problem involves two transfer matrices
−0.6
A3 =Mˆ32·Mˆ21 A1 . (16) −0.8
(cid:18)B3 (cid:19) (cid:18)B1 (cid:19)
−1
0 0.1 0.2 0.3 0.4 0.5
Surface states are defined as exponentially decaying at Optical axis inclination θ/π
infinity. Accordingly,theamplitudesA1 andB3 mustbe
set to zero. The top equation of Eq. (16) links the wave FIG. 4. Tilt Hamaker constant (21) of a parallel-plate solid
amplitudes A3 and B1 on either side of the system and slabimmersedin5CB,asafunctionoftheopticalaxistiltan-
hence defines the spatial structure of the surface mode. gleθ. Thetiltisthesameonbothsides,θ =θ . Theazimuth
1 3
The bottom equation has the form W ·B1 = 0. For a angles are ψ1 = ψ3 = 0. The Rs are referenced from their
non-vanishing B1, this implies W = 0, which yields the respective values R(0) = −14.40, −12.90, −9.21, −4.25 zJ
surface mode spectrum. Expressed via matrix elements for silica, Teflon, polystyrene, and mica, respectively. Teflon
favors homeotropic alignment, θ = 0, while other materials
of the transfer matrices, the spectrum equation is
favor the planar alignment, θ = π/2. Absolute temperature
M21M12+M22M22 =0. (17) is T =298.2 K.
32 21 32 21
6
0 0.18
Gold Silica
Silver 0.16 Teflon
−1 Copper Polystyrene
J) 0.14 Mica
z Gold
zJ) −2 0), ( 0.12 Silver
0), ( π/2, 0.1 Copper
R( −3 R(
− − 0.08
θ) ψ)
R( −4 2, 0.06
π/
R(
0.04
−5
0.02
−6 0
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Optical axis inclination θ/π Azimuth angle mismatch ψ/π
FIG.5. SameasFig.4butforseveralmetalsin5CB.Theref- FIG. 6. Tilt Hamaker constant (21) for single slabs in 5CB.
erence tilt Hamaker constants are R(0)=−103.63, −116.13, The alignment is planar on both sides of the slab, θ =θ =
1 3
−244.62zJforgold,silver,andcopper,respectively. Allmet- π/2 but the azimuth angle difference is systematically var-
als favor theplanar alignment, θ=π/2. ied. The reference values are R(π/2,0) = −14.93, −11.92,
−9.90,−5.06,−108.18,−119.87,−250.58zJforsilica,Teflon,
polystyrene, mica, gold, silver, and copper, respectively. All
explicit form of ur follows from Eq. (11). (iv) The log- materialsfavortheparallelalignmentofopticalaxes,ψ1 =ψ3
arithm is expanded in an infinite series and integration anddonotfavorthecrossalignmentψ3−ψ1 =π/2. Absolute
temperature is T =298.2 K.
overq isperformed. Theresultingexpressionforthe free
energy per unit area is
F1t =−8kπTd2 ∞ ′ ∞ m13 Z 2πd2χπ (∆23∆21)m , (19) saillsvefar,voarndthceoppplaenraarraelipgrnemseenntte.dTinheFiagb.s5o.luAtelldtihffeermenecte-
2nX=0 mX=1 0
between the planar and homeotropic orientations is 4-6
where zJ, i.e. almost an order of magnitude larger than for the
dielectrics.
