Phase transitions and unwinding of cholesteric liquid crystals

Transcription

1 Research Signpost 37/661 (2), Fort P.O., Trivandrum , Kerala, India Phase Transitions. Applications to Liquid Crystals, Organic Electronic and Optoelectronic Fields, 2006: ISBN: Editor: Vlad Popa-Nita 3 Phase transitions and unwinding of cholesteric liquid crystals Laboratoire de Physique de l' École Normale Supérieure de Lyon 46 allée d'italie, Lyon Cedex 07, France Abstract Various static and dynamic experiments about the unwinding of a cholesteric liquid crystal, and its transformation into a nematic or a smectic A phase, are described. They provide very instructive examples of phase transitions where the respective roles of the confinement (via the anchoring conditions at the boundaries) and of an external magnetic or electrical field are clearly identified Introduction A cholesteric phase is a nematic type of liquid crystals composed of optically active molecules. Its structure has a helical axis perpendicular to the preferred molecular direction characterized by a unit Correspondence/Reprint request: Dr., Laboratoire de Physique de l École Normale Supérieure de Lyon, 46 allée d'italie, Lyon Cedex 07, France. oswald@ens-lyon.fr

2 48 vector, the director n (Fig.3.1). Such a structure can also be obtained by adding a small quantity of a chiral substance to a nematic phase. In this case, the helical pitch p depends on the concentration c of the chiral substance, with p proportional to 1/c at small c. Thermodynamically, there is no phase transition between the cholesteric (Ch) and the nematic (N) phases, even if the twist can sometimes vanish and change sign at some special temperature (compensation temperature at which the pitch becomes infinite). On the other hand, it is possible to unwind a cholesteric phase, either by confining it between two surfaces which impose a director orientation incompatible with its helical structure, or by applying an external magnetic or electrical field: as a consequence, the cholesteric can transform into a nematic phase. In the following, we shall speak about the nematic-cholesteric phase transition, it being understood that we are dealing with a bifurcation driven by a competition between elasticity and an external field. From this respect, the cholesteric-nematic transition is equivalent to a Frederiks transition in a nematic liquid crystal [1]. As we shall see later, this transition can strongly change when approaching in temperature a smectic A phase. In the latter, molecules arrange into parallel fluid layers with the director perpendicular to the Figure 3.1. Structures of cholesteric (a) and smectic A (b) liquid crystal phases. In (a), the drawn planes are fictitious as there is no positional order in a cholesteric phase. The pitch p is the distance over which the director rotates by 2π. The true periodicity, corresponding to a cholesteric layer, is p/2 as n and n are equivalent (from [1]).

3 Phase transitions and unwinding of cholesteric liquid crystals 49 layers. An important property of the smectic phase is that its lamellar structure is incompatible with an helical arrangement of the molecules: twist is expulsed from the smectic A phase. So one expects a cholesteric phase tends to unwind in the vicinity of a smectic phase. This property is useful to change the pitch without changing the concentration of chiral molecules. Let us emphasize that the control parameter of the Ch-N transition is not the temperature, but an external field of mechanical, electrical or magnetic origin. More precisely, the confinement ratio C = d/p (where d is the sample thickness and p the pitch) is the main control parameter which characterizes the mechanical action of the two surfaces limiting the sample, while the magnetic field B or the electrical field E (when a voltage V is applied between the two surfaces, with E = V/d) act as additional control parameters. In this article, we review the different methods to unwind the helix and induce a nematic phase from the cholesteric phase. The plan of the article is as follows. In section 2, we describe the action of the confinement between two surfaces treated, respectively, for planar and homeotropic anchoring. The effect of the confinement ratio is first analyzed by examining the textures formed in a wedge sample. We will show that the N-Ch front is different according to the anchoring. In planar anchoring, a singular disclination line forms at the front whose structure will be described. In contrast, the director field deforms continuously at the front in homeotropic anchoring. The latter case will be analyzed in detail both experimentally and theoretically by using a representation on the unit sphere S2 of the director fields. The role of the elastic anisotropy will be emphasized. In section 3, the role of an additional magnetic or electrical field is analyzed, first in an infinite sample and then in confined geometry. Typical phase diagrams in homeotropic anchoring will be given for both liquid crystals with positive or negative dielectric anisotropy. In section 4, the problem of the unwinding transition in confined geometry is addressed when the sample is placed into a temperature gradient close to a smectic A phase. Both types of anchoring will be analyzed Unwinding at a constant temperature In this section, we assume that the sample is maintained at a constant temperature. Two types of anchoring may be used in practice: in planar anchoring, the molecules are anchored parallel to the surface and oriented in a single direction; in homeotropic anchoring, the molecules are perpendicular to the surface. A convenient way to rapidly visualize the effect of the confinement is to prepare samples of variable thickness (wedge geometry). The textures obtained in this way are first described. These observations will be then completed by a detailed analysis of the N-Ch transition in homeotropic anchoring.

4 Texture in planar sample The Grandjean-Cano wedge geometry The cholesteric is introduced by capillarity between two glass plates treated for planar anchoring. In the Grandjean-Cano geometry, the distance h between the two limiting surfaces varies along the x direction (by convention, the z axis is perpendicular to the bottom plate). In the example of Fig.3.2, the sample is sandwiched between a glass plate and a cylindrical lens. The thickness, almost zero at the contact point, increases from left to right. The photo shows a row of clearly separated stripes. It can be checked that in this experiment, the number of half-pitches changes discontinuously by one unit from one stripe to the next. In addition, simple κ disclination lines, similar to that shown in Fig. 3.3, separate the stripes. These lines are located at absxissa x m and thickness (3.1) The first line, corresponding to m = 1, separates the planar nematic phase from the cholesteric phase which develops in the second stripes, in which the number of half-pitch is equal to 1. So, from the point of view of phase transitions, Figure 3.2. a) Cholesteric texture in a Grandjean-Cano wedge between a plate and a cylindrical lens of radius R = 10 cm. Crossed polarizers; b) Arrangment of the cholesteric layers (photo by M. Brunet-Germain [2]).

