LAYER UNDULATIONS IN SMECTIC A LIQUID CRYSTALS

Transcription

1 LAYER UNDULATIONS IN SMECTIC A LIQUID CRYSTALS CARLOS J. GARCíA-CERVERA AND SOOKYUNG JOO Abstract. We investigate instabilities of smectic A liquid crystals wen a magnetic field is applied in te direction parallel to smectic layers. We use te Landau-de Gennes model of smectic A liquid crystals to caracterize te critical magnetic field. Wen smectic A liquid crystals are confined between parallel plates, we derive analytic estimates for te magnetic field strengt, at wic te undeformed state loses its stability. We also present numerical simulations to confirm te Helfric-Hurault effect due to te applied magnetic field.. Introduction. We consider smectic A liquid crystals confined in two flat plates and uniformly aligned in a way tat te smectic layers are parallel to te bounding plates. If a magnetic field is applied in a direction parallel to te smectic layers, te instability occurs above te tresold magnetic field. Wen te magnetic field reaces te critical tresold, one can see periodic perturbation of te layers. Tis penomenon is called te Helfric-Hurault effect.see [8] and [9].) We present te analytical estimate of te critical field and perform numerical simulations describing tis effect by using te Landau-de Gennes model. Liquid crystal pases form wen a material as a degree of positional or orientational ordering yet stays in a liquid state. In te nematic state, molecules tend to align temselves along a preferred direction wit no positional order of te centers of mass. Te unit vector field n, nematic director, represents te average direction of molecular alignment. Moreover, if te liquid crystal is ciral, n follows a elical pattern, wit temperature-dependent pitc. Upon lowering te temperature, or increasing concentration, according to weter te liquid crystal is termotropic or lyotropic, te nematic liquid crystal experiences a transition to te smectic A pase wit molecules arranged along equally spaced layers. Te molecules tend to align temselves along te direction perpendicular to te layers. Te Helfric-Hurault effect in a lamellar system can be caused by magnetic/electric field or by mecanical tension [4]. In tis paper, we study te magnetic field driven instabilities in smectic A liquid crystals. Helfric and Hurault proposed te model tat can explain te periodic perturbations in colesteric liquid crystals under a magnetic field or an electric field applied parallel to te elical axis [8], [9]). Tey assumed tat te layers are fixed at te cell boundaries, i.e., te undulations vanis at te boundaries. Still wit tis assumption, Stewart extended te classic Helfric-Hurault teory to tree dimensional finite samples of smectic A liquid crystals in [3]. However, e made an ansatz to simplify te problem. Experimental studies of undulations of two dimensional and tree dimensional systems were performed in [0] and [], respectively. Tey used colesteric liquid crystals wit a pitc 5µm and 50 70µm cell tickness for te optical study. Since te layer tickness of smectic A liquid crystals is in te nanometer range nm), it is too small to visualize te layer distortions. Teir experiments sow tat tere are layer undulations on te boundary of te sample. Motivated by te experimental result, Lavrentovic et al. proposed te model wit weak ancoring condition so tat te undulations are allowed to appear on te boundaries. By making an ansatz of Matematics Department, University of California, Santa Barbara, CA 9306, cgarcia@mat.ucsb.edu). Matematics Department, University of California, Santa Barbara, CA 9306.sjoo@mat. ucsb.edu)