qε2−p3 ε2−u3
∆23 ≡ qε2+p3 = ε2+u3 , (20) For the planar surface alignment (as well as for any
nonzero tilt angle), the azimuth orientation of the op-
and the same formula holds for ∆23 with 3 replaced by tical axes on the opposite sides of the slab can be dif-
1. By analogy with vdW interaction between two semi- ferent. It is of interest therefore to investigate the vdW
infinite bodies, the overall 1/d2 dependence can be iso- free energy as a function of the azimuth misalignment
lated by introducing a Hamaker-like constant R defined ψ ≡ ψ −ψ . Such dependencies are shown in Fig. 6
3 1
as Ft =R/(12πd2). One arrives at for the planar alignment θ = θ = π/2. For all di-
1 2 1 3
electrics and metals, the parallel orientation of the op-
R(θ,ψ,T)=−3k2T ∞ ′ ∞ m13Z 2πd2χπ (∆23∆21)m. tcircoasls-aoxrieesn,taψt1ion=ψψ33−, iψs1e=nerπg/et2i.caHllyowperveefer,rrtehdistoefftehcet
nX=0mX=1 0 is relatively small, on the order of 0.1 zJ. Azimuth mis-
(21)
alignment makes a small contribution of to the overall
Note that R is defined with a negative overallsign to re-
energybalance. Itis probablysmallerthanthe variation
tainthevisualappealofenergyprofiles: largenegativeR
from the uncertainty in the material parameters.
implieslowerenergyandpreferredorientation. Inthefol-
lowing, R will be referred to as “tilt Hamaker constant”
toreflectitsrelationtotheinclinationoftheopticalaxis.
Figure 4 shows tilt Hamaker constants for several di-
electric materials immersed in 5CB. The tilt angle is the
VI. TWO PARALLEL SLABS
sameonbothsidesoftheslab, θ =θ andψ =ψ =0.
1 3 1 3
For better presentation, the constants are referenced
from their respective values at homeotropic alignment Inthis Section,two parallelslabsimmersedina liquid
θ = 0. The reference values are listed in the cap- crystal are considered. First, the spectrum of surfaces
tion. Among the materials studied, Teflon favors the modesisderivedfromaproductoffourtransfermatrices.
homeotropic alignment, while all other materials favor The casesof semi-infinite slabs andfinite-thickness slabs
the planar alignment θ = π/2. Similar plots for gold, are analyzed in order.
7
A. General expression for the van der Waals energy expression for A is
The system consists of five spatial regions separated ∞ ′ ∞ 2πdχ
by four interfaces, cf. Fig. 1. Collecting scattering at A=6kT tdt ×
all four interfaces, the wave amplitudes in region 5 are nX=0 Z0 Z0 2π
expressed via the wave amplitudes in region1 as follows ln 1−∆23∆21e−2Ld2t 1−∆43∆45e−2Ld4t
nh ih i
A5 =Mˆ54·Mˆ43·Mˆ32·Mˆ21 A1 . (22) −e−2εuz3z3t ∆23−∆21e−2Ld2t ∆43−∆45e−2Ld4t .(27)
(cid:18)B5 (cid:19) (cid:18)B1 (cid:19) h ih i(cid:27)
Surface states are defined by setting A1 = B5 = 0. The Thisformulacanbeusedfornumericalcalculations. The
top equation of Eq. (22) defines the spatial structure of
inputparametersarethegeometricalfactorsd /L,d /L,
2 4
thesurfacemode. Thebottomequationdefinesthespec-
theorientationofopticalaxesθr andψr andthetemper-
trum. Developing the bottom equation via matrix ele-
ature T. The last three parameters define the quantities
ments one obtains
ur that enter via ∆ij.
M21M11+M22M21 M11M12+M12M22 + In the limit of thin slabs the integrand is nonzero
54 43 54 43 32 21 32 21
within the large interval 0<t<L/d so the second term
(cid:0)M5241M4132+M5242M4232(cid:1)(cid:0)M3221M2112+M3222M2212(cid:1)=0.(23) under the logarithm does not contribute much. Then A
(cid:0) (cid:1)(cid:0) (cid:1) converges to the quantity R defined in Section V mul-
Substituting here the explicit matrix elements from tiplied by the factor (L/d)2 that is responsible for the
Eq. (14) and (15) and cancelling common positive- difference in definitions of A and R.
