Gradient dynamics models for liquid films with soluble surfactant
GGradient dynamics models for liquid films with soluble surfactant
Uwe Thiele ∗ Institut f¨ur Theoretische Physik, Westf¨alische Wilhelms-Universit¨at M¨unster,Wilhelm Klemm Str. 9, D-48149 M¨unster, Germany andCenter of Nonlinear Science (CeNoS),Westf¨alische Wilhelms Universit¨at M¨unster,Corrensstr. 2, 48149 M¨unster, Germany
A. J. Archer
Department of Mathematical Sciences, Loughborough University,Loughborough, Leicestershire, LE11 3TU, UK
L. M. Pismen
Department of Chemical Engineering,Technion Israel Institute of Technology, Haifa 32000, Israel a r X i v : . [ phy s i c s . f l u - dyn ] S e p bstract In this paper we propose equations of motion for the dynamics of liquid films of surfactant suspen-sions that consist of a general gradient dynamics framework based on an underlying energy functional.This extends the gradient dynamics approach to dissipative non-equilibrium thin film systems with severalvariables, and casts their dynamic equations into a form that reproduces Onsager’s reciprocity relations.We first discuss the general form of gradient dynamics models for an arbitrary number of fields and dis-cuss simple well-known examples with one or two fields. Next, we develop the gradient dynamics (threefield) model for a thin liquid film covered by soluble surfactant and discuss how it automatically resultsin consistent convective (driven by pressure gradients, Marangoni forces and Korteweg stresses), diffusive,adsorption/desorption, and evaporation fluxes. We then show that in the dilute limit, the model reduces tothe well-known hydrodynamic form that includes Marangoni fluxes due to a linear equation of state. Inthis case the energy functional incorporates wetting energy, surface energy of the free interface (constantcontribution plus an entropic term) and bulk mixing entropy. Subsequently, as an example, we show howvarious extensions of the energy functional result in consistent dynamical models that account for nonlinearequations of state, concentration-dependent wettability and surfactant and film bulk decomposition phasetransitions. We conclude with a discussion of further possible extensions towards systems with micelles,surfactant adsorption at the solid substrate and bioactive behaviour. ∗ Electronic address: [email protected]; URL: . INTRODUCTION Onsager’s evolution equations [1, 2], based on the principle of detailed balance embedded inOnsager’s reciprocity relations, became a key tool for understanding the relaxational approachto equilibrium in a variety of physical processes. More recently, Doi [3] extended the range ofthis approach to processes in macroscopic soft matter systems, such as the swelling of gels andthe dynamics of liquid crystals. It is less obvious that a similar approach can also be applied toprocesses out of equilibrium in spatially extended open systems. A well known example is thedynamics of single layer thin films in the long-wave (or lubrication) approximation [4, 5] wherea single variable – the layer thickness – is sufficient for a description of the system. In this case,it is not a-priori obvious that an energy functional of thermodynamic origin exists for the system.Nevertheless, as noticed by Mitlin [6] for dewetting films and by Rosenau and Oron for thin filmsheated from below [7], the dynamic equation for the layer thickness h can be cast into a gradientdynamics form ∂ t h = ∇ · (cid:20) Q c ∇ δ F δh (cid:21) − Q nc (cid:18) δ F δh − p vap (cid:19) , (1)showing that the evolution can be derived from a certain “energy” functional F [ h ] . p vap is thepressure of the vapour phase that may instead be incorporated into F . Here and in the following ∂ t denotes partial time derivatives and ∇ is the two-dimensional (2D) spatial gradient operator.Eq. (1) is the general form in which the dynamics has both a conserved and a non-conservedcontributions with mobilities Q c ( h ) ≥ and Q nc ( h ) ≥ , respectively [8].The usual procedure of irreversible thermodynamics is thereby reversed: first comes a dynamicequation obtained through a series of simplifications, and then a suitable functional is assigned,ensuring a dissipative evolution toward a minimum of this energy. However, in the case of dewet-ting the energy functional is the “interface Hamiltonian” that is obtained via a systematic coarse-graining procedure from the microscale interaction energies [9]. Sometimes, even systems thatare permanently out of equilibrium can be accommodated, as in the case of sliding droplets on aninfinitely extended incline, where the correct thin film model can be brought into the form of agradient dynamics with an underlying energy functional that includes potential energy [10].Besides long-wave thin film equations, other examples of one-field gradient dynamics are theCahn-Hilliard equation describing the demixing of a binary mixture, i.e., a purely conserved dy-namics ( Q nc = 0 ) [11–13] and the Allen-Cahn equation that models, for instance, the purelynon-conserved dynamics ( Q c = 0 ) of the Ising model in the mean field continuum limit [13]. In3 eneral, equations of the form (1) are ubiquitous. They appear with various choices of F , notonly in the context of the dynamics of films of non-volatile and volatile liquids on solid substrates[4, 6, 14, 15], but also as evolution equations for surface profiles in epitaxial growth [8, 16–19],and, indeed, as models of one-component lipid bilayer adhesion dynamics [20]. Another fieldof application is in dynamical density functional theory (DDFT), describing the dynamics of thedensity distribution of colloidal particles [21–24].Furthermore, many hydrodynamic two- and more-field long-wave models were developed thatdescribe, e.g., the evolution of multilayer films, films of mixtures or surfactant-covered films [5].Normally, they are not written in the gradient dynamics form. However, recently, the gradientdynamics approach was extended to several two-field models, namely, for the dewetting of two-layer films [25, 26], for the coupled decomposition and dewetting of a film of a binary mixture[27, 28] and for the evolution of a layer of insoluble surfactant on a thin liquid film [29]. In allthese cases, energies with a clear physical meaning can be given that may also be obtained via thecoarse-graining procedures of statistical physics. Note though that the description of a thin twolayer-film heated from below cannot be brought into the Onsager form [30], marking the singlelayer case as a fortuitous ‘accident’. Nonetheless, certain out of equilibrium phenomena can bedescribed via the addition of appropriate potential energies to the energy functional or, as in thecase of dip coating and Langmuir–Blodgett transfer through “comoving frame terms” that accountfor a moving substrate that is withdrawn from a bath [31]. Similar two-field gradient dynamicsmodels exist for the dynamics of membranes [32, 33] or as DDFTs for mixtures [34, 35].The aim of this paper is to extend the gradient dynamics approach to describe the non-equilibrium dissipative dynamics of thin film systems with several variables, and to cast the dy-namic equations into a form that reproduces Onsager’s reciprocity relations. A further aim is toincorporate interphase exchange processes, such as evaporation and surfactant dissolution to de-rive equations combining conserved (Cahn–Hilliard-type) and non-conserved (reaction-diffusion,or Allen–Cahn-type) terms. In doing so, several limitations of the known two-field models arealleviated. The particular example treated in detail is a thin liquid film that is covered by a solublesurfactant and rests on a solid substrate. The gradient dynamics model then describes the cou-pled evolution of the film height profile, the amount of surfactant within the film and the surfaceconcentration dynamics (three field) model for the case of a thin liquid film covered by a solublesurfactant as sketched in Fig. 1.This paper is structured as follows: In the following section II we discuss the general form4 IG. 1: Sketch of the system under consideration. It consists of a film of liquid on a surface of thickness h ( x, t ) , that varies with location on the surface x and with time t . On the liquid film free surface aresurfactant molecules, with local density Γ( x, t ) . The surfactant molecules have some solubility in the liquidand the local concentration within the body of the liquid is φ ( x, t ) . We assume that φ does not vary verticallyand only varies horizontally and with t . This is equivalent to treating φ as a height-averaged concentration.Over time there is exchange of surfactant molecules between the surface of the liquid and the bulk. Therecan also be condensation or evaporation of the liquid to vapour in the air above. of gradient dynamics models, first, for an arbitrary number of fields in section II A and then insection II B we write the diffusion equation and the thin film equation as gradient dynamics anddiscuss known two-field models. Next, in section III we develop the gradient dynamics (three field)model for the case of a thin liquid film covered by a soluble surfactant and discuss in section IVspecial cases and extensions. We draw our conclusions in section V. Appendices A and B clarify anissue in the comparison of hydrodynamic long-wave approach and the present variational approachand give the variations of the energy functional in the most general case covered by the presentwork, respectively. II. GENERAL N -FIELD MODEL AND KNOWN APPLICATIONSA. General model The dynamics of a spatially extended system may be characterised by the coupled evolutionof N scalar state variable fields (order parameter fields) u = ( u , u , ..., u n ) T . Not too far fromequilibrium, the dynamics is governed by a single equilibrium free energy functional F [ u ] , i.e., itis a gradient dynamics. Using Einstein’s index notation that presumes summation over repeatedindices, the coupled evolution equations read ∂ t u a = ∇ α (cid:20) Q c ab ∇ α δ F δu b (cid:21) − Q nc ab δ F δu b (2)5 here α = 1 , , . . . , d refers to spatial coordinates and a, b = 1 , . . . , n refer to the differentorder parameter fields that might have a conserved, or non-conserved, or mixed dynamics. Here, Q c ab ( u ) and Q nc ab ( u ) represent n × n dimensional positive definite and symmetric mobility matricesfor the conserved and non-conserved parts of the dynamics, respectively. The mobilities Q c ab govern the fluxes j a = − Q c ab ∇ ( δ F /δu b ) of the conserved part of the dynamics for all orderparameters u a . These are given as linear combinations of the influences of all thermodynamicforces −∇ ( δ F /δu b ) , i.e. are linear in the thermodynamic forces. In contrast, the coefficients Q nc ab give the transition rates between fields and are also linear combinations of the thermodynamicpotentials δ F /δu a .It is straightforward to show that the free energy F [ u , . . . , u n ] is a Lyapunov functional, i.e., itmonotonically decreases in time: dd t F [ u , . . . , u n ] = (cid:90) Ω δ F δu a ∂u a ∂t d d x = (cid:90) Ω δ F δu a ∇ α (cid:20) Q c ab ∇ α δ F δu b (cid:21) d d x − (cid:90) Ω δ F δu a Q nc ab δ F δu b d d x = − (cid:90) Ω (cid:18) ∇ α δ F δu a (cid:19) Q c ab (cid:18) ∇ α δ F δu b (cid:19) d d x − (cid:90) Ω δ F δu a Q nc ab δ F δu b d d x ≤ . (3)where Ω is the domain in which the system is defined. Above we used Eq. (2) and partial integra-tion, assuming periodic or no-flux boundary conditions.A further advantage of the general formulation is the ease with which one may change thechoice of variables u a . If new order parameter fields ˜ u a are introduced via a linear transformation ˜ u a = R ab u b the kinetic equations for the new fields are ∂ t ˜ u a = ∇ α (cid:20) (cid:101) Q c ab (˜ u , . . . , ˜ u n ) ∇ α δ F δ ˜ u b (cid:21) − (cid:101) Q nc ab (˜ u , . . . , ˜ u n ) δ F δ ˜ u b (4)with (cid:101) Q i ab = R ad Q i de R be ( i = c , nc ) where we take into account that δ F /δu a = R ba δ F /δ ˜ u b . Fortwo conserved fields, similar relations were already given in Refs. [31, 36].Up to here, we have not specified the free energy F [ u , . . . , u n ] that can, in principle, be anarbitrary functional of the order parameter fields. If F is a multiple integral, Eq. (2) becomes asystem of integro-differential equations, as is often the case in DDFT for colloids [23]. However,often the kernel is expanded in derivatives of the order parameter fields and Eq. (2) correspondsto a system of partial differential equations. Examples are Phase Field Crystal (PFC) models [37]and membrane models [33, 38] where the highest order terms in the energy are ∼ (∆ u ) . Here we6 estrict our attention to a lower order and only consider models where the highest order terms are ∼ ( ∇ u ) . Then the general form is F [ u , . . . , u n ] = (cid:90) Ω (cid:20)
12 ( ∇ α u a ) Σ ab ( ∇ α u b ) + f ( u , . . . , u n ) (cid:21) d d x, (5)where we have introduced in the free energy a symmetric n × n dimensional gradient interactionmatrix Σ ab that, in principle, may itself also depend on u . The integrand may also contain metricfactors (see below).Before we come in Section III to the case of liquid films that are covered with a soluble surfac-tant, we briefly review in Section II B some basic examples where only one or two order parameterfields are involved. B. Specific known examples of gradient dynamics
1. Diffusion equation
In the dilute limit, the diffusion of a species with part-per-volume concentration c in a quiescentcarrier medium can be represented as the conserved gradient dynamics ∂ t c = ∇ · (cid:20) Q c cc ∇ δ F δc (cid:21) , (6)with the purely entropic Helmholtz free energy functional F [ c ] = kTl (cid:90) c [ln c − dV, (7)where k is Boltzmann’s constant, T is the temperature and l is a molecular length scale. Themobility function in Eq. (6) is Q c cc = (cid:101) Dc and can be obtained via Onsager’s variational principle[3, 36, 39]. Here, (cid:101) D is the molecular mobility. This corresponds to ∂ t c = −∇ · j diff whereFick’s law takes the form j diff = − (cid:101) Dc ∇ µ = − D ∇ c with the chemical potential µ = δ F /δc =( kT /l ) ln c , i.e. D = (cid:101) Dc dµ/dc = (cid:101) DkT /l .The equivalence of Eq. (6) and the standard diffusion equation has been easily shown, and nowallows one to use the advantages of the gradient dynamics form, namely, the straightforward wayto account for free energies that are not purely entropic. If, for instance, one replaces the integrandin F [ c ] of Eq. (7) by the sum of a double-well potential and a squared gradient term, one obtainsthe Cahn-Hilliard equation (then using a constant Q c cc ) [12].7 . Thin films of simple liquids As discussed above, Eq. (1) describes the evolution of the height profile of a thin liquid filmon a solid substrate for non-volatile ( Q nc ( h ) = 0 ) or volatile ( Q nc ( h ) ≥ ) liquids. Detaileddiscussions of the various physical situations treated can be found in [4, 8, 15]. In the most basiccase of mesoscopic hydrodynamics, only the influence of capillarity and wettability is considered.The corresponding free energy F [ h ] is then F [ h ] = (cid:90) (cid:104) γ ∇ h ) + g ( h ) (cid:105) d x, (8)where γ is the surface tension of the liquid and g ( h ) is a local free energy (wetting or ad-hesion energy, or binding potential), related to the Derjaguin (or disjoining) pressure Π( h ) by Π = − dg ( h ) /dh [40]. Note, that varying sign conventions are used throughout the literature. Forparticular forms of Π , see, e.g., Refs. [4, 6, 14, 40–42]. Similar expressions are obtained as “in-terface Hamiltonians” in the context of wetting transitions [9]. Therefore mesoscopic thin film (ortwo-dimensional) hydrodynamics might be considered as a gradient dynamics on the underlyinginterface Hamiltonian. Note that recently such mesoscopic wetting energies have been extractedvia parameter passing methods from different microscopic models (molecular dynamics and den-sity functional theory) [43, 44]. Without slip at the substrate, Q c ≡ Q c hh = h / η , where η is thedynamic viscosity. Different slip models can be accommodated by alternative choices of Q c hh [45].Although several functions Q nc ( h ) are discussed in the literature for the case of volatile liquids(see, e.g., [15]), often a constant is used [46]).
