Pattern Formation for a Planar Layer of Nematic Liquid Crystal

Transcription

1 Pattern Formation for a Planar Laer of Nematic Liquid Crstal David Chillingworth Department of Mathematics Universit of Southampton Southampton SO7 BJ UK Martin Golubits Department of Mathematics Universit of Houston Houston, TX USA December, 22 Abstract Using equivariant bifurcation theor, and on the basis of smmetr considerations independent of the model, we classif square and heagonall periodic patterns that tpicall arise when a homeotropic or planar isotropic nematic state becomes unstable, perhaps as a consequence of an applied magnetic or electric field. We relate this to a Landau de Gennes model for the free energ, and derive dispersion relations in sufficient generalit to illustrate the role of up/down smmetr in determining which patterns can arise as a stable bifurcation branch from either initial state. Introduction In the Landau theor of phase transitions for a liquid crstal the degree of coherence of alignment of molecules is usuall represented b a tensor order parameter, a field of smmetric 3 3 tensors Q(), R 3 with tr(q) = [5]. We thin of Q as the second moment of a probabilit distribution for the directional alignment of a rod-lie molecule. In a spatiall uniform sstem, Q is independent of R 3. When Q = the sstem is isotropic, with molecules not aligned in an particular direction. If there is a preferred

2 direction along which the molecules tend to lie (but with no positional constraints) the liquid crstal is in nematic phase. There are man other tpes of phase involving local and global structures, see [5]. In this paper we consider a thin planar laer of nematic liquid crstal where the top and bottom boundar conditions on this laer are identical. In this situation the smmetries of an liquid crstal model will include planar Euclidean smmetries E(2) as well as up/down reflection smmetr. A configuration or state of a liquid crstal is just a director field that at each point in the planar laer indicates the direction n() in R 3 along which molecules tend to align. We approimate a planar laer b a plane so our states consist of a 3-dimensional direction field n defined on R 2. In the Landau theor the direction n() is just the eigendirection corresponding to the largest eigenvalue of Q() the direction in which a molecule has the maimum probabilit of aligning. We shall also refer to Q() as the state of the sstem. In our discussion we assume an initial equilibrium state Q that is E(2)- invariant. Because of translation smmetr such states are spatiall uniform and because of rotation smmetr the have the form Q = η 2 for some nonzero η R. For η > the state Q represents a homeotropic phase (the state has constant alignment in the vertical direction), whereas for η < it represents a planar isotropic liquid crstal (a molecule is equall liel to align in an horizontal direction). The state Q is also invariant under up/down reflection, that is conjugac b the matri τ = We consider models that are determined internall b a free energ rather than eternall b, sa, a magnetic field. Thus, the smmetr group for our discussion is Γ = E(2) Z 2 (τ), since these are the smmetries of both the initial state Q and the model.. 2

3 Our aim in this paper is to stud local bifurcation from Q to states that have spatiall varing alignment along the plane. Specificall, we consider bifurcation to states ehibiting spatial periodicit with respect to some planar lattice. We use group representation theor (following [, ]) to etract information about nonlinear behavior near bifurcation that is independent of the model. There is a common approach to all lattice bifurcation problems, which we now describe. This discussion, adapted from [2], will be familiar to anone who has studied pattern formation in Bénard convection models, although there are minor differences due to the change in contet. See [, ]. Let λ be a bifurcation parameter and assume that the equations have Q as an equilibrium for all λ. Let L denote the equations linearized about Q. In the models, λ is the temperature and bifurcation occurs as λ is decreased.. A linear analsis about Q leads to a dispersion curve. Translation smmetr in a given direction implies that (comple) eigenfunctions have a plane wave factor w () = e 2πi where R 2. Rotation smmetr implies that the linearized equations have infinitedimensional eigenspaces; instabilit occurs simultaneousl to all functions w () with constant =. The number is called the wave number. Points (, λ) on the dispersion curve are defined b the maimum values of λ for which an instabilit of the solution Q to an eigenfunction with wave number occurs. 2. Often, the dispersion curve has a unique maimum, that is, there is a critical wave number at which the first instabilit of the homogeneous solution occurs as λ is decreased. Bifurcation analses near such points are difficult since the ernel of the linearization is infinite-dimensional. This difficult can be side-stepped b restricting solutions to the class of possible solutions that are doubl periodic with respect to a planar lattice L. 3. The smmetries of the bifurcation problem restricted to L change from Euclidean smmetr in two was. First, translations act on the restricted problem modulo L; that is, translations act as a torus T 2. Second, onl a finite number of rotations and reflections remain as smmetries. Let the holohedr H L be the group of rotations and reflections that preserve the lattice. The 3