definite factors [this does not affect the final force after
taking the logarithm in Eq. (3)], the spectrum equation
becomes
B. Two semi-infinite slabs
W = 1−∆23∆21e−2qd2 1−∆43∆45e−2qd4
−(cid:2)e−2εpz33zL ∆23−∆21e(cid:3)−(cid:2)2qd2 ∆43−∆45e−(cid:3)2qd4 .(24) Two semi-infinite bodies interacting via a gap L is
(cid:2) (cid:3)(cid:2) (cid:3) the basic vdW geometry. In this Section, the Hamaker
Here L=L32−L43 is the gapbetweenthe slabsandthe constant for uniaxial anisotropic media is derived from
factors ∆43 and ∆45 are defined according to Eq. (20) the general formalism and then numerical results are
with 2 replaced by 4. presented. The spectrum of surface modes is given by
In accordance with the recipe (3), the free energy per Eq. (26). Going over to polar coordinates, expanding
unit interface area is
∞ ∞
′ dqxdqy
F1 =kTnX=0 −Z∞Z (2π)2 ln[W]. (25) 6 STeilifcloan
5
Polystyrene
If the gap between the slabs is large, L ≫ d2,d4, then Mica
the second term in Eq. (24) vanishes. The first term J) 4
z
ueancdhercotrhreesploognadriinthgmtoitnheEvqd.W(25en)esrpglyitosfianntoisotwlaotedpasrltasb, A(0), ( 3
surrounded by the medium. The total energy reduces to −
)3
a sum of two terms derived in the preceding Section. θA( 2
If the slabs are thick, d ,d ≫ L, the spectrum equa-
2 4
tion (24) reduces to
1
W∞ =1−∆23∆43e−2εpz33zL =0. (26) 00 0.1 0.2 0.3 0.4 0.5
Optical axis inclination θ /π
3
It will be analyzed in Section VIB. Here the general ex-
pression (25) is adapted for numerical evaluation.
FIG. 7. Hamaker constant (28) for two semi-infinite bod-
In the integral over q, polar coordinates qx = qcosχ, ies made of different dielectric materials separated by liq-
q = qsinχ are useful. Then the Hamaker “constant” uid crystal 5CB at T = 298.2 K. The reference values are
y
A=12πL2F isintroducedtoaccountfortheusual1/L2 A∞(0) = −16.09, −16.57, −11.87, and −5.32 zJ for silica,
1
scaling of the energy. It is also convenient to change Teflon,polystyrene,andmica, respectively. Theattraction is
the integration variable from q to t = qL. The final strongest at θ3 =0.
8
50 −14.2
T = 307.9 K
45 Gold −14.4
T = 305.7 K
Silver
40 Copper J)−14.6 TT == 330030..03 KK
zJ) 35 nt (z−14.8 T = 298.2 K
0), ( 30 nsta −15
A( 25 co−15.2
θ)−320 ker −15.4
A( 15 ma−15.6
a
H
10 −15.8
5 −16
0
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Optical axis inclination θ /π Optical axis inclination θ /π
3 3
FIG. 9. Hamaker constant (28) for two semi-infinite silica
FIG. 8. Hamaker constant (28) for two semi-infinite bodies
bodiesseparated byliquidcrystal5CBforfivetemperatures.
made of gold, silver, and copper separated by 5CB at T =
298.2K.ThereferencevaluesareA∞(0)=−128.82,−144.94,
and −283.55 zJ, respectively. The attraction is strongest at
θ3 =0 for all metals. vdW free energy. At large separations L ≫ d2,d4, one
expects one-slab effects to be dominant. For all ma-
terials except Teflon the vdW energy is minimal when
the logarithm and integrating over q results in the liquid crystal is planar aligned on both sides of the
slab. In the opposite limit of very small separations,
A∞ =−3kT ∞ ′[εzz]2 ∞ 1 2πdχ(∆23∆43)m . L ≪ d2,d4, one expects the interaction effects to dom-
2 nX=0 3 mX=1m3 Z0 2π u23 inate. As discussed above the interaction energy favors
(28) the homeotropic alignment for all the materials studied.
