DOI 10.1140/epje/i2002-10164-3 Eur. Phys. J. E 12, 291–302 (2003) T HE EUROPEAN P HYSICAL JOURNAL E Monodomain and polydomain helicoids in chiral liquid-crystalline phases and their biological analogues G. De Luca and A.D. Rey a Department of Chemical Engineering, McGill University, 3610 University Street, Montreal H3A 2B2, Quebec, Canada Received 28 December 2002 and Received in final form 15 September 2003 / Published online: 11 November 2003 – c  EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag 2003 Abstract. Many natural composites exhibit an architecture known as twisted plywood which imparts to them a superior set of physical properties. The origin of this structure is complex and not yet understood. However, it is thought to involve a lyotropic chiral nematic liquid-crystalline mesophase. Indeed, striking structural similarities have been observed and reported between biological fibrous composites and ordered fluids. In this work, a mathematical model based on the Landau-de Gennes theory has been developed to investigate the role played by constraining surfaces in the structural development of a composite material that experiences a liquid-crystalline state during the early steps of its morphogenesis. The goal of this study is to verify the need for an initial constraining surface in the formation of monodomain twisted plywoods as hypothesized by Neville (Tissue & Cell 20, 133 (1988); Biology of Fibrous Composites (Cambridge University Press, 1993)). The numerical simulations qualitatively confirm this theory and highlight the important role that modelling of liquid-crystalline self-assembly plays in the study of tissue morphogenesis. PACS. 61.30.-v Liquid crystals – 61.30.Dk Continuum models and theories of liquid crystal structure – 61.30.Mp Blue phases and other defect-phases – 61.30.St Lyotropic phases 1 Introduction In contrast to man-made composite materials, Nature’s composites are assembled at ambient temperature and pressure [1–3]. In addition, despite being made of rela- tively simple constituents, they have outstanding physi- cal properties and are biodegradable. A major challenge in the field of applied material science is to manufacture synthetic equivalents to these natural materials. In order to do so, their formation process must be understood and described. Thus there is a recognized need for theories and computational studies directed to the structural formation of biological materials. Biological fibrous composites are highly organized ma- terials found in the skeletal systems of animals and plants. The most widely found architecture in these lam- inated materials is the twisted plywood, also referred to as helicoidal plywood, whose organization resembles the one used in classical industrial composites [1–3]. In the helicoidal plywood, fibrous macromolecules are oriented in parallel to form sheets stacked in a “spiral wooden staircase”. The properties of this architecture, explored through polarized and electron microscopy, have been su- perbly illustrated in the literature [1,2,4,5]. A question of fundamental interest to the morphogen- esis of natural composites is: how are the water insoluble a e-mail: alejandro.rey@mcgill.ca fibrous components precisely manipulated into the extra- cellular matrix so as to form twisted plywood assemblies? The most probable answer seems to be that the extracellu- lar matrix undergoes a liquid-crystalline mesophase which provides the required mobility for the fibrous components to self-assemble [1,2,4–6]. Liquid-crystalline mesophases are anisotropic vis- coelastic materials with properties between liquid and crystalline phases. Within a given range of temperature and concentration, they exhibit a long-range orientational order. Among the different kind of liquid-crystalline mesophases found in nature, chiral nematics are the ones which exhibit the greatest structural similarity to the supramolecular organizations of natural composites [1, 2,4–6]. In chiral nematic mesophases, molecules lie on a series of equidistant pseudo-planes, which are slightly rotated with respect to one another. This change in the molecular orientation through the structure is character- ized by a length scale called the pitch, denoted by p 0 , corresponding to the distance required by the average molecular orientation to rotate by 2π radians along the helical axis. Figure 1 shows a schematic of the structural organi- zation in the twisted plywood architecture. The rod-like constituents display the classical chiral nematic spatial or- ganization defined, in a rectangular (x, y, z) coordinate