2 C. Garcia-Cervera AND S. Joo periodic undulations, tey sow tat teir model explains te experiment better tan te classic Helfric-Hurault teory. In section, we present te model and te geometry for our problem and state te existence result of minimizers of te free energy. Te models used in previous works are based on an assumption tat te director and te layer normal vector are identical, eiter in an infinite sample [4], [8], [9]), or in a finite sample [3]. We refine te teory to allow te director and te layer normal to differ by studying te Landau-de Gennes energy of smectic A liquid crystals. In [3], motivated from te analogy to te Ginzburg Landau model for superconductivy, de Gennes introduced te complex function to describe layer structures of smectic liquid crystals. Tis model was used in [] to rigorously analyze te pase transition between te ciral nematic and smectic A liquid crystals. Recently, Lin and Pan used tis model to sow tat critical magnetic field is acieved in an arbitrary domain []. However, tey did not obtain te estimate of te critical magnetic field. In section 3, we find te analytical estimate of te critical magnetic field for layer undulations. Te teory of all previous works makes a coice of sinusoidal perturbation for te undulation in order to derive te critical magnetic field. Witout tis assumption, we derive te estimate of te tresold in terms of te cell tickness. More precisely, we prove tat tere exist universal constants 0 < c < c, suc tat, if d is te cell tickness, ten te critical field H c satisfies c K χ a dλ ) Hc c K χ a dλ ),.) were K, χ a, λ are material constants, wic will be discussed in section. Tis estimate is consistent wit te result found in te classic Helfric-Hurault teory see p.363 of [4] and [9]). We sould mention tat te scaling of te critical field in tis case is different from te tresold for Fredericks transition of nematic liquid crystals, were te critical field is proportional to /d [6]. In section 4, We perform numerical simulations to te gradient flow equations. We find undulation instabilities above te critical magnetic field.. Te model. Te total free energy density of smectic A liquid crystals consists of te nematic f n and smectic f s part. Te Oseen-Frank energy density for a nematic is given by f n = K n) + K n n) + K 3 n n) + K + K 4 )tr n) n) ), were K, K and K 3 are te splay, twist, and bend elastic constants, respectively. Te last term in f n is a null-lagrangian since its integral only depends on te boundary values of n. We consider te energy wit te one constant approximation case, K = K = K 3 = K > 0 and K 4 = 0. Ten te nematic energy density becomes K n. In order to associate smectic and nematic structure wit a state n, Ψ) we write Ψx) = ρx)e iϕx). Ten te molecular mass density is defined by δx) = ρ 0 x) + Ψx) + Ψ x)) = ρ 0 x) + ρx) cos ϕx),

3 Layer undulations 3 were ρ 0 is a locally uniform mass density, ρx) is te mass density of te smectic layers, and ϕ parametrizes te layers, so tat ϕ is in te direction normal to te layer. Finally, q is te wave number and π/q is te layer tickness. Te Landau-de Gennes energy density for smectic A is given by f s = C Ψ iqnψ + r Ψ + g Ψ 4, were C and g are positive material constants, and r may be positive or negative. In [], te energy density f n + f s was used to study te pase transition and stability of te equilibrium states. Since we investigate te smectic structure far from te nematic smectic transition, we may assume tat te magnitude of te smectic order parameter is constant, i.e., ρ is constant. Ten f s becomes f s = Cq ρ ϕ n. Tis energy density vanises wen ϕ = n, wic describes te configuration of smectic A liquid crystals. Te magnetic free energy density is given by [4], [4]) f m = χ a n H) = χ a σ n ),.) were χ a is te magnetic anisotropy, H = σ, and σ = H. We assume tat χ a > 0. Terefore, te energy.) favors molecular orientations were te director is parallel to te applied magnetic field. Collecting all contributions to te free energy, te free energy density for te one-constant approximation model becomes f = K n + B ϕ n χ a σ n).) were B = Cq ρ is called te de Gennes compressibility constant. In tis paper, we consider a two dimensional domain = [ L, L] [ d, d]. We also assume tat =, 0) so tat te magnetic field tends to make te director orient along te x direction. We impose te periodic boundary condition for φ and n in te x direction so tat we can minimize te unnecessary boundary effect, wile we assume te strong ancoring condition for n on te boundary plates, i.e., simply nx, ±d) = 0, ) for all x [ L, L]..3) However, we do not impose any boundary conditions on φ. We make te problem dimensionless by introducing new variables K B were λ =.) becomes x = x λ, ȳ = y, and ϕ = λ φ, λ is of te order of te smectic layer tickness. Ten te free energy B n + φ n) κn ) ) d x,.4)