4 smmetr group Γ L of the lattice problem is then generated b H L, T 2, as well as (in our case) Z 2 (τ). 4. The restricted bifurcation problem must be further specialized. First, a lattice tpe needs to be chosen (in this paper square or heagonal). Second, the size of the lattice must be chosen so that a plane wave with critical wave number is an eigenfunction in the space F L of matri functions periodic with respect to L. Those R 2 for which the scalar plane wave e 2πi is L-periodic are called dual wave vectors. The set of dual wave vectors is a lattice, called the dual lattice, and is denoted b L. In this paper we consider onl those lattice sizes where the critical dual wave vectors are vectors of shortest length in L. Therefore, genericall, we epect er L = R n where n is 4 and 6 on the square and heagonal lattices, respectivel. 5. Since er L is finite-dimensional, we can use Liapunov-Schmidt or center manifold reduction to obtain a sstem of reduced bifurcation equations on R n whose zeros are in : correspondence with the stead-states of the original equation. Moreover, this reduction can be performed so that the reduced bifurcation equations are Γ L -equivariant. 6. Solving the reduced bifurcation equations is still difficult. A partial solution can be found as follows. A subgroup Σ Γ L is aial if dim Fi(Σ) = where Fi(Σ) = { er L : σ = σ Σ}. The Equivariant Branching Lemma [] states that genericall there eists a branch of solutions corresponding to each aial subgroup. These solution tpes are then classified b finding all aial subgroups, up to conjugac. On general grounds, when restricting attention to bifurcations corresponding to shortest wave length vectors, we ma assume the representation (action) of Γ L to be irreducible: see [, 7]. In Section 2 we show that there are four distinct tpes of irreducible representation of Γ L that can occur in bifurcations from Q. These representations are the four combinations of (i) scalar or pseudoscalar (see [, 2, ]) and (ii) preserve or brea τ smmetr. In Section 2 we also compute the aial subgroups for each of these 4

5 representations and draw pictures of each of the relevant planforms on the square and heagonal lattices. It is an elementar et curious observation that in Landau models restricted to a planar laer, bifurcations from the homeotropic phase (η > ) do not lead to clear new patterns unless τ smmetr is broen: the τ smmetr freezes the director field to the vertical. This point is discussed in more detail in Section 2. Therefore we present pictures of the four bifurcations from the isotropic case (η < ) and onl the two bifurcations when τ acts as in the homeotropic case (η > ). The τ = + bifurcations in the homeotropic case can lead to patterns in a theor posed on a thicened planar laer. In such a theor, which goes beond what we present here, the precise form of boundar conditions on the upper and lower boundaries of the laer will determine the pattern tpes. In the other bifurcations, the contributions to pattern selection of these boundar conditions should be less important. In fact, the bifurcation theor for each of these four representations of Γ L has been discussed previousl in different contets. It is onl the interpretation of eigenfunctions in the contet of Q() that needs to be computed, along with the pictures of the resulting planforms. More specificall, when τ acts triviall on er L the scalar representation has been used in the stud of pattern formation in Raleigh-Bénard convection b Busse [4] and Buzano and Golubits [5], and the pseudoscalar representation has been studied b Bosch-Vivancos, Chossat, and Melbourne [] and also in the contet of geometric visual hallucinations b Bressloff, Cowan, Golubits, and Thomas [3, 2]. When τ acts nontriviall the two representations have the same matri generators and although the planforms are different for these two representations the bifurcation theor is identical. Indeed, this theor is just the one studied for Raleigh-Bénard convection with a midplane reflection b Golubits, Swift, and Knobloch [2]. Perhaps the most interesting patterns that appear from our analsis are the stripes or rolls (from convection studies) tpe patterns that bifurcate from the isotropic state when τ smmetr is not broen, that is τ = +. The scalar pattern is a martensite pattern whereas the pseudoscalar pattern is a chevron pattern. See Figure. The fact that such patterns do occur in liquid crstal laers is well nown: see for eample [4], from which the pictures in Figure 2 are taen. Note, however, that these patterns are eamples that bifurcate from homeotrop; moreover, the also ehibit finer periodic structures (a characteristic feature in practice) that we do not discuss in this paper. 5

