Van der Waals torque and force between dielectrically anisotropic layered media
VVan der Waals torque and force between dielectrically anisotropic layered media
Bing-Sui Lu ∗ and Rudolf Podgornik , Department of Theoretical Physics, J. Stefan Institute, 1000 Ljubljana, Slovenia and Department of Physics, Faculty of Mathematics and Physics,University of Ljubljana, 1000 Ljubljana, Slovenia. (Dated: November 7, 2018)We analyse van der Waals interactions between a pair of dielectrically anisotropic plane-layeredmedia interacting across a dielectrically isotropic solvent medium. We develop a general formalismbased on transfer matrices to investigate the van der Waals torque and force in the limit of weak bire-fringence and dielectric matching between the ordinary axes of the anisotropic layers and the solvent.We apply this formalism to study the following systems: (i) a pair of single anisotropic layers, (ii) asingle anisotropic layer interacting with a multilayered slab consisting of alternating anisotropic andisotropic layers, and (iii) a pair of multilayered slabs each consisting of alternating anisotropic andisotropic layers, looking at the cases where the optic axes lie parallel and/or perpendicular to theplane of the layers. For the first case, the optic axes of the oppositely facing anisotropic layers of thetwo interacting slabs generally possess an angular mismatch, and within each multilayered slab theoptic axes may either be the same, or undergo constant angular increments across the anisotropiclayers. In particular, we examine how the behaviors of the van der Waals torque and force can be“tuned” by adjusting the layer thicknesses, the relative angular increment within each slab, and theangular mismatch between the slabs.
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I. INTRODUCTION
Van der Waals (vdW) forces exist between any pairof bodies if their material polarizability differs from thebackground [1, 3–6]. Additionally, a vdW torque can ap-pear if these bodies display either an anisotropic shapeor are birefringent [27], i.e., their dielectric propertiesare different along different principal dielectric axes, asis typically the case with crystals such as quartz or crys-tallite structures such as kaolinite [7]. In dielectrically (oroptically) anisotropic materials, there is a special princi-pal axis called the optic axis , which coincides with theaxis of symmetry of the dielectric ellipsoid of the crys-tal (see Fig. 1). Dielectrically anisotropic materials canbe classified as either uniaxial or biaxial , depending onwhether the principal dielectric permittivities in the di-rections perpendicular to the optic axis are respectivelyidentical or distinct [7, 8].The dielectric anisotropy effects were first addressedin the Lifshitz theory of vdW interactions for isotropicboundaries and anisotropic intervening material by Kats[9, 10], while Parsegian and Weiss independently formu-lated the non-retarded Lifshitz limit for vdW torquesin the case of two uniaxial half-spaces separated byanother dielectrically anisotropic medium [11]. Laterthe complete Lifshitz result, including retardation, fortwo dielectrically anisotropic half-spaces with an inter-vening isotropic slab was obtained by Barash [12–15].The general Lifshitz theory results for the vdW inter-actions in stratified anisotropic and optically active me-dia with retardation effects are algebraically unwieldy ∗ Electronic address: [email protected] [16], not permitting any final simplification [17]. Fur-ther efforts in the investigation of the vdW torque be-tween a pair of single -layered dielectrically anisotropicslabs include a one-dimensional calculation [18], calcula-tions on two ellipsoids with anisotropic dielectric func-tion [19], a pair of dielectric slabs with different con-ductivity directions [20], and a quantum torque cal-culation for two specific uniaxial materials (barium ti-tanate and quartz or calcite) [14, 15]. Experiments havealso been proposed to measure the vdW torque usingcholesteric liquid crystals [21]. Apart from the dielectricanisotropy, morphological anisotropy has been studiedbetween anisotropic bodies [22–24] or even between sur-faces that have anisotropic decorations [25, 26] and theeffects of dielectric vs. morphological anisotropy havebeen delineated and compared [27].Dielectrically anisotropic multi -layered materials havemany examples, appearing in ceramics and clays, suchas kaolinite [30] and computations of the vdW forces for multi-layered systems are well-known in the litera-ture [16, 31–36, 38]. In addition, many common minerals,e.g. micas, serpentine and chlorite to name a few, existas different polytypes differing in layer-stacking configura-tions with repeated lateral offsets and rotations betweenthe neighboring layers [37]. These rotations of the layerorientations, implying also rotations in the principal axesof the respective dielectric tensor, implicate local long-range vdW torques between the building blocks of thelayered materials. These torques could play a stabilizingrole favoring certain type of polytype with e.g. orderedperiodic layer sequence as opposed to random stackingsequences. It is thus obviously important and relevant toinvestigate the vdW torques corresponding to such sys-tems and the role it plays in the self-assembly and stabi-lization of isolated single layers in crystallite structures, a r X i v : . [ c ond - m a t . s o f t ] J u l consisting of alternating dielectrically anisotropic crystaland isotropic (solvent) layers. From the nanoscale mate-rials engineering side it is also of interest to create mate-rials whose interactions can be tuned by suitable modi-fications of the internal structure [28, 29], as in the caseof materials composed of layers of different but knownoptical properties whose overall interaction behavior canbe controlled by changing the thicknesses and the opticalanisotropies of the individual layers.In the present Paper our objective is to investigate thevdW torque as well as the interaction force between apair of layered slabs, each composed of coplanar layersthat alternate between two distinct types of media: anoptically anisotropic material and an isotropic (solvent)material. Within each layered slab, the optic axis under-goes a constant angular increment across the anisotropiclayers, while between the slabs there is also a relativeangular difference between the optic axes of the oppo-sitely facing anisotropic layers. The way is thus paved toexplore how the behaviors of the vdW torque and forcechange as one changes the following parameters: (i) thethicknesses of the layers, (ii) the angular dielectric in-crement within each slab, and (iii) the relative dielec-tric angular difference between the slabs. Methodologi-cally, our approach is a cross-pollination of the ideas andmethods of previous approaches to determine the vdWtorque between single-layered slabs and the transfer ma-trix method of computing the vdW force between multi-layered slabs [34, 35]. We shall delimit ourselves to the non-retarded limit, i.e., the limit where the speed of lightis taken as effectively infinite. This is a good approxi-mation for slabs that are separated by distances smallerthan ∼
100 nm lengthscale. Furthermore, if the systemis at high temperature (e.g., room temperature) and theintervening isotropic solvent medium is water, at suffi-ciently large separations the zero Matsubara frequencyterm dominates over correction terms coming from re-tardation effects [1], and thus the non-retarded limit alsoprovides a good approximation for the latter regime.In Sec. II, we describe our system and develop a gen-eral formalism based on the method of transfer matrices.From Secs. III to VI, we apply our formalism to illustra-tive, specific examples, in the simplifying approximation(of Ref. [11]) that the dielectric anisotropy is weak andthe dielectric susceptibility along the ordinary axes of theuniaxial layers matches the dielectric susceptibility of thesolvent. We examine in turn a system with two singleuniaxial layers, a single uniaxial layer interacting witha multilayered slab that has all its optic axes aligned, asingle uniaxial layer interacting with a multilayered slabhaving rotating optic axes, and two interacting multilay-ered slabs each with rotating optic axes. In these systemsthe optic axes lie in the plane parallel to the layers. InSec. VII, we reconsider the previous systems but nowwith optic axes perpendicular to the plane of the layers. d b ✓ ✓ b b b b b multilayer y zx m . . . p " B x p " B y p " B z ✓ d B B FIG. 1: A model system of layered slabs. Each slab may con-sist of one or many uniaxial optically anisotropic layers, eachwith the same thickness b (cid:48) . Within each slab and betweenevery pair of adjacent anisotropic layers is a layer of interven-ing isotropic (solvent) medium of thickness b . The slabs areseparated by a gap m of width d , which has the same dielec-tric properties as the isotropic (solvent) medium. The spacecoordinates have been chosen such that the x -axis is parallelto the optic axis (shown as the green unbroken arrow) of theleft-most layer of the right slab (denoted by B , which we taketo be the reference layer), and the optic axis of the right-mostlayer of the left slab has a relative angle θ d . The optic axis un-dergoes a constant rotation of δθ within each slab. Shown onthe right is the dielectric ellipsoid corresponding to layer B .The geometric and optical anisotropies of the system shouldin no way be conflated, as the geometric axis points in the z -direction whereas the optic axes are perpendicular to z . II. THE SYSTEM
Our model system (see Fig. 1) consists of a pair of co-axial and co-planar slabs with an intervening medium m of thickness d . The (solvent) medium m is dielectrically isotropic with dielectric permittivity ε m . On the otherhand, the slabs can either be single or multi-layered.The single-layered slab is dielectrically anisotropic . Inthe multi-layered slab, there are N + 1 dielectricallyanisotropic layers (which we call type B (cid:48) ) and N isotropiclayers (which we call type B – this can be an aqueous ornon-aqueous solvent, such as water or ethanol), the layersalternating between dielectric anisotropy and isotropy.For example, the B (cid:48) -type layer could represent silicateand the B -type layer could represent water in systemssuch as kaolinite clays [30]. A. Dielectric tensor
In terms of principal dielectric axes the dielectric ten-sor for each B -type layer is given by ε (prin) B = ε W I (2.1)where ε W is the dielectric permittivity of the isotropicmedium (which, e.g., for water has a static value of ∼ ε at 293 K, and ε is the vacuum permittivity), and I is the identity matrix. Written in terms of principalaxes, the dielectric tensor for the reference B (cid:48) -type layeris given by ε (prin) B (cid:48) = ε B (cid:48) x ε B (cid:48) y
00 0 ε B (cid:48) z (2.2)where the x and y directions lie in the plane of the layer,and z is perpendicular to the plane of the layer, see Fig.1 (where layer B is taken to be the reference layer). Tak-ing the dielectrically anisotropic material to be uniaxial and defining the optic axis to be parallel to the x -axis,then ε B (cid:48) x (cid:54) = ε B (cid:48) y = ε B (cid:48) z . If layer i is of type B (cid:48) (i.e.,anisotropic) and its optic axis is rotated relative to theoptic axis of the reference layer by an angle θ i , we canexpress the corresponding dielectric tensor as ε ( i ) ( θ i ) = (2.3) ε B (cid:48) x cos θ i + ε B (cid:48) y sin θ i ( ε B (cid:48) x − ε B (cid:48) y ) sin θ i cos θ i ε B (cid:48) x − ε B (cid:48) y ) sin θ i cos θ i ε B (cid:48) x sin θ i + ε B (cid:48) y cos θ i
00 0 ε B (cid:48) z If the i -th layer is type B (i.e., isotropic) then ε ( i ) = ε W I . B. van der Waals interaction free energy
To calculate the free energy of vdW interaction we em-ploy the van Kampen-Nijboer-Schram method [1, 39],in which the electromagnetic field is represented asan ensemble of harmonic oscillators described by theHelmholtz free energy F ( T ) = k B T (cid:88) { ω j } ln(2 sinh( β (cid:126) ω j / T is the temperature, k B is Boltzmann’s constant, β = 1 /k B T is the inverse temperature, and (cid:126) = h/ π where h is Planck’s constant. As the vdW interactionarises from correlations of electromagnetic surface fluc-tuational modes [4], the sum only include those modefrequencies ω j that obey the dispersion relation, whichis in general a nonlinear equation in ω j . The task ofcomputing F is however drastically simplified by the useof the Argument Principle [1, 3, 39], via which the freeenergy can be transformed to the following more man-ageable form: F ( T ) = k B T ∞ (cid:88) n =0 (cid:48) ln D ( iξ n ) , (2.5)where the sum is over Matsubara frequencies, ξ n =(2 πk B T / (cid:126) ) n , the prime denotes that we have to mul-tiply the n = 0 term by a factor 1 /
