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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 nonretarded 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 developed. 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 orientation 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 dielectric 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 expression (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 materials 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 crystal. 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 observed. 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 paper, 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 1E. Verwey and J. T. G. Overbeek, Theory of the Stability of 12J. Mahanty and B. W. Ninham, Dispersion Forces (Academic LyophobicColloids(DoverPublicationsInc.,Mineola,NewYork, Press,London, NewYork,1976). 1999). 13V. A. 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