3. Two-field models
In the context of thin film hydrodynamics, two-field gradient dynamics models were presentedand analysed (i) for dewetting two-layer films on solid substrates, i.e., staggered layers of twoimmiscible fluids [25, 26, 30], (ii) for decomposing and dewetting films of a binary liquid mixture(with non-surface active components) [27, 28], and (iii) for the dynamics of a liquid film thatis covered by an insoluble surfactant [29]. In all three cases, the model has the form (2) with a, b = 1 , and all Q nc ab = 0 (purely conserved dynamics). The conserved fields u and u representin case (i) the lower layer thickness h and overall thickness h , respectively [25, 26, 30] or thelower and upper layer thickness [47] (the transformation between the two formulations followsfrom the discussion around Eq. (4)). In case (ii), u and u represent the film height h and the8 ffective solute height ψ = ch , respectively, where c is the height averaged concentration. Finally,in case (iii), u and u represent the film height h and the surfactant coverage (cid:101) Γ (that is projectedon the cartesian substrate plane), respectively [29].As already emphasised, a crucial point in cases (ii) and (iii) is the choice of the two fields thatcan be varied independently of each other. This is not the case if, e.g., film height h and heightaveraged concentration c are used in case (ii), since then a variation in the height for fixed particlenumber per substrate area implies that c varies [27]. In case (iii), the projected coverage (cid:101) Γ has to beused since the surfactant coverage Γ on the free surface and the height profile h are not independent[29]: If the slope of h changes locally, the surface area changes and so also Γ . Therefore, for afixed local number of surfactant molecules, the local concentration changes without any surfactanttransport. If one uses dependent fields, one is not able to employ the general form (2). Note that inRefs. [48, 49] case (ii) has been treated employing a gradient dynamics for h and c . For a furthercomparison with the approach employed in [27, 28], see Ref. [15]. In all three cases (i) to (iii) theunderlying free energy functionals have a clear thermodynamic significance. They may be seenas extensions of the interface Hamiltonian for a single adsorbed layer, and the individual termsmay be obtained from equilibrium statistical physics. As expected, the mobility matrices Q c arepositive definite and symmetric [25, 28, 29, 47]. All their entries are low order polynomials in therespective fields u and u . In particular, in cases (ii) and (iii), one has Q c = = 13 η u qu u qu u ru u + 3 (cid:101) Dηu . (9)where (cid:101) D is a respective molecular mobility related to diffusion, and q = r = 1 in case (ii) and q = 3 / , r = 3 in case (iii). Actually, in the parametrisation of Ref. [26], the mobility matrix Q c of case (i) also agrees with case (iii) if the diffusion term (cid:101) Dηu is replaced by η r u where η r isthe viscosity ratio of the two layers.Note in particular that cases (ii) and (iii) in the respective low concentration limit give theknown hydrodynamic thin film equations coupled to the equation for the solute / surfactant asdiscussed in detail in Refs. [27] and [29], respectively. It recovers also a number of other specialcases and can be employed to devise models that incorporate various energetic cross-couplings ina thermodynamically consistent manner. Examples include wetting energies that depend on soluteor surfactant concentration, effects of surface rigidity for surfactant covered films, free energiesof mixing/decomposition including gradient contributions, etc. It also allows one to discuss the9 nfluence of solutes / surfactants on evaporation.Note that the discussion above mixes the possible extensions in cases (ii) and (iii) that areseparately discussed in [27] and [29], respectively. It was noted in [28] that the two-field modelfor a film of a mixture cannot accommodate a solutal Marangoni effect by simply incorporat-ing a concentration-dependent surface tension since this breaks the gradient dynamics structure.Another disadvantage of the two-field model is that most surfactants are soluble, a situation thatcannot be treated via case (iii). In the following, we develop a three-field model that alleviates allthe mentioned problems. III. SOLUBLE SURFACTANT - GRADIENT DYNAMICS MODELA. Energy functional
We consider a thin film of liquid of thickness h on a solid substrate with a free surface thatis covered by a soluble surfactant, i.e., part of the surfactant is dissolved in the bulk of the filmand part is adsorbed at the free surface – see Fig. 1. We neglect adsorption at the solid-liquidinterface and micelle formation but discuss in teh conclusion how they can be incorporated. Thesurfactant concentration φ within the film represents a height-averaged concentration, i.e., it isassumed that the concentration is nearly uniform over the film layer thickness. The system isconsidered in relaxational situations, i.e., the boundary conditions do not sustain energy or massfluxes. Therefore, we expect the system dynamics to follow a pathway that approaches a staticequilibrium. In the absence of evaporation and surfactant exchange between the interface andthe bulk solution, the approach to equilibrium can be described by gradient dynamics for three independent fields : the film thickness h ( r , t ) , the local amount of dissolved surfactant ψ ( r , t ) = h ( r , t ) φ ( r , t ) , and the surfactant concentration at the interface projected onto a cartesian referenceplane (cid:101) Γ( r , t ) . The surfactant concentration on the interface is given by Γ = (cid:101) Γ / √ a where a is thedeterminant of the surface metric tensor (see below). Here r = ( x, y ) are “horizontal” coordinatesin the substrate plane. The fields φ and Γ are expressed as volume fraction and area fractionconcentrations, respectively, i.e., they are both dimensionless. As emphasised in section II B 3 forthe two-field cases, variations in h , φ and Γ are not independent, whilst variations with respect to h , ψ and (cid:101) Γ are independent. 