Drag coefficient of a rigid spherical particle in a near-critical binary fluid mixture, beyond the regime of the Gaussian model

Abstract

The drag coefficient of a rigid spherical particle deviates from the Stokes law when it is put into a near-critical fluid mixture in the homogeneous phase with the critical composition. The deviation (∆γ d ) is experimentally shown to depend approximately linearly on the correlation length far from the particle (ξ ∞ ), and is suggested to be caused by the preferential attraction between one component and the particle surface. In contrast, the dependence was shown to be much steeper in the previous theoretical studies based on the Gaussian free-energy density. In the vicinity of the particle, especially when the adsorption of the preferred component makes the composition strongly off-critical, the correlation length becomes very small as compared with ξ ∞ . This spacial inhomogeneity, not considered in the previous theoretical studies, can influence the dependence of ∆γ d on ξ ∞ . To examine this possibility, we here apply the local renormalized functional theory, which was previously proposed to explain the interaction of walls immersed in a (near-)critical binary fluid mixture, describing the preferential attraction in terms of the surface field. The free-energy density in this theory, coarse-grained up to the local correlation length, has much complicated dependence on the order parameter, as compared with the Gaussian free-energy density. Still, a concise expression of the drag coefficient, which was derived in one of the previous theoretical studies, turns out to be available in the present formulation. We show that, as ξ ∞ becomes larger, the dependence of ∆γ d on ξ ∞ becomes distinctly gradual and close to the linear dependence.

Suppression of viscosity enhancement around a Brownian particle in a near-critical binary fluid mixture

“…Similar calculations can be found in deriving (2.17) of Fujitani (2007) and in deriving (3.20) of Yabunaka & Fujitani (2020). Equations (2.11) and (2.18) give…”

“…The kernel above for J = 1 is equal to 1/30 multiplied by Γ R of Okamoto et al (2013). Similar calculations are there in Fujitani (2018) and Yabunaka & Fujitani (2020). As mentioned above (2.23), we can delete Π J and T J from the last two terms of (2.13).…”

“…The right-hand side above is related to the fraction appearing in (2.8b) because of the Lorentz reciprocal theorem (Lorentz 1896), as shown in appendix B of Fujitani (2018) and mentioned at (A2) of Yabunaka & Fujitani (2020). We thus arrive at…”

“… where is the rate-of-strain tensor. Here, a low Reynolds number is assumed, as discussed in § 2 of Yabunaka & Fujitani (2020). The no-slip boundary condition is imposed at the particle surface, while tends to zero and approaches a constant, denoted by , far from the particle.…”

“…In some combinations of the mixture and particle material, one of the components is preferentially attracted by the particle surface and the preferred component is remarkably adsorbed near the particle surface because of the near-criticality (Beysens & Leibler 1982; Beysens & Estève 1985). The particle motion deforms the adsorption layer, which affects the force exerted on the particle (Lee 1976; Omari, Grabowski & Mukhopadhyay 2009; Okamoto, Fujitani & Komura 2013; Fujitani 2018; Tani & Fujitani 2018; Yabunaka & Fujitani 2020). In other combinations exhibiting negligible preferential adsorption, the particle motion remains still influenced by the near-criticality because of the critical enhancement of the viscosity (Ohta 1975; Ohta & Kawasaki 1976).…”