2, and D ( ω j ) = 0is the dispersion relation whose solutions are the modefrequencies ω j .In principle D ( ω ) is calculated from the full set ofMaxwell equations, but here we delimit ourselves to thecase of c → ∞ which corresponds to the non-retarded case as discussed in detail in Refs. [11, 39]. (The retar-dation actually enters only for separations on the orderof 10-100 nm, which is not the case we are interested in.)The frequency summation comes from the poles of theln(2 sinh( βω/ non-retardation limit, i.e. c → ∞ , which wehave taken, and (ii) the zero frequency limit, ω →
0. Bothlimits lead to vanishing right-hand sides in the Maxwellequations Eqs. (2.6). On the other hand, the zero fre-quency limit leads to a static value for the dielectric func-tion, and thus a dispersion relation with no frequencydependence that can be reduced to (spatial) fluctuationdeterminant of the field modes [2], whereas the dielectricfunction (and correspondingly the dispersion relation) re-tains its frequency dependence in the c → ∞ limit. Fordetails, see the discussion in e.g. Ref. [1] C. Dispersion relation
The dispersion relation for the two-slab system can bederived from the boundary conditions that the electro-magnetic surface modes have to satisfy. For a source-and current-free system, Maxwell’s equations are givenby ∇ × H = 1 c ∂ D ∂t , ∇ · D = 0 , (2.6a) −∇ × E = 1 c ∂ B ∂t , ∇ · B = 0 (2.6b)where D = ε · E and B = µ · H , where ε and µ are thedielectric and magnetic tensors. We assume that the di-electric properties are anisotropic but the magnetic prop-erties are isotropic, so we will set µ = I , where I is theunit matrix. For the case of slab geometry, it is knownthat there are two sets of solutions to Maxwell’s equa-tions, viz., the TM and TE modes, describing the twodifferent polarizations of the electromagnetic wave. Inwhat follows, we shall consider only the TM mode con-tribution to the vdW free energy and neglect the TEmode contribution, as the effect of dielectric anisotropyis present only in the former [40].In the non-retarded regime, c → ∞ , and the equationsgoverning the electric field become ∂ a ( ε ab E b ) = 0 , (2.7a) (cid:15) abc ∂ b E c = 0 . (2.7b)Here a, b, c = 1 , , x, y, z ) are Cartesian indices la-beling the directions in space, (cid:15) abc is the completely an-tisymmetric tensor, ∂ a ≡ ∂/∂x a where x = x , x = y and x = z . Solving these equations subject to boundaryconditions in the slab geometry leads to the TM mode.Owing to the curl-free condition we can also representthe electric field as the gradient of a scalar potential ϕ : E = −∇ ϕ, (2.8)whence we obtain ∂ a ( ε ab ∂ b ϕ ) = 0 . (2.9)This has to be solved with respect to the boundary con-ditions that both ϕ and ( ε · E ) z are continuous across theinterface between every pair of adjacent layers. As thetranslational symmetry is broken along the 3-directionwe can represent ϕ in terms of a two-dimensional Fouriertransform: ϕ i ( x ⊥ , z ) = (cid:90) du dv (2 π ) e i ( ux + vy ) f i ( z ) , (2.10)where x ⊥ = ( x, y ), u, v are the momenta in the x and y directions, and the subscript i labels the layer [39].Plugging this into Eq. (2.9) gives ∂ z f i ( z ) − ρ i ( θ i ) f i ( z ) = 0 . (2.11)If layer i is an isotropic medium then ρ i = √ u + v . Onthe other hand, if layer i is dielectrically anisotropic with its optic axis lying in the plane of the layer, then ρ i ( θ i ) ≡ ε ( i )11 u + 2 ε ( i )12 uv + ε ( i )22 v ε B (cid:48) z (2.12)= ε B (cid:48) x ε B (cid:48) z ( u cos θ i + v sin θ i ) + ε B (cid:48) y ε B (cid:48) z ( v cos θ i − u sin θ i ) , where ε ( i ) ab denotes the ab element of the dielectric tensorin Eq. (2.3). The solution to Eq. (2.11) is given by f i ( z ) = A i e ρ i z + B i e − ρ i z . (2.13)Continuity of ϕ and ( ε · E ) z at the interface betweenlayers i and i + 1 demands f i +1 ( (cid:96) i,i +1 ) = f i ( (cid:96) i,i +1 ) , (2.14) ε ( i +1)33 ∂ z f i +1 ( (cid:96) i,i +1 ) = ε ( i )33 ∂ z f i ( (cid:96) i,i +1 ) (2.15)where z = (cid:96) i,i +1 is the position of the interface. Theseequations lead to (cid:18) A i +1 B i +1 (cid:19) = − (cid:18) ε ( i )33 ρ i ε ( i +1)33 ρ i +1 (cid:19) (cid:18) e − ( ρ i +1 − ρ i ) (cid:96) i,i +1 ¯∆ i +1 ,i e − ( ρ i +1 + ρ i ) (cid:96) i,i +1 ¯∆ i +1 ,i e ( ρ i +1 + ρ i ) (cid:96) i,i +1 e ( ρ i +1 − ρ i ) (cid:96) i,i +1 (cid:19) (cid:18) A i B i (cid:19) , (2.16)where we have defined a reflection coefficient describingthe dielectric discontinuity across the interface betweenlayers i and i + 1:¯∆ i +1 ,i ≡ ε ( i +1)33 ρ i +1 − ε ( i )33 ρ i ε ( i +1)33 ρ i +1 + ε ( i )33 ρ i . (2.17)Denoting the left-most layer of the slab on the left by theindex L , and the right-most layer of the slab on the rightby the index R , the dispersion relation is obtained fromthe condition that A R = B L = 0. We are thus at libertyto ignore the prefactor on the right-hand side (RHS) ofEq. (2.16) and redefine amplitudes such that (cid:18) (cid:101) A i +1 (cid:101) B i +1 (cid:19) = (cid:18) e ρ i +1 (cid:96) i,i +1 e − ρ i +1 (cid:96) i,i +1 (cid:19) (cid:18) A i +1 B i +1 (cid:19) (2.18) (cid:18) (cid:101) A i (cid:101) B i (cid:19) = (cid:18) e ρ i (cid:96) i − ,i e − ρ i (cid:96) i − ,i (cid:19) (cid:18) A i B i (cid:19) . (2.19)We can write (cid:18) (cid:101) A i +1 (cid:101) B i +1 (cid:19) = e ρ i ( (cid:96) i,i +1 − (cid:96) i − ,i ) M i +1 ,i · (cid:18) (cid:101) A i (cid:101) B i (cid:19) (2.20) where M i +1 ,i ≡ (cid:18) − ¯∆ i,i +1 e − ρ i ( (cid:96) i,i +1 − (cid:96) i − ,i ) − ¯∆ i,i +1 e − ρ i ( (cid:96) i,i +1 − (cid:96) i − ,i ) (cid:19) (2.21)We can further decompose M i +1 ,i into the product oftwo matrices: M i +1 ,i = D i +1 ,i · T i , (2.22)where D i +1 ,i ≡ (cid:18) − ¯∆ i +1 ,i − ¯∆ i +1 ,i (cid:19) , (2.23) T i ≡ (cid:18) e − ρ i ( (cid:96) i +1 ,i − (cid:96) i,i − ) (cid:19) , (2.24)We can practically ignore the prefactor e ρ i ( (cid:96) i,i +1 − (cid:96) i − ,i ) in subsequent calculations because it does not affect thedispersion relation. By induction we can relate the coef-ficients (cid:101) A R and (cid:101) B R of the right-most layer to the coeffi-cients (cid:101) A L and (cid:101) B L (= 0) of the left-most layer, viz., (cid:18) (cid:101) A R (cid:101) B R (cid:19) = Θ · (cid:18) (cid:101) A L (cid:19) , (2.25)where the overall transfer matrix Θ is given by Θ ≡ D R,P N − (cid:89) i =0 T i +1 D i +1 ,i , (2.26) L corresponds to the i = 0 layer, and we have assumedthat there is a total of N + 1 layers in the system, ofwhich the end layers on the left and the right are semi-infinite. If we consider the effective interaction betweentwo layered slabs separated by a gap of isotropic mediumof width d , then the dispersion relation is given by D ( d, ω ) = Θ ( d, ω )Θ ( d → ∞ , ω ) = 0 , (2.27)where Θ is the 11 component of the transfer matrix.This follows since both media (L) and (R) are semi-infinite and the fields should decay far away from thedielectric boundaries, and thus (cid:101) A R = 0 and (cid:101) B L = 0. Thiscan only happen if Θ ≡
0, i.e., if Eq. (2.27) is valid. InEq. (2.27) we have normalized the dispersion relation byits value for infinitely separated layers. According to thedefinition of the vdW free energy, Eq. (2.5), this amountsto the same thing as subtracting the bulk contributionfrom the complete free energy, with the remainder obvi-ously being just the vdW interaction free energy . D. Anisotropy factor
By writing u = Q cos ψ, v = Q sin ψ, (2.28)we can rewrite ρ i (cf. Eq. (2.12)) in the simpler form: ρ i = Q g i ( θ i − ψ ) , (2.29)where the effects of dielectric anisotropic are now con-tained inside the anisotropy factor g i . For isotropic, B -type media, g i = 1, whilst for anisotropic, B (cid:48) -type media,it is given by g i ( θ i − ψ ) ≡ (cid:114) ε B (cid:48) y ε B (cid:48) z + ε B (cid:48) x − ε B (cid:48) y ε B (cid:48) z cos ( θ i − ψ ) (2.30)The two-dimensional integral measure becomes du dv = Q dQ dψ. (2.31)The preceding formal discussions will be fleshed out morefully in the following sections, where we apply our formal-ism to concrete examples.
III. TWO INTERACTING SINGLEANISOTROPIC LAYERS
We consider two co-axial parallel single layers B and B composed of an anisotropic material (for example, d m W b B B b W FIG. 2: A pair of single anisotropic layers B and B of thesame thickness b (cid:48) interacting across an intervening isotropicsolvent medium m of thickness d . The layers B and B arealso bounded on the left and the right respectively by thesame solvent W . silicate), each being of thickness b (cid:48) , separated by an in-tervening isotropic medium (for example water) of width d and dielectric permittivity ε W , and the media to theleft of B and the right of B are also isotropic and of thesame dielectric permittivity ε W (see Fig. 2). The opticaxis of B is however rotated relative to the optic axis of B by an angle θ B − θ B . Using transfer matrices, wecan express this set-up by (cid:18) (cid:101) A R (cid:101) B R (cid:19) = Θ (ss) · (cid:18) (cid:101) A L (cid:19) (3.1)where the overall transfer matrix is given by Θ (ss) ≡ D W B T B D B W T m D W B T B D B W (3.2)and the boundary condition that A R = 0 can be enforcedvia the requirement that Θ (ss)11 = 0. The matrices aregiven by D W B ≡ (cid:18) − ¯∆ W B − ¯∆ W B (cid:19) , (3.3) D W B ≡ (cid:18) − ¯∆ W B − ¯∆ W B (cid:19) , (3.4) T B ≡ (cid:18) e − Qb (cid:48) g B (cid:19) , (3.5) T B ≡ (cid:18) e − Qb (cid:48) g B (cid:19) , (3.6) T m ≡ (cid:18) e − Qd (cid:19) , (3.7)and D B W ( D B W ) corresponds to D W B ( D W B ) with¯∆ W B ( ¯∆ W B ) replaced by − ¯∆ W B ( − ¯∆ W B ). The re-flection coefficients are given by¯∆ W B ≡ − g B g B , ¯∆ W B ≡ − g B g B , (3.8)¯∆ W B = − ¯∆ B W , ¯∆ W B = − ¯∆ B W . (3.9)We assume that the anisotropic layers have the samedielectric properties (apart from the orientation of theoptic axis) and are uniaxial (i.e., the dielectric propertyalong the optic axis is different from the dielectric proper-ties along the other two principal axes, and the dielectricproperties along the latter axes are identical).In addition, following Ref. [11], we adopt the simpli-fying assumption that the dielectric permittivity alongeach of the non-optic principal (i.e., ordinary ) axes isequal to the dielectric permittivity of the isotropic media: ε B ,y = ε B ,z = ε B ,y = ε B ,z = ε W and ε B ,x = ε B ,x .A possible realization where such dielectric matching be-tween the ordinary axes of the anisotropic layer and thesolvent holds approximately is a stack of LiNbO layersimmersed in water at room temperature [42]; both staticdielectric constants of the solvent and the anisotropiclayer along the ordinary axes are approximately 80 atroom temperature [43].Let us also define a quantity that characterizes theanisotropy between the principal dielectric permittivities,viz., γ n ≡ ε B ,x ( iξ n ) /ε B ,z ( iξ n ) − , (3.10)where the subscript n reflects the dependence of the di-electric anisotropy on the frequency. The anisotropy fac-tors are then expressible by g B = (cid:112) γ n (cos( θ B − ψ )) , (3.11a) g B = (cid:112) γ n (cos( θ B − ψ )) . (3.11b)By using the matrices above, we can readily compute the11 element of the overall transfer matrix. Its value isgiven in Eq. (A1) of App. A. A. Interaction between isotropic layers
As a check of consistency, let us consider layers that aredielectrically isotropic (i.e., g B = g B = 1) and ¯∆ W B =¯∆ W B = ¯∆ W B = ¯∆ W B ≡ ¯∆; in this case, we have(using Eq. (A1))Θ (ss)11 ( d, ω ) = (1 − ¯∆ e − Qb (cid:48) ) − ¯∆ e − Qd (1 − e − Qb (cid:48) ) (3.12)Using Eqs. (2.5) and (3.12), we find that the interactionfree energy per unit area is given by G ss = k B T π (cid:48) (cid:88) n (cid:90) ∞ dQ Q ln Θ (ss)11 ( d, iξ n )Θ (ss)11 ( d → ∞ , iξ n ) (3.13)= k B T π (cid:48) (cid:88) n (cid:90) ∞ dQ Q ln (cid:20) − ¯∆ e − Qd (1 − e − Qb (cid:48) ) (1 − ¯∆ e − Qb (cid:48) ) (cid:21) . This agrees with previous results [1, 3, 36] on an inter-acting pair of dielectrically isotropic layers immersed inan isotropic solvent.
