Behaviour of a Sm C disclination line s = + 1/2 with a partial twist under a magnetic field

2014 The theoretical results of Allet, Kléman and Vidal [15] for the behaviour of the Sm C disclination s = + 1/2 with a partial twist are extended to the case of a magnetic field applied along the cylinder axis. The solution has shown that a critical magnetic field exists above which the disclination with a jump is transformed into a usual edge-disclination without twist. This makes possible the experimental determination of the 2 A21 A11 and 2 A12 A11 Sm C coefficient combination when A11 0 and A11 2 A21 and A11 2 A12 when A11 > 0. J. Physique Lett. 45 (1984) L-185 L-191 15 FEVRIER 1984, Classification Physics Abstracts 61.30J 47.15 The elastic energy of the smectic C (Sm~) liquid crystal (LC) consists of three parts. The first part was obtained for the first time by Salupe [1] and includes all the possible reorientations of the Sm C director around the normal to the Sm C plane : where lp is the angle between the projection of the molecular director on the Sm C plane and the axis Y, B1, B2, B3 and B13 are the elastic coefficients describing the resistance of the Sm C LC to all the possible reorientations of the Sm C director. The second part of the Sm C elastic energy, obtained for the first time by the Orsay Liquid Crystal Group [2], includes the curvature elasticity and the dilation (or compression) of the Sm C layers : Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01984004504018500 L-186 JOURNAL DE PHYSIQUE LETTRES where u is the displacement of the Sm.C layers in the direction of the Z axis (i.e. normal to the Sm C plane), All, A 12 and A21 are the elastic coefficients describing the resistance of the Sm C to the curvature of the Sm C layers and B is the dilation elastic coefficient. The third part of the Sm C elastic energy, also obtained by the Orsay Liquid Crystal Group [2], describes the coupling between the layer curvature and the director field : In addition, a number of inequalities about various combinations of the Sm C elastic coefficients are obtained from the considerations of stability : All the elastic coefficients, with the exception of the dilation modulus B, are very similar to those already known for the case of nematics and cholesterics and are expected to be in the range of 10 6 dyne [3]. The value of the Sm C elastic coefficients can be obtained, for instance, by the appropriate performance of all the possible kinds of the usual Fredericksz transition or by other, more complicated, deformations of the Sm C LC which include not only reorientations of the Sm C director but also the curvature of the Sm C layers. Although some authors have dealt with the possible Fredericksz transition of the Sm C [4-7], experimental results are very scarce [810] and they mainly give the value of the characteristic Sm C tilt and its temperature dependence. Recently, only Schiller and Pelzl have estimated the mean value of the elastic coefficients B1 ~~ B2 for the case of 5-n-heptyl-2(4-n-nonyloxyphenyl)-pyrimidine to be in the range of 6 x 10 -’ dyne [ 11 ]. In addition, the measurements of the elastic coefficients All, A I 2 and A 21 which describe the curvature of the Sm C layers, are much more complex since the electric and magnetic fields cannot ensure one eventual optically-observable deformation of the Sm C layers by the usual Fredericksz transition [4]. Another useful way for determining these coefficients is connected with measurements of the periodicity of the Sm C layer undulations [12-14]. The aim of this communication is to present theoretical results about the behaviour of the Sm C disclination line s = + 1/2 with a partial twist under the action of a magnetic field applied in a suitable manner. We have chosen this Sm C disclination since, first, its energy is given in an explicit form [15] and permits a simple accounting of the magnetic field action on the orientation of the Sm C director and second, the theoretical calculations can be easily verified by the performance of eventual measurements. Indeed, we shall see that the theoretical calculations clearly show interesting phase transitions which might be useful for the determination of some combinations of the Sm C elastic coefficients. ’ The density of the total energy of the disclination s = + 1/2 with a partial twist under the action of a magnetic field consists of two parts. The elastic energy, as noted, has been already given by Allet, Kleman and Vidal (see the relation (1) in Ref [15]). The type of magnetic field energy depends crucially on the direction of the magnetic field. It is known from the theoretical results of Allet et al [15] that w = 7r/2 and co = 0 (where (?c/2 w) is the angle between the projection of the molecular axis on the Sm C plane t and the cylinder axis) are minima for the disclination under study in the case of a positive sign of the elastic coefficient A I I Let us for convenience denote these cases by a) and b), respectively. On the