4 4 C. Garcia-Cervera AND S. Joo were te dimensionless parameters are given by κ = χ aσ B, = [ L, L] [, ], = d λ, L = L λ. Since is te ratio of te cell tickness to te layer tickness, we may assume tat. In fact, te values d = mm and λ = 0Å are employed in [4]. Ten = In tis paper, we will consider. From n =, we can introduce te scalar variable θ, wit 0 θ < π, suc tat n = sin θ, cos θ). Ten te free energy.4) becomes, dropping te bar notation, Fθ, φ) = φ x sin θ) + φ y cos θ) + θ κ sin θ) dy dx.5) and te corresponding boundary condition on θ is te omogeneous Diriclet boundary condition on te top and te bottom of te plate. Tis energy.5) as a trivial critical point, θ = 0, φ = y, wic describes te undeformed state were te layers are parallel to te boundary plates and te directors are aligned in te y direction. Te second variation of te energy at te undeformed state, φ 0 = y, and θ 0 = 0, gives d D Fθ 0 + tθ, φ 0 + tφ) := t=0 dt Fθ 0 + tθ, φ 0 + tφ) = φ x θ) + φ y + θ κ θ ) dxdy..6) Te undeformed state, θ 0, φ 0 ), is stable if te second variation is nonnegative at θ 0, φ 0 ). Setting Gθ, φ) := φ x θ) + φ y + θ ) dxdy,.7) te critical field κ c is defined by κ c = Here, te admissible set A is given by inf Gθ, φ)..8) θ,φ) A A = {θ, φ) W, ) W, ) : θ L ) =, θx, ±) = 0 for all x, θ and φ periodic in te x direction}. Tus, te undeformed state, θ 0, φ 0 ), is stable if κ κ c and unstable if κ > κ c. In section 3 of [], tey proved tat te critical field κ c is acieved wen Diriclet boundary condition is imposed on. Following teir work, we consider φ as a function of θ and apply te standard calculus of variations to prove te existence of a minimizer of G. For te reader s convenience, we present a sort proof ere. Proposition.. Tere exists a minimizer for G in A. Proof. If θ, φ) is a minimizer, ten φ can be expressed in terms of θ. Tat is, we write κ c = inf{gθ, φ θ ) : θ W, ), θ L ) =, θx, ±) = 0 for all x, θ L, y) = θl, y) for all y},

5 Layer undulations 5 were φ θ is te solution of φ θ = θ x, φ θ = 0, φ θ x, ±) = 0 for all x, φ L, y) = φl, y) for all y. y.9) Now we take a minimizing sequence {θ j } for G and write φ θj = φ j. Since θ j is bounded in W, ), it follows, for a subsequence, still labeled θ j tat θ j θ in W, ), θ j θ almost everywere in. For te L convergence on te boundary, we used te estimate, from te proof of Teorem.5..0 in [7], u dσ C){ u L ) u L ) + u L ) }, valid for all u W, ). We also ave ) φj x θ j + ) φj C y for some constant C. Ten togeter wit φ j = 0, we ave φ j W, ) C. Te elliptic estimates on.9) wit θ = θ j gives φ j W, ) C and tus it follows, for a subsequence, still labeled φ j tat φ j φ in W, ), φ j φ in W, ), and φ satisfies.9) wit θ = θ, i.e., φ = φ θ. Tus θ, φ ) A and Gθ, φ θ ) lim inf j Gθ j, φ θj ) = inf θ A Gθ, φ θ). 3. Caracterization of te critical field. In tis section, we prove tat te critical field in.8) satisfies κ c /. In oter words, we prove Teorem 3.. Assume tat L. Ten, tere exist universal constants c and c suc tat c κ c c. Note tat tis statement is equivalent to.), were te estimate is expressed in terms of te real parameters. We prove tis teorem in two steps: In te first step, we prove a lower bound for te energy wit te above-mentioned scaling. In te second step, we prove a matcing upper bound. Te periodic boundary conditions allow us to use te Fourier series representation. θx, y) = n= θ n y)e iµnx and φx, y) = n= φ n y)e iµnx,

6 6 C. Garcia-Cervera AND S. Joo were µ n = πn/l. Ten.7) becomes ) Gθ, φ) = L θ n + µ n θ n + θ n iµ n φ n + φ n ) dy. 3.) n= We define κµ) = inf θ,φ) B := inf Fθ, φ, µ), θ,φ) B θ + µ θ + θ iµφ + φ ) dy 3.) were B = {θ, φ) W, 0, ) W,, ) : θy) dy = }. As in [], it follows tat Gθ, φ) L L = κ c n= n= κµ n ) θ n y) dy κ c θ n y) dy θ dx dy for all θ W, ). Immediately we see tat κ c = inf κµ). 3.3) µ 3.. Lower bound. In following lemma we prove tat tere exists a constant c suc tat κµ) c / for any µ. Tis implies κ c c / due to 3.3). As a result, te undeformed state is stable if κ c /. Lemma 3.. Let > and µ be given positive constants. Ten tere exists a universal positive constant c suc tat θ iµφ + µ θ + θ + φ ) dy c θ dy 3.4) for all θ, φ) B. Proof. We may assume tat µ 0, oterwise te inequality 3.4) is trivial wit c =. By using te claim 3.4) is equivalent to ỹ = y, and ϕ = iµφ, θ + µ ϕ + µ θ + θ ϕ ) dy c θ dy. 3.5)

