Metric approach for sound propagation in nematic liquid crystals
aa r X i v : . [ c ond - m a t . s o f t ] J a n A metric approach for sound propagation in nematic liquid crystals
E. Pereira ∗ Instituto de F´ısica, Universidade Federal de Alagoas,Campus A.C. Sim˜oes, 57072-900, Macei´o, AL, Brazil.
S. Fumeron
Laboratoire d’ ´Energ´etique et de M´ecanique Th´eorique et Appliqu´ee,CNRS UMR 7563, Nancy Universit´e, 54506, Vandœuvre Cedex, France.
F. Moraes
Departamento de F´ısica, CCEN, Universidade Federal da Para´ıba,58051-970, Caixa Postal 5008, Jo˜ao Pessoa, PB, Brazil.
In the eikonal approach, we describe sound propagation near to topological defects of nematicliquid crystal as geodesics of a non-euclidian manifold endowed with an effective metric tensor.The relation between the acoustics of the medium and this geometrical description is given byFermat’s principle. We calculate the ray trajectories and propose a diffraction experiment to retrieveinformations about the elastic constants.
PACS numbers: 43.20.Wd, 43.20.Dk, 43.20.El, 61.30.Jf
I. INTRODUCTION
Number of systems in condensed matter physics can bedescribed by the same mathematical structures as thosedescribing the gravitational field in General Relativity:this is the core of the analogue gravity (or effective ge-ometry) programme [1]. In General Relativity, gravita-tion is accounted by the distortions of spacetime, whichis modeled by a pseudo-riemannian manifold [2] of sig-nature (-,+,+,+). The properties of this spacetime areencompassed in the metric tensor g (or simply metric )and other quantities derived from it, such as the Riemanntensor or the Ricci scalar. Free-falling particles followtrajectories corresponding to the geodesics (curves of ex-tremal length) of the manifold. In particular, light pathscorrespond to the null geodesics of spacetime ds = g µν dx µ dx ν = 0 , (1)where ds is the square of elementary length along thegeodesic and the g µν are the covariant components of g (the Einstein summation convention is assumed, wheregreek indices refer to spacetime coordinates and latin in-dices are related to spatial ones).As light trajectories can also be curved when crossinga refractive medium, Fermat’s principle can be stated interms of an effective geometry [2, 3]. For a given refrac-tive index n ( ~r ), where ~r is the position vector of the lightwavefront, the light trajectories are interpreted as thegeodesics of a non-Euclidean space with the line elementdΣ d Σ = n ( ~r )( dx + dy + dz ) , (2) ∗ Corresponding author: erms@fis.ufal.br
Despite (2) represents only a three-dimensional non-Euclidean manifold, the corresponding geodesics areequal to the null ones in a static four-dimensional pseudo-riemannian manifold with the time coordinate [4]. Sincewe will deal with static refractive indexes, whenever con-sidering the time, the time coordinate appears in the lineelement ds = − c dt + d Σ , (3)where c is the speed of the light in vacuum and d Σ rep-resents the spatial line element obtained by the methodthat will be exposed in this paper. For the study that willbe developed here, c is the velocity of the sound in thematerial of interest when it is homogeneous and isotropic.In this paper, we will apply this gravitational anal-ogy to describe the sound propagation in liquid crystals[5–7], materials raising a considerable interest as theyare both theoretically challenging [8–10] and of practi-cal interest (PVA screens, q-plates or cellphones [11]).They are made of anisotropic molecules (disc-like or rod-like) that present meta-states between the crystalline andliquid phases. One of these meta-states is the nematicphase: the molecules’ axis are oriented on average alonga specific direction called director (represented by theversor ˆ n ) whereas their centers of mass are randomly lo-cated. Once in the nematic phase, the refractive index,the sound velocity and other macroscopic properties be-come anisotropic. Moreover, the nematic phase can ex-hibit birefringence, as the director plays the role of theoptical axis [5, 6, 