10 he general expression for the energy includes surface and bulk contributions: F = F s + F b = (cid:90) ( L s + L b ) dxdy, (10) L s = (cid:104) κ s a αβ ( ∂ α Γ)( ∂ β Γ) + f s (Γ) (cid:105) √ a + g ( h ) , (11) L b = h (cid:104) κ |∇ φ | + f ( φ ) (cid:105) . (12)The interfacial terms in Eq. (11) depend on the surface metric tensor a αβ = δ αβ + ∂ α h∂ β h [wherewe exclude overhangs in order to use a Monge representation h ( x, y ) ] and its inverse a αβ ; a is thedeterminant of a αβ and determines the extension of the interface, and δ αβ is the Cartesian metricof the planar substrate or a planar surface h = const. Distinction between lower (covariant) andupper (contravariant) indices is essential for a non-Euclidean surface metric. The wetting potential g ( h ) in Eq. (11) describes the interactions with the substrate that determine the Derjaguin (ordisjoining) pressure Π( h ) = − dg ( h ) /dh (cf. Section II B 2). The first terms in parentheses inEqs. (11) and (12) contain the interfacial and bulk rigidity coefficients κ s and κ , respectively, andpenalize surfactant concentration gradients. The second terms in the parentheses in each case takesaccount of molecular interactions. f s (Γ) contains the free energy contribution due to the presenceof surfactant molecules at the interface. In the limit Γ → , then this is just the pure liquid-vapoursurface tension, i.e. f s (Γ →
0) = γ , but more generally f s (Γ) = γ + kTl s Γ[ln Γ −
1] + f exs (Γ) . (13)The second term is the contribution to the free energy when the amount of surfactant on the surfaceis low enough that interactions between molecules are negligible and can be treated as a 2D ideal-gas. l s is a molecular length scale related to the size of the adsorbed surfactant molecules ( l s isthe area on the surface occupied by a surfactant molecule). As the surface coverage Γ increases,then the excess free energy f ex s (Γ) gives an increasing contribution. For example, treating thesurfactant on the surface via a lattice-gas approximation, one would write f ex s (Γ) = kTl s [Γ + (1 − Γ) ln(1 − Γ)] − b (14)where the first (entropic) excluded volume term comes from assuming only one surfactantmolecule can occupy a site of area l on the surface and the final term is a simple mean-fieldterm coming from the attraction between pairs of neighbouring surfactant molecules. If the attrac-tion strength parameter b > is sufficiently large, then surface phase transitions may occur. An11 lternative approximation might be f ex s (Γ) = f hd (Γ) − b Γ / , where f hd is the hard-disk excessfree energy – see for example the approximations in Refs. [50, 51].Similarly, the bulk free energy in Eq. (12) can be written as: f ( φ ) = kTl φ [ln φ −
1] + f exb ( φ ) , (15)where l is a molecular length scale related to the surfactant molecules in solution ( l is the volumeoccupied by a surfactant molecule). The simplest approximation is to assume l s = l . f exb ( φ ) is the bulk excess contribution which in general may be written as a virial expansion f exb ( φ ) = (cid:80) ∞ i =2 c i φ i , with coefficients c i that depend on the temperature. Alternatively one may approximate,e.g. by assuming a lattice-gas free energy f exb = kTl [ φ + (1 − φ ) ln(1 − φ )] − b b φ , (16)where b b > is an inter-surfactant molecule attraction strength parameter. Or, instead one couldassume f exb ( φ ) = f cs ( φ ) − b b φ / , where f cs ( φ ) is the Carnahan-Starling approximation for thehard-sphere excess free energy [52]. Specific cases for the excess contributions f ex s (Γ) and f ex ( φ ) will be discussed below in Section III E 2. B. Pressures, chemical potentials and surface stress
The expression for pressure p = δ F /δh is obtained by calculating the variation of Eq. (10)with respect to h for fixed (cid:101) Γ , ψ . The variation of F s depends on the surface metric and uses therelations δa = a a αβ δa αβ = − a a αβ δa αβ , gδa αβ = gδ ( ∂ α h∂ β h ) = − [ ∂ α ( g∂ β h ) + ∂ β ( g∂ α h )] δh, (17)where g is an arbitrary function of the surface coordinates. Also note that a αβ a αβ = δ αα = 2 . Asmentioned above, Γ changes with surface extension or contraction, so that before the variation of F s is computed one needs to replace Γ = (cid:101) Γ / √ a , where (cid:101) Γ is a reference surfactant coverage of aplanar interface, or coverage per substrate area [29]. Similarly, one must replace φ → ψ/h beforethe variation of F b is computed [27]. This yields p = δ F δh = − ∂ α (cid:0) √ a σ αβ ∂ β h (cid:1) − Π( h ) + p b , (18) p b = δ F b δh = p osm + κ (cid:20) |∇ φ | + φh ∇ · ( h ∇ φ ) (cid:21) , p osm = f ( φ ) − φf (cid:48) ( φ ) . (19)12 here p osm is the bulk osmotic pressure. With solely the ideal-gas (entropic) terms in Eq. (15),it becomes p osm = − kT φ/l . Note too that ∇ is the 2D gradient operator, and ∇ is the 2DLaplacian. The second term in Eq. (18) is the disjoining pressure, while the first term contains theinterfacial stress σ αβ = δ F s δa αβ = 12 a αβ [ f s (Γ) − Γ f (cid:48) s (Γ)] − κ s a αβ a γδ ∂ γ Γ ∂ δ Γ+ κ s Γ4 a αβ (cid:2) ∂ γ (cid:0) √ a a γδ ∂ δ Γ (cid:1) + ∂ δ (cid:0) √ a a γδ ∂ γ Γ (cid:1)(cid:3) . (20)In particular, the standard surface tension is defined as γ (Γ) = a αβ ( σ αβ ) κ s → = f s (Γ) − Γ f (cid:48) s (Γ) . (21)The function f s in Eq. (13) with Eq. (14) for b = 0 results then in what is sometimes called theLangmuir equation of state [53, 54] or the Von