Decomposition driven interface evolution for layers of binary mixtures. III. Two-dimensional steady films with flat and modulated surfaces

Bribesh, Fathi; Fraštia, Ľubor; Thiele, Uwe

doi:10.1063/1.4727888

Cited by 13 publications

(10 citation statements)

References 42 publications

(88 reference statements)

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“…Such boundaries may result in a very rich surface phase behavior (see e.g. figure 6 in [71] for a phase diagram, figures 3-14 in [53] for bifurcation diagrams and 1D concentration profiles, and [58] for 2D results). Incorporating such boundaries into the present study will most likely add a new level of complexity to some aspects of the bifurcation diagrams, allowing one to investigate the interplay between different bulk (liquid-gas, demixing, crystallization) and surface (wetting, pre-wetting, surface freezing, pre-melting) phase transitions in finite systems and the corresponding transition towards the TL.…”

Section: Discussionmentioning

confidence: 99%

First order phase transitions and the thermodynamic limit

et al. 2019

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We consider simple mean field continuum models for first order liquid-liquid demixing and solidliquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn-Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid-liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit.However, in a system of finite size or when a finite time horizon is considered, metastable states often play an important role and even unstable states may be crucial, as transient states, for extended time periods. The full set of states and their dependence on the various control parameters is conveniently presented in the form of bifurcation diagrams, well known in the context of dynamical systems and pattern formation theory [3][4][5]. The place of thermodynamic phase diagrams is taken by 'morphological phase diagrams' or state diagrams and stability diagrams [3,4,6]. The notion of a Maxwell point is often used in the context of pattern formation in nonconserved systems [7][8][9][10][11] to indicate equal energy states because of its dynamical significance [12,13]. In this context this notion applies equally to finite and infinite systems [7,8] although for finite systems it lacks the thermodynamic relevance as the condition for phase coexistence. In the context of buckling the corresponding concept is the Maxwell load [14].In this paper we show and discuss how the discontinuities in the TL represented by the Maxwell construction arise from the bifurcation diagrams relating stable, metastable and unstable steady states in finite-size systems. We focus on two systems: (i) phase decomposition of a binary liquid mixture and (ii) the liquid to crystalline solid phase transition. We investigate the transitions that occur in the context of the most basic mean-field continuum models for these two different phase transitions, namely, the Cahn-Hilliard equation [15][16][17] and the phase field crystal (PFC) model (or conserved Swift-Hohenberg equation) [18][19][20].Some aspects related to this question have been considered previously, in particular in relation to the nature of some of the states that can arise in finite-size systems in the two-phase region. References [21-24] describe theory and computer simulation results for atomistic models exhibiting gas-liquid, liquid-hexatic and hexaticsolid phase transitions that indicate how the Maxwell construction develops as the system size increases or the temperature decreases. For example, figures 3-5 of [23] compare Monte-Carlo computer simulation results in a finite three-dimensional domain (see also figures 2 and 3 of [22]) with the results from a capillary drop type model, also in a finite domain, with a mean-field expression for the chemi...

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Section: Discussionmentioning

confidence: 99%

First order phase transitions and the thermodynamic limit

et al. 2019

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“…A future avenue for improvements of our mesoscopic hydrodynamic model is to incorporate solute-solute and solute-solvent interaction. A related thin film model for a layer of a decomposing non-volatile binary mixture has recently been derived by Náraigh and Thiffeault 104 as a long-wave approximation to model-H. [105][106][107] A general approach of deriving such thin film evolution equations in the context of nonequilibrium thermodynamics (taking the form of a gradient dynamics based on a free energy functional), was recently proposed. 108,109 This then naturally allows one to incorporate effects like solute-dependent wettability and capillarity (including solutal Marangoni effects), and the dependence of evaporation on the osmotic pressure.…”

Section: Discussionmentioning

confidence: 99%

Modelling the formation of structured deposits at receding contact lines of evaporating solutions and suspensions

2012

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When a film of a liquid suspension of nanoparticles or a polymer solution is deposited on a surface, it may dewet from the surface and as the solvent evaporates the solute particles/polymer can be deposited on the surface in regular line patterns. In this paper we explore a hydrodynamic model for the process that is based on a long-wave approximation that predicts the deposition of irregular and regular line patterns. This is due to a self-organised pinning-depinning cycle that resembles a stick-slip motion of the contact line. We present a detailed analysis of how the line pattern properties depend on quantities such as the evaporation rate, the solute concentration, the Péclet number, the chemical potential of the ambient vapour, the disjoining pressure, and the intrinsic viscosity. The results are related to several experiments and to depinning transitions in other soft matter systems.

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“…In §5 we proceed by example and consider the Kuramoto-Sivashinsky equation with periodic boundary conditions. For the computation of steady branches and branches of traveling waves, this requires two phase conditions of type (14) and (15), and in §5 we explain how to modify these for the computation of Hopf orbits, i.e., for standing waves and modulated traveling waves.…”

Section: Hopf Bifurcation and Time Periodic Orbitsmentioning

confidence: 99%

“…The second continuation parameter is given by the fictitious advection speed ε 1 which is kept at zero by simultaneously fulfilling the phase condition as given in eq. (15). The branches of localized states snake upwards in an intertwined manner, adding a pair of bumps per pair of saddlenode bifurcations and finally terminate again on the branch P when the localized structure fills the available space.…”

Section: Continuationmentioning

confidence: 99%

“…employing the energy functional (5) with f (φ ) = − 1 2 φ 2 , and the nongradient flux term of type (ii) with n = 2 ( §1), i.e., j ng c = − 1 2 ∂ x (φ 2 , 0) T . Equation (51) with periodic BC is translationally invariant, it has the boost invariance φ (x,t) → φ (x − st) + s, and thus we need the two phase conditions (14) and (15). We therefore modify (51) to…”

Section: The Kuramoto-sivashinsky Equationmentioning

confidence: 99%

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Continuation for Thin Film Hydrodynamics and Related Scalar Problems

Engelnkemper

Gurevich

Uecker

et al. 2018

Computational Methods in Applied Sciences

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This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution of a single scalar field like a density or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn-and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dipcoating. Through the different examples we illustrate how to employ the numerical tools provided by the packages AUTO07P and PDE2PATH to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues.Published as: Engelnkemper, S., Gurevich, S.V.,

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Decomposition driven interface evolution for layers of binary mixtures. III. Two-dimensional steady films with flat and modulated surfaces

Cited by 13 publications

References 42 publications

First order phase transitions and the thermodynamic limit

First order phase transitions and the thermodynamic limit

Modelling the formation of structured deposits at receding contact lines of evaporating solutions and suspensions

Continuation for Thin Film Hydrodynamics and Related Scalar Problems

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