12].The gravitational analogy will be based on Fermat’sprinciple, once it can also be used to determine soundtrajectories [13]. The local speed of sound plays the roleof the speed of light in General Relativity and soundpropagates along the null geodesics of the acoustic met-ric. The interest of using this analogy (or, more properly,this geometrical approach) is that it helps understandingfrom a different point of view several problems dealingwith acoustics (for example, tomographic image recon-struction via ray tracing [14], flight time in acousticalwaveguides [15], problems using variational method [16])or nematoacustics [17].We will consider the sound propagation around a punc-tual (called hedghog ) and a linear (called disclination )topological defect of the nematic phase of a liquid crys-tal with rod-like molecules [5–7, 18]. Sound trajectorieswill be exhibited and the diffraction patterns will be de-termined in the perspective of retrieving the elastic con-stants of the liquid crystalline medium. II. ANALOG MODEL OF ANISOTROPICLIQUID CRYSTALSA. Acoustic aspects
The specific structure of the liquid crystal in nematicphase allows two kinds of acoustic waves to exist [5, 6, 19],similarly for sound in solid crystals [20, 21]: the ordi-nary wave, that behaves as inside an isotropic medium,and the extraordinary one that depends on the angle be-tween the direction of the propagation and the directorˆ n . For each kind of wave, one usually defines two char-acteristic velocities: the phase velocity, which points inthe direction of the wave vector ~k , and the group velocity that is oriented similarly to the acoustic Poynting vector ~S (in general, not parallel to ~k ) and indicates the direc-tion of energy propagation. The phase velocity v p of theextraordinary wave, for a liquid crystal in the nematicphase with the director ˆ n orientated on the z-direction,can be measured by the pulse-superposition technique[12], producing the expression v p ( α ) = (cid:0) C + ( C − C ) cos α (cid:1) ρ , (4)where ρ is the density, α is the angle between the wavevector ~k and ˆ n [5, 6]. This relation expresses that a liquidcrystal can be viewed as a simple solid with linear elasticconstants in the x- and y- directions both equal to C and in the z-direction equal to C [5, 19]. Additionally,the smallest portion of liquid crystal where the directorcan be defined presents local cylindrical symmetry andtherefore, we will consider (4) to be locally valid for anyconfiguration of the director ˆ n (even in the presence ofdefects as in sections III A and III B).The procedure to obtain N g ( β ) relies firstly on the de-termination of v p ( α ) and phase refractive index N p ( α ).To determine the latter, one replaces (4) in N p = v v p ( α ) ,where v is the velocity of sound in the isotropic phase ofthe liquid crystal, obtaining N p = v ρC sin α + C cos α , or N p sin α vρC + N p cos α vρC = v. (5)Following [7], we define the refractive index vector ~N p ≡ v ˆ kv p and denote by index ⊥ (resp. // ) components of vec-tors that are orthogonal (resp. parallel) to director ˆ n .Thus, the components of ~N p are N p ⊥ = N p sin α and N p // = N p cos α . For acoustic waves in anisotropic crys-tals (which is the case of a liquid crystal in its nematicphase [20, 22]), it is known that ~v g · ˆ k = v p = ⇒ ~v g · ~N p = v. (6)Identifying (6) with (5), we found the components of ~v g v g ⊥ = N p ⊥ vρC ; v g // = N p // vρC . (7)Introducing β as the angle between ~v g and ˆ n , then alter-nate expressions for the group velocity components are v g ⊥ = v/N g sin β and v g // = v/N g cos β . Thus, the useof (7) in (6) results in v = ~v g · ~N p = vN g (cid:0) N p ⊥ sin β + N p // cos β (cid:1) = v N g (cid:18) vρC sin β + vρC cos β (cid:19) . Finally, we obtain N g ( β ) = v ρC sin β + v ρC cos β. (8)In the next section, we will discuss about the connec-tion between the acoustic group index and the effectivemetric experienced by sound in the nematics. B. Geometric aspects