B. Interaction between anisotropic layers
We return to Eq. (A1) and consider weak anisotropy,for which γ n (cid:28)
1, as is the case in materials with weak dielectric anisotropies that include amongst others thecalcite, whose static dielectric susceptibility along the or-dinary axes is 8.5 and that along the optic axis is 8 atroom temperature [14]. To leading order we haveΘ (ss)11 ≈ − γ n e − Qb (cid:48) × (cid:0) e − Qd sinh ( Qb (cid:48) ) cos ( θ B − ψ ) cos ( θ B − ψ )+ cos ( θ B − ψ ) + cos ( θ B − ψ ) (cid:1) (3.14)The corresponding interaction free energy per unit areais given by G ss = k B T π (cid:48) (cid:88) n (cid:90) du dv ln Θ (ss)11 ( d )Θ (ss)11 ( d → ∞ ) (3.15)For weak anisotropy, this leads to G ss ≈ − γ k B T π (cid:90) π dψ (cid:90) ∞ dQ Q e − Q ( b (cid:48) + d ) sinh ( Qb (cid:48) ) × cos ( θ B − ψ ) cos ( θ B − ψ )= − γ k B T π (1 + 2 cos ( θ B − θ B )) × (cid:20) d − d + b (cid:48) ) + 1( d + 2 b (cid:48) ) (cid:21) , (3.16)where we have defined γ ≡ (cid:80) (cid:48) n γ n . For large separa-tion ( d (cid:29) b (cid:48) ), we have G ss ≈ − γ k B T (1 + 2 cos ( θ B − θ B ))( b (cid:48) ) πd , (3.17)which corresponds to an attractive force per unit areathat decays with d − , viz., F ss ≈ − γ k B T (1 + 2 cos ( θ B − θ B ))( b (cid:48) ) πd . (3.18)For small separation ( d (cid:28) b (cid:48) ), we have G ss ≈ − γ k B T (1 + 2 cos ( θ B − θ B ))2048 πd , (3.19)which corresponds to an attractive force per unit areathat decays with d − , viz., F ss ≈ − γ k B T (1 + 2 cos ( θ B − θ B ))1024 πd . (3.20)This is in agreement with the result obtained byParsegian and Weiss [11] in the case of non-retarded in-teraction. This agreement comes about because two lay-ers that are separated by a distance much smaller thantheir thicknesses effectively resemble a pair of thick slabs.Also, notably, the cos ( θ B − θ B ) dependence on the rel-ative anisotropy angle change, signals the material dielec-tric anisotropy effect [27]. C. van der Waals torque
We can now straightforwardly derive the vdW torqueper unit area τ , by applying the general definition τ = − ∂G∂θ d , (3.21)where θ d ≡ θ B − θ B . We consider the weak anisotropyregime and multi-layered slabs with a large number of B (cid:48) -type layers. From Eq. (3.16) we obtain the torqueper unit area for two interacting single layered slabs, τ ss : τ ss = − γ k B T sin 2 θ d π (cid:20) d − d + b (cid:48) ) + 1( d + 2 b (cid:48) ) (cid:21) . (3.22)For d (cid:29) b (cid:48) , the torque is approximately given by τ ss ≈ − γ k B T ( b (cid:48) ) πd sin(2 θ d ) , (3.23)while for d (cid:28) b (cid:48) , the torque is approximated by τ ss ≈ − γ k B T πd sin(2 θ d ) . (3.24)This latter limit is the same as that obtained byParsegian and Weiss [11] in the limit of non-retardation,as two single layers separated by a distance much smallerthan their individual thicknesses is approximately equiv-alent to two thick slabs.Thus for both the vdW interaction free energy andtorque, there is a crossover in the scaling behaviorwith separation from d − to d − as the separation in-creases beyond a lengthscale set by the thickness of eachanisotropic layer. The vdW force is always attractive,owing to G ss being always negative (and growing in mag-nitude as the separation decreases). On the other hand,the vdW torque can change sign depending on θ d . For0 < θ d < π/ π < θ d < π/ τ ss <
0, which impliesthat the configuration in which the optic axes of the twoanisotropic layers are aligned (or anti-aligned) is stable :any deviation from alignment will generate an attractivetorque that tends to restore the two layers to the alignedconfiguration. For π/ < θ d < π and 3 π/ < θ d < π , τ ss >
0, which implies that the configuration in whichthe optic axes are perpendicular is unstable , as a slightdeviation will bring about a repulsive torque that drivesthe layers away from their initial angular configuration. IV. SINGLE ANISOTROPIC LAYERINTERACTING WITH MULTILAYER HAVINGALIGNED OPTIC AXES
Next, we consider a set-up in which we have a semi-infinite slab of the isotropic B -type medium, bounded onthe right by a single layer B made of anisotropic medium B (cid:48) (of thickness b (cid:48) ), followed by a gap m of width d whichis composed of an isotropic B -type medium, and this is . . . b d W m B B W b b B . . . b d L W m B B W b b N B B FIG. 3: A single anisotropic layer B of thickness b (cid:48) interactswith a slab composed of a sequence of alternating B (cid:48) -type(anisotropic) and B -type (isotropic) layers, of thicknesses b (cid:48) and b respectively, across an intervening isotropic medium m of thickness d . The layer B and the slab are bounded on theleft and the right respectively by isotropic media W . followed in turn by N repeats of the B (cid:48) -type layer (ofthickness b (cid:48) ) and B -type layer (of thickness b ), with a fi-nal B (cid:48) layer that is followed by a semi-infinite slab of the B -type medium (see Fig. 3). The optic axis of the B (cid:48) -type single layer on the left is oriented at an angle θ d withrespect to the optic axis of the left-most B (cid:48) -type layer ofthe multilayered slab (the latter axis being our referenceaxis), and the optic axis of every B (cid:48) -type layer insidethe multilayered slab has the same orientation. Mathe-matically, this set-up is represented by the sequence oftransfer matrices: (cid:18) (cid:101) A R (cid:101) B R (cid:19) = Θ (sm) · (cid:18) (cid:101) A L (cid:19) (4.1)where Θ (sm) is given by Θ (sm) ≡ A N D W B (cid:48) T B (cid:48) D B (cid:48) W T m D W B T B D B W . (4.2)Here the matrix A describes a bilayer consisting of a B -type layer and a B (cid:48) -type layer: A ≡ D W B (cid:48) T B (cid:48) D B (cid:48) W T B . (4.3)Here D W B (cid:48) and D W B describe the dielectric disconti-nuity at the interfaces between the B and B (cid:48) -type layers,and are given by D W B (cid:48) ≡ (cid:18) − ¯∆ W B (cid:48) − ¯∆ W B (cid:48) (cid:19) , (4.4) D W B ≡ (cid:18) − ¯∆ W B − ¯∆ W B (cid:19) , (4.5)¯∆ W B ≡ − g B ( θ d − ψ )1 + g B ( θ d − ψ ) = − ¯∆ B W , (4.6)¯∆ W B (cid:48) ≡ − g B (cid:48) ( ψ )1 + g B (cid:48) ( ψ ) = − ¯∆ B (cid:48) W , (4.7)where g B and g B (cid:48) are anisotropy factors. For a systemin which ε B (cid:48) y = ε B (cid:48) z = ε W , this is given by g B ≡ (cid:112) γ n cos ( θ d − ψ ) , (4.8a) g B (cid:48) ≡ (cid:112) γ n cos ψ, (4.8b)where we have defined γ n ≡ ε B (cid:48) x ( iξ n ) /ε B (cid:48) y ( iξ n ) − . (4.9)The matrices T B and T B (cid:48) are related to the thicknessesof the B -type and B (cid:48) -type layers respectively, and aregiven by T B ≡ (cid:18) e − Qb (cid:19) , (4.10) T B (cid:48) ≡ (cid:18) e − Qb (cid:48) g B (cid:48) (cid:19) (4.11)Similarly T B and T m are related to the thicknesses ofthe layer B and the gap m : T B ≡ (cid:18) e − Qb (cid:48) g B (cid:19) , (4.12) T m ≡ (cid:18) e − Qd (cid:19) , (4.13)where g B ≡ (cid:112) γ n cos ( θ d − ψ ) . (4.14)As before the boundary condition that A R = 0 can beenforced via the dispersion relation: Θ (sm)11 = 0. The ele-ments of the matrix A are given by Eqs. (B1) of App. B.The matrix product A N can be found using Abel`es’formula (see, e.g., Ref. [7]), which gives A N = (cid:32) A ( N )11 A ( N )12 A ( N )21 A ( N )22 (cid:33) (4.15)where A ( N )11 ≡ (cid:18) A (cid:112) | A | U N − − U N − (cid:19) | A n | N/ (4.16a) A ( N )12 ≡ A U N − | A | ( N − / (4.16b) A ( N )21 ≡ A U N − | A | ( N − / (4.16c) A ( N )22 ≡ (cid:18) A (cid:112) | A | U N − − U N − (cid:19) | A | N/ (4.16d)Here U N are the Chebyshev polynomials, with U = 1and U N = 0 for N <
0, and for
N > U N − = sinh N ξ sinh ξ (4.17)where ξ ≡ cosh − (cid:18) A + A (cid:112) | A | (cid:19) . (4.18)The above representation for Chebyshev polynomials isvalid for ξ >
1, which is the case as can be verified easily.For weak anisotropy, we can approximate ξ by ξ ≈ Q ( b + b (cid:48) ) + γ n Qb (cid:48) cos ( θ B (cid:48) − ψ ) . (4.19) The vdW interaction free energy per unit area is thengiven in the non-retardation limit by G sm = k B T π (cid:48) (cid:88) n (cid:90) π dψ (cid:90) ∞ dQ Q ln Θ (sm)11 ( d )Θ (sm)11 ( d → ∞ )= k B T π (cid:48) (cid:88) n (cid:90) π dψ (cid:90) ∞ dQ Q ln(1 − ¯∆ W B ¯∆ (eff) e − Qd ) . (4.20)Here we have defined an effective dielectric reflection co-efficient to characterize the alternating B (cid:48) - and B -typelayers to the right of the intervening medium m :¯∆ (eff) ≡ s + s ¯∆ W B (cid:48) + s ¯∆ W B (cid:48) t + t ¯∆ W B (cid:48) + t ¯∆ W B (cid:48) × e − Qb (cid:48) g B sinh( Qb (cid:48) g B )1 − ¯∆ W B e − Qb (cid:48) g B , (4.21)where coefficients in the numerator are given in Eqs. (B3)and coefficients in the denominator are given in Eqs. (B4)in App. B.The formulas Eqs. (4.20) and (4.21) are exact, whichcan be used to determine the interaction free energy be-havior for arbitrary anisotropy strengths γ n and numberof layers N . Equation (4.20) is also formally equivalentto a vdW free energy of two interacting planar slabs, inwhich the effect of the multi-layeredness of the secondslab only enters through the effective reflection coeffi-cient, ¯∆ (eff) . The logarithmic form of the free energy im-plies that it accounts for microscopic many-body effectsto all orders. An expansion of the logarithm to quadraticorder in reflection coefficients would correspond to mak-ing a Hamaker pairwise-summation approximation. A. van der Waals interaction free energy
For N (cid:29)
1, we have U N − ≈ e ( N − ξ , and thus w ≈ e − ξ . For the case of weak anisotropy ( γ n (cid:28) W B ≈ − γ n cos ( θ d − ψ ) / W B (cid:48) ≈− γ n (cos ψ ) /
4, i.e., ¯∆
W B (cid:48) is of the order of γ n and wecan thus expand ¯∆ (eff) in powers of γ n . To leading orderwe find¯∆ (eff) ≈ − γ n cos ψ e − Q ( b (cid:48) − b ) sinh ( Qb (cid:48) )2 sinh( Q ( b + b (cid:48) )) . (4.22)From Eq. (4.20) we then find that the interaction freeenergy per unit area is given to the same order by G sm ≈ − k B T π (cid:48) (cid:88) n (cid:90) π dψ (cid:90) ∞ dQ Q ¯∆ W B ¯∆ (eff) e − Qd = − γ k B T (1 + 2(cos θ d ) )2048 π ( b + b (cid:48) ) (cid:20) ψ (1) (cid:18) db + b (cid:48) (cid:19) − ψ (1) (cid:18) d + b (cid:48) b + b (cid:48) (cid:19) + ψ (1) (cid:18) d + 2 b (cid:48) b + b (cid:48) (cid:19) (cid:21) (4.23) � � � � � �� - ��� - ��� - ��� - ��� - ������ � / � � � � ( �� - � � � / γ � � � � ) �� = ��� = ���� = �� � � � � � � � - ��� - ��� - ��������������� θ � τ � � ( �� - � � � / γ � � � � ) FIG. 4: A single anisotropic layer interacting with a multi-layer (weak anisotropy and large N ): behavior of free energyper unit area G sm (Eq. (4.23)) with separation d for θ d = 0,and (inset) behavior of torque per unit area τ sm (Eq. (4.29))with θ d for d = 10 b . Both behaviors are plotted for the follow-ing values of b (cid:48) : (i) b (cid:48) = b (blue), (ii) b (cid:48) = 2 b (green, dashed),and (iii) b (cid:48) = 5 b (red, dotted). where γ ≡ (cid:80) (cid:48) n γ n , ψ (1) ( z ) ≡ ∂ψ ( z ) /∂z is thepolygamma function of order unity, and ψ ( z ) ≡ Γ (cid:48) ( z ) / Γ( z ) is the digamma function. We plot the be-havior of G sm in Fig. 4. The interaction is less attractiveif the thickness b of the B -type layers is larger, but be-comes more attractive if the thickness b (cid:48) of the B (cid:48) -typelayers is larger. We can understand this as a manifesta-tion of the vdW attraction being generated by the dielec-tric contrast between adjacent media. We have assumedthat the dielectric permittivity of the B -type mediumis identical to two of the principal permittivities of theanisotropic, B (cid:48) -type medium, whilst the latter mediumhas an extra principal permittivity with a different value.Thus increasing b ( b (cid:48) ) implies a weakening (strengthen-ing) of the effect of the dielectric contrast, and thereforethe vdW attraction.In the large separation limit ( d (cid:29) b + b (cid:48) ) we find G sm ≈ − γ k B T (1 + 2 cos θ d )( b (cid:48) ) π ( b + b (cid:48) ) d , (4.24)which corresponds to a force per unit area that decayswith d − and is given by F sm ≈ − γ k B T (1 + 2 cos θ d )( b (cid:48) ) π ( b + b (cid:48) ) d . (4.25)On the other hand, at small separations ( d (cid:28) b + b (cid:48) ) wefind G sm ≈ − γ k B T (1 + 2 cos θ d )2048 πd . (4.26)This corresponds to a force per unit area which decayswith d − and is given by F sm ≈ − γ k B T (1 + 2 cos θ d )1024 πd . (4.27) As expected Eq. (4.26) agrees with Eq. (3.19), which ef-fectively approximates the interaction behavior of twovery thick anisotropic B (cid:48) -type slabs. In contrast to thecase of interacting single-layer B (cid:48) -type plates, the behav-ior of G sm crosses over from d − decay at small separa-tion to d − decay at large separation. Thus, for a single-layer B (cid:48) -type plate interacting with a multilayered slab,the attraction at large separation is stronger than thatfor a pair of interacting single-layer B (cid:48) -type plates. B. van der Waals torque