5 Phase transitions and unwinding of cholesteric liquid crystals 51 Figure 3.3. κ disclination line. The nail representation has been used to describe the director field. Tilted molecules are represented by nails, proportional in length to the director projection in the plane of the drawing. Around the Burgers circuit, the director turns by an angle Ω = 2πm. Here Ω = π so this κ disclination is of strength m = 1/2. it marks the limit of coexistence between the two phases. It must be noted that we cannot pass continuously from one phase to the other because of the topological constraints imposed by the strong anchoring of the molecules on the surfaces. For that reason, the N-Ch transition is always discontinuous (first order) in this geometry. This can be seen experimentally by changing the confinement ratio C, for instance by changing the temperature (the pitch is very sensitive to this parameter, especially close to a transition toward a smectic A phase, where it diverges, see section 4). This experiment shows that both phases remain metastable in a large range of confinement ratios and transform into each other (in the limit of very strong anchoring) only by the nucleation of κ disclination lines from the edge of the sample. We emphasize that the homogeneous (and even heterogeneous) nucleation of looped κ disclination lines is very difficult because of the high energy of these lines (for an estimate see the next subsection). The situation is different when the planar anchoring is sufficiently weak and can be broken. The director can then rotate on the surfaces, which renders much easier the passage from one phase to the other (note here that the phase is always slightly twisted, even at small C: so, strictly speaking, there is no nematic phase). This phenomenon is hysteretic, showing that the phase transition is still first order. This has been observed experimentally [3] and analyzed theoretically for different shapes of the anchoring potential [4, 5]. Finally we note that in very strong anchoring, the planar nematic phase destabilizes in the middle plane of the sample when the confinement ratio is

6 52 increased and develops spontaneously a helical modulation parallel to the anchoring direction. The onset of instability C + and the corresponding wavelength λ + of this modulation can be obtained by minimizing the Frank elastic energy of the cholesteric phase: (3.2) where n is the director, q the equilibrium twist of the cholesteric phase, and K 1, K 2, and K 3 the splay, twist, and bend elastic constants, respectively. A straightforward calculation gives [6]: and (3.3) (3.4) For C C +, this instability should lead to a structure with the helical planes perpendicular to both the surfaces and the anchoring direction. This transition is obviously different from that described before as the helical axis is now parallel to the glass plate. A non-linear analysis would be required to determine the nature (first-order or second-order?) of this continuous transition. Nevertheless, preliminary observations of A. Dequidt [6] indicate clearly that this transition is first order. Indeed, cholesteric fingers, analogous to those described in the next subsection 2.2 devoted to the Ch-N transition in homeotropic anchoring, nucleate from dust particles when C < C + and progressively invade the sample. The director field inside these fingers is given in Fig. 3.9c, except that it must be rotated by π/2 along the finger axis in order to match the boundary conditions. In the following subsection, we return to planar κ lines and give their internal structure. Structure of planar κ lines Planar means that we assume the director remains everywhere parallel to the limiting surfaces. This is a strong assumption which, in practice, only applies to the first line (m = 1). A way to force the other lines (m = 2, 3, ) to remain planar would be to use a liquid crystal with a negative dielectric anisotropy and to apply a large enough electrical field perpendicular to the surface.

7 Phase transitions and unwinding of cholesteric liquid crystals 53 Figure 3.4. Planar κ line of order m = 2 in a constant thickness cholesteric slice (from [1]). The structure of the line is obtained by minimizing the Frank energy (2) with n = (cos φ, sin φ, 0) and φ(x, z) the angle between the director and the x axis. In many systems, K 1 and K 3 are almost equal and larger than K 2. Assuming K 1 = K 3 = K K 2 and putting ξ = (K 2 /K) 1/2, Malet [7] was able to solve exactly the problem for a planar κ disclination line sandwiched between two parallel plates (Fig. 3.4). The solution reads: (3.5) where h is the sample thickness. Integration over x and z gives the energy per unit length of the line: (3.6) where r c is a core radius of molecular size, E c a core energy of the order of K and C the Catalan constant (C 0.916). Note that κ lines can group together to form double lines separating adjacent regions in which the number of layers differ by two units. The structure of non-planar κ lines and the conditions in which they group in a wedge sample are described in detail in Ref. [1].

8 Texture in homeotropic sample In this section, we describe the N-Ch phase transition when the molecules are strongly anchored perpendicularly to the limiting surfaces. Textures observed under the microscope are completely different from those observed in planar anchoring. The reason is that, contrary to what happens in planar anchoring, there is no orientation of the cholesteric helix compatible with the boundary conditions. The cholesteric phase is thus always frustrated. This frustration may induce a complete unwinding of the helix, leading to a homeotropic nematic phase. It turns out that the passage from one phase to the other occurs by a continuous distortion of the director field. After a description of the textures observed in a wedge sample, we shall examine the order of the transition, experimentally and theoretically. We shall show, using a representation on the unit sphere S2 of the director field, that the transition may be first or second order, depending on the anisotropy of the elastic constants. Observation in the wedge geometry Fig. 3.5 shows the three different textures observed in homeotropic anchoring in a wedge sample and the existence of two particular values of the confinement ratio for which one passes from one texture to the other. More precisely, one sees that: 1. For C C c 1.2, the cholesteric is completely unwound, with the director normal to the plates (homeotropic nematic). 2. For C c C C *, a fingerprint texture forms, composed of the socalled cholesteric fingers (CF) separated by homeotropic nematic. 3. For C C * 1.6, we still observe a fingerprint texture, but the fingers adhere together. Figure 3.5. Textures in homeotropic anchoring. From left to right (i.e., at increasing thickness) one can distinguish a black zone (homeotropic nematic), then fingers separated by homeotropic nematic and finally adhering fingers. Crossed polarizers (from Ref. [8]).

