Computer Simulation of Liquid Crystals CARE, C

Computer simulation of liquid crystals CARE, C. M. and CLEAVER, D. J. <http://orcid.org/0000-0002-4278-0098> Available from Sheffield Hallam University Research Archive (SHURA) at: http://shura.shu.ac.uk/869/ This document is the author deposited version. You are advised to consult the publisher's version if you wish to cite from it. Published version CARE, C. M. and CLEAVER, D. J. (2005). Computer simulation of liquid crystals. Reports on progress in physics, 68, 2665-2700. Copyright and re-use policy See http://shura.shu.ac.uk/information.html Sheffield Hallam University Research Archive http://shura.shu.ac.uk REVIEW Computer simulation of liquid crystals C M Care and D J Cleaver Materials and Engineering Research Institute, She±eld Hallam University, Howard Street, She±eld, S1 1WB, UK E-mail: [email protected], [email protected] Article Reference: ROP/93935/REV Computer simulation of liquid crystals 2 Abstract. A review is presented of molecular and mesoscopic computer simulations of liquid crystalline systems. Molecular simulation approaches applied to such systems are described and the key ¯ndings for bulk phase behaviour are reported. Following this, recently developed lattice Boltzmann (LB) approaches to the mesoscale modelling of nemato-dynamics are reviewed. The article concludes with a discussion of possible areas for future development in this ¯eld . Computer simulation of liquid crystals 3 1. Introduction In this article, we review molecular and mesoscopic computer simulations of liquid crystalline (LC) systems. Owing to their ability to form LC mesophases, the molecules of LC materials are often called mesogens. Following a scene setting introduction and a brief description of the key points of LC behaviour we ¯rst review the application of molecular simulation approaches to these mesogenic systems; we only consider bulk behaviour and do not report work on con¯ned or inhomogeneous systems. This section is largely broken down by model-type, rather than area of application, and concentrates on the core characteristics of the various models and the results obtained. In contrast, in Section 4, we give relatively detailed descriptions of a series of recently developed lattice Boltzmann (LB) approaches to LC modelling and nemato-dynamics - following a period of relatively rapid development, a unifying review of this area is particularly timely. Finally, in Section 5, we identify a number of key unresolved issues and suggest areas in which future developments are likely to make most impact. Note that we do not review results obtained using conventional solvers for the continuum partial di®erential equations of LC behaviour. Whilst it might legitimately be argued that the mesoscopic technique considered in this review, LB, is simply an alternative method of solving the macroscopic equations of motion for the LCs, this particular method is perhaps best thought of as lying on the boundary between macroscopic and molecular methods. Additionally, it is straightforward to adapt the LB method to include additional physics (eg the moving interfaces found in LC colloids); as such, a clear distinction can be drawn between the LB approaches reviewed in Section 4 and conventional continuum solvers. Previous reviews of LC simulation include the general overviews by Allen & Wilson (1989) and Crain & Komolkin (1999) and a number of other works which concentrate on speci¯c classes of model. For hard-particle models, the papers by Frenkel (1987) and Allen (1993) o®er accessible alternatives to the all-encompassing Allen et al. (1993). Zannoni's (1979) very early review of lattice models of LCs has now been signi¯cantly updated by Pasini et al. (2000a) in a NATO ARW proceedings (Pasini et al. 2000b) which contains some other useful overview material, while Wilson (1999) has summarised work performed using all-atom models. Finally, there are two accessible accounts of work performed using generic models - Rull's (1995) summary of the studies that enabled initial characterisation of the Gay-Berne mesogen and the more recent overview by Zannoni (2001) which cogently illustrates subsequent developments and diversi¯cations. 1.1. The role of computer simulation in liquid crystal research Computer simulation is simply one of the tools available for investigating of mesogenic behaviour. It is, however, a relative newcomer compared with the many experimental and theoretical approaches available, and its role is often complementary; there is little to be gained from simulating a system which is already well characterised by more established routes. That said, appropriately focussed computer simulation studies can yield a unique insight into molecular ordering and phase behaviour and so inform the development of new experiments or theories. Most obviously, molecular simulations can provide systematic structure property information, through which links can be established between molecular properties and macroscopic behaviour. Alternatively, simulation can be used to test the validity of various theoretical assumptions. For example, by applying both theory and simulation to the same underlying model, uncertainties regarding the treatment of many-body e®ects in the former can be quanti¯ed by the latter. One of the strengths of simulation in the context of LCs arises in situations for which there are (spatial or temporal) gradients in key quantities such as the order tensor or the composition pro¯le. Such gradients are often di±cult to resolve experimentally and are only accessible to relatively coarse- grained theories. In comparison, computer simulation of `gradient regions' can often be achieved at the same computational cost as that needed to treat ùniform regions'; in these contexts, therefore, simulation is becoming the lead complementary technique for improved understanding of behaviours which are not fully accessible to experimental investigation. Computer simulation of liquid crystals 4 At continuum length scales, LCs are characterised by a large number of experimentally observable parameters: viscosities, determined by the Leslie Coe±cients; orientational elasticity, controlled by the Frank constants; substrate-LC orientational coupling, governed by anchoring coe±cients and surface viscosities. Given a full set of these parameters, mesoscale simulations are now able to incorporate much of this complex behaviour into models of real devices. Thus, subject to the usual provisos concerning continuum models, these approaches are now starting to gain the status of design and optimisation tools for various LC device applications. Before presenting the details of this review, we ¯rst ask the rather fundamental question - why perform computer simulations of LCs? LCs are fascinating systems to study because, like much of soft-condensed matter, their behaviour is characterised by the interplay of several very di®erent e®ects which operate over a wide range of time- and length-scales. These e®ects range from changes in intramolecular con¯gurations, through molecular librations to many-body properties such as mass flow modes and net orientational order to the fully equilibrated director ¯eld observed at the continuum length and time scales. The extent of the associated time- and length-scale spectra dictate that no single computer simulation model will ever be able to give a full àtom-to-device' description for even the simplest mesogen. Moreover, since these di®erent phenomena are, in general, highly coupled (eg intramolecular con¯gurations are influenced, in part, by the local orientational order) they represent a multi-level feedback system rather than a simple linear chain of independent links. Thus, addressing each phenomenon with its own model and simply collating the outputs of a series of stand- alone studies will again fail to achieve a full description. In partial recognition of this, most of the methods and models currently used to simulate LCs seek to explore only a subset of the spectrum of behaviours present in a real system. In some cases, this pragmatic approach is entirely appropriate: for certain bulk switching applications, for example, a continuum description can prove perfectly adequate (explaining the popularity of Ericksen-Leslie theory) and the molecular basis of LCs can be neglected. Alternatively, a well de¯ned generic model approach can provide the cleanest route to establishing relationships between molecular characteristics and bulk properties such as the Frank constants or the Leslie coe±cients. Returning to the original question, then, we can reply that appropriately focussed simulation studies have certainly provided a sound understanding of many of the processes underlying bulk mesogenic behaviour and the operation of some simple switching devices. The role of computer simulations in studying LCs is relatively well established; as noted above and shown in more detail in the following Sections, successful approaches have now been developed for many of the regions in the atom-to-device spectrum. As such, several of the fundamental problems in this ¯eld are now essentially solved. Given these achievements, it is now appropriate to raise the supplementary question - why continue to perform computer simulations of LCs? There is little of note to be gained by simply re¯ning existing approaches and exploiting Moore's law to incrementally enlarge the scope of, say, 3d bulk simulations of generic LC models of conventional thermotropic behaviour. Many of the outstanding challenges in LC science and engineering call, instead,

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