The van der Waals torque per unit area is given by τ sm = − ∂G sm ∂θ d (4.28)In the weak anisotropy regime and for large N , we cancompute τ sm from Eq. (4.23), obtaining τ sm ≈ − γ k B T sin(2 θ d )1024 π ( b + b (cid:48) ) (cid:20) ψ (1) (cid:18) db + b (cid:48) (cid:19) − ψ (1) (cid:18) d + b (cid:48) b + b (cid:48) (cid:19) + ψ (1) (cid:18) d + 2 b (cid:48) b + b (cid:48) (cid:19) (cid:21) (4.29)For large separation ( d (cid:29) b + b (cid:48) ), the torque is approxi-mately given by τ sm ≈ − γ k B T ( b (cid:48) ) sin(2 θ d )512 π ( b + b (cid:48) ) d . (4.30)For small separation ( d (cid:28) b + b (cid:48) ), we find τ sm ≈ − γ k B T sin(2 θ d )1024 πd . (4.31)In Fig. 4, we plot the behavior of the vdW torque asa function of θ d , for three different thicknesses b (cid:48) . Wesee that the torque becomes enhanced as b (cid:48) increases.This enhancement can be understood as originating fromthe greater extent of dielectric contrast that we have dis-cussed above. V. SINGLE ANISOTROPIC LAYERINTERACTING WITH A MULTILAYERHAVING ROTATING OPTIC AXES
Having considered a single anisotropic layer interact-ing with a multilayer in which the optic axes of all itsanisotropic layers are aligned, we now turn to the casewhere the optic axes of the multilayer undergo angular in-crements of δθ on moving across the anisotropic, B (cid:48) -typelayers (see Figs. 1 and 5). In the language of transfermatrices, this system is described by (cid:18) (cid:101) A R (cid:101) B R (cid:19) = Θ (sr) · (cid:18) (cid:101) A L (cid:19) , (5.1)0 . . . b d W m B B W b b N B . . . b d W m B B b B ... ... W j L j R N R N L a a B FIG. 5: A single anisotropic layer B , oriented at angle θ d ,is of thickness b (cid:48) and interacts with a slab composed of asequence of alternating B (cid:48) -type and B -type layers, of thick-nesses b (cid:48) and b respectively, across an intervening isotropicmedium m of thickness d . The layer B and the slab arebounded on the left and the right respectively by isotropicmedia W . The numbers 1, 2 , . . . , N refer to the BB (cid:48) bilay-ers, with optic axes oriented at δθ , 2 δθ, . . . Nδθ respectively.The optic axis of the B (cid:48) -type layer immediately to the left ofbilayer 1 are oriented at zero angle. where Θ (sr) ≡ N (cid:89) j =1 A ( j ) D (0) W B (cid:48) T (0) B (cid:48) D (0) B (cid:48) W T m D W B T B D B W , (5.2) A ( j ) ≡ D ( j ) W B (cid:48) T ( j ) B (cid:48) D ( j ) B (cid:48) W T B , (5.3) D ( j ) W B (cid:48) ≡ (cid:32) − ¯∆ ( j ) W B (cid:48) − ¯∆ ( j ) W B (cid:48) (cid:33) , (5.4)¯∆ ( j ) W B (cid:48) ≡ − g ( j ) B (cid:48) g ( j ) B (cid:48) , (5.5) g ( j ) B (cid:48) ≡ (cid:112) γ n cos ( jδθ − ψ ) (5.6) T ( j ) B (cid:48) ≡ (cid:18) e − Qb (cid:48) g ( j ) B (cid:48) (cid:19) , (5.7)As before, γ n ≡ ε B (cid:48) x ( iξ n ) /ε B (cid:48) y ( iξ n ) −
1. The matrixproduct is ordered in the following manner N (cid:89) j =1 A ( j ) ≡ A ( N ) · A ( N − · · · A (1) , (5.8)and D W B , T B , T m are defined by Eqs. (4.5), (4.12)and (4.13). We set the orientation of the optic axis ofthe B (cid:48) -type layer immediately to the right of medium m to be at zero angle, so θ d is the relative orientation ofthis axis with respect to the axis of slab B . The valuesof the matrix elements of A ( j ) are given in Eqs. (C1) ofApp. C. As in the case of interacting single-layered slabs,the vdW interaction free energy per unit area is given by G = k B T π (cid:48) (cid:88) n (cid:90) dQ Q (cid:90) dψ ln Θ (sr)11 ( d, iξ n )Θ (sr)11 ( d → ∞ , iξ n ) (5.9)The matrix Θ (sr) involves a product over N matrices A ( j ) , each of which depends on the specific orientationof the optic axis of the layer. For a slab with a large number of layers, it is probablynot possible to obtain an exact closed-form result for thefree energy per unit area. We also do not have the bene-fit of Abel`es’ formula which is valid only for the productof N identical matrices. However it is still possible toobtain a relatively simple expression for the case of weakanisotropy ( γ (cid:28) δθ = 0, wewill obtain a free energy expression completely in termsof analytic functions, and equivalent to the free energyexpression we already obtained in Sec. IV.In what follows, we consider the weak anisotropyregime. In this regime, we can approximate Eqs. (4.5),(4.14), (5.5) and (5.6) by¯∆ W B ≈ − γ n θ d − ψ )) (5.10) g B ≈ γ n θ d − ψ )) (5.11)¯∆ ( j ) W B (cid:48) ≈ − γ n jδθ − ψ )) (5.12) g ( j ) B (cid:48) ≈ γ n jδθ − ψ )) (5.13)Similarly, we can approximate A ( j ) ≈ A + δ A ( j ) , (5.14)where A and δ A ( j ) are of zeroth and linear order in γ n respectively, and we can make a corresponding linear-order approximation to the matrix product N (cid:89) j =1 A ( j ) ≈ A N + B , (5.15)where B is a matrix of linear order in γ , and encodes theeffect of the dielectric anisotropy: B ≡ N (cid:88) j =1 A N − j δ A ( j ) A j − (5.16)= δ A ( N ) A N − + A δ A ( N − A N − + · · · + A N − δ A (2) A + A N − δ A (1) The matrix elements of A , δ A ( j ) and B can be com-puted and are given by Eqs. (C3), (C4) and (C5) ofApp. C. We can rewrite Θ rr in Eq. (5.2) in the followingform: Θ (sr) = Θ ( R ) · T m · Θ ( L ) , (5.17)where Θ ( R ) ≡ N (cid:89) j =1 A ( j ) D (0) W B (cid:48) T (0) B (cid:48) D (0) B (cid:48) W ; (5.18a) Θ ( L ) ≡ D W B T B D B W (5.18b)The matrix elements of Θ ( R ) and Θ ( L ) are given inEqs. (C8) in App. C. The matrix element Θ (sr)11 is givenby Θ (sr)11 = Θ ( L )11 Θ ( R )11 + Θ ( L )21 Θ ( R )12 e − Qd (5.19)1and the normalized dispersion relation is given byΘ (sr)11 ( d, iξ n )Θ (sr)11 ( d → ∞ , iξ n ) = 1 − ¯∆ (eff) W R ( iξ n ) ¯∆ (eff) W B ( iξ n ) e − Qd , (5.20)where the effective reflection coefficients correspondingto the interfaces of the left and of the right slabs withthe intervening solvent medium are given by¯∆ (eff) W R ≡ Θ ( R )12 Θ ( R )11 , ¯∆ (eff) W B ≡ − Θ ( L )21 Θ ( L )11 . (5.21)In the weak anisotropy limit, these coefficients can befurther approximated by¯∆ (eff) W R ≈ − γ n e − Qb (cid:48) sinh( Qb (cid:48) ) cos ψ − γ n e − Qb (cid:48) sinh( Qb (cid:48) )8(cosh(2 Q ( b + b (cid:48) )) − cos(2 δθ )) × (cid:2) cos(2( δθ − ψ )) − e − Q ( b + b (cid:48) ) cos(2 ψ )+ e − N +1) Q ( b + b (cid:48) ) cos(2( N δθ − ψ )) − e − NQ ( b + b (cid:48) ) cos(2(( N + 1) δθ − ψ )) (cid:3) − γ n e − Q (( N +1) b +( N +2) b (cid:48) ) sinh( Qb (cid:48) ) × sinh( N Q ( b + b (cid:48) ))sinh( Q ( b + b (cid:48) )) (5.22a)¯∆ (eff) W B ≈ − γ n e − Qb (cid:48) sinh( Qb (cid:48) ) cos ( θ d − ψ ) . (5.22b) A. van der Waals free energy
The interaction free energy per unit area is then givenby G sr = k B T π (cid:48) (cid:88) n (cid:90) π dψ (cid:90) ∞ dQ Q ln Θ (sr)11 ( d, iξ n )Θ (sr)11 ( d → ∞ , iξ n )= k B T π (cid:48) (cid:88) n (cid:90) dQ Q (cid:90) dψ ln(1 − ¯∆ (eff) W B ¯∆ (eff) W R e − Qd )= k B T π (cid:48) (cid:88) n (cid:90) dQ Q (cid:90) dψ ln(1 − ¯∆ W B ¯∆ (eff) e − Qd )(5.23) δθ = � δθ = π / �� δθ = π / � � � �� �� �� - ��� - ��� - ��� - ������ � / �� � � � ( �� - � ( � � ) � / γ � � � � ) � � � � � � � - � - � - ����� θ � τ � � ( �� - � ( � � ) � / γ � � � � ) FIG. 6: Single anisotropic layer interacting with multilayerhaving rotating optic axes (weak anisotropy, large N , and b (cid:48) = b ): behavior of G sr (Eq. (5.28)) with d for θ d = 0, and(inset) behavior of τ sr (Eq. (5.32)) with θ d for d = 10 b (cid:48) , for thefollowing values of δθ : (i) δθ = 0 (blue), (ii) δθ = π/
10 (green,dashed), and (iii) δθ = π/ τ ss for θ d = 0 (black, dot-dashed line; cf.Eq. (3.22)). Using Eqs. (5.22) and (5.10) we find for ¯∆ (eff) ¯∆ (eff) = − γ n e − Qb (cid:48) (sinh Qb (cid:48) ) (cos ψ ) − γ n (sinh Qb (cid:48) ) Q ( b + b (cid:48) )) − cos 2 δθ ) × (cid:2) e − Qb (cid:48) cos(2 δθ − ψ ) − e − Q ( b +2 b (cid:48) ) cos 2 ψ + e − Q (( N +1) b +( N +2) b (cid:48) )) cos(2 N δθ − ψ ) − e − Q ( Nb +( N +1) b (cid:48) ) cos(2( N + 1) δθ − ψ ) (cid:3) − γ n e − Q (( N +1) b +( N +3) b (cid:48) ) × (sinh Qb (cid:48) ) sinh( N Q ( b + b (cid:48) ))sinh( Q ( b + b (cid:48) )) (5.24)In the large N limit, the above simplifies to¯∆ (eff) (5.25) ≈ − γ n e − Qb (cid:48) (sinh Qb (cid:48) ) (cos ψ ) − γ n (sinh Qb (cid:48) ) Q ( b + b (cid:48) )) − cos 2 δθ ) × (cid:2) e − Qb (cid:48) cos(2 δθ − ψ ) − e − Q ( b +2 b (cid:48) ) cos 2 ψ (cid:3) − γ n e − Q ( b +3 b (cid:48) ) (sinh Qb (cid:48) ) Q ( b + b (cid:48) )) (5.26)For the case δθ = 0, the above expression for ¯∆ (eff) re-duces to Eq. (4.22), as we expect. For weak anisotropywe can also approximate the free energy per unit area toleading order by G sr ≈ − k B T π (cid:48) (cid:88) n (cid:90) dQ Q (cid:90) dψ ¯∆ W B ¯∆ (eff) e − Qd (5.27)2In the large N limit, we find G sr ≈ G ss + δG sr (5.28)where G ss is the interaction free energy per unit area oftwo thin single-layered slabs (cf. Sec. III), given by G ss ≡ − γ k B T (1 + 2(cos θ d ) )2048 π × (cid:20) d − d + b (cid:48) ) + 1( d + 2 b (cid:48) ) (cid:21) , (5.29)and δG sr is the correction from the additional layers ofthe second slab, given by δG sr ≡ − γ k B T π ( b + b (cid:48) ) (cid:20) ψ (1) (cid:18) d + b + b (cid:48) b + b (cid:48) (cid:19) − ψ (1) (cid:18) d + b + 2 b (cid:48) b + b (cid:48) (cid:19) + ψ (1) (cid:18) d + b + 3 b (cid:48) b + b (cid:48) (cid:19) (cid:21) − γ k B T π (cid:90) dQ Q (cid:20) e − Qb (cid:48) (sinh Qb (cid:48) ) cos(2( δθ − θ d ))cosh(2 Q ( b + b (cid:48) )) − cos 2 δθ − e − Q ( b +2 b (cid:48) ) (sinh Qb (cid:48) ) cos(2 θ d )cosh(2 Q ( b + b (cid:48) )) − cos 2 δθ (cid:21) e − Qd (5.30)For the case where the optic axis of each layer in thesecond slab are oriented at the same angle (i.e., δθ =0), the interaction free energy per unit area admits of aclosed-form expression: G sr ≈ − γ k B T (1 + 2(cos θ d ) )2048 π × (cid:20) d − d + b (cid:48) ) + 1( d + 2 b (cid:48) ) (cid:21) − γ k B T (1 + 2(cos θ d ) )2048 π ( b + b (cid:48) ) (cid:20) ψ (1) (cid:18) d + b + b (cid:48) b + b (cid:48) (cid:19) − ψ (1) (cid:18) d + b + 2 b (cid:48) b + b (cid:48) (cid:19) + ψ (1) (cid:18) d + b + 3 b (cid:48) b + b (cid:48) (cid:19) (cid:21) (5.31)As expected, this is in fact equivalent to Eq. (4.23),i.e., the interaction free energy per unit area of a sin-gle anisotropic layer interacting with a multilayer hav-ing optic axes all aligned. On the other hand, as weprogressively increase the thickness of the isotropic, B -type layers within the multilayered slab (i.e., let b → ∞ ),we expect to recover the interaction energy of two thinsingle-layered slabs; this is indeed the case as we see fromEq. (5.30), in which the correction δG sr →
0, and thus G sr → G ss . B. van der Waals torque
