Dynamical effects and phase separation in cooled binary fluid films

Abstract

We study phase separation in thin films using the Navier-Stokes Cahn-Hilliard equations in the lubrication approximation, modeling substrate-film interactions with a van der Waals potential. We investigate the thin-film equations numerically and compare them with experimental results. We find that the model captures the qualitative features of real phase-separating fluids, in particular the tendency of concentration gradients to produce film thinning and surface roughening. The ultimate outcome of the phase separation depends strongly on the dynamical backreaction of concentration gradients on the flow, as we demonstrate when a shear stress is applied at the film's surface. When the backreaction is small, the phase domain boundaries align with the direction of the imposed stress, while as the backreaction is made larger, the domains begin to align in the perpendicular direction.PACS numbers: 47.15.gm, 47.55.-t, 64.75.+g When a binary fluid is cooled below the critical temperature, the homogeneous state is energetically unfavourable and the system spontaneously phase-separates and forms domains rich in either fluid component [1,2]. Due to the relevance of phase-separating thin films in industrial applications [3], many experiments and numerical simulations focus on understanding how phase separation is altered if the binary fluid is confined in a thin layer. We propose a lubrication approximation based on the coupled Navier-Stokes Cahn-Hilliard equations to explain the main features of these studies.Several recent experiments have clarified the different regimes of domain growth in a binary thin film. Wang and Composto [4] have identified early, intermediate, and late stages of evolution. The early stage comprises threedimensional domain growth, while the intermediate stage is characterized by the formation of wetting layers at the film boundaries, the thinning of the middle layer, and significant surface roughening. Due to the thinning of the middle layer, the sandwich-like structure breaks up and matter from the wetting layer flows back into the bulk. Thus, a late stage is reached, consisting of bubbles coated by thin wetting layers. This characterization of the evolution has been seen in other experiments [5,6], although clearly a variety of behaviors is possible, depending on the wetting properties of tnhe mixture. Our model captures the essential features of this evolution, in particular the tendency for concentration gradients to promote film rupture and surface roughening.In a series of papers, Das et al. [7,8] investigate the behaviour of binary fluids with wetting. In [7] they specialize to ultra-thin films. In bulk mixtures, where one of the fluid components is preferentially attracted to the boundary, a layer rich in that component may be established there, followed by depletion layer, and so on. This so-called spinodal wave propagates into the bulk [8]. In * Electronic address: jeanluc@imperial.ac.uk ultra-thin films, the film thickness is less than a single spinodal wavelength and the spinodal wav...

Nonlinear dynamics of phase separation in thin films

“…By promoting the constant C = νρD/ (h 0 σ 0 δ 2 ) to O (1), and by taking q > 1, this equation reduces to P = −C −1 H xx , as in Eq. (18). Similarly, the normal-stress condition reduces to νρ (∂u/∂z) h = ∂σ/∂x, which in dimensionless form reads…”

“…In the present case, the reduction yields two equations: one for the free surface, and one for the Cahn-Hilliard concentration. The resulting thin-film Stokes Cahn-Hilliard equations have already been introduced by the authors in [18], although the focus there was on control of phase separation and numerical simulations in three dimensions. Here we confine ourselves to the two-dimensional case: we derive the thin-film equations from first principles, present analysis of the resulting equations, and highlight the impossibility of film rupture, once a regularizing potential is prescribed.…”

“…Equations of Cahn-Hilliard type have been proposed to explain the qualitative features of self-assembly [20,25], and knowledge of variations in the film height could enhance these models. Indeed in [18] the authors use the present thinfilm Cahn-Hilliard model in three dimensions to control phase separation, a useful tool in applications where it is necessary for the molecules in the film to form a given structure.…”

Fundamentals of Phase Separation in Polymer Blend Thin Films

“…The Clarke model, as it will be referred to in this chapter, utilised non-equilibrium thermodynamics based upon a free energy functional, demonstrating that phase separation generally couples to dewetting [68]: the Clarke model is explained more in section 5.1.2. A model based on the Navier-Stokes Cahn-Hilliard equations in the lubrication approximation showed that concentration gradients can create a roughened pattern that mirrors the underlying phase separation [78]. However, the film composition has no vertical dependence in these models, which means that a meaningful preferential surface attraction of blend components cannot be included.…”

Pattern Formation in Polymer Blend Thin Films: Surface Roughening Couples to Phase Separation

“…The "Clarke model" (name introduced here) utilized nonequilibrium thermodynamics based upon a free energy functional, demonstrating that phase separation generally couples to dewetting [12]. A model based on the Navier-Stokes Cahn-Hilliard equations in the lubrication approximation showed that concentration gradients can create a roughened pattern that mirrors the underlying phase separation [14]. However, the film composition has no vertical dependence in these models, so a meaningful preferential surface attraction of blend components cannot be included.…”