7 Te Euler-Lagrange equation for ϕ is Layer undulations 7 µ ϕ + ϕ = θ, ϕ ±) = ) We will prove te inequality 3.5) for all θ and ϕ satisfying 3.6). We suppose tat tis inequality is false, i.e., tere exists {C n } wit lim n C n = 0, for wic we can find n, µ n, ϕ n H, ) satisfying 3.6) and θ n H0, ) wit θ n = for eac n =,,... suc tat θ n + n n µ ϕ n + n µ n θ n + n θ n ϕ n ) dy C n. 3.7) n Ten we ave two small parameters, n µ n C n and / n C n, 3.8) were te second inequality follows from te Poincaré inequality. We integrate te equation 3.6) to obtain ϕ n = θ n 3.9) were ϕ n and θ n are averages of ϕ n and θ n, respectively. In te following proof, we denote by C a universal positive constant wic may differ from line to line. Also, by C n we denote C C n. From te second and te last terms in 3.7) and te Poincaré inequality, we ave C n n θ n ϕ n ) = θ n θ n ) dy θ n θ n ϕ n ϕ n )) dy 3.0) θ n θ n ) dy C ϕ n ϕ n ) dy θ n θ n ) dy n µ nc n. ϕ n) dy Tus, we ave ) θ n θ n ) dy C n + n µ n. 3.) n Using 3.), togeter wit 3.0), we obtain Since θ n L =, we ave ) ϕ n ϕ n ) dy C n + n µ n. 3.) n θ n θ n ) dy = θ n.

8 8 C. Garcia-Cervera AND S. Joo Togeter wit 3.), we see tat θ n / as n. By taking a subsequence, still labeled wit n, we may assume θ n / as n. Terefore, from 3.8), 3.9), and 3.), we ave η n W C, n were η n = ϕ n /. Tis implies ) + n µ n C n, 3.3) n sup η n C n. 3.4) [,] Setting f n = θ n η n and ε n = / n C n, we rewrite te first and te last terms of 3.7) as ε n f n ) dy + ε n f n + η n) dy C n, 3.5) were η n satisfies te estimate 3.3). Te part on te left and side is an Allen-Can functional wit a single well potential. From te first integral in 3.5), one can see tat for eac n, tere exists a n, ) suc tat / < f n a n ) <. Finally we get, from 3.5), C n an f n f n + η n dy f n f n dy f n η n dy f n ) f n dy f n L η n L, were we used te triangle inequality and Hölder s inequality. Doing te cange of variables r = f n y), it follows from 3.3) and 3.5) tat fna n) η n) r ) dr C n Cn ε n C n. 3.6) Tis is a contradiction, since / < f n a n ) < and η n ) C n, wic follows from 3.4). In fact, computing te integral in 3.6) we obtain C n + C n ε n ) f n a n ) + η n ))f n a n ) + η n ) ) ) C n C n ), wic is impossible since C n Upper bound. We find an upper bound for te critical field by evaluating.) for an appropriate test function. Specifically, we prove tat κ c / for some c > 0. Tis implies tat if κ > c /, te undeformed state, θ 0 0, φ 0 = y), is no longer stable. It follows from 3.3) tat it suffices to find an upper bound for te one-dimensional problem. Tat is te content of te following lemma.

9 Layer undulations 9 Lemma 3.3. Let L. Tere exists a constant c suc tat F θ, φ, µ) c θ dy 3.7) for some θ, φ) B and for some constant µ. Proof. Define θ, φ and µ by µ = π L n 0 π L θ = cos π y, [ L ], φ = i µ cos π y, were [ ] denotes te greatest integer function. Since L, we may see tat n 0 and ten A simple computation sows tat F θ, φ, µ) 4π + n 0 L n 0 + ) 4n 0. ) 4 π + θy) dy. 4 Te lemma is proved wit c = 4π + /4 + π /4. Teorem 3. is a consequence of te two lemmas 3. and Numerical simulations. We ave carried out two-dimensional simulations wit model.) to study te Helfric-Hurault effect. Te energy is minimized by solving te gradient flow: φ t = φ n, 4.) n t = Π n n + φ n + κn )), were Π n f) = f n, f)n is te projection onto te plane ortogonal to n, and n, f) denotes te usual L inner product. Tis projection appears as a result of te constraint n =. For numerical purposes, it is more convenient to write tis term as n t = n n n + φ n + κn ))). 4.) Written in tis way, te equation resembles te Landau-Lifsitz equation of micromagnetics in te ig damping limit [5], and te eat-flow of armonic maps [5]. For te initial condition, we take a small perturbation from te undeformed state. More precisely, for all x, y), nx, y, 0) = ɛu, + ɛu ) ɛu, + ɛu ), φx, y, 0) = y + ɛφ 0,