Figure 7 shows the Hamaker constant A∞ for several It is of interest therefore to study the evolution of the
dielectricmaterialsseparatedby5CB,asafunctionofthe optimal orientation of the middle liquid crystal and to
tilt angle of liquid crystal’s optical axis. For the conve- follow the interplay between the single-slab and interac-
nienceofpresentation,theHamakerconstantshavebeen tion effects.
referenced from their values at θ = 0. The reference Consider the 5CB-silica-5CB-silica-5CB system as an
3
values are listed in the caption. All the materials show example. A single silica slab in 5CB favors the planar
preference of homeotropic alignment θ3 =0. Among the alignment on both surfaces. Accordingly, one sets θ1 =
materials studied, Teflon has shown the largest differ- θ5 = π/2, and φ1 = ψ3 = ψ3 = 0. The tilt angle of
ence between the homeotropic and planar vdW energies the middle LC sectionθ3 remains variableto include the
(about 5.5 zJ). possibility of homeotropic and other alignments. The
Metals possess qualitatively similar angle variations slabsareassumedtobe ofthesamethickness,d2 =d4 ≡
of vdW energy, as shown in Fig. 8. The attraction is d. The overallvdW energy is studied as a function of θ3
stronger for the homeotropic alignment. However, the for different ratios d/L.
absolutescaleofthevariationisaboutoneorderofmag- ResultsofnumericalcalculationsareshowninFig.10.
nitude larger than for dielectric materials. A stronger For large d/L > 5, the vdW energy is dominated by
vdWattractionforhomeotropicalignmentmaybeagen- the interaction across the gap, and the θ3 dependence is
eral feature of 5CB and perhaps of any positive liquid virtuallythe sameasinthesemi-infinitecase. (Compare
crystal. the d/L = 50 plot in Fig. 10 with the T = 298.2 K plot
One expects the inclination dependence to go away as inFig.9.) Astheslabsgetthinner,thesingle-slabeffects
thetemperatureincreasesandthemediumbecomesopti- growstrongerandeventuallydominate. Accordingly,the
cally isotropic. The temperature dependence of A∞(θ3) planarorientationθ3 =π/2becomestheabsoluteenergy
for the silica-5CB system is shown in Fig. 9. minimum at d/L ≤ 0.5. At intermediate thicknesses,
0.5 < d/L < 1.0, the energy has two local minima, at
θ =0 and θ =π/2, as can be seen in the figure.
3 3
C. Two finite-thickness slabs
To analyze a system of two slabs oriented parallel to VII. SUMMARY AND DISCUSSION
each other, the full four-transfer matrix solution (27) is
needed. The present study is focused on finding an op- Collective behaviorof colloidalparticles in anisotropic
timal orientation of the optical axes that minimizes the media is a fascinating and complex subject. This be-
9
except Teflon, favor the planar alignment of the optical
2 axis. If the real anchoring orientation is different from
1.5 the optimal one, the dependence of the vdW energy on
1 thetiltanglewillgenerateavdWtorquethatneedstobe
taken into consideration in determining the equilibrium
0.5
J) orientation of the slab. The torque disappears as the
z
0), ( 0 temperatureisraisedabovethe nematic-isotropictransi-
A( −0.5 tion.
θ)−3 −1 dd//LL == 510.0 In the case of planar alignment and other non-homeo-
A( −1.5 d/L = 0.8 tropic surface alignments, the optical axes may have dif-
d/L = 0.7 ferentazimuthorientationoneithersideoftheslab. The
−2 d/L = 0.6 vdWenergyisingeneralafunctionofthe azimuthangle
d/L = 0.5
−2.5 mismatch. ThiseffecthasbeeninvestigatedinSectionV
d/L = 0.4
and found to be numerically small. All the materials
−3
0 0.1 0.2 0.3 0.4 0.5 studied favor parallel orientation of the optical axes, i.e.
Optical axis inclination θ /π
3 equal azimuth angles on both surfaces.