Using Eqs. (5.28), (5.29) and (5.30), we obtain thetorque per unit area τ sm for a single layered slab inter-acting with a multi-layered slab: τ sr = τ ss + δτ sr , (5.32) where δτ sr ≡ − γ k B T π (cid:90) dQ Q e − Qb (cid:48) − Qd (5.33) × (sinh Qb (cid:48) ) cos 2 δθ − cosh(2 Q ( b + b (cid:48) )) × [sin(2( δθ − θ d )) + e − Q ( b + b (cid:48) ) sin(2 θ d )] . In Fig. 6, we consider the effect of changing δθ on thevdW torque (inset) and the interaction free energy forthe case where θ d = 0. The vdW attraction is strongestand the torque has the maximum amplitude for δθ = 0,progressively becoming weaker as δθ increases to π/ δθ = 0 (shown asthe blue curve) is stronger than that for two interactingsingle anisotropic layers (shown as the black dot-dashedcurve), as we increase δθ the vdW torque weakens andcan in fact become smaller than that for the two singlelayers, as we see from the behavior for δθ = π/ δθ = 0, every anisotropic layer in the multi-layer will experience the same deviation of θ d and hencea torque with the same sign, so the overall torque thatacts on the multilayer is enhanced relative to that actingon a single anisotropic layer. Conversely, for δθ = π/
2, aperturbation of θ d will cause half the anisotropic layers inthe multilayer to experience an attractive torque and theother half to experience a repulsive torque, so the overalltorque acting on the multilayer as a whole will be smallerthan that acting on a single layer. This overall torque is not however equal to zero, because the magnitude of thetorque is different for layers at different positions, be-coming smaller for the ones that are farther away. For δθ = 0 and δθ = π/
2, the stable (unstable) angular con-figurations are those for which θ d = nπ ( θ d = ( n + ) π ),where n is integer. Here we define the stable (unstable)angular configuration as one for which the torque is zero,and the multilayer experiences an attractive (repulsive)torque when θ d is perturbed. On the other hand, as weincrease δθ from 0 to π/
2, the torque amplitude decreasesand additionally there is a “phase shift” as the angularpositions of the stable and unstable configurations takeon values different from nπ and ( n + ) π . VI. TWO INTERACTING MULTILAYERSWITH ROTATING OPTIC AXES
We can straightforwardly generalize our results tothe case of two multilayered slabs interacting across anisotropic medium. Let us consider a system consisting oftwo slabs separated by a solvent medium of thickness d .The left (right) slab has N L + 1 ( N R + 1) B (cid:48) -type lay-ers each of thickness a (cid:48) ( b (cid:48) ) and N L ( N R ) solvent layerseach of thickness a ( b ). The optic axis of the B (cid:48) -typelayer in the left (right) slab immediately adjacent to theintervening medium is oriented at an angle θ d (0). For3 . . . b d L W m B B W b b N B . . . b d W m B B b B ... ... W j L j R N R N L a a FIG. 7: Two multilayered slabs, each consisting of alternat-ing B (cid:48) -type and B -type layers, interacting across an isotropicmedium m of thickness d . The left (right) slab has N L + 1( N R + 1) B (cid:48) -type layers, and the solvent (blue) and B (cid:48) -type(brown) layers in the left (right) slab have thicknesses a and a (cid:48) ( b and b (cid:48) ) respectively. The label j L ( j R ) is an index for B (cid:48) -type layers in the left (right) slab. In the right slab, theoptic axis of the layer at j R = 0 is oriented at zero angle andthe the orientation of the optic axis of each successive layeron the right is j R δθ . In the left slab, the optic axis of thelayer at j L = 0 is oriented at θ d and the the orientation ofthe optic axis of each successive layer on the left is θ d − j L δθ .The two slabs thus have the same chirality, i.e., the optic axisrotates clockwise as one moves from left to right. simplicity, we assume that the dielectric properties of the B (cid:48) -type layers in both slabs are the same, and representthe system in terms of transfer matrices, viz., (cid:18) (cid:101) A R (cid:101) B R (cid:19) = Θ (rr) · (cid:18) (cid:101) A L (cid:19) . (6.1)Here Θ (rr) ≡ N R (cid:89) j =1 A ( j ) D (0) W B (cid:48) T (0) B (cid:48) D (0) B (cid:48) W T m · (cid:101) D (0) W B (cid:48) (cid:101) T (0) B (cid:48) (cid:101) D (0) B (cid:48) W N L (cid:89) j =1 (cid:101) A ( j ) (6.2)with (cid:101) A ( j ) ≡ T B (cid:101) D ( j ) W B (cid:48) (cid:101) T ( j ) B (cid:48) (cid:101) D ( j ) B (cid:48) W , (6.3) (cid:101) D ( j ) W B (cid:48) ≡ (cid:32) − (cid:101) ∆ ( j ) W B (cid:48) − (cid:101) ∆ ( j ) W B (cid:48) (cid:33) , (6.4) (cid:101) ∆ ( j ) W B (cid:48) ≡ − (cid:101) g ( j ) B (cid:48) (cid:101) g ( j ) B (cid:48) , (6.5)and (cid:101) g ( j ) B (cid:48) ≡ (cid:112) γ n cos ( θ d − jδθ − ψ ) (6.6) (cid:101) T ( j ) B (cid:48) ≡ (cid:18) e − Qa (cid:48) (cid:101) g ( j ) B (cid:48) (cid:19) , (6.7) (cid:101) T B ≡ (cid:18) e − Qa (cid:19) . (6.8)Above, A ( j ) , D ( j ) W B (cid:48) , ¯∆ ( j ) W B (cid:48) , g ( j ) B (cid:48) and T ( j ) B (cid:48) are given byEqs. (5.3), (5.4), (5.5), (5.6) and (5.7), respectively. The quantities D W B , T B , T m are defined by Eqs. (4.5),(4.12) and (4.13).The matrix product (cid:81) N R j =1 A ( j ) is ordered as inEq. (5.8), whereas (cid:81) N L j =1 (cid:101) A ( j ) is ordered in the contrarydirection, viz., N L (cid:89) j =1 (cid:101) A ( j ) ≡ (cid:101) A (1) · (cid:101) A (2) · · · (cid:101) A ( N L ) . (6.9)The matrix elements of (cid:101) A ( j ) are given in Eqs. (D1) ofApp. D. As in Sec. V we consider the weak anisotropyregime. We can thus approximate (cid:101) ∆ ( j ) W B (cid:48) ≈ − γ n θ d − jδθ − ψ )) (6.10) (cid:101) g ( j ) B (cid:48) ≈ γ n θ d − jδθ − ψ )) . (6.11)Similarly, we can approximate (cid:101) A ( j ) ≈ (cid:101) A + δ (cid:101) A ( j ) , (6.12)where (cid:101) A and δ (cid:101) A ( j ) are of zeroth and linear order in γ n respectively, and we can make a corresponding linear-order approximation to the matrix product N L (cid:89) j =1 (cid:101) A ( j ) ≈ (cid:101) A N L + (cid:101) B , (6.13)where (cid:101) B is a matrix of linear order in γ n : (cid:101) B ≡ N L (cid:88) j =1 (cid:101) A j − δ (cid:101) A ( j ) (cid:101) A N L − j (6.14)= δ (cid:101) A (1) A N L − + (cid:101) A δ (cid:101) A (2) (cid:101) A N L − + · · · + (cid:101) A N L − δ (cid:101) A ( N L − (cid:101) A + (cid:101) A N L − δ (cid:101) A ( N L ) The matrix elements of (cid:101) A are given by Eq. (D2), whilst δ (cid:101) A ( j ) and (cid:101) B are given by Eqs. (D3) and (D4). We canrewrite Θ rr in Eq. (6.2) as the following matrix product: Θ (rr) = Θ ( R ) · T m · Θ ( L ) , (6.15)where Θ ( R ) ≡ N R (cid:89) j =1 A ( j ) D (0) W B (cid:48) T (0) B (cid:48) D (0) B (cid:48) W ; (6.16a) Θ ( L ) ≡ (cid:101) D (0) W B (cid:48) (cid:101) T (0) B (cid:48) (cid:101) D (0) B (cid:48) W N L (cid:89) j =1 (cid:101) A ( j ) . (6.16b)The matrix elements of Θ ( R ) and Θ ( L ) are given inEqs. (D6) in App. D. The matrix element Θ (rr)11 is givenby Θ (rr)11 = Θ ( L )11 Θ ( R )11 + Θ ( L )21 Θ ( R )12 e − Qd (6.17)4 δθ = � δθ = π / �� δθ = π / � � � �� �� �� - � - � - � - � � / �� � �� ( �� - � ( � � ) � / γ � � � � ) � � � � � � � - ��� θ � τ �� ( �� - � ( � � ) � / γ � � � � ) FIG. 8: Two interacting multilayers with rotating optic axes(weak anisotropy, large N , and a (cid:48) = a = b (cid:48) = b ): behaviorof G rr (Eq. (6.21)) with d for θ d = 0, and (inset) behavior of τ rr (Eq. (6.30)) with θ d for d = 10 b (cid:48) , for the following valuesof δθ : (i) δθ = 0 (blue), (ii) δθ = π/
10 (green, dashed), and(iii) δθ = π/ τ ss for θ d = 0 (black, dot-dashed line; cf. Eq. (3.22)). and the dispersion relation is given byΘ (rr)11 ( d, iξ n )Θ (rr)11 ( d → ∞ , iξ n ) = 1 − ¯∆ (eff) W R ( iξ n ) ¯∆ (eff) W B ( iξ n ) e − Qd (6.18)where¯∆ (eff) W R ≡ Θ ( R )12 ( iξ n )Θ ( R )11 ( iξ n ) , ¯∆ (eff) W B ≡ − Θ ( L )21 ( iξ n )Θ ( L )11 ( iξ n ) . (6.19)In the weak anisotropy regime, we find that ¯∆ (eff) W R and¯∆ (eff)
W B have the approximate values given by Eqs. (D7)of App. D. In the limit of large N L and N R , these coef-ficients further simplify to¯∆ (eff) W R ≈ − γ n e − Qb (cid:48) sinh( Qb (cid:48) ) cos ψ − γ n e − Qb (cid:48) sinh( Qb (cid:48) )8(cosh(2 Q ( b + b (cid:48) )) − cos(2 δθ )) × (cid:2) cos(2( δθ − ψ )) − e − Q ( b + b (cid:48) ) cos(2 ψ ) (cid:3) − γ n e − Q ( b +2 b (cid:48) ) sinh( Qb (cid:48) )8 sinh( Q ( b + b (cid:48) )) (6.20a)¯∆ (eff) W B ≈ − γ n e − Qa (cid:48) sinh( Qa (cid:48) ) cos ( θ d − ψ ) − γ n e − Qa (cid:48) sinh( Qa (cid:48) )8(cosh(2 Q ( a + a (cid:48) )) − cos(2 δθ )) × (cid:2) cos(2( δθ − θ d + ψ )) − e − Q ( a + a (cid:48) ) cos(2( θ d − ψ )) (cid:3) − γ n e − Q ( a +2 a (cid:48) ) sinh( Qa (cid:48) )8 sinh( Q ( a + a (cid:48) )) . (6.20b) A. van der Waals interaction free energy
The interaction free energy per unit area of the inter-acting multilayered slabs is given by G rr = k B T π (cid:48) (cid:88) n (cid:90) dQ Q (cid:90) dψ ln(1 − ¯∆ (eff) W B ¯∆ (eff) W R e − Qd ) ≈ − k B T π (cid:48) (cid:88) n (cid:90) dQ Q (cid:90) dψ ¯∆ (eff) W B ¯∆ (eff) W R e − Qd = − γ k B T π (cid:90) dQ Q e − Qd (cid:2) h + h cos(2 θ d )+ h cos(2( δθ − θ d )) + h cos(2(2 δθ − θ d )) (cid:3) , (6.21)where as before γ ≡ (cid:80) (cid:48) n γ n , and h , h , h and h aregiven by Eqs. (D8) of App. D.For the case where the optic axes within each multi-layer are all aligned (i.e., δθ = 0), the free energy perunit area simplifies to G mm ≡ G rr ( δθ = 0) = − γ k B T π (1 + 2 cos θ d ) J, (6.22)with J ≡ (cid:90) ∞ dQ Q e − Qd f ( Q ) , (6.23a) f ( Q ) ≡ e Q ( a + b ) sinh( Qa (cid:48) ) sinh( Qb (cid:48) )sinh( Q ( a + a (cid:48) )) sinh( Q ( b + b (cid:48) )) . (6.23b)To find the asymptotic behavior of G mm , we note thatas Q → ∞ , f ( Q ) →
1, and as Q → f ( Q ) → a (cid:48) b (cid:48) / (( a + a (cid:48) )( b + b (cid:48) )), i.e., f ( Q ) is always of the orderof unity (assuming that the thicknesses a , a (cid:48) , b and b (cid:48) areof comparable order). The variation of the integrand of J is thus determined by the behavior of Q exp( − Qd ),which is peaked at Q = 1 / d . The dominant contribu-tion to the integral J thus comes from the modes with Q ∼ / d .Hence for large d , the dominant mode contributioncomes from Q ∼
0, where J can be approximated by J ≈ (cid:90) ∞ dQ Q e − Qd a (cid:48) b (cid:48) ( a + a (cid:48) )( b + b (cid:48) ) = a (cid:48) b (cid:48) a + a (cid:48) )( b + b (cid:48) ) d , (6.24)and thus G mm ≈ − γ k B T (1 + 2 cos θ d ) a (cid:48) b (cid:48) π ( a + a (cid:48) )( b + b (cid:48) ) d , (6.25)which corresponds to a force per unit area that decayswith d − , viz., F mm ≈ − γ k B T (1 + 2 cos θ d ) a (cid:48) b (cid:48) π ( a + a (cid:48) )( b + b (cid:48) ) d . (6.26)In the other limit where d is much smaller than the layerthicknesses, f ( Q ) is dominated by Q ∼ / d , and thus5we can approximate f ( Q ) ≈ e a + b d sinh( a (cid:48) d ) sinh( b (cid:48) d )sinh( a + a (cid:48) d ) sinh( b + b (cid:48) d ) ≈ , J ≈ d , (6.27)and thus G mm ≈ − γ k B T (1 + 2 cos θ d )2048 πd . (6.28)This also corresponds to a force per unit area that decayswith d − , which is given by F mm ≈ − γ k B T (1 + 2 cos θ d )1024 πd . (6.29)Thus in both the large and small d limits, G mm decays as d − , and in the small d limit, G mm approximates to thevalue of G ss , which is the free energy per unit area of twothick anisotropic slabs, which is physically reasonable.The behavior of the free energy per unit area for twointeracting multilayers is shown in Fig. 8. We see thatthe attraction is strongest when δθ = 0, and weak-est when δθ = π/
2. Comparing with the analogouscurves in Fig. 6, we also note that the vdW attractionfor two interacting multilayers is stronger than that fora single anisotropic layer interacting with a multilayer.This behavior is consistent with the idea that additionalanisotropic layers (in the presence of a multilayer) in-crease the extent over which dielectric contrast occurs,thus contributing to an increase in the vdW attraction.