9 Phase transitions and unwinding of cholesteric liquid crystals 55 Nature of the cholesteric-nematic phase transition: Experimental determination The nature of the N-Ch transition in homeotropic anchoring can be determined by studying the nucleation and growth of the cholesteric fingers as a function of the confinement ratio [9]. The experiment consists of preparing a sample of a given thickness between two parallel electrodes and of applying an electrical field. If the dielectric anisotropy of the liquid crystal is positive, the field tends to unwind the helix and can be used to induce the nematic phase, provided it is large enough (see the next section for the calculation of the critical field). By turning off the field, the cholesteric phase nucleates and grows inside the nematic phase. Subsequent evolution strongly depends on the value of C: 1. If C < C c, the nematic phase is stable. 2. If C c C C *, cholesteric fingers (CF) nucleate from dust particules and invade the whole sample. Two growth modes are observed experimentally (Fig. 3.6). Close to C c, fingers grow from their two tips while undulating. It must be noted that the two tips are different, one of them being more pointed than the other. This comes from the absence of mirror symmetry in cholesterics. At larger values of C the pointed tip of the finger becomes unstable and splits continuously whereas the other remains unchanged. This leads to the formation of a circular domain. At the end of the growth process, the sample shows a fingerprint texture with fingers separated by homeotropic nematic. 3. If C C *, the nematic phase is absolutely unstable. Immediately after the field is switched off, the bulk of the sample lights up (in a fraction of a second) and presents a transient, translationally invariant configuration (TIC). This texture then slowly relaxes (over a few minutes) toward a periodic structure composed of adhering fingers, without homeotropic nematic in between (Fig. 3.7). These observations show that the N-Ch transition is first order, Figure 3.6. The two usual types of domains observed for C c C C *. a) Isolated fingers growing from its two tips; b) Flowerlike growth of a periodic finger pattern. Note the rounded tips split continuously contrary to the pointed tip which remains unchanged (from Ref. [9]).

10 56 Figure 3.7. Nucleation of a translationally invariant configuration (TIC) immediately after the field is turned off. This configuration then slowly relaxes toward a fingerprint texture with adhering fingers (from Ref. [9]). C * corresponding to the spinodal limit of the nematic homeotropic phase. To conclude this section, we recall that this transition was first observed in 1974 by Brehm et al. [10] (for a review see Ref. [11, 12]). Second, we mention that the fingers forming the fingerprint texture are stable and match continuously the nematic phase (there is no visible discontinuity of the director field between the two phases, contrary to what happens in planar anchoring). Finally, we mention that we recently observed the nucleation very close to C * of metastable textures. Two such textures are shown in Fig. 3.8 [13]. In the Figure 3.8. Two germs filled with metastable textures. They form only when C is very close to C * (from Ref. [13]).

11 Phase transitions and unwinding of cholesteric liquid crystals 57 first case, concentric fingers of a new type grow parallel to the boundary of the germ. In the second case, the germ is filled with a square-pattern texture. Both textures are metastable and slowly transform into the stable fingerprint texture when the sample is annealed during many days. The molecular structure of these two metastable states is still unknown. In the next subsection, we use a representation on the unit sphere S2 of the director field to describe the TIC and the cholesteric fingers. This method was first used by Thurston and Almgren [14] in nematics and was generalized to cholesterics by Lequeux et al. [15]. Translationally Invariant Configuration (TIC) and Cholesteric Fingers (CF): Structure and description on the unit sphere S2 In order to find a relevant order parameter to describe the N-Ch transition within the general Landau-Ginzburg formalism, a representation on the unit sphere S2 of the TIC and of the CF s is very useful. It consists of associating a director orientation with a point on S2. When the director orientation changes in the real space, the representative point describes a trajectory on S2. The simplest case is the homeotropic nematic: the director has the same orientation everywhere (parallel to the z axis) and the entire director field is represented by a unique point on S2, conventionally the North pole (Fig. 3.9a). For the TIC, it is sufficient to give the image of a line perpendicular to the plates (parallel to the z axis) as the configuration is invariant by translation in the (x, y) plane (Fig. 3.9b). The image on S2 is thus a close curve going through the North pole at z = 0 and z = d in order to satisfy the homeotropic anchoring on the plates. A point P on this curve is defined by the two angles α(z) and β(z). At height z the director has components: (3.7) These formula establish the correspondence between curve C in S2 and the director field in real space. The case of a CF parallel to the x axis is more complicated as its description requires us to give all the S2 images of all the straight lines parallel to the y axis that cross the finger. For simplicity, we assume that these curves are circles whose centers lie on the curve describing the TIC. One thus goes continuously from the TIC to the finger by opening up circles representing the director field n (y, z) around each point P of the curve C giving n (z) in the TIC (Fig. 3.9c). Let γ (z) be the half-opening angle of the circles, assumed to be traced at constant velocity. The director field reads:

12 58 Figure 3.9. Representation on the S2 sphere (right) and in real space (left). a) Homeotropic nematic; b) TIC; c) CF in the general case (adhering fingers); d) Homeotropically edged CF. Each trajectory drawn in thick line on S2 is the image of the oriented straight line superimposed to the director field (from Ref. [1]). (3.8) with λ the finger width and k = 2π/λ. Obviously, these equations reduce to those of TIC for γ = 0. As for the fingers which are edged with homeotropic