6 SCALARP /ROLLS Perturbation from(,, 2) PSEUDOSCALARP /ROLLS Perturbation from(,, 2) Figure : Rolls from isotropic (η < ) state with τ = + representations: scalar martensite (left); pseudoscalar chevron (right). In Section 3 we introduce free energies that illustrate that all four representations can be encountered as λ is decreased, although in our model onl two of them can be the first bifurcation from homeotrop while onl the other two can be the first bifurcation from isotrop. (It is liel that different models will allow other variations.) It then follows from the Equivariant Branching Lemma that each of the aial equilibrium tpes that we describe in Section 2 is an equilibrium solution to the nonlinear model equations. 2 Spatiall Periodic Equilibrium States In this section we list the aial subgroups for each of the four representations of Γ L on the square and heagonal lattices, and then plot the planforms for the associated bifurcating branches from both the isotropic (η < ) and homeotropic (η > ) states. We emphasize that these results depend onl on smmetr and can be obtained independentl of an particular model. First, we describe the form of the eigenspaces for each of these four representations. Second, we discuss the group actions and the aial subgroups for each of these representations. Finall, we plot the associated direction fields. Linear Theor Let L denote the linearization of the governing sstem of differential equations at Q (for the free energ model with free energ F we have L = d 2 F (Q )). 6

7 Figure 2: Rolls (left) and chevrons (right) bifurcating from homeotrop. (Pictures courtes of J.-H. Huh.) Bifurcation occurs at parameter values where L has nonzero ernel. We prove that genericall, at bifurcation to shortest dual wave vectors, er L has the form given in Theorem 2.. Let Q ++ = Q + = a b a b Q + = Q = i i i i (2.) In the double superscript on Q, the first ± refers to scalar or pseudoscalar representation and the second ± refers to the action of τ. In Table we also fi the generators of the lattice and its dual lattice. Lattice l l 2 2 Square (, ) (, ) (, ) (, ) Heagonal (, 3 2 ) (, 3 ) (, ) (, 3) 2 Table : Generators for the planar lattices and their dual lattices. 7

8 Theorem 2.. On the square lattice, let ξ be rotation counterclocwise b π. Then, in each irreducible representation, er L is four-dimensional and 2 its elements have the form z e 2πi Q ±± + z 2 e 2πi 2 ξ Q ±± + c.c. (2.2) for z, z 2 C, where Q ±± is the appropriate matri specified in (2.), ξ Q denotes ξqξ, and c.c denotes comple conjugate. On the heagonal lattice, let ξ be rotation counterclocwise b π 3. Then, in each irreducible representation, er L is si-dimensional and its elements have the form z e 2πi Q ±± + z 2 e 2πi 2 ξ 2 Q ±± + z 3 e 2πi 3 ξ 4 Q ±± + c.c. (2.3) for z, z 2, z 3 C. Proof. Let V and V C denote the space of (respectivel) real and comple 3 3 smmetric matrices with zero trace. Planar translation smmetr implies that eigenfunctions (nullvectors) of L are linear combinations of matrices that have the plane wave form e 2πi Q + c.c. (2.4) where Q V C is a constant matri and R 2 is a wave vector. For fied let W = {e 2πi Q + c.c. : Q V C } (2.5) be the ten-dimensional L-invariant real linear subspace consisting of such functions. Rotations and reflections γ O(2) Z 2 (τ) O(3) act on W b γ(e 2πi Q) = e 2πi(γ) γqγ. (2.6) When looing for nullvectors we can assume, after rotation, that = (, ). We can also rescale length so that the dual wave vectors of shortest length have length ; that is, we can assume that =. Bosch Vivancos, Chossat, and Melbourne [] observed that reflection smmetries can further decompose W into two L-invariant subspaces. To see wh, consider the reflection κ(,, z) = (,, z). 8

9 Note that the action (2.6) of κ on W (dropping the +c.c.) is κ ( e 2πi Q ) = e 2πiκ() κqκ = e 2πi κqκ. Since κ 2 =, the subspace W itself decomposes as W = W + W (2.7) where κ acts triviall on W + and as minus the identit on W, and each of W + and W are L invariant. We call functions in W + even and functions in W odd. Bifurcations based on even eigenfunctions are called scalar and bifurcations based on odd eigenfunctions are called pseudoscalar. A further simplification in the form of Q can be made. Consider ρ SO(2) O(3) given b (,, z) (,, z). Since (dropping the + c.c. ) ρ(e 2πi Q) = e 2πiρ ρqρ = e 2πi ρqρ = e 2πi ρqρ the associated action of ρ on V C is related to the conjugac action b ρ(q) = ρqρ. (2.8) Since L commutes with ρ and ρ 2 =, the subspaces of the ernel of L where ρ(q) = Q and ρ(q) = Q are L-invariant. Therefore, we can assume that Q is in one of these two subspaces. Note moreover that translation b implies 4 that if e 2πi Q is an eigenfunction then ie 2πi Q is a (smmetr related) eigenfunction. It follows from (2.8) that if ρ acts as minus the identit on Q, then ρ acts as the identit on iq. Thus we can assume without loss of generalit that up to translational smmetr Q is ρ-invariant, that is Q has the form Q = a g ic g b ih ic ih a b where a, b, c, g, h R. Therefore we have proved Lemma 2.2. Up to smmetr eigenfunctions in W have the form e 2πi Q + c.c. where Q is nonzero, ρ-invariant, and either even or odd. 9

