Critical adsorption profiles around a sphere and a cylinder in a fluid at criticality: Local functional theory

Abstract

We study universal critical adsorption on a solid sphere and a solid cylinder in a fluid at bulk criticality, where preferential adsorption occurs. We use a local functional theory proposed by Fisher et al. [M. E. Fisher and P. G. de Gennes, C. R. Acad. Sci. Paris Ser. B 287, 207 (1978); M. E. Fisher and H. Au-Yang, Physica A 101, 255 (1980)PHYADX0378-437110.1016/0378-4371(80)90112-0]. We calculate the mean order parameter profile ψ(r), where r is the distance from the sphere center and the cylinder axis, respectively. The resultant differential equation for ψ(r) is solved exactly around a sphere and numerically around a cylinder. A strong adsorption regime is realized except for very small surface field h_{1}, where the surface order parameter ψ(a) is determined by h_{1} and is independent of the radius a. If r considerably exceeds a, ψ(r) decays as r^{-(1+η)} for a sphere and r^{-(1+η)/2} for a cylinder in three dimensions, where η is the critical exponent in the order parameter correlation at bulk criticality.

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

“…The order-parameter fluctuation is significant only on length scales smaller than ξ . On larger length scales, the probability distribution of the order-parameter profile should have a sharp peak around its maximum, and thus the most probable profile is regarded as observed without fluctuation (Okamoto & Onuki 2012;Yabunaka, Okamoto & Onuki 2013;Yabunaka & Onuki 2017). Hence, in the renormalized local functional theory, we can obtain this profile by minimizing (3.1), which is regarded as the grand-potential functional with the chemical potential µ, conjugate to ψ, being Drag coefficient of a particle in a binary fluid mixture 886 A2-7 put equal to zero; µ vanishes because of the critical composition (ψ = 0) far from the particle.…”

“…The equilibrium profile can deviate from zero near the particle surface because of the preferential adsorption. We here assume that the preferential adsorption is represented by only the surface field and neglect higher-order terms with respect to ψ, such as the second-order term involving the surface enhancement (Bray & Moore 1977;Diehl 1986Diehl , 1997Cardy 1996), to study how the adsorption influences the drag coefficient, as in the previous studies of the renormalized local functional theory or the deviation of the drag coefficient (Okamoto & Onuki 2012;Furukawa et al 2013;Yabunaka et al 2013Yabunaka et al , 2015Yabunaka & Onuki 2017;Fujitani 2018).…”

“…Asĥ increases in figure 2(a), the inhomogeneity appears at smaller ξ ∞ /a and the plateau value is smaller, which is expected because the preferential adsorption causes the inhomogeneity. At the critical point (ξ ∞ = ∞), the equilibrium profile around a spherical particle is calculated in Yabunaka & Onuki (2017), where ξ 1 /a is found to be given by 6 −1/3ĥ−2/3 forĥ 1. This theoretical result, respectively, gives ξ 1 /a = 6.6, 3.6 and 1.9 × 10 −2 forĥ = 24, 60 and 150.…”

Isothermal transport of a near-critical binary fluid mixture through a capillary tube with the preferential adsorption

“…( 28) and (29), which are substituted into Eqs. (33) and (34) to yield the expressions of the Onsager coefficients of Eq. (35).…”

Diffusiophoresis in a Near-critical Binary Fluid Mixture

“…In the renormalized local functional theory [26,27], the free-energy functional (FEF) is coarse-grained up to the local correlation length, ξ. After coarse-grained, the equilibrium profile of the composition minimizes the FEF because many profiles different only on smaller length scales are unified into much fewer profiles [27,28]. The hydrodynamics formulated from the coarse-grained FEF [29,30] is used in our previous work [25].…”