13 Phase transitions and unwinding of cholesteric liquid crystals 59 nematic, we must take α = γ in order that all circles go through the North pole (Fig. 3.9d). The following, more tedious, step is to calculate and then minimize the Frank energy per unit area in the (x, y) plane. This procedure which is detailed in Ref [1, 9] shows that the trivial solution α = γ = 0 corresponding to the homeotropic nematic phase becomes absolutely unstable as long as where K 32 = K 3 /K 2. So C * defines the spinodal limit for the nematic phase. For larger value of C the solution reads: (3.9) (3.10) where Z = z/d. The couple (α o, γ o ) thus provides a natural choice for the order parameter, as in the nematic phase α o = γ o = 0 while the cholesteric phase corresponds to α o γ o 0. Role of the elastic anisotropy on the first- or second-order character of the transition: Theoretical predictions In order to determine the type of solution which develops above C *, we need to calculate the elastic free energy F per unit surface area in the horizontal plane. A straightforward calculation yields after minimization with respect to k: with (3.11) (3.12) and K 12 = K 1 /K 2 and K 32 = K 3 /K 2. It can be checked from Eq. 3.9 that the transition is first order for B < 0 or D + 2B > 0 and second order for B > 0 and D + 2B > 0.

14 60 Let us examine the case corresponding to a second-order phase transition. Above the critical threshold (C > C * = C c or ν < 0), the solutions belong to one of the two types: 1. Two solutions (which are in fact identical) with the same energy (3.13) describing the same TIC (with k = 0), or; 2. One solution of the finger type with homeotropic edges for which (3.14) and of energy (3.15) (3.16) These formula show that the fingers are more favorable than the TIC for D 2B < 0. Conversely, the TIC is preferred for D 2B > 0. This discussion can be graphically summarized in the plane of the elastic parameters (K 12, K 32 ) (Fig. 3.10). Three curves must be considered: 1. Curve 1, corresponding to B = 0 of equation: 2. Curve 2, corresponding to D + 2B = 0, of equation: 3. Curve 3, corresponding to D 2B = 0, of equation: (3.17) (3.18) (3.19) These three curves divide space (K 12, K 32 ) in three domains. Below lines 1 and 2 (hence below line 2) the transition is second order. In this half space, two regions

15 Phase transitions and unwinding of cholesteric liquid crystals 61 Figure Nature of the N-Ch transition as a function of the elastic anisotropy in the vicinity of C *. must be considered: region (II) in which fingers with homeotropic edges develop and region (III) in which the TIC is more stable. Above line 2, in region (I), the transition is first order. The solution still consists of fingers with homeotropic edges which must be determined numerically. In the following section we analyze the influence of a magnetic or an electrical field Unwinding in a magnetic or an electrical field This problem was first tackled simultaneously by P.-G de Gennes [16] and B. Meyer [17] in an infinite medium. We first recall the main results obtained in this limit Infinite sample: The de-gennes-meyer model Let us consider a cholesteric phase with a positive diamagnetic anisotropy κ a > 0. If a magnetic field is applied perpendicularly to the helix, it unwinds. This process leads to the formation of unwound zones in which the director is parallel to the field separated by π-walls in which the twist concentrates (Fig. 3.11). In that case, the structure wavelength Λ diverges above a critical field of expression: Below this field, the wavelength is given by (3.20) (3.21)

16 62 where k is solution of the equation: (3.22) Here, K(k) and E(k) denote the complete elliptical integrals of the first and of the second kind, respectively. Figure Schematic representation of the cholesteric helix for 0 < B < B c (from [1]) Planar sample: Pitch divergence measurement The divergence of the pitch was observed experimentally simultaneously by B. Meyer [18] and G. Durand et al. [19] by placing the cholesteric in a Cano wedge. When submitting the sample to a magnetic field parallel to the anchoring direction, the Grandjean lines move out of the wedge till their complete disappearance when B > B c. Agreement between theory and experiment is excellent. The situation is more complex in homeotropic samples when they are submitted to a electrical field (easier to produce than a magnetic field). This problem is discussed in the next subsection Homeotropic sample We now analyze the behavior of a cholesteric sample sandwiched between two parallel electrodes treated for strong homeotropic anchoring. The control parameter are now the confinement ratio C and the electrical field (or, equivalently, the applied voltage U). Two cases must be distinguished, depending on the sign of the dielectic anisotropy a.

17 Phase transitions and unwinding of cholesteric liquid crystals 63 Phase diagram for a cholesteric of positive dielectric anisotropy Let us first consider the case of a material with a > 0. The applied field tends to align the molecules perpendicularly to the electrodes, so favoring the homeotropic nematic phase. Experiments are performed under AC electrical field in order to avoid electrohydrodynamic instabilities. The experiment shows that the cholesteric-nematic phase transition is first order in practice. The phase diagram of Fig has been obtained with a mixture of 8CB with the chiral molecule ZLI811 from Merck (0.46 wt% and T = 39 C). The texture at equilibrium consists of fingers below the coexistence line V 2 and of homeotropic nematic above. Lines V 0 and V 3 correspond to the spinodal limits of the nematic and the fingers, respectively. A linear stability analysis gives the following formula for V 0 : (3.23) The thin line V 1 gives the limit between isolated fingers (region III and Fig. 3.6a) and periodic fingers (in region IV the fingers are separated by homeotropic nematic as in Fig. 3.6b), whereas in region V they adhere together as observed in Fig. 3.7c (see also Fig. 3.5 on the right-hand side). Very similar phase diagrams were obtained from the Landau theory described in section by using the representation on the unit sphere S2 of the fingers [9]. Figure Phase diagram for the 8CB/ZLI811 mixture (from Ref. [1]). Phase diagram for a cholesteric of negative dielectric anisotropy The diagram of Fig corresponds to a material of negative dielectric anisotropy (Roche wt% ZLI811 from Merck) [20]. In this case, the molecules tend to align perpendicularly to the electrical field. For this reason,