10 Lemma 2.2 implies that tpicall eigenfunctions in W lie in one of the 2-dimensional subspaces V +, V of W +, W that have the form V + = {ze 2πi Q + : z C} V = {ze 2πi Q : z C} where Q + = a ic b ic a b and Q = g g hi hi (2.9) with the specific values a, b, c, g, h R being chosen b L (cf. [, 5.7]). Moreover, since L commutes with τ we can further split V + = V ++ V + and V = V + V into subspaces on which τ acts triviall and b minus the identit, and each of these subspaces is L-invariant. Since a g ic τqτ = g b ih ic ih a b we see that V ±± = {ze 2πi Q ±± : z C}, where the matrices Q ±± are as given in (2.). Finall, note that er L is invariant under the action of ξ. It follows that on the square lattice er L = V ±± ξ ( ) V ±± whereas on the heagonal lattice er L = V ±± ξ ( ) 2 V ( ) ±± ξ 4 V ±± thus verifing (2.2),(2.3) and completing the proof of Theorem (2.). Aial Subgroups The scalar and pseudoscalar actions of E(2) on the eigenfunctions on the square and heagonal lattices are computed in [2]. The results are given in Table 2 in terms of the coefficients z j in (2.2) and (2.3).

11 D 4 Action D 6 Action (z, z 2 ) (z, z 2, z 3 ) ξ (z 2, z ) ξ (z 2, z 3, z ) ξ 2 (z, z 2 ) ξ 2 (z 3, z, z 2 ) ξ 3 (z 2, z ) ξ 3 (z, z 2, z 3 ) κ ɛ(z, z 2 ) ξ 4 (z 2, z 3, z ) κξ ɛ(z 2, z ) ξ 5 (z 3, z, z 2 ) κξ 2 ɛ(z, z 2 ) κ ɛ(z, z 3, z 2 ) κξ 3 ɛ(z 2, z ) κξ ɛ(z 2, z, z 3 ) κξ 2 ɛ(z 3, z 2, z ) κξ 3 ɛ(z, z 3, z 2 ) κξ 4 ɛ(z 2, z, z 3 ) κξ 5 ɛ(z 3, z 2, z ) [θ, θ 2 ] (e 2πiθ z, e 2πiθ 2 z 2 ) [θ, θ 2 ] (e 2πiθ z, e 2πiθ 2 z 2, e 2πi(θ +θ 2 ) z 3 ) Table 2: (Left) D 4 + T 2 action on square lattice; (right) D 6 + T 2 action on heagonal lattice. Here [θ, θ 2 ] = θ l + θ 2 l 2 as in Table. For scalar representation ɛ = +; for pseudoscalar representation ɛ =. The aial subgroups for each of the four irreducible representations of Γ L are given in Table 3, together with generators (z, z 2 ) C 2 or (z, z 2, z 3 ) C 3 (fied vectors) of the corresponding -dimensional fied-point subspaces (aial eigenspaces) in er L, and descriptions of the associated patterns (planforms). The results in Table 3 summarize nown results for scalar actions with and without the midplane reflection [5, 2] and the less well nown results for pseudoscalar actions [, 2]. See also []. More precisel, on the heagonal lattice, the scalar + action is identical to the action studied in Bénard convection [4, 5] and the scalar action is identical to the one studied in Bénard convection with the midplane reflection [2]. The pseudoscalar + action is identical to that studied in [, 2], whereas the pseudoscalar action is again the same as the one in Bénard convection with the midplane reflection but with different isotrop subgroups, as Figures 5 and 6 show.