We intent to study the paths followed by the extraor-dinary ray. This can be achieved by applying Fermat’sprinciple [23], for which the path followed by sound be-tween points A and B is the one that minimizes the in-tegral F = Z BA N g ( β ) dl, (9)where l is the arc length of the path. As pointed outby Joets and Ribotta [24] for light inside an anisotropiccrystal, the use of Fermat’s principle to determine soundpaths is equivalent to calculating the null geodesics of amanifold with line element d Σ = P i,j g ij dx i dx j , wherethis null geodesic is the curve that minimizes the integral Z BA d Σ . (10)The mathematical resemblance of the two previous ideasin the application of (9) and (10) suggests one to makethe identification N g ( β ) dl = X i,j g ij dx i dx j = d Σ , (11)where dl is recognized as the Euclidean line element.Still according to Joets and Ribotta, the resulting spaceis a Finslerian one. However, due to the symmetry ofthe liquid crystal molecules and the spatial configurationof the director ˆ n in the applications where this metricapproach will be applied, the Riemannian geometry isenough to describe the sound propagation [25, 26].Thus, substituting (11) in (3), we found the generalexpression of the line element experienced by the soundwave in cylindrical coordinates ds = − c dt + N g ( β ) (cid:0) dr + r dθ + dz (cid:1) (12)and the covariant components of the acoustic metric re-lated to the line element (12) are g = − c N g ( β ) 0 00 0 r N g ( β ) 00 0 0 N g ( β ) . (13)Form the standpoint of sound, an anisotropic mediumcan thus be mimicked by an effective gravitational field.As for light, the WKB (or short wavelength) approxima-tion enables to identify sound paths to the null geodesicsof the effective metric g . In pseudo-Riemannian geome-try [2, 4, 27], the curve that minimizes the line element ds is obtained from the geodesic equation d x µ dτ + Γ µνσ dx µ dτ dx σ dτ = 0 , (14)with τ a parameter along the geodesic and Γ µνσ the com-ponents of the Riemannian connectionΓ µνσ = 12 g µξ (cid:18) ∂g ξσ ∂x ν + ∂g νξ ∂x σ − ∂g νσ ∂x ξ (cid:19) . (15)In the next section, we will apply these equations tostudy sound propagation near defects occurring in thenematic phase of liquid crystals. III. APPLICATIONSA. Sound paths near a defect in nematics
When a liquid crystal transits from the liquid (orisotropic) phase to nematic one, defects (points or lines)can arise spontaneously, causing a reorientation of the di-rector. We will focus on two different topological defects
FIG. 1. Hedgehog and ( k = 1 , c = 0)-disclination defects.The arrows represent the director ˆ n . of the nematic phase: thepunctual defect called hedgehog ,with director ˆ n = ˆ r written in spherical coordinates, andthe linear defect called disclination with director ˆ n = ˆ ρ written in cylindrical coordinates (Fig. 1).To determine the acoustic metric associated to anhedgehog defect, the strategy is to express cos β and sin β in terms of the laboratory frame coordinates, by the Eu-clidean line element dl = dr + r (cid:0) dθ + sin θdφ (cid:1) , sub-stitute them in (8) and we obtain the g ij ’s by (11).In Frenet-Serret frame [28], the position vector alongthe sound path ~R = r ˆ r and the tangent vector ~T ( l ) arerelated by ~T ( l ) = d ~Rdl = d ( r ˆ r ) dl = drdl ˆ r + r d ˆ rdl . (16)As ~T ( l ) has the same direction as ~v g , then ~T · ˆ n = cos β = drdl ≡ ˙ r. (17)From the modified Euclidean line element1 = ˙ r + r (cid:16) ˙ θ + sin θ ˙ φ (cid:17) , one identifies sin β = r r (cid:16) ˙ θ + sin θ ˙ φ (cid:17) . (18)The replacement of (17) and (18) in (8) results in the lineelement d Σ = ρv C dr + ρv C r (cid:0) dθ + sin θdφ (cid:1) , which gives the effective metric for sound in the vicinityof a hedgehog defect. The rescaling of the radial coordi-nate of this equation by ˜ r ≡ q ρC vr produces d Σ = d ˜ r + b ˜ r (cid:0) dθ + sin θdφ (cid:1) , (19)where b ≡ C C . Therefore sound paths are the geodesicsof (19), that is the spatial part of line element of a global FIG. 2. Sound trajectories on the equatorial plane of a hedge-hog in a liquid crystal with b = 0 .