18 64 Figure Phase diagram for the Roche2860/ZLI811 mixture (from Ref. [1]). applying an electrical field allows us to recover the cholesteric phase under the form of fingers or TIC from the homeotropic nematic when the sample thickness is less than pk 32 /2. The experimental phase diagram now exhibits three different thickness ranges, separated by two particular points P and P. To the left of P the transition is second order and leads directly to TIC. Between P and P, the transition is still second order, but the solution consists of periodic fingers. To the right of P, the transition becomes first order with the formation of fingers. As before V 2 (C) is the coexistence line, V 0 (C) is the spinodal limit of the nematic phase, V 3 (C) is the spinodal limit of the fingers and V 1 (C) separates two types of finger growth (isolated below this line and forming circular germs above it). Note that fingers are separated by homeotropic nematic between V 2 (C) and V 0 (C) and adhere above V 0 (C). It can be shown that the spinodal limit of the nematic phase is: (3.24) Point P, where the transition changes order, is a tricritical point. Its position is solution of the following equation [20]: (3.25) Point P is a triple point where the nematic phase, the fingers and the TIC coexist. Its position is given by the following equation [20]:

19 Phase transitions and unwinding of cholesteric liquid crystals 65 (3.26) These two equations can be solved numerically. The result is in excellent agreement with the experimental data Unwinding in a temperature gradient close to the smectic A phase In this section, we analyze the unwinding transition in the vicinity of a smectic A phase. To simplify, we will assume that the nematic used to prepare the cholesteric mixture has a second-order phase transition toward a smectic A phase. This is the case, for instance, of 8CB (4-n-4 cyanobiphenyl) within a very good approximation [21]. In such a material the elastic constants K 2 and K 3 diverge at the nematic-smectic A transition temperature T NA because of the divergence of the smectic correlation lengths ξ and ξ [22] parallel and perpendicular to the director: t ν ξ and ξ t ν, where t is the reduced temperature (T T NA )/T NA and ν and ν are two critical exponents, equal to 2/3 within the 3DXY model, but generally different experimentally (ν whereas ν in most materials [23]). Because K 2 ξ 2 / ξ and K 3 ξ 2ν + ν [24], we have the following scaling laws for the elastic constants: K 2 t and K 3 t ν. In the following, we successively analyze the role of the chirality on the order of the cholesteric-smectic A phase transition, first when the sample is infinite and second, when it is confined between two surfaces treated for planar or homeotropic anchoring which unwinds the helix. Then, we shall describe the structure of the front separating the nematic (unwound cholesteric) from the cholesteric phase when the sample is placed into a temperature gradient The bulk smectic A-cholesteric phase transition We first consider the case of the smectic A-nematic phase transition. Following de Gennes, the pertinent order parameter to describe this transition reads: (3.27) This quantity represents the complex amplitude of the density modulation in the smectic phase: (3.28)

20 66 where ρ(r ) is the densitiy, q o = (2π/a o ) n the wave vector of the density modulation and a o the thickness of the smectic layers at equilibrium. In the smectic phase Ψ 0, whereas in the nematic phase Ψ = 0. The Landau-de Gennes free energy describing the phase transition reads (omitting the gradient terms and the coupling with the director fluctuations): (3.29) In this expression, α = α o (T T NA )/T NA with α o > 0. Assuming β > 0 ensures a second-order phase transition. Minimizing g gives Ψ = in the smectic phase (T < T NA ) and Ψ = 0 in the nematic phase (T > T NA ). We now consider the cholesteric-smectic A phase transition. The natural way to describe the cholesteric phase is to add a twist term to the free energy which becomes [25, 26]: (3.30) This expression generalizes Eq and allows us to calculate the energy of the two phases. In the cholesteric phase, Ψ = 0 and n.curln = q so that: In the smectic phase, n.curln = 0, which gives: (3.31) (3.32) This expression generalizes Eq as it contains the energy (1/2)K 2 q 2 necessary to unwind the cholesteric phase. We can now calculate the transition temperature T ChA between the cholesteric and the smectic A phases. From g A / Ψ = 0 one obtains Ψ 2 = α/β and g A = α 2 /4β + (1/2)K 2 q 2. The transition temperature is given by setting g A = 0 (the energy of the two phases are equal at the transition), which yields: (3.33) This expression shows that the transition temperature decreases (T ChA < T NA ) when the nematic phase is doped with a chiral agent and transforms into a cholesteric phase.