12 Lattice Planform Aial Isotrop Subgroup Fied Vector Scalar Representation (ɛ = +); τ = + Square squares D 4 (κ, ξ) Z 2 (τ) (, ) rolls Z 2 2(κξ 2, τ) O(2)[θ 2, κ] (, ) Heagonal heagons + D 6 (κ, ξ) Z 2 (τ) (,, ) heagons D 6 (κ, ξ) Z 2 (τ) (,, ) rolls Z 2 2(κξ 3, τ) O(2)[θ 2, κ] (,, ) Square Pseudoscalar ( Representation (ɛ = ); τ = + squares D 4 κ[ Z (τ) (, ) rolls Z 2 2(κξ 2 [, ], τ) O(2)[θ 2 2, κ[, ]] (, ) 2 Heagonal heagons Z 6 (ξ) Z 2 (τ) (,, ) triangles D 3 (κξ, ξ 2 ) Z 2 (τ) (i, i, i) rectangles Z 3 2(κ, ξ 3, τ) (,, ) rolls Z 2 2(κξ 3 [, ], τ) O(2)[θ 2 2, κ[, ]] (,, ) 2 Scalar Representation (ɛ = +); τ = Square squares D 4 (κ, ξ) Z 2 (τ[, ]) 2 2 (, ) rolls Z 2 2(κξ 2, τ[, ]) O(2)[θ 2 2, κ] (, ) Heagonal heagons D 6 (κ, ξ) (,, ) triangles D 6 (κ, τξ) (i, i, i) rectangles Z 3 2(τκ, ξ 3, τ[, ]) (,, ) 2 rolls Z 2 2(κξ 3, τ[, ]) O(2)[θ 2 2, κ] (,, ) Pseudoscalar Representation (ɛ = ); τ = Square squares D 4 (τκ, ξ) Z 2 (τ[, ]) 2 2 (, ) rolls Z 2 2(κξ 2 [, ], 2 τ[, ]) O(2)[θ 2 2, κ[, ]] (, ) 2 Heagonal heagons D 6 (τκ, ξ) (,, ) triangles D 6 (τκ, τξ) (i, i, i) rectangles Z 3 2(κ, ξ 3, τ[, ]) (,, ) 2 rolls Z 2 2(κξ 3 [, ], 2 τ[, ]) O(2)[θ 2 2, κ[, ]] (,, ) 2 Table 3: Summar of aial subgroups. On the heagonal lattice in the scalar case with τ = + the points (,, ) and (,, ) have the same isotrop subgroup (D 6 (κ, ξ) Z 2 (τ)) but are not conjugate b an element of Γ L. Therefore, the associated eigenfunctions generate different planforms. 2

13 The Planforms We now consider 2-dimensional patterns b disregarding the z-coordinate in (but not in Q) and restricting attention to equilibrium states that are periodic with respect to a square or heagonal lattice in the -plane. To visualize the patterns of bifurcating solutions we assume a laer of liquid crstal material in the -plane that to first order has the form Q() = Q + εe() where E is an aial eigenfunction, ε is small, and Q is either isotropic (η = ) or homeotropic (η = +). At each point (, ) we choose the eigendirection corresponding to the largest eigenvalue of the smmetric 3 3 matri Q() at = (, ) and we plot onl the, components of that line field. In this picture, a line element that degenerates to a point corresponds to a vertical eigendirection, so the initial solution loos lie at arra of points. Suppose that Q is homeotropic. Then in our simulations no pattern will appear in bifurcations for which Q() is fied b the action of τ. For, if E() V ++ or V + then [ ] A Q() = b where A is a 2 2 bloc and b is a scalar. Since b is close to 2 (the largest eigenvalue of Q ) it is also the largest eigenvalue for Q() for ε small. Hence, the leading eigendirection (corresponding to the largest eigenvalue) is alwas vertical and no patterns appear that are determined b changes in eigendirection. However, since variation in the vertical eigenvalue of Q() represents variation in the propensit of molecules to align verticall it is plausible that indistinct patterns could nevertheless be observed in practice. In Figures 3 and 4 we plot solutions corresponding to scalar and pseudoscalar square lattice patterns. In Figures 5 we plot those for a heagonal lattice. Observe the dislocations that occur where the leading eigendirection changes discontinuousl. The two competing directions are necessaril orthogonal in R 3. 3 Free Energ Models These results impl that for a planar liquid crstal there are four tpes of stead-state bifurcations, scalar, pseudoscalar and τ = ± of each tpe, that 3