B. van der Waals torque
From Eq. (6.21) we obtain the vdW torque per unitarea for two interacting multi-layered slabs, τ mm : τ rr = − k B T π (cid:90) dQ Q e − Qd (cid:2) h sin(2 θ d ) (6.30)+ h sin(2( θ d − δθ )) + h sin(2( θ d − δθ )) (cid:3) . For the case where the optic axes within each multilayerare all aligned (i.e., δθ = 0), the vdW torque per unitarea simplifies to τ mm ≡ τ rr ( δθ = 0) = − ∂G mm ∂θ d ≈ − γ k B T sin(2 θ d ) J π , (6.31)where J is defined by Eq. (6.23a). Similar to the case of G mm , we can again determine the asymptotic behaviorof τ mm . For large d , we find τ mm ≈ − γ k B T sin(2 θ d ) a (cid:48) b (cid:48) π ( a + a (cid:48) )( b + b (cid:48) ) d , (6.32)whereas for small d , we find τ mm ≈ − γ k B T sin(2 θ d )1024 πd . (6.33) The behavior of the vdW torque is shown in Fig. 8. Anal-ogous to what we have observed for the vdW torquebetween a single anisotropic layer and a multilayer inSec. V B, the vdW torque between two multilayers is alsostrongest for δθ = 0 and weakest for δθ = π/
2, and suchbehavior can be similarly understood using the qualita-tive explanations given in that section. If we comparewith the curves for the vdW torque in Fig. 6, we see thatthe vdW torque for two multilayers is enhanced relativeto that for a single layer interacting with a multilayer if δθ = 0, and relatively reduced if δθ = π/
2. Finally, wenote that the “phase shift” is more pronounced for thecase of two multilayers.
VII. OPTIC AXIS PERPENDICULAR TOPLANE OF ANISOTROPIC LAYERS
We now turn our attention to the case where the opticaxis of each dielectrically anisotropic, uniaxial layer is perpendicular (rather than parallel) to the plane of thelayer, and the layers are “stacked” co-axially as before.In this case the optic axes of the anisotropic layers areall parallel, and there is no vdW torque. The dielectrictensor in diagonal form is given by ε (prin) B (cid:48) = ε ⊥ ε ⊥
00 0 ε || , (7.1)where ε ⊥ ( ε || ) is the dielectric permittivity in a directionperpendicular to (parallel with) the optic axis, and n labels the Matsubara frequencies, ξ n = 2 πk B T n/ (cid:126) .As in Sec II B we start from the Laplace equationEq. (2.9) with Eq. (7.1), obtaining Eq. (2.11) with ρ i = √ u + v ≡ Q if layer i is the solvent, and ρ i = ρ B (cid:48) ifthe layer is B (cid:48) -type, where ρ B (cid:48) is now given by ρ B (cid:48) ≡ (cid:114) ε ⊥ ε || Q (7.2)The reflection coefficient for the dielectric discontinu-ity at the solvent- B (cid:48) -type interface can be found fromEq. (2.17), where now¯∆ W B (cid:48) = ε W Q − ε || ρ B (cid:48) ε W Q + ε || ρ B (cid:48) = 1 − g n ( ε ⊥ /ε W )1 + g n ( ε ⊥ /ε W ) ≡ ¯∆ n , (7.3)where g n ≡ ( ε || /ε ⊥ ) / = ( ε B (cid:48) zz /ε B (cid:48) xx ) / . Similarto Ref. [11], we make the simplifying assumption that ε ⊥ ,n = ε W , from which we obtain¯∆ n = 1 − g n g n . (7.4)6We can write g n = 1 + δg n , where δg n ≡ ( ε || /ε ⊥ ) / − δg n (cid:28) n can be approximated by¯∆ n ≈ − δg n . (7.5) A. Two interacting single layers
We first consider the case of two single uniaxial layersof the same thickness b (cid:48) , interacting across a solvent layerof thickness d . In this case the analogue of Eq. (3.2) isgiven by Θ (ss) ≡ D W B (cid:48) T B (cid:48) D B (cid:48) W T m D W B (cid:48) T B (cid:48) D B (cid:48) W (7.6)In the above the matrices are given by D W B (cid:48) ≡ (cid:18) − ¯∆ n − ¯∆ n (cid:19) , (7.7)where T B (cid:48) ≡ (cid:18) e − Qb (cid:48) g − n (cid:19) , T m ≡ (cid:18) e − Qd (cid:19) . (7.8)We findΘ (ss)11 ( iξ n ) = (1 − ¯∆ n e − Qb (cid:48) /g n ) − ¯∆ n (1 − e − Qb (cid:48) /g n ) e − Qd (7.9)The free energy per unit area is given by G ⊥ = k B T π (cid:48) (cid:88) n (cid:90) ∞ dQ Q ln Θ (ss)11 ( d, iξ n )Θ (ss)11 ( d → ∞ , iξ n ) . (7.10)Specializing to the weak anisotropy regime, i.e., δg n (cid:28) G ⊥ ≈ − δg k B T π (cid:20) d − d + b (cid:48) ) + 1( d + 2 b (cid:48) ) (cid:21) , (7.11)where δg ≡ (cid:80) (cid:48) n δg n , and (cid:80) (cid:48) n is a sum running from n = 0 to n = ∞ , but with the n = 0 multiplied byan additional factor of 1 /
2. The corresponding force perunit area is given by F ⊥ ≈ − δg k B T π (cid:20) d − d + b (cid:48) ) + 1( d + 2 b (cid:48) ) (cid:21) . (7.12)The decay behavior of G ⊥ is essentially the same as thatof two single layers with optic axes parallel to the planeof the layers (cf. Eq. (3.15)).If we consider the high temperature limit, such that iξ n → ε || , ε ⊥ →
1) for n (cid:54) = 0, then we canmake a convenient comparison with the case of two in-teracting single anisotropic layers whose optic axes liein the plane of the layers, viz., Eq. (3.16). Let uswrite α ≡ ( ε opt /ε nopt ) / , where ε opt ( ε nopt ) denotes theprincipal dielectric permittivity along (perpendicular to) the optic axis, so that in the limit of weak anisotropy, α ≈
1. For a system where the optic axis is paral-lel to the x -axis of (and parallel to the plane of) thereference B (cid:48) -type layer, α = ( ε B (cid:48) xx, /ε B (cid:48) zz, ) / , and γ → γ = ( α − ≈ α − . On the other hand,for a system where the optic axis is perpendicular to theplane of the layer, which we take to be parallel to the z -axis, α = ( ε B (cid:48) zz, /ε B (cid:48) xx, ) / , and δg → δg = ( α − .Taking G ss from Eq. (3.16) for the former system and G ⊥ from Eq. (7.11) for the latter system, we find G ⊥ G ss → δg γ (1 + 2 cos θ d ) = 81 + 2 cos θ d (7.13)For weak anisotropy and at high temperature, two singleanisotropic layers thus attract each other more stronglywhen their optic axes are oriented perpendicular to theplane of the layers than when the optic axes are parallelto the plane. B. Single layer interacting with multilayer
Next, we consider a single anisotropic layer interactingwith a stack of N + 1 anisotropic layers across an in-tervening isotropic medium. The matrices Θ (sm) and A are still expressed by the formulas Eqs. (4.2) and (4.3),where now the matrices are given by D W B (cid:48) = D W B ≡ (cid:18) − ¯∆ n − ¯∆ n (cid:19) , (7.14a) D B (cid:48) W = D B W ≡ (cid:18) n ¯∆ n (cid:19) , (7.14b) T B ≡ (cid:18) e − Qb (cid:19) , (7.14c) T B (cid:48) = T B ≡ (cid:18) e − Qb (cid:48) g − n (cid:19) , (7.14d) T m ≡ (cid:18) e − Qd (cid:19) . (7.14e)The elements of the matrix A n are given by A = 1 − e − Qb (cid:48) g − n ¯∆ n , (7.15a) A = e − Qb (1 − e − Qb (cid:48) g − n ) ¯∆ n , (7.15b) A = − (1 − e − Qb (cid:48) g − n ) ¯∆ n , (7.15c) A = e − Qb ( e − Qb (cid:48) g − n − ¯∆ n ) (7.15d)and the determinant is | A | = (1 − ¯∆ n ) e − Q ( b + b (cid:48) g − n ) (7.16)The matrix product A N can be found using Abel`es’ for-mula (see, e.g., Ref. [1]), which gives A N = (cid:32) A ( N )11 A ( N )12 A ( N )21 A ( N )22 (cid:33) (7.17)7where A ( N )11 ≡ (cid:18) A (cid:112) | A | U N − − U N − (cid:19) | A | N/ , (7.18a) A ( N )12 ≡ A U N − | A | ( N − / , (7.18b) A ( N )21 ≡ A U N − | A | ( N − / , (7.18c) A ( N )22 ≡ (cid:18) A (cid:112) | A | U N − − U N − (cid:19) | A | N/ (7.18d)Here U N are the Chebyshev polynomials that we alreadyencountered in Eq. (4.17). For weak anisotropy, we canapproximate ξ (defined in Eq. (4.18)) by ξ ≈ Q ( b + b (cid:48) ) − δg n Qb (cid:48) . (7.19)We can compute Θ (sm)11 from Eq. (4.2). For weakanisotropy ( ¯∆ n ≈ − δg n / (cid:28)
1) we findΘ (sm)11 ( d, iξ n )Θ (sm)11 ( d → ∞ , iξ n ) (7.20) ≈ − δg n sinh ( Qb (cid:48) ) e − Qb (cid:48) − Qd − sinh(( N − Q ( b + b (cid:48) ))sinh( NQ ( b + b (cid:48) )) e − Q ( b + b (cid:48) ) × (cid:34) e − Q ( b + b (cid:48) ) − e − Q ( b + b (cid:48) ) sinh(( N − Q ( b + b (cid:48) ))sinh( N Q ( b + b (cid:48) )) (cid:35) In the limit of large N , the above simplifies toΘ (sm)11 ( d, iξ n )Θ (sm)11 ( d → ∞ , iξ n ) ≈ − δg n sinh ( Qb (cid:48) ) e − Qb (cid:48) − Qd − e − Q ( b + b (cid:48) ) . (7.21)In the regime of weak anisotropy and for large N , theinteraction free energy per unit area is thus given by G ⊥ = k B T π (cid:48) (cid:88) n (cid:90) ∞ dQ Q ln Θ (sm)11 ( d, iξ n )Θ (sm)11 ( d → ∞ , iξ n ) ≈ − δg k B T π ( b + b (cid:48) ) (cid:104) ψ (1) (cid:18) db + b (cid:48) (cid:19) − ψ (1) (cid:18) d + b (cid:48) b + b (cid:48) (cid:19) + ψ (1) (cid:18) d + 2 b (cid:48) b + b (cid:48) (cid:19) (cid:105) , (7.22)where (cid:80) (cid:48) n is a sum running from n = 0 to n = ∞ , butwith the n = 0 multiplied by an additional factor of 1 / δg ≡ (cid:80) (cid:48) n δg n . The corresponding force per unitarea is given by F ⊥ ≈ δg k B T π ( b + b (cid:48) ) (cid:104) ψ (2) (cid:18) db + b (cid:48) (cid:19) − ψ (2) (cid:18) d + b (cid:48) b + b (cid:48) (cid:19) + ψ (2) (cid:18) d + 2 b (cid:48) b + b (cid:48) (cid:19) (cid:105) , (7.23)where ψ (2) ( z ) is the second derivative of the digammafunction ψ ( z ). The decay behavior of G ⊥ is qualitativelythe same as that for the corresponding system with optic axes all parallel to the plane of the layers, in which the op-tic axes in the multilayer are all aligned (cf. Eq. (4.23)).In the high temperature limit we can compare thestrengths of the van der Waals attraction for the cases ofoptic axes aligned parallel to the plane of the anisotropiclayers, viz., G sm from Eq. (4.23), and optic axes alignedperpendicular to the plane of the layers, viz., G ⊥ from Eq. (7.22). In this case, defining again α ≡ ( ε opt /ε nopt ) / , we have γ → γ ≈ α − and δg → δg = ( α − . We find G ⊥ G sm → δg γ (1 + 2 cos θ d ) = 81 + 2 cos θ d , (7.24)i.e., the same value that we found in Eq. (7.13) for thecase of two interacting single anisotropic layers. C. Two interacting multilayers