21 Phase transitions and unwinding of cholesteric liquid crystals 67 More important is that the phase transition becomes first order. This can be easily seen by calculating the entropy of the two phases at T ChA. In the cholesteric phase, S Ch = 0, but in the smectic A phase, S A = dg A /dt = 2 α o q K2 /(2 β T NA ). This entropy jump, to which is associated the latent heat (3.34) is a clear indication of a first-order transition. Note that the smectic order 2 2 parameter also jumps at the transition from Ψ A = 2 Kq 2 / β in the smectic A phase, to Ψ 2 ch = 0 in the cholesteric phase. To conclude this section, we can make crude estimates of the elastic constant K 2, of the cholesteric pitch p and of the latent heat H at the transition temperature T ChA. These quantities (in particular K 2 and q) are difficult to measure experimentally. To fix ideas, let us consider as a typical example the mixture 8CB + 1 wt.% ZLI811 (from Merck). For pure 8CB, T NA 33 C. For the mixture, the transition temperature decreases: T ChA 32 C. Measurements in a Cano-wedge gives q NI 10 4 cm 1 close to the nematic-isotropic phase transition [27]. The Landau coefficients (assumed to be the same in the mixture and in pure 8CB) are given by Lelidis and Durand [28]: α o = erg 1 cm 3 and β = erg cm 3. Finally, the twist constant has been measured by Madhusudana et al. [29]: K 2 (T NI ) dyn. Assuming the product k = K 2 q is independent of the temperature because it is mainly determined by the molecular forces creating the cholesteric twist [30], we obtain k dyn/cm, independently of the temperature. From Eq. 3.33, we calculate: (3.35) The cholesteric twist, q, no longer diverges at the transition temperature and is given by (3.36) This calculation shows that the pitch varies from 6.3 µm close to T NI to 210 µm at T ChA. Finally, we can calculate the latent heat of the mixture. Eq gives: (3.37)

22 68 These theoretical predictions are compatible with very recent calorimetric measurements of Jamée et al. [31] performed with a similar mixture (8OCB + CB15 which is another chiral dopant from Merck). As for 8CB, the nematicsmectic A phase transition in 8OCB (octyloxycyanobiphenyl) is second order experimentally (undetectable latent heat by adiabatic scanning calorimetry). The temperature decrease T is close to 1 C when 1 wt.% of CB15 is added to 8OCB, and the latent heat measured in this mixture is close to 2 J/kg, which is of the same order of magnitude as that estimated before for the mixture 8CB + 1 wt.% of ZLI811. These results show that the cholesteric-smectic A transition is first order. First-order vs. second-order: Role of the confinement The previous analysis shows that the cholesteric pitch does not diverge at the transition toward the smectic A phase. In addition, the cholesteric helix can completely unwind (leading to a nematic phase) if the sample is confined between two parallel surfaces imposing a strong planar or homeotropic anchoring of the molecules. Let p be the equilibrium pitch (p = 2π/q) measured in a bulk sample at the transition temperature T ChA and let C c be the critical confinement ratio (by definition C = d/p where d is the sample thickness) below which the helix unwinds at this temperature. In planar anchoring, C c = 1/4 while C c 1 for homeotropic anchoring (its value depends on the anisotropy of the Frank elastic constants: it is not calculable analytically but an upper limit is given by K 32 /2). Let us first consider the case of a sample treated for planar anchoring. Three cases must be considered: 1. The sample is strongly confined (C < C c = 1/4). The cholesteric phase is unwound at the transition. In these conditions, one must observe a secondorder nematic-smectic A phase transition at T = T NA since q = 0 at the transition. The confinement (which acts as an external field which opposes to the internal field created by the chiral molecules) thus drives the transition from first order to second order. 2. The sample is moderately confined (1/4 < C < 3/4). In that case, an halfpitch develops at the transition in the sample thickness. One thus must observe a first-order cholesteric-smectic A phase transition. Nevertheless, there is an important difference with respect to the case of an infinite sample, as the pitch is now fixed by the boundary conditions: p = 2d. According to the previous theory, the free energies of the two phases, respectively, read at the transition temperature: (3.38)

23 Phase transitions and unwinding of cholesteric liquid crystals 69 (3.39) Equaling these two expressions give the transition temperature T ChA (d) [or, equivalently, the temperature shift T(d) = T NA T ChA (d)] as a function of the sample thickness: (3.40) As expected, T(p/4) = 0 and T(p/2) = T( ) (in that case, the cholesteric has the same pitch as in an infinite medium, so that the anchoring does not play any role). Finally T(p/2) = 8/9 T( ), which is close to T( ). 3. The sample is slightly confined (C > 3/4). One must then consider the intervals (2n 1)/4 < C < (2n + 1)/4 with n = 2, 3,. In the n th interval, the number of half-pitch is equal to n. The transition temperature can be calculated as before, which gives: (3.41) From this equation we calculate the transition temperatures at the two limits of the n th interval: (3.42) while in the middle of the interval: (3.43) (3.44) As expected, T(d) tends to T( ) when n. The normalized transition temperature is plotted as a function of the confinement ratio in Fig This graph shows clearly that the transition temperature is very close to T ChA as long as the confinement ratio is larger than 1/2. A different behavior may be expected in homeotropic anchoring in spite of the fact that the calculations cannot be done analytically due to the absence of analytical expression for C c. Nevertheless, we know that the spinodal limit for the homeotropic nematic phase is given by C * = K 32 /2. As K 32 scales like 2( ) (T T NA ) ν ν and because ν > ν experimentally [23], it follows that C * must

24 70 Figure Normalized transition temperature shift as a function of the confinement ratio in planar anchoring. diverge at T NA. If the ratio C * /C c remains finite, which seems reasonable from our point of view, C c also diverges at T NA and the cholesteric phase must always unwind whatever the sample thickness in homeotropic anchoring. The first-order cholesteric-smectic A transition is thus systematically driven toward a second-order nematic-smectic A phase transition in homeotropic sample. This is a prediction which deserves an experimental verification. We emphasize that this behavior differs from that predicted in planar anchoring. This difference is due to the fact that the cholesteric phase is more frustrated in homeotropic anchoring (always incompatible with the helical structure of the phase) than in planar anchoring (perfectly compatible with the helical structure of the phase each time the sample thickness is equal to a multiple of p/2). Unwinding in a planar sample: Position and Herring instability of the first simple κ-disclination line In this subsection, we assume the sample is sandwiched between two parallel glass plates treated for planar anchoring. In addition, the anchoring directions are the same on the two plates. The sample is placed in a temperature gradient taken along the x axis, parallel or perpendicular to the anchoring direction. The sample is observed under a polarizing microscope. The confinement ratio (calculated by taking the value of the pitch at T ChA ) is supposed to be less than 1/4 so that the cholesteric phase completely unwinds in the vicinity of the cholesteric (nematic)-smectic A front. In practice, the sample straddles two ovens separated by a gap of a few mm. The oven temperatures are chosen in order that the front lies approximately in the middle

