Periodic buckling of smectic-A tubular filaments in an isotropic phase Masayoshi Todorokihara, Yosuke Iwata, and Hiroyoshi Naito * Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan (Received 28 January 2003; published 13 August 2004) Periodic buckling of smectic-A tubular filaments in an isotropic phase has been investigated both experi- mentally and theoretically in the binary mixture of octyloxycyanobiphenyl and dodecyl alcohol. As the mixture is cooled, straight filaments become unstable and continuously buckle by elongating at a constant periodicity. An analytical solution to the minimization of the curvature elastic energy of the smectic-A filament has been shown, and is consistent with the shapes of the buckled filaments. In addition, the dependence of the load on the length of the filament has been numerically calculated, and the critical buckling load of the smectic-A filament has been found to be of the order of 1 pN. DOI: 10.1103/PhysRevE.70.021701 PACS number(s): 61.30.Cz, 02.40.Hw, 46.32.+x, 62.20.Fe I. INTRODUCTION A smectic-A phase exhibits a variety of spatial patterns during its growth process from an isotropic phase [1–8]. One of the most interesting examples of the processes is the for- mation of smectic-A filamentally structures and their ther- motemporal evolution [4–7]. These filaments have a tubular structure (a cylindrical structure with an isotropic core) re- flected by the smectic-A layer structure with well-defined spacing [4–6]. Smectic-A straight filaments elongate when liquid crystal (LC) materials are cooled from an isotropic phase. Upon further cooling, these straight filaments become unstable and buckle continuously [4–6]. We note that the buckled filaments grow at constant periodicity (Fig. 1), and the periodically buckled filaments retain their shapes at a constant temperature. Such a buckling phenomenon is similar to the buckling of elastic columns known in classical mechanics [9]. A column subjected to compression undergoes displacements trans- verse to the load when the increasing load reaches the critical buckling load (the Euler load)[9]. Recently, layer buckling in bulk smectic-A LC has been reported [10,11]. However, the periodic buckling of smectic-A filaments reported in this paper is apparently different from the layer buckling in bulk smectic-A LC. In addition, no theoretical studies have been made on the growth of smectic-A filaments from the view- point of the buckling of elastic tubes. In this paper, we study the periodic buckling of smectic-A filaments both experimentally and theoretically. We first de- scribe observations of the pattern formation of periodic buckling of smectic-A tubular filaments in an isotropic phase. Second, we describe the curvature elastic energy of the smectic-A filament. We then derive the shape equations for the smectic-A filament from the minimization of the curva- ture elastic energy with the periodic boundary condition. We find that an analytical solution to the shape equations well describes shapes of the periodically buckled filaments ob- served in the experiments. The dependence of the load on the length of the filament is numerically calculated based on the principle of virtual work using the analytical solution, and found to decrease with the lateral displacement of the buck- led filament. We show that the critical buckling load is of the order of 1 pN. II. EXPERIMENTAL SETUP AND RESULTS Dodecyl alcohol (DODA) was mixed with octyloxycy- anobiphenyl (8OCB)(a molar ratio of 8OCB to DODA is 4 to 6) to suppress a nematic phase and to observe a smectic-A phase in an isotropic phase [5]. Liquid crystal cells with dimensions of 10 mm  10 mm  100 m were filled with the mixture. The LC cells were cooled at 0.05°C/min from the isotropic phase to the coexisting region of the smectic-A and isotropic phases to observe smectic-A fila- ments. The temperature of the LC cells was controlled in a *Electronic address: naito@pe.osakafu-u.ac.jp FIG. 1. Periodic buckling of a smectic-A filament observed at 41.0° C with polarizer and analyzer crossed. The cooling rate was -0.05°C/min. The bar indicates 50 m. PHYSICAL REVIEW E 70, 021701 (2004) 1539-3755/2004/70(2)/021701(6)/$22.50 ©2004 The American Physical Society 70 021701-1