Asymmetric oscillations during phase separation under continuous cooling: A simple model

Abstract

We investigate the phase separation of binary mixtures under continuous cooling using the Cahn-Hilliard equation including the effect of gravity. In our simple model, sedimentation is accounted for by instantaneously "removing" droplets from the supersaturated mixture into the coexisting phase once the droplets have reached a defined maximum size. Our model predicts an oscillatory variation of turbidity. Depending on the composition, either both phases oscillate (symmetric oscillations) or only one of the phases oscillates (asymmetric oscillations). In the asymmetric case, droplet sedimentation from the majority phase into the minority phase reduces supersaturation in the minority phase. This inhibits droplet formation in the minority phase. The cooling rate dependence of the period agrees with experimental results.

A test-tube model for rainfall

“…The cell is placed in a computer controlled thermostat, and its temperature is varied smoothly away from the phase coalescence temperature T c , at a rate de-scribed by a variable ξ which has dimensions of inverse time (defined by equation (4) below). Several examples of this type of experiment have been reported: [5][6][7][8][9]. The experiments show similar quantitative results: both layers show a variation in turbidity, which is approximately periodic, with period ∆t.…”

“…Discussion. -It has been argued that the periodic precipitation phenomenon which has been described in several works [5][6][7][8][9][10][11] is analogous to an atmospheric precipiation cycle in a stable atmosphere. Consider whether the same mechanisms are relevant to the growth of real rain droplets.…”

“…The Ostwald ripening process may allow this gap to be bridged, because it allows droplets to grow by diffusive transfer of material from smaller droplets, without the necessity for collisions. Note that the growth law, given by equation (9), is independent of the density of the droplets. The physical parameters which are required to estimate the growth law are, at 10 • C, diffusion constant for water vapour in air, D = 2.4 × 10 −5 m 2 s −1 , surface tension σ = 7.4 × 10 −2 Nm −1 , molar volume V m = 1.8 × 10 −5 m 3 , saturation volume fraction Φ 0 = 1.2 × 10 −5 .…”

Particle tracking for polydisperse sedimenting droplets in phase separation

“…The black squares denote the average length for each radius bin and the solid red line is the sedimentation boundary. (Krishnamurthy and Goldburg (1980); Sagui and Grant (1999); Vollmer (2008)), coarsening (Aarts et al (2005); Vollmer et al (2007)) and sedimentation (Hayase et al (2008)) causes oscillations of the supersaturation and turbidity, modeled by Vollmer et al (2007) and Benczik and Vollmer (2010). They can also clearly be identified in false color plots of the droplet size distribution displayed in Fig.…”

Pattern formation in phase separating binary mixtures

“…1). [9][10][11][12][13][14] They are caused by repeated cycles of nucleation, growth and sedimentation, i.e. they are dominated by thermodynamics.…”