25 Phase transitions and unwinding of cholesteric liquid crystals 71 of the gap. Due to the temperature gradient and providing that the cholesteric pitch p measured at high temperature is smaller than d/4, a simple κ disclination line forms in the vicinity of the front. The question which then arises concerns the position of the line. At a small temperature gradient (G 0), it is simply given by x l = G(T T NA ) with p(t) = d/4. Note that we have taken x = 0 at T = T NA. It turns out that this result fails when G 0 because the line energy depends on the temperature via the Frank elastic constants. Let E(T) be the energy of the line. In order to calculate its position x l with respect to the smectic front, we calculate the total energy of the cholesteric phase. It reads: where L is the size of the sample. Minimizing with respect to x l gives: (3.45) (3.46) In principle, this equation allows us to calculate the position (or equivalently, the temperature) of the line in the temperature gradient. Experimentally, x l can be measured under the microscope, which gives the temperature of the line T l as a function of the temperature gradient. The temperature T l is a solution of the following equation, equivalent to eq. 3.46: (3.47) where p(t l ) is the pitch at temperature T l. So, plotting T l as a function of the temperature gradient G, and then extrapolating this curve to G = 0, gives the pitch at the extrapolated temperature T o = T l (G 0): (3.48) This method has been used recently to measure the pitch close to T NA [6]. In practice, it works well only when the line is rectilinear, which is the case when it is parallel to the anchoring direction. Indeed, rotating the sample by 90 in the temperature gradient leads to a zigzagging disclination line (Fig. 3.15). This instability is similar to the Herring instability [32] which sometimes occurs at the solid-liquid interface [33]. It develops when the line tension becomes negative [22]. Indeed, let α be the angle between the line and the anchoring direction. Due to the elastic anisotropy, the line energy per unit length E depends on α and is different from the line tension τ of the expression:

26 72 Figure Zigzag instability of the first simple κ-disclination line. The anchoring direction is parallel to the temperature gradient α = π/2. G = 2 C/cm, d = 10 µm and p(t NI ) 14 µm (8CB wt% ZLI811). Note the continuous passage from the smectic to the nematic phase, showing that this transition is second order (or very weakly first order) (courtesy A. Dequidt). (3.49) As the line is stable when α = 0 and unstable when α = π/2 we can conclude that: (3.50) Preliminary calculations indicate that the instability develops when the main distortion close to the line core is of the bend-twist type rather than of the splay-twist type. This indeed happens when α = π/2. The energy increase in the latter case is due to the divergence of the bend constant K 3, the splay constant K 1 remaining finite at the transition. As a consequence, the Herring instability of the line is due to the divergence of the anisotropy ratio K 31 = K 3 /K 1. To our knowledge, this instability has never been reported before. Calculations are in progress to find the critical value of K 31 above which the instability develops. Unwinding in a homeotropic sample: static and dynamic properties of the cholesteric-nematic interface As we already pointed out, the cholesteric phase unwinds in a temperature gradient in the vicinity of the smectic A phase. This phenomenon was reported in Refs. [1, 11, 15] and attracted much attention recently in the dynamical regime [13]. In this section, we recall the main experimental results obtained in this geometry. If the sample is sufficiently thick, i.e., if its thickness d is larger than the pitch measured at high temperature, a nematic-cholesteric front spontaneously develops in the sample at some distance of the nematic-smectic A front. The structure of this front has been studied at rest, but also in the

27 Phase transitions and unwinding of cholesteric liquid crystals 73 Figure Directional melting experiment. The sample is moved at constant velocity v in the temperature gradient. The cholesteric-nematic front then moves at velocity v with respect to the glass plates and is observed under the microscope between crossed polarizers. dynamical regime when the cholesteric phase grows at velocity v into the nematic phase. The experiment consists of moving the sample in the temperature gradient G at velocity v from the cold oven toward the hot oven and of observing the front under the microscope (directional melting experiment, Fig. 3.16). The experiment shows that the front temperature and its morphology in the stationary regime depend on both the imposed velocity v and the confinement ratio C calculated from the pitch measured at high temperature (in practice close to the cholesteric-isotropic phase transition temperature). The experiment has been performed with the mixture 8CB+0.45wt% ZLI811. The pitch measured at high temperature is close to 14 µm. Fig shows the front behavior in a thin sample (C < 1.8). The front at rest consists of cholesteric fingers with their rounded tips at the front ( cellular regime). This morphology does not change up to a critical velocity above which the fingers spontaneously rotate by π. The front then consists of fingers with their pointed tips at the front ( dendritic regime). It must be noted that in this regime, the wavelength is larger than in the previous regime. The structure of the front remains essentially unchanged until a second critical velocity above which the fingers are progressively replaced by the TIC. In this high-velocity regime, the TIC relaxes behind the front by forming adhering fingers containing a large density of edge dislocations. It must be noted that the width of the nematic band which forms between the smectic and the cholesteric phases linearly increases as a function of the velocity in the cellular regime, then abruptly decreases when the fingers rotate by π for again increasing

