* Correspondence address: Institute for Energy Technology, P.O. Box 40, N-2027 Kjeller, Norway. Fax: #47-63-81-09-20. E-mail address: arne.skjeltorp@ife.no (A.T. Skjeltorp).  Present address: SINTEF, Blindern N-0314, Oslo, Norway. Journal of Magnetism and Magnetic Materials 226 } 230 (2001) 534}539 Ferro#uids, complex particle dynamics and braid description Arne T. Skjeltorp*, Sigmund Clausen  , Geir Helgesen Institute for Energy Technology, P.O. Box 40, N-2027, Kjeller, Norway Physics Department, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway Abstract Finely divided magnetic matter is important in many areas of science and technology. A special sub-class of systems are made up of freely moving particles suspended in a carrier liquid where the magnetic interactions play an important role in the actual structure formation and dynamical behaviour. These include ferro#uids, which are colloids of magnetic particles dispersed in carrier #uids, magnetic micro-beads, which are micrometer sized plastic beads loaded with iron oxide, and nonmagnetic particles dispersed in ferro#uids, forming the so-called `magnetic holesa. How, in a simple and forceful way, is it possible to characterise the dynamics of systems with several moving components like dispersed magnetic particles subjected to external magnetic "elds? The methods based on the theory of braids may provide the answer. Braid theory is a sub-"eld of mathematics known as topology. It involves classifying di!erent ways of tracing curves in space. The topological description of braids thus provides a simple and concise language for describing the dynamics of a system of moving particles as if they perform a complicated dance as they move about one another, and the braid encodes the choreography of this dance.  2001 Elsevier Science B.V. All rights reserved. Keywords: Ferro#uids; Critical phenomena*dynamics; Fluctuations*classical; Magnetic alignment; Magnetic colloids; Magnetic #uids; Particles *ferro#uid 1. Introduction Magnetic materials and systems have, in addition to their vast technological importance, also been tradition- ally important model systems and metaphors for explor- ing co-operative phenomena and complex processes. To a large extent, the scienti"c enterprise is concerned with the description of phenomena associated with one level in the hierarchy in terms of the particles and interactions associated with lower, or shorter length scale, levels. Insights obtained by studying systems on one length scale can be helpful in understanding processes occurring at smaller or larger length scales. The part of the hier- archy that is the subject of this review provides an impor- tant link between macroscopic phenomena and interactions on atomic and molecular length scales. For this reason, processes associated with these `colloidala length scales are of considerable practical importance and scienti"c interest. For an extensive review of the applications of experimental and numerical models to the physics of multiparticle systems, see [1]. In recent years, the development of a variety of methods for producing relatively well-characterized col- loidal particles with a narrow size distribution and uni- form composition has stimulated experimental work, new theoretical ideas and the development of simple computer models. An improved understanding of collo- idal systems including magnetic ones, resulting from these developments is also having a bene"cial impact on the study of processes occurring on neighbouring length scales. In particular, experiments carried out using relatively large (micron range) particles provide a dramatic visualisation of processes occurring in sys- tems comprising much smaller particles (down to atomic sizes) that are less easily observed. In this respect, experi- ments carried out with essentially monodisperse mi- cron}sized small particles can be regarded as analogue simulations. 0304-8853/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0304-8853(00)00929-X