When two slabs are brought close together, they at-
FIG. 10. Hamaker“constant” A,Eq. (27), of twosilica slabs
tract via a vdW force that is a function of the optical
in 5CB for several slab thicknesses. The alignment of liquid
axisdirectionofthe intermediatemedium. Ageneralso-
crystalontheoutsidesurfacesisplanar,θ =θ =π/2,while
1 5 lution to this problem has been developed in this paper,
the alignment in the gap is varied. The reference values are
−16.09, −39.21, −54.25, −67.33, −87.77, −122.12, −186.06 cf. Section VIA. It has been found that the vdW force
zJ,(ford/Lgoingfromlargetosmall). Theabsolutetemper- is strongest for the homeotropic orientation of 5CB for
ature is T =298.2 K. all the materials analyzed. 5CB is a positive liquid crys-
tal. Thus the mainresultsuggeststhe vdWattractionis
strongest when the surface normal is parallel to the line
havioris determinedby the balanceofsurface alignment of largest polarizability of the medium molecules.
energy, bulk elastic energy, electrostatic forces, van der The last observation might have important implica-
Waals forces, and thermal fluctuations. Given the tech- tions forthe stability ofcolloidsin liquid crystals. Imag-
nological importance of both liquid crystals and non- ineapairofsphericalparticlesinapositiveliquidcrystal
aqueous colloids it would be desirable to have a com- under weak anchoring conditions. When the center-to-
prehensive theory of colloidal stability in liquid crystals center line is parallelto the optical axis the vdW attrac-
of the same clarity as the classical theory of colloidal tion will be stronger than when it is perpendicular to
stability in isotropic fluids.1 Such a theory does not yet the optical axis. Since the electrostatic repulsion from
exist. double-layer overlap also depends on the dielectric con-
The main purpose of the present work has been to stant, the balance between the attractive and repulsive
demonstrate that even a single component in this mix, forces will depend on the mutual orientation of the par-
the vander Waalsforce,is complex andcanleadto non- ticles. As a result, the particles may attract along some
trivial consequences. Due to the complexity of the gen- directions but repel along others, which could lead to
eralproblem,onlytheplanegeometryinthenonretarded chainformation. Thisintriguingpossibilitywarrantsfur-
limithasbeenanalyzed. ThevdWfreeenergycanbeob- ther investigation.
tained in this case from the spectrum of electromagnetic Finally,theeffectsoffiniteslabthicknesshavebeenin-
surface modes relatively easily. Unlike previous works, vestigated. Using the general solution, Eq. (27), smooth
the focus here has been the vdW energy dependence on evolution of the vdW energy from the gap dominated
the inclination angle of the optical axis. limit to the slab-thickness dominated limit has been ob-
served. At least for some materials (such as silica in
A significant barrier for any realistic calculation of
5CB) this implies that the planar alignment in the gap
vdWforcesis the lackofreliableparameterizationofthe
is preferred for thin slabs and large gaps, while the
dielectricfunctionontheentireimaginaryfrequencyaxis.
homeotropic alignment is preferred for thick slabs and
In liquid crystals,this is further complicated by birefrin-
small gaps.
gence and a strong temperature dependence. In this pa-
per, a three-oscillator temperature-dependent model of
5CB has been constructed based on the real-frequency
data of Wu and co-workers.20,22–24 More work will be ACKNOWLEDGMENTS
neededtofurthervalidateandrefinethemodelpresented
in Section III. ThisworkgrewoutofaprojectatHewlett-Packardon
In an anisotropic fluid, the vdW energy of a parallel- the dynamicsofchargedcolloidsinnonaqueoussolvents.
plate slab becomes a function of the tilt angle between The author wishes to thank Susanne Klein and Vladek
the optical axis and the surface normal. Energy profiles Kasperchikfor illuminating discussionsonthe subjectof
havebeencalculatedinSectionV. Allstudiedmaterials, this paper.
10
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