Virial Expansions for Low Dimensional Ferrofluids
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Virial Expansions for Low Dimensional Ferrofluids M. Widom and H. Zhang Department of Physics, Carnegie-Mellon University, Pittsburgh, Pa. 15213 Abstract Ferrofluids are colloidal suspensions of magnetic particles which we model as a dilute gas of dipolar hard spheres. At low particle density the osmotic pressure and magnetic susceptibility may be expanded in a virial series. We evaluate the second virial coefficients for fluids confined to a line, or to a plane, as well as ordinary three dimensional fluids in ellipsoidal containers. We focus our attention on anisotropy of the magnetic susceptibility and point out the role of particle chaining in determining the magnetic anisotropy of low dimensional ferrofluids. Introduction Dipolar hard sphere fluids are a widely studied model in statistical mechanics [1]. The motivations for this study include the application to the study of fluids of molecules [2] whose interactions are dominated by electric dipole moments 1 at large separation, and the application to ferrofluids [31 and electro-rheological fluids [4]. A further motivation is the anisotropy of the dipolar interaction 0(r 12 ) =
3(tj.
P12 )(• 2r 3 r 12 ) -
p41 2
(1)
which ranges from ferromagnetic when r 12 11 A to antiferromagnetic when r 12 1 1,. In three dimensions there are additional subtleties since the interaction is long range, so that physical properties become dependent on the shape of the container in which the fluid is held. One consequence of the anisotropy of the dipolar interaction (1) is a tendency for spheres to align into chains, with each dipole moment pointing towards a nearest neighbor along the chain (see Figure 1.) The degree of chain formation depends both on the number density p of particles, and on the strength of the dipolar interaction. A reduced dipolar coupling constant may be defined 3 (2) kBTa where a is the hard sphere diameter. For typical dipolar molecules this coupling is not large and ranges from A z 0.06 for CHC13 to A 2 0.5 for H2 0 at their Boyle temperatures [2]. For room temperature ferrofluids, however, it is quite possible to achieve A ;-t 1 - 10, and for electrorheological fluids A may reach 1000. These strongly coupled colloidal materials thus represent a parameter range less widely investigated than the weaker couplings characteristic of molecules. In particular, chain formation is pronounced in the colloidal materials, but not in the molecules. In tandem with the stronger coupling, however, we may expect interesting phenomena even at fairly low particle density. This paper addresses the magnetic anisotropy resulting from the onset of chain formation in low dimensional ferrofluids. Our principal conclusion is that correlations among particle positions serve to enhance susceptibility to fields in the parallel (infinite length) directions because chains can respond to the field as a coherent unit. At the same time, susceptibility to fields in the perpendicular (finite length) directions is diminished because the long chains cannot rotate into alignment with the f
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