PHYSICAL REVIEW E 88, 012511 (2013) Orientational energy of anisometric particles in liquid-crystalline suspensions S. V. Burylov 1,* and A. N. Zakhlevnykh 2,† 1 Institute of Transport Systems and Technologies, Ukrainian National Academy of Science, 5 Pisargevsky St., Dnepropetrovsk 49005, Ukraine 2 Physics of Phase Transitions Department, Perm State University, 15 Bukirev St., Perm 614990, Russia (Received 14 March 2013; published 30 July 2013) We obtain a general expression for the orientational energy of an individual anisometric particle suspended in uniform nematic liquid crystals when the main axis of the particle rotates with respect to the nematic director. We show that there is a qualitative and quantitative analogy between the internal and external problems for cylindrical volumes of nematic liquid crystals, and on this basis we obtain an estimate of the orientational energy of a particle of cylindrical (rodlike, needlelike, or ellipsoidal) shape. For an ensemble of such particles we propose a modified form of their orientational energy in the nematic matrix. This orientational energy has the usual second-order term, and additional fourth-order term in the scalar product of the nematic director and the vector which characterizes an average direction of the main axes of the particles. As an example we obtain the expression for the free energy density of ferronematics, i.e., colloidal suspensions of needlelike magnetic particles in nematic liquid crystals. Unlike previous models, the free energy density includes the proposed modified form of the particle orientational energy, and also a contribution describing the surface saddle-splay deformations of the liquid crystal matrix. DOI: 10.1103/PhysRevE.88.012511 PACS number(s): 61.30.Dk, 61.30.Hn, 61.30.Pq, 75.50.Mm I. INTRODUCTION Soft condensed matter, which includes such materials as liquid crystals, colloids, foams, gels, polymer melts, and solutions is one of the important areas of modern physics. The presence of internal degrees of freedom and high sensitivity of these materials to external influences leads to a wide variety of observable physical effects in them. Important examples of soft condensed matter are colloidal suspensions of anisometric particles in liquid crystals. The unique physics of these systems is due to the mutual influence of the anisotropic properties of spontaneously ordered liquid-crystalline environment and prolate or oblate particles of the solid phase embedded in it. Therefore, the physical properties of these composite materials are significantly richer than the properties of the constituent components. The collective response of liquid- crystalline suspensions to external fields leads to the existence of many new physical phenomena that are interesting from both fundamental point of view and applied perspectives. Apparently, the first historical examples of liquid-crystalline suspensions are ferronematics [1,2]. Ferronematics (FNs) are magnetic suspensions on the basis of nematic liquid crystals (NLCs). The solid phase of FNs consists of single-domain needlelike ferri- or ferromagnetic particles with length L and diameter d ∼ (L/10). Due to the shape anisotropy the particles have a magnetic rigidity at which the magnetic moments of needlelike ferroparticles are always directed along their main axes [3,4]. The length L of the particles and the diameter d are large in comparison with the nematic molecule size a, i.e., the particles represent the mesoscopic objects suspended in a liquid crystal. The volume fraction f of a solid phase is rather small (10 −7 –10 −2 ), therefore the magnetic impurity in FNs can be considered as an ideal gas of noninteracting magnetic grains. The existence of a small ferroparticle additive does not change the nature * burylov@westa-inter.com † anz@psu.ru of orientational order of a nematic matrix, therefore, FNs having high magnetic susceptibility otherwise behave like usual NLCs. More than a hundred papers are devoted to theoretical and experimental studies of FNs; their review is given in [5]. Below we discuss the main theoretical investigations devoted to the construction of FN continuum theory, as well as the results of the experiments which influenced the development of theoretical views about the internal structure and behavior of such liquid crystal materials. The idea of creation of magnetic suspensions was proposed by Brochard and de Gennes in their classical work [1] which initiated the beginning of development of the FN continuum theory. Later FNs were synthesized on a basis of both thermotropic [6–8], and lyotropic [9,10] nematic liquid crystals. The performed experiments [6–10] showed that values of the magnetic fields necessary for reorientation of a liquid crystal matrix of real magnetic suspensions, at least, are two orders of magnitude lower than fields of reorientation of usual nematics. This circumstance expands the use of magnetic liquid crystal materials in applications. The high magnetic susceptibility of FNs theoretically predicted by Brochard and de Gennes and confirmed exper- imentally, is caused by an orientational interaction between ferroparticles and a nematic matrix. For magnetized FN in which the magnetic moments of ferroparticles coincide in the direction at each local point of the sample, the natural variables responsible for the orientational order of nematic and magnetic subsystems are the director n( r ) and magnetization vector M( r ) averaged at scales much greater than L. Therefore the orientational interaction has to describe the local correlation of spatial distributions of n and M. This interaction makes the corresponding contribution to the total free energy of FN which we call “orientational energy of ferroparticles” or simply “orientational energy.” The detailed theoretical description of orientational inter- action of anisometric particles with a nematic matrix is given by Brochard and de Gennes [1]. The analysis of their theory allows us to conclude that the general expression for the 012511-1 1539-3755/2013/88(1)/012511(16) ©2013 American Physical Society