9. This result is similar tosound trajectories on z = C st plane near to a ( k = 1 , c = 0)-disclination lying on the z direction with b = 0 . monopole [29], and they are depicted on Fig. (2) for b = 0 . k and c [7], whichare related to director in cylindrical coordinates by ~n = (cos ( kφ + c ) , sin ( kφ + c ) , . (20)Repeating the same procedure as for hedgehog defects,we obtain a generalized effective metric, similar to onefound in [26] d Σ = (cid:18) ρ C cos α + ρ C sin α (cid:19) dr + (cid:18) ρ C sin α + ρ C cos α (cid:19) r dφ (21) − (cid:20) (cid:18) ρ C − ρ C (cid:19) sin α cos α (cid:21) rdrdφ, where α = kφ + c . In the case of ( k = 1 , c = 0)-disclination, the geodesics are represented on Fig. 2.Equations (19) and (21) are examples of effective met-rics experienced by the sound close to topological defectsin nematic liquid crystals. In the next section, they willserve as basis for calculations beyond the WKB approx-imation to determine scattering sections. B. Acoustical diffraction by defects in nematics
The connection between geometric and wave optics iswell-known: in the WKB approximation, light rays iden-tify with curves that are tangent at each point to thePoynting vector [22]. Similarly, since the direction of theenergy velocity for sound waves defines the direction ofthe sound rays [21], we use the partial wave method [30]to examine the scattering of sound waves by the previ-ous hedgehog defect and indicate its diffraction pattern(for disclinations, a similar analysis can be found in [31]).Sound plane waves obey d’Alembert’s wave equation forscalar fields: ∇ µ ∇ µ Φ ≡ √− g ∂ µ (cid:0) √− gg µν ∂ ν Φ (cid:1) = 0 . (22) Here, we denote by ∂ µ ≡ ∂∂x µ , g µν are the components ofthe metric g (determinant g ) given by eq. (19). We havealso used the convention of the repeated indexes for sum-mations. Solutions are the usual harmonic plane wavesof the form Φ = Φ( t, r, θ, φ ) = e − iωt ψ ( r, θ, φ ), where ω isthe wave pulsation.The spherical symmetry of the problem allow us towrite ψ ( r, θ, φ ) = ψ ( r, θ ) = P ∞ l =0 a l R l ( r ) P l (cos θ ), where P l (cos θ ) are the Legendre polynomials of order l and a l are worthless constants for the partial wave method.Applying ψ ( r, θ ) in Φ( t, r, θ, φ ) and expanding (22), weobtain the radial equation R ′′ l ( r ) + 2 R ′ l ( r ) r + (cid:20) ω − l ( l + 1) b r (cid:21) R l ( r ) = 0 , (23)where R ′ l ( r ) ≡ ∂ r R l ( r ). Solutions to this equation areBessel functions of the first kind R l ( r ) = J n ( l ) ( r ), where n ( l ) = 1 b "(cid:18) l + 12 (cid:19) − − b . When b = 1, we recover the flat space and the last equa-tion becomes l + . The phase shift δ l ( b ) of the scatteredwave is δ l ( b ) = π (cid:18) l + 12 − n ( l ) (cid:19) = π l + 12 − b "(cid:18) l + 12 (cid:19) − − b . (24)The angular distribution of the scattered sound is givenby [30] the differential scattering cross section σ ( θ ) σ ( θ ) ≡ | f ( θ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) iω ∞ X l (2 l + 1)( e iδ l − P l (cos θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (25)where f ( θ ) is called scattering amplitude. The sphericalsymmetry of the scattering generates an annular diffrac-tion pattern, a ring of sound. The angular location of themaximum of sound is given by eq. (25), as it is shown inFig. (3).An analytical expression for the location of the lightdiffraction ring scattered by a global monopole defect inthe real spacetime is found in [32] and a similar one isdeveloped here. The idea is to expand the scatteringamplitude about b ≈ f ( θ ) = f (0) ( θ ) + f (1) ( θ ) + . . . , (26)and to analyze the angular behavior of the first twoterms. For the phase shift (24), we make ζ ≡ l + and a ≡ − b , obtaining δ l ( b ) = π ζ − ζb s − a ζ ! . FIG. 3. Differential scattering cross section, eq. (25), for ahedgehog defect with ( b = 0 . , ω = 1) truncated at l = 600. We expand this last equation about a ≈ b ≈ δ l ( b ) ≈ π (cid:20)(cid:18) − b (cid:19) ζ + a bζ + O ( a ) (cid:21) . (27)Substituting (27) in the scattering amplitude, we obtainthe first two terms of the expansion, f (0) ( θ ) and f (1) ( θ ).They can be written in terms of the generating functions, h ( θ, α ), of the Legendre polynomials h ( θ, α ) = ∞ X l =0 e πiαζ ( l ) P l (cos θ ) = 1 p πα − cos θ ) , where α = 1 − b . The zero-order term, f (0) ( θ ), can bewritten considering the derivative of h ( θ, α ) f (0) ( θ ) = 12 √ ω sin πα (cos πα − cos θ ) / , (28)and the first-order term, f (1) ( θ ), can be written consid-ering h ( θ, α ) f (1) ( θ ) = πα bω p πα − cos θ ) . (29)The equations (28) and (29) diverge when θ = θ = πα ,that is the analytical expression for the angular loca-tion of the diffraction ring. For example, when b = 0 . θ = π (1 − / . ≈ .