Szyckowski equation [55] γ = γ + kTl ln(1 − Γ) , (22)i.e., for Γ (cid:28) one has γ ≈ γ − kT Γ /l = γ − γ Γ Γ , where we introduced the Marangonicoefficient γ Γ = kT /l for the resulting linear solutal Marangoni effect. Note that with b (cid:54) = 0 inEq. (14) one obtains the Frumkin equation of state as given in [53] and further discussed below insection III E 2.The surface chemical potential µ s is obtained by varying Eq. (10) with respect to (cid:101) Γ : µ s = δ F δ (cid:101) Γ = df s d Γ − κ s (cid:2) ∂ α (cid:0) a αβ ∂ β Γ (cid:1) + ∂ β (cid:0) a αβ ∂ α Γ (cid:1)(cid:3) . (23)Finally, the bulk chemical potential is [56] µ = δ F δψ = f (cid:48) ( φ ) − κh − ∇ · ( h ∇ φ ) . (24)The mechanical interaction between the surfactant layer and the bulk liquid is carried by thebalance of the interfacial stress and the viscous stress in the bulk fluid proportional to the normalderivative of the velocity v α tangential to the interface and the bulk viscosity η : σ αβ ; β = ηv α ; n , (25)where the semicolon denotes the covariant derivative necessary when vectors defined on a curvedinterface are involved. This equation reduces to the commonly used tangential stress balanceincluding the Marangoni force when the rigidity κ s is neglected.13 . Thin film hydrodynamics The above general expressions for the surface stress and pressure can be simplified in the casewhen the curvature and inclination are small so that the long-wave or lubrication approximationcan be made. To this end, we scale ∂ α ∼ O ( (cid:15) ) , v α ∼ O ( (cid:15) ) , ∂ t ∼ O ( (cid:15) ) and retain terms up to thelowest relevant order in (cid:15) (cid:28) . With this scaling, a αβ differs from the Cartesian surface metric δ αβ by O ( (cid:15) ) , so that a αβ = δ αβ + (cid:15) ∂ α h∂ β h and its inverse is, to leading order, a αβ = δ αβ − (cid:15) ∂ α h∂ β h .Then, the above expressions can be rewritten using Cartesian coordinates x α spanning the plane ofthe substrate, whereby the distinction between covariant and contravariant tensors disappears (sothat all indices can be written as subscripts) and covariant derivatives are replaced by usual partialderivatives. Retaining the leading order terms only, Eqs. (18) – (23) become p = δ F δh = −∇ · [( γ − p s ) ∇ h ] − Π( h ) + p b , (26) p s = γ − γ (Γ) − κ s (cid:18) Γ ∇ Γ − |∇ Γ | (cid:19) (27) µ s = δ F δ (cid:101) Γ = f (cid:48) s (Γ) − κ s ∇ Γ , (28)where we have used σ αβ = δ αβ ( γ − p s ) and where p s is the surface pressure that captures thedifference between reference surface tension without surfactant γ and the full concentration-dependent expression (including rigidity). Further, p b and p osm remain as in Eq. (19) while µ is still given by Eq. (24).The bulk flow field is computed by solving the modified Stokes equation, also called the mo-mentum equation of model-H [57, 58]. Its relevant components are parallel to the substrate plane: η v (cid:48)(cid:48) ( z ) = ∇ p + φ ∇ µ. (29)where v is the 2D vector of the velocities parallel to the substrate plane. An alternative form ofequation (29) can be obtained using the relation ∇ p b = f (cid:48)(cid:48) ( φ ) ∇ φ − κ ∇ (cid:20) |∇ φ | + φh ∇ · ( h ∇ φ ) (cid:21) = − φ ∇ µ + κh ∇ ( h |∇ φ | ) , (30)which reduces the right-hand side of the Stokes equation (29) to ∇ ( p − p b ) + ∇ p b + φ ∇ µ = ∇ (cid:98) p + κh ∇ ( h |∇ φ | ) , (31)where (cid:98) p is the effective pressure excluding p b . This shows that osmotic pressure p osm does notaffect hydrodynamic flow (as ∇ p osm = − φ ∇ µ ), while the contribution of the bulk rigidity isexpressed by the last term in the above relation.14 olving Eq. (29) in the lubrication approximation with the no-slip boundary condition at thesubstrate plane z = 0 and the momentum balance condition (25) at the interface z = h yields v = − zη (cid:104) ∇ p s + (cid:16) h − z (cid:17) ( ∇ p + φ ∇ µ ) (cid:105) . (32)Integrated over the local film thickness, this leads to the convective fluid flux J conv = (cid:90) h v dz = − h η ∇ p s − h η ( ∇ p + φ ∇ µ ) , (33)and the interfacial velocity v s = v ( h ) . Then the volume conservation condition ∂ t h = −∇ · J conv (34)results, to leading order, in the evolution equation of the film thickness ∂ t h = ∇ · (cid:20) h η ( ∇ p + φ ∇ µ ) + h η ∇ p s (cid:21) − J ev ( h, Γ , φ ) , (35)where we now incorporated the evaporation flux J ev . The leading-order equations expressing thesurface and bulk surfactant conservation laws are ∂ t Γ = ∇ · (cid:18) h η Γ( ∇ p + φ ∇ µ ) + hη Γ ∇ p s + M s (Γ) ∇ µ s (cid:19) + (cid:101) J ad (Γ , φ ) , (36) ∂ t ψ = ∇ · (cid:18) h η ψ ( ∇ p + φ ∇ µ ) + h η ψ ∇ p s + hM ( φ ) ∇ µ (cid:19) − J ad (Γ , φ ) , (37)where M s (Γ) and M ( φ ) are general surface and bulk mobility functions and ˜ J ad = J ad /l s is thenet surfactant adsorption flux; surface distortions contribute to Eq. (36) as O ( (cid:15) ) terms only. In thedilute limit, the mobilities can be expressed as M s (Γ) = D s l Γ kT and M ( φ ) = Dl φkT (38)where D s and D are surface and bulk diffusivities, respectively. The lengths in the diffusion termsare introduced for convenience. They ensure that the diffusivities D ’s have units m / s as usual fordiffusion constants. The conserved dynamics in Eqs. (36) and (37) have the form of conservationlaws ∂ t Γ = −∇ · (Γ v s + J Γdiff ) and (39) ∂ t ( φh ) = −∇ · ( φ J conv + J φ diff ) , (40)respectively. We also take into account the relation ∇ p s = Γ f (cid:48)(cid:48) s (Γ) ∇ Γ + κ s ∇ (cid:18) Γ ∇ Γ − |∇ Γ | (cid:19) = Γ ∇ f (cid:48) s (Γ) − κ s Γ ∇∇ Γ = Γ ∇ µ s , (41)that allows us to replace the gradient of the surface pressure in Eq. (35) by ∇ µ s . Gradient dynamics formulation Eqs. (36) – (37) can be now presented in the general gradient dynamics form (2) with a, b =1 , , for three fields as ∂ t h = ∇ · (cid:18) Q hh ∇ δ F δh + Q h Γ ∇ δ F δ (cid:101) Γ + Q hψ ∇ δ F δψ (cid:19) − β evap (cid:18) δ F δh − p vap (cid:19) , (42) ∂ t Γ = ∇ · (cid:18) Q Γ h ∇ δ F δh + Q ΓΓ ∇ δ F δ (cid:101) Γ + Q Γ ψ ∇ δ F δψ (cid:19) − β ψ Γ (cid:18) l s δ F δ (cid:101) Γ − δ F δψ (cid:19) , (43) ∂ t ψ = ∇ · (cid:18) Q ψh ∇ δ F δh + Q ψ Γ ∇ δ F δ (cid:101) Γ + Q ψψ ∇ δ F δψ (cid:19) − β ψ Γ (cid:18) l s δ F δψ − δ F δ (cid:101) Γ (cid:19) , (44)The mobility matrix for the conserved dynamics reads Q c = Q c hh Q c h Γ Q c hψ Q cΓ h Q cΓΓ Q cΓ ψ Q c ψh Q c ψ Γ Q c ψψ = h η h Γ2 η h ψ ηh Γ2 η h Γ η + M s (Γ) hψ Γ2 ηh ψ η hψ Γ2 η hψ η + hM ( φ ) . (45)Note that Q c is symmetric and positive definite, corresponding to Onsager relations between thefluxes and positive entropy production, respectively. Also the mobility matrix Q nc = Q nc hh Q nc h Γ Q nc hψ Q ncΓ h Q ncΓΓ Q ncΓ ψ Q nc ψh Q nc ψ Γ Q nc ψψ = β evap β ψ Γ l s − β ψ Γ − β ψ Γ l s β ψ Γ . (46)for the non-conserved dynamics is symmetric and positive definite. Note that the mobility func-tions that involve Γ have a dimension different from the other terms; the same applies to thevariations. However, the overall contributions to the respective fluxes of course have the samedimensions.The final non-conserved terms in Eqs. (42) to (44) correspond to − J evap , (cid:101) J ad = J ad /l s , and − J ad , respectively. We discuss below in Section III E 2 that in the limit of a flat surface and withoutrigidity terms they give exactly the expressions for adsorption/desorption most often derived in theliterature from kinetic considerations [59, 60]. However, in contrast to these considerations, ourformulation also naturally captures the influence of surface modulations and rigidity effects.Comparing the three conserved fluxes in Eqs. (42)-(44) to the conservation laws Eqs. (34), (39),and (40) one notes that only Q c1 = Q c hh , Q c2 = Q c h Γ and Q c3 = Q cΓΓ are independent, the othermobility functions can be derived from the relation between J conv and φ J conv , i.e., the mobility16 atrix is Q c = Q c1 Q c2 φQ c1 Q c2 Q c3 φQ c2 φQ c1 φQ c2 φ Q c1 (47)This structure ensures that for any f ( φ ) the osmotic pressure in the bulk film p osm does not con-tribute to the convective flux J conv . However, it does have an influence on evaporation (see sec-tion III E). Without slip, one has Q c1 = h / η , Q c2 = Γ h / η and Q c3 = Γ h/η , but slip can beeasily incorporated. E. Non-conserved fluxes
The general gradient dynamics form in Eqs. (42)-(44) incorporates conserved and non-conserved fluxes. The considered non-conserved fluxes include an evaporation/condensation flux J ev that only enters the equation for the film height (42) and an adsorption/desorption flux J ad thatenters the equations for the bulk and surface concentrations (43) and (44). If the respective fluxesare zero the exchange processes are at equilibrium, i.e., the evaporation and condensation of thesolvent balance as well as adsorption and desorption of the solute. In the following we discuss thefluxes individually.
1. Evaporation and condensation
Assuming that the solute does not influence the film height the evaporation flux is given by J ev ( h, Γ , φ ) = β evap (cid:18) δ F δh − p vap (cid:19) . (48)With (26) and (19) this becomes J ev ( h, Γ , φ ) = β evap (cid:18) ∇ · ( p s ∇ h ) − Π( h ) + p osm − κ (cid:20) |∇ φ | + φh ∇ · ( h ∇ φ ) (cid:21) − p vap (cid:19) , (49)where as before p osm = f ( φ ) − φf (cid:48) ( φ ) and p vap is the partial vapour pressure in the ambient air.Besides the known Kelvin effect (first term on the r.h.s., here with the full dependence p s (Γ) ) [61],wettability (second term on the r.h.s.) and osmotic pressure (third term) influence evaporation asdoes the bulk rigidity (fourth term). Normally, even on mesoscopic scales, the dominant termis that involving the vapour pressure (fifth term) and this term largely controls the evaporation17 ate – see Ref. [15] for further discussion on this. However, the other terms do matter close tocontact lines, for nanodroplets and at diffuse interfaces of dense and dilute phases. Note thatsuch thermodynamically consistent relations for J ev are also obtained for all the model extensionsdiscussed below in section IV. Also note that the rate β evap is not necessarily constant. It maydepend on film height, e.g., β evap = E/ ( K + h ) when incorporating effects of latent heat [62–64](see [15] for more details).Problems may arise in the limit of very high bulk concentrations of the solute, since the physicalfilm height can then be virtually identical to the effective solute height in contradiction to the modelassumption that the effective solute height is small as compared to the effective solvent height thatis identified with the film height. This issue may be resolved through a solvent-solute symmetricmodel as proposed in [36] in the two-field case. This case of high solute concentrations will bepursued elsewhere.
2. Adsorption and desorption
