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Kinetics of Spinodal Phase Separation in Unstable Thin Liquid Films

We study universality in the kinetics of spinodal phase separation in unstable thin liquid films, via simulations of the thin film equation. It is shown that in addition to morphology and free energy,the number density of local maxima in the film profile can also be used to identify the early, intermediate and late stages of spinodal phase separation. A universal curve between the number density of local maxima and rescaled time describes the kinetics of early stage in d = 2, 3. The Lifshitz-Slyozov exponent of -1/3 describes the kinetics of the late stage in d = 2 even in the absence of coexisting equilibrium phases.

The kinetics of spontaneous phase separation, usually referred to as spinodal decomposition, is of interest in far-from-equilibrium systems in diverse areas ranging from materials science and biological physics to cosmology and astrophysics [1]. Spinodal  and has been successfully explained by Lifshitz-Slyozov (LS) theory [2] and its variants.
In contrast, the dynamics of the early and intermediate stages remains poorly understood.
For example, it is not easy to study the time-dependence of the domain size as given by correlation functions or structure factors, in the absence of well-defined phases. Experimental access to these stages suffers from the added difficulty of detecting low-amplitude and multi-modal fluctuations. The present letter investigates the early and intermediate stages of spinodal phase separation in the technologically and scientifically important system of supported unstable thin liquid films [3]. Our study is simulations of a model thin film equation [4]. We discuss the unique universal features of these stages and present a useful marker to track their kinetics. The LS growth scenario for the thin film system is also validated for the late stage. Interestingly, many different growth laws have been reported for the late stage [5,6]. The similarity between the thin film equation and the Cahn-Hilliard (CH) equation [7] indicates that these unique features are likely to be shared by other spinodally phase-separating systems.
As in all spinodal processes, random fluctuations in the free surface of initially flat, supported thin liquid films (< 100 nm) grow and evolve into two distinct phases (viz., a lowcurvature and thinner flat film phase and a thicker high-curvature droplet phase) whenever ∆G shows a minimum and the spinodal parameter, ∂ 2 ∆G/∂h 2 | h=h 0 < 0 (Fig. 1). Here, ∆G is the excess intermolecular free energy (per unit area), h is the film thickness, and h 0 is the average thickness [8,9]. Notice that the double-tangent construction for ∆G in Fig. 1 shows that the film phase-separates into phases with h = h m and h = ∞. This should be contrasted with the usual phase-separation problems such as segregation of binary mixtures which are described by a double-well potential, i.e., there are two possible values for the equilibrium composition [10]. Actually, the droplet phase remains bounded due to overall volume conservation and so does the maximum thickness. One way to address this is to treat the droplet phase as a defect with changing thickness rather than a true equilibrium phase.
We study the thin film equation which models the spatio-temporal evolution of the film's surface in supported thin liquid films. This is derived by considering a thickness-dependent excess intermolecular energy ∆G(h), and simplifying the equations of motion under the lubrication approximation [4]. The resulting equation can be written as a CH equation with a thickness-dependent effective mobility [7], M(h) = h 3 /(3µ) (corresponding to Stokes flow with no slip). The total free energy is F s denotes the net excess free energy and F i denotes the interfacial free energy. In the above expressions, γ and µ refer to surface tension and viscosity of the liquid film, respectively.

The corresponding CH equation is
where all gradients are taken in the plane of the substrate. Simulations were done for a variety of forms of ∆G(h) to uncover universal features of the early stage, if they exist.
Here, a long-range van der Waals attraction due to the substrate, and a comparatively short-range van der Waals repulsion provided by a nano-coating on the substrate [11], is chosen to illustrate the results. The corresponding ∆G is shown in Fig. 1, and has the form Here, A s and A c (= RA s ) are the effective Hamaker constants for the system, which consists of the fluid bounding the film from the top, film fluid and a solid substrate (s) or coating material (c). The thickness of the nano-coating is δ.
Equation (1) is solved in the following non-dimensional form to reduce parameters: Here, H is the non-dimensional local film thickness scaled with the mean thickness h 0 ; the coordinates along the substrate are scaled with the characteristic length-scale for the van der The excess energy term is completely non-dimensionalized as where D = δ/h 0 is the non-dimensional coating thickness. The linear stability analysis of Eq.
(2) predicts a dominant spinodal wave of wavelength, We numerically solve Eq. (2) Fig. 3) only explains the beginning of the early stage. So, while the spinodal wave itself is given by linear theory, the kinetics leading to its emergence can only be described by nonlinear analysis.
These stages can be readily identified in the evolution of the free energy also (Fig. 4). In the early stage, there is not much change in F s , F e and F i . This stage ends with a sharp increase in F i and a corresponding decrease in F e . In the late stage, there is a smooth decrease in both F i and F e . A master curve to describe all these results will strongly indicate a universal early-stage kinetics. How does one rescale the results to arrive at the master curve? It is known that initial fluctuations greatly influence the kinetics for film thicknesses at the edge of the spinodal regime or in the defect sensitive spinodal region (DSSR), as opposed to thicknesses in the deep inside spinodal region (DISR). [12]. A new evolution coordinate found by dividing T by a time, T i , at which different films are at a given morphological event during their evolution, will account for the different initial fluctuations. There can be many such choices of events, with the emergence of the flat-film phase being the most obvious one. However, the increasing dominance of the repulsive force field slows down the dynamics near this event, leading to uncertainty in estimating the corresponding times. Another event, which provides a much sharper estimate, is found by tracking the minimum thickness of the evolving film: (2). They also show the same universality but with an exponent closer to −1/3. This is consistent with results from the phase separation of mixtures [1]. The early formation and coalescence of domains is known to have a dependence on dimensionality.
In summary, we conclude that the number density of local maxima (defects) and not the usual measures like structure factors and correlation functions can be used to identify the early, late and intermediate stages of spinodal phase separation in thin films. The kinetics of the early stages for a wide range of potentials and parameters can be described by a universal curve between the number density of defects and a rescaled time in d = 2, 3. This process has strong similarities with the usual phase-separation processes except for the absence of coexisting phases. Still, LS exponent of −1/3 governs the late stages of domain growth. Our results in this letter will facilitate the understanding of many experimental results for phase separation in thin films. We also hope that our predictions of universality in early-stage dynamics will be put to experimental test.    We superpose data for a wide range of potential and parameters, as described in the text. The inset shows the variation of minimum thickness with T which is used in defining the evolution coordinate.