14 SCALARP /SQUARES Perturbation from(,, 2) PSEUDOSCALARP /SQUARES Perturbation from(,, 2) SCALARM /SQUARES Perturbation from(,, 2) PSEUDOSCALARM /SQUARES Perturbation from(,, 2) Figure 3: Square lattice bifurcations from isotropic (η < ) liquid crstal to square patterns: (upper left) scalar τ = +; (upper right) pseudoscalar τ = +; (lower left) scalar τ = ; (lower right) pseudoscalar τ =. Corresponding rolls patterns can be found in Figures and 7-. can occur from a spatiall homogeneous equilibrium to spatiall periodic equilibria. Whichever bifurcation occurs, then genericall all of the planforms that we listed in the relevant section of Table 3 will be solutions. We have not discussed the difficult issue of stabilit of these solutions since these are model dependent results, whereas the classification of equilibria that we have given is independent of the model. What remains is to complete a linear calculation to determine when a stead-state bifurcation occurs and whether it is scalar or pseudoscalar. The outline of such a calculation goes as follows. We first compute a dispersion curve for both scalar and pseudoscalar eigenfunctions. That is, for each wave 4

15 SCALARM /SQUARES Perturbation from(,,2) PSEUDOSCALARM /SQUARES Perturbation from(,,2) Figure 4: Square lattice bifurcations from homeotropic (η > ) to squares with τ = : (left) scalar; (right) pseudoscalar. Corresponding rolls patterns can be found in Figures 5-6. length = we determine the first value λ of the bifurcation parameter λ where L has a nonzero ernel. The curve (, λ ) is called the dispersion curve. We then find the minimum value λ = λ on the dispersion curve; the corresponding wave length is the critical wave length. We epect the first instabilit of the spatiall homogeneous equilibrium to occur at the value λ of the bifurcation parameter. A bifurcating branch can consist of stable solutions onl if the branch emanates from the first bifurcation (at λ ). As an illustration we now carr out these calculations for a Landau de Gennes tpe model with appropriate planar smmetr. Related calculations were carried out for bifurcation from the 3-dimensional isotropic phase in [3]. In this model we show that there are bifurcations corresponding to each of the four irreducible representations of Γ L, and which of them occurs first depends on the action of τ. Dispersion Curves for a 2-Dimensional Landau de Gennes Model The free energ F is epressed as an integral per unit volume of a free energ densit F which has two principal components F and F d corresponding to bul terms and deformation terms respectivel: we write F accordingl as F = F + F d. For a sstem in 3 dimensions these tpicall (see e.g. [3]) 5

16 SCALARM /ROLLS Perturbation from(,,2) SCALARM /HEXAGON Perturbation from(,,2) SCALARM /TRIANGLE Perturbation from(,,2) SCALARM /RECTANGLE Perturbation from(,,2) Figure 5: Heagonal lattice bifurcations from homeotropic (η > ) with scalar τ = representation: (upper left) rolls; (upper right) heagons; (lower left) triangles; (lower right) rectangles. tae the form F (Q) = 2 λ Q 2 3 B tr Q3 + 4 C Q 4 F d (Q) = c Q 2 + c 2 Q 2 + c 3 Q Q respectivel, where R 2 denotes the sum of the squares of the coefficients of the tensor R. The epression for F represents the simplest SO(3)-invariant function on V ehibiting nontrivial interaction of local minima close to Q =, while F d consists of those SO(3)-invariant terms of at most order 2 in spatial first derivatives (the chiral term Q Q is not reflection-invariant). For a 2-dimensional problem this choice of free energ function is not full appropriate: the relevant smmetr group is now Γ = E(2) Z 2 (τ). 6

17 PSEUDOSCALARM /ROLLS Perturbation from(,,2) PSEUDOSCALARM /HEXAGON Perturbation from(,,2) PSEUDOSCALARM /TRIANGLE Perturbation from(,,2) PSEUDOSCALARM /RECTANGLE Perturbation from(,,2) Figure 6: Heagonal lattice bifurcations from homeotropic (η > ) with pseudoscalar τ = representation: (upper left) rolls; (upper right) heagons; (lower left) triangles; (lower right) rectangles. Consequentl a wider range of terms can appear in F, while the Q Q term will no longer appear in F d. We are interested in planforms that bifurcate from either the bul homeotropic state or isotropic state, represented b Q with η > or η < respectivel. An eample of a bul term with E(2) Z 2 (τ) invariance is (Q Q) 2, and a candidate for a deformation term to replace the chiral term is Q 2 representing longer range interactions of molecules. Accordingl we consider a free energ densit F = F + F d where now F (Q) = 2 λ Q 2 3 B tr Q3 + 4 C Q D(Q Q) 2 F d (Q) = c Q 2 + c 2 Q 2 + c 4 Q 2. 7