Besides evaporation, the non-conserved part of the gradient dynamics (42)-(44) also describesthe dynamics of exchange of surfactant molecules between the liquid bulk and the free surface.When (cid:101) J ad > this corresponds to an adsorption flux of molecules attaching to the free surface,while when (cid:101) J ad < there is desorption from the free surface, i.e., it is an influx into the bulk.Overall, the exchange between the bulk and the free surface is mass conserving, i.e., it suffices todiscuss (cid:101) J ad , then J ad = l s (cid:101) J ad . Within the gradient dynamics it is given by (cid:101) J ad ( h, Γ , φ ) = β ψ Γ (cid:18) δ F δψ − l s δ F δ (cid:101) Γ (cid:19) (50) = β ψ Γ (cid:18) µ − l s µ s (cid:19) (51) = β ψ Γ (cid:20) dfdφ − κh − ∇ · ( h ∇ φ ) − l s (cid:18) df s d Γ − κ s ∇ Γ (cid:19)(cid:21) , (52)where we have used Eqs. (24) and (28). Note that the bulk rigidity ( κ (cid:54) = 0 ) introduces an explicitfilm height dependence. Without rigidity influences ( κ, κ s = 0 ), the flux is (cid:101) J ad = β ψ Γ (cid:104) dfdφ − l s df s d Γ (cid:105) ,and one may now consider several particular cases.In the dilute limit for the bulk concentration φ we have f exb = 0 and Eq. (15) becomes f ( φ ) = kTl [ φ (ln φ − . (53)18 his implies that when solely entropic surface packing effects are included in f s (Γ) , i.e., Eqs. (13)and (14) with inter-molecular attraction parameter b = 0 , we obtain (cid:101) J ad = β ψ Γ kTl ln (1 − Γ) φ Γ , (54)where we also assume l = l s (otherwise φ → φ l /l ). An expression identical to (54) is given insection 2.3 of [65] where a free energy approach is followed to study the kinetics of surfactantadsorption (set β = 0 in Eq. (2.14) to recover the purely entropic case). For a full agreement with[65] one needs β ψ Γ = (cid:102) M φ where (cid:102) M is a molecular mobility. The approximation discussed nextmakes it likely that there is actually a typo in [65] and it should read β ψ Γ = (cid:102) M Γ .In many cases, the surfactant isotherms that relate equilibrium surface concentration Γ eq andequilibrium bulk concentration φ eq are introduced based on kinetic arguments of equal desorptionand adsorption fluxes (see, e.g., Refs. [59, 60]). However, the isotherm is an equilibrium propertyand may be directly obtained from the free energy. In the present context, one has at equilibrium (cid:101) J ad = 0 , i.e., φ eq = Γ eq / (1 − Γ eq ) or Γ eq = φ eq / (1+ φ eq ) corresponding to the Langmuir adsorptionisotherm [59]. To obtain the kinetics when the system is out-of but still close to equilibrium weexpand the logarithm in Eq. (54) about the equilibrium state and obtain (cid:101) J ad ≈ β ψ Γ kT Γ l [(1 − Γ) φ − Γ] . (55)This expression for the effective adsorption flux (adsorption minus desorption) agrees for β ψ Γ = (cid:102) M Γ up to normalisation factors with Eqs. (6) of Ref. [53] that result from kinetic considerations.One may also go beyond purely entropic interactions, e.g., by using Eq. (14) or other forms of f ex s (Γ) . With b > in Eq. (14) one introduces a simple attraction between surfactant molecules atthe free surface. Then (cid:101) J ad = β ψ Γ kTl ln (1 − Γ) φ l /l Γ + β ψ Γ b Γ l s , (56)where this time we retain the general l (cid:54) = l s .At equilibrium (cid:101) J ad = 0 , i.e., φ eq = (cid:18) Γ eq − Γ eq (cid:19) ( l/l s ) e − ˜ b Γ eq (57)where ˜ b = bl /kT l s , or in an implicit form Γ eq = φ ( l s /l ) eq e ˜ b ( l s /l ) Γ eq φ ( l s /l ) eq e ˜ b ( l s /l ) Γ eq . (58)19 oth are common in the literature [66], in particular, for l = l s they are known as the Frumkinisotherm [59, chap. 2.N]: φ eq = (cid:18) Γ eq − Γ eq (cid:19) e − ˜ b Γ eq (59)or Γ eq = φ eq e − ˜ b Γ eq + φ eq . (60)The kinetic adsorption equation given in [67] is obtained by expanding (56) about this equilibriumstate (59). The linearised flux is (cid:101) J ad = β ψ Γ kTl Γ e ˜ b Γ (cid:104) (1 − Γ) φ − Γ e − ˜ b Γ (cid:105) (61)that has the same form as Eq. (16) of Ref. [67] and implies certain Γ -dependencies of their mobili-ties α and β or of our mobility β ψ Γ . Note that the case of adhesion (their K < ) here correspondsto ˜ b > .The expression in Eq. (49), that is linear in the thermodynamic potentials (variations of F )must be linearised about the equilibrium state (cid:101) J ad = 0 to obtain the expressions obtained in theliterature based on kinetic considerations. This may imply that these kinetic considerations onlycapture a linearised picture of the process. Alternatively, one may introduce expressions such as ( φ − Γ) / (log φ − log Γ) into the mobility β ψ Γ as proposed in [68] in the context of gradient dynam-ics formulations of reaction-diffusion dynamics. However, for the more complicated free energiesdiscussed here this seems inadequate. Another option is to go beyond linear nonequilibrium ther-modynamics, i.e., beyond the expression linear in the thermodynamic potentials in Eq. (49). Foractivated processes, activation barriers have to be overcome and Arrhenius-type exponential fac-tors may be appropriate. For instance, an adsorption flux (cid:101) J ad ( h, Γ , φ ) = ˆ β ψ Γ (cid:18) exp (cid:20) − a kT δ F δψ + a kT l s δ F δ (cid:101) Γ (cid:21) − (cid:19) (62)( a is a microscopic length scale) with appropriately defined mobility ˆ β ψ Γ results in the same ex-pressions for the flux as obtained via kinetic considerations.We end this section with a side remark on the general adsorption isotherm. Using the standarddefinition of the surface tension given in Eq. (21), we obtain dγ = − Γ eq f (cid:48)(cid:48) s d Γ eq = − Γ eq f (cid:48)(cid:48) s d Γ eq d (ln φ eq ) d (ln φ eq ) (63)20 n the dilute limit of the bulk surfactant concentration, i.e. for f ( φ ) = kTl φ (ln φ − , the adsorptionisotherm is ( kT l s /l ) ln φ eq = f (cid:48) s , i.e., d (ln φ eq ) /d Γ eq = ( l /kT l s ) f (cid:48)(cid:48) s implying that the Gibbsadsorption isotherm dγ = − kT l s l Γ eq d (ln φ eq ) (64)is valid for any form of f (cid:48)(cid:48) s (Γ) . However, this is not the case for more complicated expressions for f ( φ ) or, indeed, when rigidity effects are included. Then Eq. (52) with (cid:101) J ad = 0 provides a generalrelation valid for heterogeneous equilibria. IV. SOLUBLE SURFACTANT - SPECIAL CASES AND EXTENSIONS
In this section we explore further the general gradient dynamics model (42)-(44). In particular,we first show that well known hydrodynamic long-wave models are recovered as limiting cases.We then discuss extensions incorporating physical effects of interest that can be described withinthe present framework.