10 0 C. Garcia-Cervera AND S. Joo 5 κ= 5 κ= κ=. 5 κ= Fig. 4.. Contour plots of φ, te solution of te system 4.). were a small number ɛ = 0.00 and u, u, and φ 0 are arbitrarily cosen. As described in section, we impose strong ancoring condition for te director field,.3), and natural boundary condition on φ at te top and te bottom plates, φ ν y=± = n ν y=±. Periodic boundary conditions are imposed for bot n and φ on eac side of te domain. We use a Fourier spectral discretization in te x direction, and second order finite differences in te y direction. Te fast Fourier transform is computed using te FFTW libraries [6]. We use te fourt order Runge-Kutta metod to solve te corresponding initial value problem. We take te domain size L = 00 and = 5. A more pysically relevant value for in smectic A liquid crystals is However, te layer undulations can be observed if. In fact, te undulations in colesteric liquid crystals occur wit 0 []). Te numbers of grid points in te x and y directions are 56 and 8, respectively. In Fig. 4. we sow te layer structures in response to te various magnetic field strengts κ. Te pictures are contour maps of φ since te level sets of φ represent te layer. One can see tat te undeformed state is stable before te magnetic field κ reaces te tresold κ c. If κ increases and reaces κ c, te layer undulations occur. As κ increases beyond κ c, te displacement amplitude increases as in te Fig. 4.. Te maximum undulation occurs in te middle of te cell y = 0) and te displacement amplitude decreases as approacing te boundary y = ±). In te classic Helfric- Hurault teory, te layers are fixed at te boundary, i.e., no undulations at y = ±. However, Fig. 4. indicates tat te undulations do not vanis at te boundary even toug we impose te strong ancoring condition.

11 Layer undulations 5. Acknowledgements. Te work of Carlos J. García-Cervera is supported by NSF grant DMS REFERENCES [] P. Bauman, M. Carme Calderer, C. Liu and D. Pillips Te Pase Transition between Ciral Nematic and Smectic A* Liquid Crystals, Arc. Rational mec. Anal ), [] P. Bauman and D. Pillips and Q. Tang Stable nucleation for te Ginzburg-Landau system wit an applied magnetic field, Arc. Rational mec. Anal ), 43. [3] P.G. De Gennes An analogy between superconductivity and smectics A, Solid State Commum. 0, 97), [4] P.G. De Gennes and J. Prost Te Pysics of Liquid Crystals, nd Edition. Clarendon Press, Oxford, 993. [5] W. E and X. Wang Numerical metods for te Landau-Lifsitz equation, SIAM J. Numer. Anal. 38, 000), no. 5, [6] M. Frigo and S. G Jonson Te design and implementation of FFTW3, Proc. of te IEEE 93, 005), no., 6 3, special issue on Program Generation, Optimization, and Platform Adaptation [7] P. Grisvard Elliptic Problems in Nonsmoot Domains, Pitman Advanced Publising Program, 985. [8] W. Helfric Electroydrodynamic and dielectric instabilities of colesteric liquid cyrstals, J. Cem. Pys. 55, 97), [9] J.P. Hurault Static distortions of a colesteric planar structure induced by magnetic or ac electric-fields, J. Cem. Pys. 59, 973), [0] T. Isikawa and O.D. Lavrentovic Undulations in a confined lamellar system wit surface ancoring, Pys. Rev. E 63 00), 03050R). [] F.H. Lin and X.B. Pan Magnetic field-induced instabilities in liquid crystals, Siam J. Mat. Anal ), no. 5, [] B.I. Senyuk, I.I. Smalyuk and O.D. Lavrentovic Undulations of lamellar liquid crystals in cells wit finite surface near and well above te tresold, Pys. Rev. E ), 07. [3] I.W. Stewart Layer undulations in finite samples of smectic A liquid crystals subjected to uniform pressure and magnetic fields, Pys. Rev. E. 58, 998), no. 5, [4] I.W. Stewart Te static and dynamic continuum teory of liquid crystals, Taylor & Francis, 004. [5] M. Struwe Heat-flow metods for armonic maps of surfaces and applications to free boundary problems, Springer, Lect. Notes Mat. Berlin, 34, 988), [6] E.G. Virga Variational teories for liquid crystals, Capman & Hall, 994.