18 SCALARP /HEXAGON Perturbation from(,, 2) SCALARP /HEXAGONM Perturbation from(,, 2) Figure 7: Heagonal lattice bifurcations from isotropic (η < ) with scalar τ = + representation: rolls in Figure ; (left) heagons + ; (right) heagons. Equilbrium states are critical points of F, and for F we have df (Q)R = λq R BQ 2 R + C Q 2 Q R + 6 D(Q Q)(Q R) for arbitrar 3 3 real smmetric matrices Q, R ; thus restricted to Q with trace zero we have df (Q) = when λq B(Q 2 3 Q 2 I) + C Q 2 Q + 6 D(Q Q)Q = and we easil verif the following df (Q ) = λ Bη + (6C + D)η 2 =. (3.) Observe that df (Q)R = automaticall for an spatiall-periodic state R with zero mean, as the integral of an epression linear in R or its derivatives remains bounded as the volume tends to infinit. Therefore (3.) is the condition for Q to be an equilibrium state in our free energ model. To stud stabilit of the state Q we evaluate the second derivative of the free energ at Q. For R V we find d 2 F (Q )R 2 = λ R 2 2BQ R 2 + C ( 2(Q R) 2 + Q 2 R 2) + 6 D(Q R) 2 and (integrating over unit area) d 2 F d (Q )R 2 = c R 2 + c 2 R 2 + c 4 R 2 8

19 SCALARM /ROLLS Perturbation from(,, 2) SCALARM /HEXAGON Perturbation from(,, 2) SCALARM /TRIANGLE Perturbation from(,, 2) SCALARM /RECTANGLE Perturbation from(,, 2) Figure 8: Heagonal lattice bifurcations from isotropic (η < ) with scalar τ = representation: (upper left) rolls; (upper right) heagons; (lower left) triangles; (lower right) rectangles. since Q is spatiall constant and terms linear in R integrate to zero. We have alread seen from Theorem 2. that the E(2) Z 2 (τ) invariance of the free energ implies that genericall the eigenfunctions of d 2 F (Q ) on the space of L-periodic matri functions are linear combinations of functions belonging to one of the four subspaces V ±± and their rotations under π 2 (square lattice) or ± 2π (heagonal lattice). We net see dispersion relations 3 for each of the spaces V ±± in turn. When R = e 2πi Q + c.c. it is eas to chec that R 2 = 2 2 Q 2, 4π 2 where =. 4π 2 R 2 = 2 Q 2, 6π 4 R 2 = 2 4 Q 2 Without loss of generalit we can tae = (,, ) and 9

20 PSEUDOSCALARP /ROLLS Perturbation from(,, 2) PSEUDOSCALARP /HEXAGON Perturbation from(,, 2) PSEUDOSCALARP /TRIANGLE Perturbation from(,, 2) PSEUDOSCALARP /RECTANGLE Perturbation from(,, 2) Figure 9: Heagonal lattice bifurcations from isotropic (η < ) with pseudoscalar τ = + representation: (upper left) rolls; (upper right) heagons; (lower left) triangles; (lower right) rectangles. then after rescaling b a factor of 2π the evaluations of d 2 F (Q )R 2 and d 2 F d (Q )R 2 are given in Table 4. If we normalize b choosing D so that (3.) is satisfied b η = (corresponding to homeotrop) then we find the conditions for a zero eigenvalue in each of the last three (-dimensional) eigenspaces are respectivel λ B + 6C + (c + 2 c 2) 2 + c 4 4 = λ + 2B + 6C + (c + c 2 2) 2 + c 4 4 = λ B + 6C + c 2 + c 4 4 = (3.2) with the analogous epressions for η = (bifurcation from 2-dimensional isotrop) obtained b merel reversing the sign of B in these equations. 2

21 PSEUDOSCALARM /ROLLS Perturbation from(,, 2) PSEUDOSCALARM /HEXAGON Perturbation from(,, 2) PSEUDOSCALARM /TRIANGLE Perturbation from(,, 2) PSEUDOSCALARM /RECTANGLE Perturbation from(,, 2) Figure : Heagonal lattice bifurcations from isotropic (η < ) with pseudoscalar τ representation: (upper left) rolls; (upper right) heagons; (lower left) triangles; (lower right) rectangles. Stationar values of λ as a function of occur where 2 = (c + 2 c 2)/2c 4 for V + and V +, or for V, giving values λ = 2 = c /2c 4 B 6C + (c + c 2 2) 2 /4c 4 for V + 2B 6C + (c + c 2 2) 2 /4c 4 for V + B 6C + c 2 /4c 4 for V (3.3) 2