28 74 Figure Nematic-cholesteric front observed in directional melting at different velocities in a thin sample (d = 15µm) (from [13]). linearly as a function of the velocity in the dendritic regime, but more slowly than in the cellular regime. Finally, the width of the nematic band tends to strongly saturate at large velocities when the TIC forms at the interface. We emphasize that in the first two regimes, the nematic phase is metastable, whereas in the third regime, the nematic becomes unstable ahead of the front. The front morphology changes in samples of intermediate thicknesses (1.8 < C < 2.5). In that case, we successively observe at increasing velocity, first, fingers with their rounded ends at the front, second, fingers parallel to the front and, finally, TIC which relaxes first toward a square pattern and then toward adhering fingers (Fig. 3.18). The experiment shows that the nematic phase is metastable in the three first regimes (where the width of the nematic band increases linearly as a function of the velocity), whereas the nematic phase becomes unstable in the fourth regime (where the width of the nematic band strongly saturates). It is important to note that both the parallel fingers and the square patterns correspond to metastable states of the cholesteric phase as they disappear slowly to be replaced by usual fingers when the sample is annealed. Note that all these solutions have also been observed in free growth (see Figs. 6-8).

29 Phase transitions and unwinding of cholesteric liquid crystals 75 Figure Nematic-cholesteric front observed in directional melting at different velocities in a sample of intermediate thickness (d = 40µm) (from Ref. [13]). The situation is still different in thick samples (C > 2.5). In that case, only two regimes are observed: the usual one, observed in all samples at small velocities, consisting of fingers with their rounded tips at the front ( cellular regime), and, above some critical velocity, the TIC regime with the TIC relaxing toward adhering fingers behind the front. Again it must be pointed out that the nematic is metastable in the first regime (in which the width of the nematic band increases linearly as a function of the velocity), whereas it becomes unstable in the second regime (where the width of the nematic band saturates). All these observations are summarized in the phase diagram of Fig Two regions Figure Phase diagram observed in directional melting at the nematic-cholesteric front. c gives the orientation of the underlying TIC from which each texture is formed (from Ref. [13]).

30 76 must be distinguished (outside of the region corresponding to very thin samples where the cholesteric phase cannot develop), in which the underlying TICs leading to the various structures observed at the front and behind it, have different orientations. The TIC orientation is given by the vector c which is drawn in the phase diagram. The orientation of vector c is given in Fig in the two cases of the cellular and the dendritic regimes. It must also be emphasized that below lines 1 and 2 of the phase diagram, the nematic phase is Figure (a) Representation in the real space on the unit sphere S2 of the TIC; (b) orientation of the c director in a segment of a cholesteric finger with respect to the coordinate system (x, y, z); (c) (d) orientation of the c director in the cell regime (c) and in the dendritic regime (d) (from Ref. [13]).

31 Phase transitions and unwinding of cholesteric liquid crystals 77 metastable, whereas above, the nematic becomes unstable ahead of the interface. Thus, this system provides an unique example of directional growth of one phase ( the solid ) into another ( the liquid ) below and above the spinodal limit of the liquid. It is clear that many questions concerning this experiment are still open, one of the most intriguing being the reversal of the TIC associated to the cellto-dendrite transition observed in thin samples. Another unsolved problem concerns the formation and the structure of the metastable phases observed in the samples of intermediate thicknesses. From our point of view, all these questions can only be solved numerically, in view of the complexity of the dynamical equations, especially in anisotropic elasticity. Concerning this point, we refer to the Ph.D. thesis of J. Baudry, in which the reversal of the TIC is treated (and observed) numerically in isotropic elasticity [27], when the TICnematic phase transition is second order (see Fig. 3.10). Bibliography 1. P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments, (Taylor and Francis, CRC Press, London, 2005). 2. M. Brunet-Germain, D.Sc. Thesis, University of Sciences and Techniques of Languedoc, Montpellier, 1972; see also M. Brunet-Germain, C. R. Acad. Sci. (Paris), B274 (1972) H. Zink, V.A. Belyakov, Mol. Cryst. Liq. Cryst., 265 (1995) V.A. Belyakov, E.I. Kats, JETP, 91 (2000) V.A. Belyakov, P. Oswald, E.I. Kats, JETP, 96 (2003) A. Dequidt, Rapport de stage de DEA, Ecole Normale Supérieure de Lyon, G. Mallet, D.Sc. Thesis, University of Sciences and Techniques of Languedoc, Montpellier, S. Pirkl, Cryst. Res. Technol., 26 (5) (1991) K P. Ribière, P. Oswald, J. Physique (France), 51 (1990) M. Brehm, H. Finkelmann, H. Stegemeyer, Ber. Bunsenges. Phys. Chem., 78 (1974) P. Oswald, J. Baudry, S. Pirkl, Phys. Rep., 337 (2000) S. Pirkl, P. Oswald, Sci. Pap. Univ. Pardubice, Ser. A 10 (2004) P. Oswald, J. Baudry, T. Rondepierre, Phys. Rev. E, 70 (2004) R.N. Thurston, F.J. Almgren, J. Physique (France), 42 (1981) P. Oswald, J. Bechhoefer, A. Libchaber, F. Lequeux, Phys. Rev. A, 36 (1987) P.-G. de Gennes, Sol. Stat. Com., 6 (1968) R.B. Meyer, Appl. Phys. Lett., 12 (1968) R.B. Meyer, Appl. Phys. Lett., 14 (1969) G. Durand, L. Léger, F. Rondelez, M. Veyssie, Phys. Rev. Lett., 22 (1969) P. Ribière, P. Oswald, S. Pirkl, Phys. Rev. A, 44 (1991) A. Yethiraj, J. Bechhoefer, Phys. Rev. Lett., 84 (2000) 3642.