35 rad, as indicated by Fig. (3).For ( k = 1 , c = 0) disclinations, the same result is ob-tained [31] and they are in agreement with the calcula-tions derived by Grandjean [33]. IV. CONCLUSION AND PERSPECTIVES
We have developed an effective geometry approachto investigate properties of sound propagation inside anisotropic media such as nematic liquid crystals. In thisframework, the influence of each defect is represented byan effective metric: metrics for hedgehogs are similar tothose of global monopoles, whereas (k, c)-disclinationshave more complex forms. Sound trajectories and diffrac-tion patterns are thus modified by the non-trivial met-ric. In particular, sound scattering by hedgehog defectsis identical to light scattering by a global monopole.It’s interesting to note the relation between the met-ric approach shown here and Katanaev-Volovich’s the-ory [34]. While Katanaev uses an affine transformationto deform and curve the elastic medium, generating theanisotropic properties, we start from the anisotropic ve-locity to derive the effective metric.We could reinterpretate the obtained results by Cos-mology’s view. Once sound waves can be understood aspropagating perturbations in the effective metric gener-ated by the director field ˆ n ( ~r ), they are analog to gravi-tational waves propagating in a true gravitational metricgenerated by a cosmic defect . Thus, the annular diffrac-tion pattern due to defects in a liquid crystal can alsobe expected for gravitational waves coming on cosmicstrings or global monopoles. In other words, the diffrac-tion pattern is a gravitational signature of the presenceof cosmic defects, and its properties may help retriev-ing information about them. Furthermore, consideringthe weak effects of gravitational waves on possible detec-tors, we have presented an algebraic result for the angleof the maximum intensity of the scattered gravitationalwaves. Since the scattered wave on this angle has ampli-tude greater than incident wave’s one, our results indi-cate the angular position where detectors of gravitationalwaves must be place to improve their power of detection.Another possible extension in this way is the possibil-ity to test a cosmological mechanisms predicted in thepaper of T. Damour and A. Vilenkin [35]: from our anal-ogy, production of sound waves should occur by cusps ofdisclinations in liquid crystals. If required by the referee,we can insert these comments in our article.In the language of cohomology [36], the Volterra pro-cess for a cosmic string is obtained by a removing of acylindrical solid angle, whereas the Volterra process ofthe corresponding antistring is obtained by an additionof a cylindrical solid angle. For example, in nematics,such pair will consist in a ( k = 1 , c = 0)-disclination,with b < k = 1 , c = π/ b > [1] C. Barcel´o, S. Liberati, and M. Visser, Living Rev. Rel-ativity , 12 (2005).[2] S. Weinberg, Gravitation and Cosmology: Principles andApplications of the General Theory of Relativity (JohnWiley, New York, 1972).[3] P. M. Alsing, Am. J. Phys. , 779 (1998).[4] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Grav-itation (W. H. Freeman and Company, San Francisco,1973).[5] M. J. Stephen and J. P. Straley, Rev. Mod. Phys. , 617(1974).[6] P. G. de Gennes and J. Prost, The Physics of LiquidCrystals , 2nd ed. (Claredon Press, Oxford, 1992).[7] M. Kleman and O. D. Lavrentovich,
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