22 R d 2 F (Q )R 2 d 2 F d (Q )R 2 V ++ λ(a 2 + b 2 2ab) 2 2 (c 2 a 2 + B(a 2 + b 2 + ab)η +6C ( 7(a 2 + b 2 ) + ab ) η 2 (a 2 + b 2 + (a + b) 2 )(c + c 4 2 )) V + 4(λ Bη + 6Cη 2 ) (4c + 2c 2 ) 2 + 4c 4 4 = 4Dη 2 b (3.) V + 4(λ + 2Bη + 6Cη 2 ) (4c + 2c 2 ) 2 + 4c 4 4 V 4(λ Bη + 6Cη 2 ) 4c 2 + 4c 4 4 = 4Dη 2 Finall, if R V ++ a, b is [ Table 4: Computation of d 2 F (Q )R 2. then the matri for d 2 F (Q )R 2 as a quadratic form in λ B + 42C λ 5B + 3C ] λ 5B + 3C λ B + 42C and for d 2 F d (Q )R 2 is [ 4c 2 + 2c c 4 4 2c 2 + 2c 4 4 2c 2 + 2c 4 4 4c 2 + 4c 4 4 and so d 2 F (Q ) V ++ has a nontrivial ernel when the determinant of the sum of these two matrices vanishes. With c 2 = (that is, in phsical terms, with no energ cost to the molecules for spla ) the algebra simplifies to ield the dispersion relation λ = 2B 6C + c2 4c 4. (3.4) From (3.2) and (3.4) we therefore see that with c 2 = the values of λ for V ±± depend onl on the second ±, that is on whether bifurcating solutions have vertical reflection smmetr (τ = +) or not (τ = ) and are the same for the scalar and the pseudoscalar representations. Moreover, as λ decreases, the first bifurcation from the homeotropic state (η > ) has τ = while the first bifurcation from the isotropic state (η < ) has τ = +. These statements remain true for sufficientl small c 2. Acnowledgments. We are grateful to Tim Slucin for instructive conversations on the mathematical phsics of liquid crstals. We also wish to 22 ]

23 than Greg Forest, Steve Morris, and Ian Stewart for helpful conversations. The research of MG was supported in part b NSF Grant DMS-7735 and ARP Grant References [] I. Bosch Vivancos, P. Chossat, and I. Melbourne. New planforms in sstems of partial differential equations with Euclidean smmetr. Arch. Rational Mech. Anal. 3 (995) [2] P.C. Bressloff, J.D. Cowan, M. Golubits, and P.J. Thomas. Scalar and pseudoscalar bifurcations motivated b pattern formation on the visual corte, Nonlinearit 4 (2) [3] P.C. Bressloff, J.D. Cowan, M. Golubits, P.J. Thomas, and M.C. Wiener. Geometric visual hallucinations, Euclidean smmetr, and the functional architecture of striate corte. Phil. Trans. Roal Soc. London B 356 (2) [4] F.H. Busse. Das Stabilitätsverhalten der Zellaronvetion bei endlicher Amplitude, PhD Thesis, U Munich 962. [5] E. Buzano and M. Golubits. Bifurcation involving the heagonal lattice and the planar Bénard problem, Phil. Trans. Ro. Soc. London A38 (983) [6] D.R.J. Chillingworth, E. Vicente Alonso, and A.A. Wheeler. Geometr and dnamics of a nematic liquid crstal in a uniform shear flow. J. Phs. A 34 (2) [7] P. Chossat and R. Lauterbach. Methods in Equivariant Bifurcations and Dnamical Sstems, World Scientific, Singapore 2. [8] P.G. de Gennes. Short range order effects in the isotropic phase of nematics and cholesterics. Mol. Crst. Liq. Crst. 2 (97) [9] P.G. de Gennes. The Phsics of Liquid Crstals, Clarendon Press, Oford

24 [] M. Golubits and I. Stewart. The Smmetr Perspective: From Equilibrium to Chaos in Phase Space and Phsical Space. Progress in Mathematics 2, Birhäuser, Basel, 22. [] M. Golubits, I. Stewart, and D.G. Schaeffer. Singularites and Groups in Bifurcation Theor, II, Springer-Verlag, New Yor 988. [2] M. Golubits, J.W. Swift and E. Knobloch. Smmetries and pattern selection in Raleigh-Benard convection, Phsica D (984) [3] H. Grebel, R.M. Hornreich, and S. Shtriman. Phs. Review A 28 (983) [4] J.-H. Huh, Y. Hidaa, A.G. Rossberg and S. Kai. Pattern formation of chevrons in the conduction regime in homeotropicall aligned liquid crstals. Phs. Rev. E 6 (2) [5] T.J. Slucin. The liquid crstal phases: phsics and technolog, Contemporar Phsics 4 (2)