basis of these results and also from considerations of symmetry we decided to choose the simplest case when the magnetic field L-187 Sm C DISCLINATION LINE UNDER A MAGNETIC FIELD is applied along the cylinder axis (see Fig. 7 in Ref [15]). In accordance with Meirovitch et al. [6] we can write the following density of the magnetic energy : where AX is the value of the magnetic anisotropy susceptibility, assumed to be positive and ~ is the characteristic tilt angle of the Sm C under study for the case a) when co = n/2 is the only minimum. It is clear that in the second case b), when (JJ = 0 is the only minimum, the density of the magnetic energy has the following form : For simplicity, in the relations (5) and (6) we have neglected the usual magnetic biaxiality of the Sm C LC [4]. On the other hand, it is well known that the characteristic tilt angle / is nearly temperature independent for the Sm C LCs which are formed from the N phase under cooling and strongly temperature dependent for those Sm C LCs which are formed from the Sm A phase after cooling. Before giving the detailed theoretical results let us initially stress that in all the following equations and relations the elastic terms have been already obtained by Allet, Kleman and Vidal [15]. Minimizing the total elasto-magnetic energy with respect to OJ leads to the following differential equation : where the upper sign ( + ) corresponds to the case a) and the lower sign ( ) to the case b). Let us in accordance with Allet et al. [15] define the easy directions which minimize the elastomagnetic energy for the case of (dc~/d9) = 0. Then the energy has the following form : The extrema are obtained for where the upper sign ( ) corresponds to the case a) and the lower sign ( + ) to the case b), i.e. for a) Wt = Tc/2 (t along the cylinder axis, i.e. here the molecular axis is parallel to the anchoring direction on the glass plate) b) W2 = 0 (the molecular axis is in the right section of the cylinder) _, which is a convenient form for the case a) L-188 JOURNAL DE PHYSIQUE LETTRES which is a convenient form for the case b). The minima are obtained for d2 fmldw2 > 0 : where the upper sign ( ) corresponds to the case a) and the lower sign ( + ) to the case b). According to relations (4) the elastic coefficients A12 and A21 must be positive. On the other hand, A 11 is the only Sm C elastic coefficient which can be either positive or negative. It is clear from equation (12) that when A 11, All 2 A 12, and All 2 A 21 are positive then co = 7r/2 (A,1 > 0, All 2 A 21 > 0) or co = 0 (A,1 > 0, All 2 A 12 > 0) are the only minima (as noted these cases were designated by a) and b), respectively), whereas co3 is necessarily a maximum of fm, if it exists. On the other hand, for the case of a vanishing magnetic field and negative sign of the elastic coefficient All ro 1 = ~/2 (case a) and co = 0 (case b) are maxima, Co 3 exists and it is the only minimum of fm. We shall see, however, that this is true also for small values of the magnetic field whereas for large values that are above a certain critical magnetic field the only minimum is either w 1 = Tr/2 (case a) or co = 0 (case b). Let us consider these cases : I. All 0. One looks for a first integral of (7) (in which we shall make Bl = B2 = B) satisfying the boundary condition dro/d8 = 0 for co = ro3. This yields : which is a convenient form for the case a) and for the case b). It seems that the solution of these differential equations consists of a complex combination of elliptic integrals and we shall not deal with this problem. For our aim, it is sufficient to point out that for small values of the magnetic field C03 oscillates in a very complex manner in the range ((~3, + (03) passing either through c~ = n/2 or o. On the other hand, the equations (10) and (11 ) clearly show that the raising of the magnetic field up to the value He which is determined either from the relation : L-189 Sm C DISCLINATION LINE UNDER A MAGNETIC FIELD for the case a) or from the relation : for the case b), lead to a distinct first order phase transition and the disclination s = + 1/2 with a partial twist is transformed with a jump into an usual wedge-disclination without twist It is clear that above this critical magnetic field the Sm C LC molecules prefer to stay in a position which minimizes the magnetic energy, i.e. the application of a magnetic field with a value which is above the critical one replaces the minimum Q)3 by Q) = 0 or tu = n/2. In the case a) all the molecules go to Tr/2 whereas in the case b) they go to Q) = 0. Furthermore, the relations (15) and (16) can be used for the determination of the 2 A21 All and 2 A 12 A 11 combinations of the Sm C elastic coefficients for the case of a negative value of A 11. For this purpose one should estimate the value of the core radius ro which according to Allet et al. [15] is of the order of a Sm C layer thickness. II. A 11 > 0. One finds different first integrals, according to whether the chosen minimum is Q) = 0 :