For the case of two interacting multi-layered slabs sep-arated by a solvent layer of thickness d , the correspondingtransfer matrix is given by Θ (mm) = Θ ( R ) · T m · Θ ( L ) , (7.25)where Θ ( R ) ≡ A N R D W B (cid:48) T B (cid:48) D B (cid:48) W , (7.26a) Θ ( L ) ≡ D W B (cid:48) (cid:101) T B (cid:48) D B (cid:48) W (cid:101) A N L , (7.26b) A ≡ D W B (cid:48) T B (cid:48) D B (cid:48) W T B , (7.26c) (cid:101) A ≡ (cid:101) T B D W B (cid:48) (cid:101) T B (cid:48) D B (cid:48) W (7.26d)The matrices D W B (cid:48) , D B (cid:48) W , T B , T B (cid:48) and T m are intro-duced in Eqs. (7.14), and the matrices (cid:101) T B and (cid:101) T B (cid:48) aredefined by (cid:101) T B ≡ (cid:18) e − Qa (cid:19) , (7.27a) (cid:101) T B (cid:48) ≡ (cid:18) e − Qa (cid:48) g − n (cid:19) (7.27b)The elements of matrix A are given in Eqs. (7.15), whilstthose of matrix (cid:101) A are given by (cid:101) A = 1 − ¯∆ n e − Qa (cid:48) g − n ; (7.28a) (cid:101) A = ¯∆ n (1 − e − Qa (cid:48) g − n ); (7.28b) (cid:101) A = − ¯∆ n (1 − e − Qa (cid:48) g − n ) e − Qa ; (7.28c) (cid:101) A = ( e − Qa (cid:48) g − n − ¯∆ n ) e − Qa (7.28d)The corresponding determinant is | (cid:101) A | = (1 − ¯∆ n ) e − Q ( a + a (cid:48) g − n ) . (7.29)As in Sec. VI, the matrix element Θ is given byΘ (mm)11 = Θ ( L )11 Θ ( R )11 + Θ ( L )21 Θ ( R )12 e − Qd , (7.30)8from which we deduceΘ (mm)11 ( d, iξ n )Θ (mm)11 ( d → ∞ , iξ n ) = 1 − ¯∆ (eff) W R ( iξ n ) ¯∆ (eff) W B ( iξ n ) e − Qd , (7.31)where ¯∆ (eff) W B and ¯∆ (eff) W R are the effective reflection coeffi-cients for the left and right slabs, defined by¯∆ (eff)
W R ≡ Θ ( R )12 Θ ( R )11 , ¯∆ (eff) W B ≡ − Θ ( L )21 Θ ( L )11 . (7.32)For weak anisotropy, these coefficients can be approxi-mated by¯∆ (eff) W R ≈− δg n e − Qb (cid:48) sinh( Qb (cid:48) )1 − e − Q ( b + b (cid:48) ) sinh(( N R − Q ( b + b (cid:48) ))sinh( N R Q ( b + b (cid:48) )) (cid:20) e − Q ( b + b (cid:48) ) − e − Q ( b + b (cid:48) ) sinh(( N R − Q ( b + b (cid:48) ))sinh( N R Q ( b + b (cid:48) )) (cid:21) , (7.33a)¯∆ (eff) W B ≈− δg n e − Qa (cid:48) sinh( Qa (cid:48) )1 − e − Q ( a + a (cid:48) ) sinh(( N L − Q ( a + a (cid:48) ))sinh( N L Q ( a + a (cid:48) )) (cid:20) e − Q ( a + a (cid:48) ) − e − Q ( a + a (cid:48) ) sinh(( N L − Q ( a + a (cid:48) ))sinh( N L Q ( a + a (cid:48) )) (cid:21) . (7.33b)For large N L and N R , we can approximate¯∆ (eff) W R ≈ − δg n e Qb sinh( Qb (cid:48) )2 sinh( Q ( b + b (cid:48) )) , (7.34a)¯∆ (eff) W B ≈ − δg n e Qa sinh( Qa (cid:48) )2 sinh( Q ( a + a (cid:48) )) . (7.34b)The interaction free energy per unit area is given by G ⊥ (7.35)= k B T π (cid:48) (cid:88) n (cid:90) ∞ dQ Q ln(1 − ¯∆ (eff) W B ¯∆ (eff) W R e − Qd ) ≈ − δg k B T π (cid:90) ∞ dQ Q e Q ( a + b ) − Qd sinh( Qa (cid:48) ) sinh( Qb (cid:48) )sinh( Q ( a + a (cid:48) )) sinh( Q ( b + b (cid:48) )) . Up to a prefactor this is the same free energy expressionas that for two interacting multilayers with optic axesparallel to the plane of the layers, in which the opticaxes in a given multilayer are all aligned (cf. Eq. (6.22)).Thus using similar arguments G ⊥ decays with d − , andthe corresponding force F ⊥ decays with d − .For the case of high temperature, we can again com-pare the van der Waals interaction strengths for thecases of optic axes aligned perpendicular to the planeof the anisotropic layers, i.e., G ⊥ from Eq. (7.35), andoptic axes aligned parallel to the plane of the layers,viz., G mm from Eq. (6.22). Using α ≡ ( ε opt /ε nopt ) / , γ → γ ≈ α − and δg → δg = ( α − , we find G ⊥ G mm → δg γ (1 + 2 cos θ d ) = 81 + 2 cos θ d , (7.36) which is the same result that we found for the case oftwo interacting single anisotropic layers, Eq. (7.13), andthe case of a single anisotropic layer interacting with ananisotropic multilayer whose optic axes are all aligned,Eq. (7.24). VIII. SUMMARY AND CONCLUSION
In this Paper we have studied the behavior of the vander Waals (vdW) torque and interaction free energy ofdielectrically anisotropic layered media, in the regime ofweak dielectric anisotropy, no retardation, and where thedielectric coefficients of the ordinary axes of the uniax-ial crystal layers match the dielectric permittivity of thesolvent medium. In particular we have examined the be-havior of the following three systems: (i) two interactingsingle anisotropic layers, (ii) a single anisotropic layerinteracting with an anisotropic multilayered slab, and(iii) two interacting anisotropic multilayered slabs. Wehave considered these systems in the following two cases:(a) the optic axes lie in the plane of the layers, and (b) theoptic axes are perpendicular to the plane of the layers.For case (a), we have considered two further scenarios,one where all the optic axes of the anisotropic layers ina given multilayer are aligned, and the other where theoptic axes undergo constant angular increments δθ acrossthe multilayer.We summarize our results for case (a) as follows. Wefound that increasing the thicknesses of the anisotropiclayers has the effect of increasing the magnitude of thevdW interaction free energy and torque. Moreover, wefound that the vdW attraction is strongest for two inter-acting multilayers and weakest for two interacting singleanisotropic layers, for all values of δθ . On the other hand,the amplitude of the vdW torque is largest for two mul-tilayers and smallest for two single layers when δθ = 0,but the torque amplitude is smallest for two multilay-ers and largest for two single layers when δθ = π/
2. Inaddition, the angle θ d (the relative orientation betweenthe optic axes of the oppositely facing anisotropic layersof the two interacting layered media) at which the lay-ered media are in a stable (unstable) configuration of zerooverall torque is an integer (half-integer) factor of π for δθ = 0 and δθ = π/
2, but moves away from these valuesas we tune δθ from 0 to π/
2. We have also determinedthe asymptotic behaviors of the vdW free energy andtorque for the three systems in the case where δθ = 0.For separations that are much larger than the thicknessof each anisotropic layer, we found that the free energyand torque decay as d − in the case of system (i), d − in the case of (ii), and d − in the case of (iii). On theother hand, if the separation is much smaller than thelayer thicknesses, the free energy and torque of all threesystems approach those corresponding to two very thickanisotropic layers, decaying with d − .For case (b) (optic axes directed perpendicular to theplane of the layers), we have found that the free energies9have the same decay behaviors as those in case (a). Inthe high temperature limit, we have found that the freeenergies for (b) are larger than those for (a) by the samefactor, viz., 8 / (1 + 2 cos θ d ).Although the vdW torque has been analysed and cal-culated in different setups, a direct experiment - thoughin principle feasible - is still sorely lacking [44, 45]. Webelieve that multilayered systems, of the type analysedhere, are probably the most straightforward option for anexperimental confirmation of this less commonly appre-ciated feature of vdW interactions. In particular, liquidcrystalline arrays of the smectic C* type should provepotentially relevant for this endeavour as they can self-assemble from the solution and their properties can becontrolled by macroscopic fields. In smectic C* arraysthe director makes a tilt angle with respect to the smecticlayer that furthermore rotates from layer to layer form-ing a helix, implying furthermore also rotating optic axesthat could be controlled by temperature or other exter-nal fields and fine tuned for the different experimentalsetups. This multilayer configuration would be in addi-tion directly describable by the formalism derived anddeveloped above.The approach and analysis described in this workopens up further possible avenues of investigation. Anobvious extension would be to include the effects of re-tardation, though formally this could be quite demanding[16] as even two semi-infinite layers lead to very unwield-ing formulae [12, 13, 17]. Another line of inquiry is toexplore the effects of random disorder in the alignmentsof the optic axes on the vdW torque and interaction ingeneral, along similar lines as for the case of a disor-dered isotropic dielectric function [46]. Yet another areaof research can be to study the nanolevitation of plane-parallel multilayers caused by vdW repulsion in real sys-tems that could be controlled by macroscopic externalfields that would manipulate the degree of anisotropy. Inall these listed cases the compendium of results describedin this work would be of significant value. IX. ACKNOWLEDGMENTS
BSL thanks J. Munday, J. F. Dobson, B. Guizal, A.Gambassi, and V. Esteso for constructive discussions.The authors acknowledge support from the SloveneAgency of Research and Development (ARRS) throughGrant No. P1-0055.
Appendix A: The element Θ (ss)11 Here we show the explicit formula for the element Θ (ss)11 :Θ (ss)11 (A1)= 1 + e − Qb (cid:48) g B ¯∆ B W ¯∆ W B +[ e − Qd ¯∆ B W + e − Q ( d + b (cid:48) g B ) ¯∆ W B ] ¯∆ W B + (cid:2) e − Q ( d + b (cid:48) g B ) ¯∆ B W + e − Qb (cid:48) g B ¯∆ W B + e − Q ( d + b (cid:48) ( g B + g B )) ¯∆ W B + e − Q ( b (cid:48) ( g B + g B )) × ¯∆ B W ¯∆ W B ¯∆ W B (cid:3) ¯∆ B W Appendix B: Single anisotropic layer interactingwith multilayer having aligned optic axes
The elements of the matrix A defined in Eq. (4.3) arefound to be A = 1 − e − Qb (cid:48) g B (cid:48) ¯∆ W B (cid:48) , (B1a) A = e − Qb (1 − e − Qb (cid:48) g B (cid:48) ) ¯∆ W B (cid:48) , (B1b) A = − (1 − e − Qb (cid:48) g B (cid:48) ) ¯∆ W B (cid:48) , (B1c) A = e − Qb ( e − Qb (cid:48) g B (cid:48) − ¯∆ W B (cid:48) ) , (B1d)and the determinant is | A | = (1 − ¯∆ W B (cid:48) ) e − Q ( b + b (cid:48) g B (cid:48) ) (B2)The coefficients in the numerator of ¯∆ (eff) in Eq. (4.21)are given by s ≡ − e − Qb (cid:48) g B (cid:48) A , (B3a) s ≡ w (cid:112) | A | − A ) e − Qb (cid:48) g B (cid:48) sinh( Qb (cid:48) g B (cid:48) ) , (B3b) s ≡ A (B3c)and coefficients in the denominator are given by t ≡ w (cid:112) | A | − A , (B4a) t ≡ A e − Qb (cid:48) g B (cid:48) sinh( Qb (cid:48) g B (cid:48) ) , (B4b) t ≡ ( A − w (cid:112) | A | ) e − Qb (cid:48) g B (cid:48) (B4c)In the above w ≡ U N − /U N − . Appendix C: Single anisotropic layer interactingwith multilayer having rotating optic axes
The elements of the matrix A ( j ) in Eq. (5.3) are givenbelow: A ( j )11 = 1 − e − Qb (cid:48) g ( j ) B (cid:48) ( ¯∆ ( j ) W B (cid:48) ) , (C1a) A ( j )12 = e − Qb (1 − e − Qb (cid:48) g ( j ) B (cid:48) ) ¯∆ ( j ) W B (cid:48) , (C1b) A ( j )21 = − (1 − e − Qb (cid:48) g ( j ) B (cid:48) ) ¯∆ ( j ) W B (cid:48) , (C1c) A ( j )22 = e − Qb ( e − Qb (cid:48) g ( j ) B (cid:48) − ( ¯∆ ( j ) W B (cid:48) ) ) (C1d)0and the determinant is | A ( j ) | = (1 − ( ¯∆ ( j ) W B (cid:48) ) ) e − Q ( b + b (cid:48) g ( j ) B (cid:48) ) (C2)The matrix elements of A , δ A ( j ) and B introducedEqs. (5.14) and (5.15) are given below: A , = 1 , A , = A , = 0 , A , = e − Q ( b + b (cid:48) ) (C3) δA ( j )11 = 0 , (C4a) δA ( j )12 = − γ n jδθ − ψ )) e − Q (2 b + b (cid:48) ) sinh( Qb (cid:48) ) , (C4b) δA ( j )21 = γ n jδθ − ψ )) e − Qb (cid:48) sinh( Qb (cid:48) ) , (C4c) δA ( j )22 = − γ n (cos( jδθ − ψ )) Qb (cid:48) e − Q ( b + b (cid:48) ) (C4d) B = 0 , (C5a) B = − γ n e Qb (cid:48) − NQ ( b + b (cid:48) ) sinh( Qb (cid:48) ) × V N ( − Q ( b + b (cid:48) ) , δθ, ψ ) , (C5b) B = γ n e − Qb (cid:48) sinh( Qb (cid:48) ) V N ( Q ( b + b (cid:48) ) , δθ, ψ ) , (C5c) B = − γ n Qb (cid:48) e − NQ ( b + b (cid:48) ) ( N + P N ( δθ )) (C5d)In the above, the functions V N ( t, δθ, ψ ) and P N ( δθ ) aredefined by V N ( t, δθ, ψ ) ≡ N (cid:88) j =1 (cos( jδθ − ψ )) e − N − j ) t = 14(cosh 2 t − cos 2 δθ ) (cid:2) e t cos(2( N δθ − ψ )) − cos(2(( N + 1) δθ − ψ )) − e − N − t cos 2 ψ +2 e − ( N − t (cosh 2 t − cos 2 δθ ) sinh N t sinh t + e − Nt cos(2( δθ − ψ )) (cid:3) , (C6) P N ( δθ ) ≡ N (cid:88) j =1 (cos( jδθ − ψ )) − N = cos(( N + 1) δθ − ψ ) sin N δθ sin δθ (C7) V N can be verified e.g. by setting δθ = 0 and ψ = 0 andevaluating the sum on the LHS and the formula on theRHS, and seeing that they agree.The transfer matrix elements in Eqs. (5.18) are given byΘ ( R )11 = 1 − B ¯∆ (0) W B (cid:48) + ( B − ¯∆ (0) W B (cid:48) ) ¯∆ (0)
W B (cid:48) e − Qb (cid:48) g (0) B (cid:48) ;(C8a)Θ ( R )12 = ¯∆ (0) W B (cid:48) (1 − B ¯∆ (0) W B (cid:48) ) + ( B − ¯∆ (0) W B (cid:48) ) e − Qb (cid:48) g (0) B (cid:48) ;(C8b)Θ ( R )21 = B − ( A N , + B ) ¯∆ (0) W B (cid:48) (C8c)+ ¯∆ (0)
W B (cid:48) ( A N , + B − B ¯∆ (0) W B (cid:48) ) e − Qb (cid:48) g (0) B (cid:48) ;Θ ( R )22 = ( A N , + B − B ¯∆ (0) W B (cid:48) ) e − Qb (cid:48) g (0) B (cid:48) + ¯∆ (0) W B (cid:48) ( B − ( A N , + B ) ¯∆ (0) W B (cid:48) ) (C8d)Θ ( L )11 = 1 − ¯∆ W B e − Qb (cid:48) g B ; (C8e)Θ ( L )12 = (1 − e − Qb (cid:48) g B ) ¯∆ W B ; (C8f)Θ ( L )21 = − (1 − e − Qb (cid:48) g B ) ¯∆ W B ; (C8g)Θ ( L )22 = e − Qb (cid:48) g B − ¯∆ W B (C8h) Appendix D: Two interacting multilayers withrotating optic axes
The matrix elements of (cid:101) A ( j ) in Eq. (6.3) are given by (cid:101) A ( j )11 = 1 − e − Qa (cid:48) (cid:101) g ( j ) B (cid:48) ( (cid:101) ∆ ( j ) W B (cid:48) ) , (D1a) (cid:101) A ( j )12 = (1 − e − Qa (cid:48) (cid:101) g ( j ) B (cid:48) ) (cid:101) ∆ ( j ) W B (cid:48) , (D1b) (cid:101) A ( j )21 = − e − Qa (1 − e − Qa (cid:48) (cid:101) g ( j ) B (cid:48) ) (cid:101) ∆ ( j ) W B (cid:48) , (D1c) (cid:101) A ( j )22 = e − Qa ( e − Qa (cid:48) (cid:101) g ( j ) B (cid:48) − ( (cid:101) ∆ ( j ) W B (cid:48) ) ) (D1d)The elements of the matrix (cid:101) A in Eq. (6.12) are given by (cid:101) A , = 1 , (cid:101) A , = (cid:101) A , = 0 , (cid:101) A , = e − Q ( a + a (cid:48) ) , (D2)whilst those of matrices δ (cid:101) A ( j ) (Eq. (6.12)) and (cid:101) B (Eq. (6.14)) are given by Eqs. (D3) and (D4): δ (cid:101) A ( j )11 = 0 , (D3a) δ (cid:101) A ( j )12 = − γ n θ d − jδθ − ψ )) e − Qa (cid:48) sinh( Qa (cid:48) ) , (D3b) δ (cid:101) A ( j )21 = γ n θ d − jδθ − ψ )) e − Q (2 a + a (cid:48) ) sinh( Qa (cid:48) ) , (D3c) δ (cid:101) A ( j )22 = − γ n (cos( θ d − jδθ − ψ )) Qa (cid:48) e − Q ( a + a (cid:48) ) (D3d)1 (cid:101) B = 0 , (D4a) (cid:101) B = − γ n e − Qa (cid:48) sinh( Qa (cid:48) ) × V N L ( Q ( a + a (cid:48) ) , δθ, θ d − ψ ) , (D4b) (cid:101) B = γ n e Qa (cid:48) − N L Q ( a + a (cid:48) ) sinh( Qa (cid:48) ) × V N L ( − Q ( a + a (cid:48) ) , δθ, θ d − ψ ) , (D4c) (cid:101) B = − γ n Qa (cid:48) e − N L Q ( a + a (cid:48) ) ( N L + (cid:101) P N L ( δθ ))(D4d)In the above, the functions V N ( t, δθ, ψ ) is defined byEq. (C6), and (cid:101) P N ( δθ ) is defined by (cid:101) P N ( δθ ) ≡ N (cid:88) j =1 (cos( θ d − jδθ − ψ )) − N = cos(2 θ d − ( N + 1) δθ − ψ ) sin N δθ sin δθ (D5)The matrix elements of Θ ( R ) and Θ ( L ) in Eqs. (6.16) aregiven byΘ ( R )11 = 1 − B ¯∆ (0) W B (cid:48) + ( B − ¯∆ (0) W B (cid:48) ) ¯∆ (0)
W B (cid:48) e − Qb (cid:48) g (0) B (cid:48) ;(D6a)Θ ( R )12 = ¯∆ (0) W B (cid:48) (1 − B ¯∆ (0) W B (cid:48) ) + ( B − ¯∆ (0) W B (cid:48) ) e − Qb (cid:48) g (0) B (cid:48) ;(D6b)Θ ( R )21 = B − ( A N R , + B ) ¯∆ (0) W B (cid:48) + ¯∆ (0)
W B (cid:48) ( A N R , + B − B ¯∆ (0) W B (cid:48) ) e − Qb (cid:48) g (0) B (cid:48) ; (D6c)Θ ( R )22 = ( A N R , + B − B ¯∆ (0) W B (cid:48) ) e − Qb (cid:48) g (0) B (cid:48) + ¯∆ (0) W B (cid:48) ( B − ( A N R , + B ) ¯∆ (0) W B (cid:48) ) (D6d)Θ ( L )11 = 1 + (cid:101) B (cid:101) ∆ (0) W B (cid:48) − (cid:101) ∆ (0) W B (cid:48) ( (cid:101) B + (cid:101) ∆ (0) W B (cid:48) ) e − Qa (cid:48) (cid:101) g (0) B (cid:48) ;(D6e)Θ ( L )12 = (cid:101) B + (cid:101) ∆ (0) W B (cid:48) ( (cid:101) A N L , + (cid:101) B ) − (cid:101) ∆ (0) W B (cid:48) ( (cid:101) B (cid:101) ∆ (0) W B (cid:48) + (cid:101) A N L , + (cid:101) B ) e − Qa (cid:48) (cid:101) g (0) B (cid:48) (D6f)Θ ( L )21 = − (cid:101) ∆ (0) W B (cid:48) (1 + (cid:101) B (cid:101) ∆ (0) W B (cid:48) ) + ( (cid:101) B + (cid:101) ∆ (0) W B (cid:48) ) e − Qa (cid:48) (cid:101) g (0) B (cid:48) (D6g)Θ ( L )22 = ( (cid:101) A N L , + (cid:101) B + (cid:101) B (cid:101) ∆ (0) W B (cid:48) ) e − Qa (cid:48) (cid:101) g (0) B (cid:48) − (cid:101) ∆ (0) W B (cid:48) ( (cid:101) B + ( (cid:101) A N L , + (cid:101) B ) (cid:101) ∆ (0) W B (cid:48) ) (D6h) In the above, the values of B , B and B are given byEqs. (C5) with N → N R . In the weak anisotropy regime,the effective reflection coefficients ¯∆ (eff) W R and ¯∆ (eff)
W B de-fined by Eq. (6.19) can be approximated by¯∆ (eff) W R ≈ − γ n e − Qb (cid:48) sinh( Qb (cid:48) ) cos ψ − γ n e − Qb (cid:48) sinh( Qb (cid:48) )8(cosh(2 Q ( b + b (cid:48) )) − cos(2 δθ )) × (cid:2) cos(2( δθ − ψ )) − e − Q ( b + b (cid:48) ) cos(2 ψ )+ e − N R +1) Q ( b + b (cid:48) ) cos(2( N R δθ − ψ )) − e − N R Q ( b + b (cid:48) ) cos(2(( N R + 1) δθ − ψ )) (cid:3) − γ n e − Q (( N R +1) b +( N R +2) b (cid:48) ) sinh( Qb (cid:48) ) × sinh( N R Q ( b + b (cid:48) ))sinh( Q ( b + b (cid:48) )) (D7a)¯∆ (eff) W B ≈ − γ n e − Qa (cid:48) sinh( Qa (cid:48) ) cos ( θ d − ψ ) − γ n e − Qa (cid:48) sinh( Qa (cid:48) )8(cosh(2 Q ( a + a (cid:48) )) − cos(2 δθ )) × (cid:2) cos(2( δθ − θ d + ψ )) − e − Q ( a + a (cid:48) ) cos(2( θ d − ψ ))+ e − N L +1) Q ( a + a (cid:48) ) cos(2( N L δθ − θ d + ψ )) − e − QN L ( a + a (cid:48) ) cos(2(( N L + 1) δθ − θ d + ψ )) (cid:3) − γ n e − Q (( N L +1) a +( N L +2) a (cid:48) ) sinh( Qa (cid:48) ) × sinh( N L Q ( a + a (cid:48) ))sinh( Q ( a + a (cid:48) )) (D7b)The coefficients h , h , h and h of G rr in Eq. (6.21)are given by2 h ≡ πe Q ( a + b ) sinh( Qa (cid:48) ) sinh( Qb (cid:48) )32 sinh( Q ( a + a (cid:48) )) sinh( Q ( b + b (cid:48) )) , (D8a) h ≡ πe − Q ( a (cid:48) + b (cid:48) ) sinh( Qa (cid:48) ) sinh( Qb (cid:48) )( e Q ( a + a (cid:48) ) − δθ ))( e Q ( b + b (cid:48) ) − δθ ))64(cos(2 δθ ) − cosh(2 Q ( a + a (cid:48) )))(cos(2 δθ ) − cosh(2 Q ( b + b (cid:48) ))) , (D8b) h ≡ πe − Q ( a (cid:48) + b (cid:48) ) ( e Q ( a + a (cid:48) ) + e Q ( b + b (cid:48) ) − δθ )) sinh( Qa (cid:48) ) sinh( Qb (cid:48) )64(cos(2 δθ ) − cosh(2 Q ( a + a (cid:48) )))(cos(2 δθ ) − cosh(2 Q ( b + b (cid:48) ))) , (D8c) h ≡ πe − Q ( a (cid:48) + b (cid:48) ) sinh( Qa (cid:48) ) sinh( Qb (cid:48) )64(cos(2 δθ ) − cosh(2 Q ( a + a (cid:48) )))(cos(2 δθ ) − cosh(2 Q ( b + b (cid:48) ))) . (D8d)For the case where the optic axes within each slab are allaligned (i.e., δθ = 0), we obtain h + h + h = πe Q ( a + b ) sinh( Qa (cid:48) ) sinh( Qb (cid:48) )64 sinh( Q ( a + a (cid:48) )) sinh( Q ( b + b (cid:48) )) (D9) which combined with Eq. (6.21), yields Eq. (6.22). [1] V. A. Parsegian, Van der Waals Forces (Cambridge Uni-versity Press, Cambridge, 2006).[2] A. Naji, M. Kanduˇc, J. Forsman, and R. Podgornik,”Perspective: Coulomb fluids – weak coupling, strongcoupling, in between and beyond.”
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