A. Hydrodynamic formulation in dilute limit
The standard hydrodynamic long-wave model employed for thin films with a soluble surfactantthat is dilute within the film and also has a low coverage at the film surface [4, 5] is recoveredfrom the general gradient dynamics form (42)-(44) for zero rigidity ( κ = 0 , κ s = 0 ), and withonly the low-concentration entropic (ideal-gas) terms in the energy – i.e. neglecting the nonlinearinteraction terms in the energies. Then, Eqs. (13) and (15) become f s (Γ) = γ + kTl Γ(ln Γ − , and f ( φ ) = kTl φ (ln φ − (65)respectively, where γ is a constant. The energy functional (10) in the long-wave approximation is F = (cid:90) [ hf ( φ ) + f s (Γ) ξ + g ( h )] dx dy, (66)where ξ = 1 + ( ∇ h ) . Note that in (66) one has to write φ = ψ/h and Γ = (cid:101) Γ /ξ to obtain thevariations w.r.t. the independent fields h , ψ and (cid:101) Γ , as discussed at the begin of section III A. The21 ariations are p = δ F δh = − ∂ x ( γ (Γ) ∂ x h ) − Π( h ) − kTl φ,µ s = δ F δ (cid:101) Γ = kTl ln Γ ,µ = δ F δψ = kTl ln φ, (67)where γ (Γ) = f s − Γ f (cid:48) s = γ − kT Γ /l = γ − γ Γ Γ , i.e., purely entropic low-concentrationcontributions to the free energy result in a linear equation of state. As a result, the evolutionequations (42)-(46) become ∂ t h = ∇ · (cid:18) h η ∇ [ −∇ · ( γ ∇ h ) − Π( h )] + γ Γ h η ∇ Γ (cid:19) − β evap (cid:18) ˆ µ − ∇ · ( γ ∇ h ) − Π( h ) − kTl φ (cid:19) , (68) ∂ t Γ = ∇ · (cid:18) h Γ2 η ∇ [ −∇ · ( γ ∇ h ) − Π( h )] + (cid:18) γ Γ h Γ η + D s (cid:19) ∇ Γ (cid:19) + βl (ln φ − ln Γ) , (69) ∂ t ψ = ∇ · (cid:18) h ψ η ∇ [ −∇ · ( γ ∇ h ) − Π( h )] + γ Γ hψ η ∇ Γ + Dh ∇ φ (cid:19) − β (ln φ − ln Γ) , (70)where we have assumed l s = l , used the mobility functions (38) and introduced β = β ψ Γ kT /l and ˆ µ = − p vap . Note, that in the capillary terms γ = γ (Γ) is often replaced by γ and that β maystill depend on the concentrations.The model can be related to the standard hydrodynamic long-wave models for films with sol-uble surfactants found in the literature. In the simple case without solvent evaporation ( β evap = 0 and without wettability ( Π = 0 ), it corresponds to Eqs. (117-119) of the review [5] if the expres-sion ln φ − ln Γ in our adsorption flux is replaced by the linearised φ − Γ as already discussedin section III E 2. Eqs. (21) of [69] [also cf. Eqs. (4.29a-c) of the review [4]] further neglect allLaplace pressure contributions (equivalent to γ ≈ , but keeping Marangoni flows) and adds per-meability of the substrate for the surfactant. In Ref. [70] the case of a volatile solvent is studied fora surfactant-covered film on a heated substrate. Their Eqs. (50-52) add thermal Marangoni flowsto our Eqs. (68) - (70), have a linearised adsorption flux and an evaporation flux ∼ / ( h + K ) that in our equation corresponds to β evap ∼ / ( h + K ) and a ˆ µ that is much larger than the otherevaporation terms. 22 . Mixture of liquids without surfactant Another important limit is the case of a liquid film of a binary mixture that consists of compo-nents that change the surface tension without forming a proper monolayer of surfactant moleculesat the free surface. Refs. [27, 28] presented a two-field gradient dynamics model for the evolutionof a film of a liquid binary mixture on a solid substrate that allows for the description of coupleddewetting and decomposition processes for arbitrary bulk (mixing) energies including bulk rigid-ity terms, capillarity and wetting energies that may depend on the film height and concentration.The two fields are the film height h and the effective solute layer height ψ . The model recovers,for instance, the long-wave limit of model-H (Navier-Stokes Cahn-Hilliard equations) as derivedin [71], but also goes far beyond as it allows for a number of other systematic extensions [27, 28].However, this two-field model has an important shortcoming: in Ref. [28] it was noted thatno obvious way exists to incorporate a concentration-dependent surface tension into the modelwithout breaking the gradient dynamics structure. This implies that introducing a Marangoni flowcaused by the solutal Marangoni effect into the hydrodynamic two-field thin-film model for a mix-ture could break the thermodynamic consistency: If one incorporates a concentration-dependentsurface tension directly into the energy functional [ γ ( φ ) in Eq. (1) of [28]] that only depends on theheight-averaged bulk concentration φ and film height h , a Marangoni-like flux term is obtained,however, with the wrong prefactor in the mobility function. Therefore the use of the model inRef. [28] is limited to cases where surface activity can be neglected.Here, in the context of the three-field model, this issue is resolved in the following way. Weshow that one may take the full gradient dynamics model for soluble surfactants introduced abovein Section III D and consider the limit of very fast (instantaneous) adsorption/desorption. This limitcorresponds to β ψ Γ (cid:29) in Eqs. (43) and (44) implying that the non-conserved fluxes equilibratefast. As a result, on the slower time scale of the conserved fluxes one has J ad ≈ (cf. Eq. (50)-(52)) and the surfactant concentration at the free surface is slaved to the one in the bulk film. Thedependence corresponds to the equilibrium relations discussed in section III E 2.For example, in the case without rigidity one has f (cid:48) ( φ ) = f (cid:48) s (Γ) /l s and in the limit of lowconcentrations φ (cid:28) and Γ (cid:28) one obtains l ln φ = l ln Γ implying Γ = φ ( l s /l ) . For l s = (1 + ε ) l and ε (cid:28) , Γ = φ + 3 εφ ln φ + O ( ε ) . Assuming Γ ≈ φ (i.e., l s ≈ l , the governingequations (42)-(45) with the mobility functions (38), can be simplified by multiplying Eq. (43) by l and adding it to Eq. (44). As a result, an evolution equation for (cid:101) ψ = ψ + l Γ = ( h + l ) φ ≈ hφ = ψ s obtained where we use h (cid:29) l . Dropping the tilde and approximating the mobilities accordingto h (cid:29) l , the equation reads ∂ t ψ = ∇ · (cid:20) h ψ η ∇ δFδh + (cid:18) ψ η + D s l φkT (cid:19) ∇ δFδ (cid:101) Γ + (cid:18) hψ η + Dl ψkT (cid:19) ∇ δFδψ (cid:21) . (71)The film height equation (42) becomes ∂ t h = ∇ · (cid:20) h η ∇ δFδh + h φ η ∇ δFδ (cid:101) Γ + h ψ η ∇ δFδψ (cid:21) − β evap (cid:18) δFδh − p vap (cid:19) . (72)As we are in the dilute limit for f s , the second term in the conserved part of (72) becomes γ Γ h ∇ φ/ with γ Γ = kT /l corresponding to the standard form of the Marangoni flux. Thehydrodynamic form of Eq. (72) is then ∂ t h = ∇ · (cid:20) − h η ∇ ( γ ∆ h + Π( h )) + γ Γ h η ∇ φ (cid:21) (73)while Eq. (71) becomes (again with h (cid:29) l and approximating γ by the reference value γ in thecapillary term) ∂ t ( φh ) = ∇ · (cid:20) − h φ η ∇ ( γ ∆ h + Π( h )) + (cid:18) γ Γ h φ η + Dh (cid:19) ∇ φ (cid:21) (74)with the bulk diffusion constant D . Eqs. (73) and (74) correspond exactly to the hydrodynamicthin film equations employed, e.g., in the study of coalescence and non-coalescence of sessiledrops of mixtures in Ref. [72, 73]. We emphasise that as shown here they may be derived fromthe full three-field gradient dynamics model in the dilute limit. Remarkably, the resulting modelcan not be brought into the form of a two-field gradient dynamics. This poses the intriguing ques-tion whether there exist circumstances (consistent with the employed approximations) where thebroken gradient dynamics structure can result in unphysical behaviour. This merits further con-sideration. We finally remark that the proposed reduction from the three-field gradient dynamicsmodel to a two-field model also works for other choices of the energies (also with rigidities) – theyonly have to be consistent between bulk and surface. C. Nonlinear equation of state
In the literature, thin film dynamics is sometimes studied in the case of soluble surfactantswith equations similar to Eqs. (68) to (70) but employing nonlinear equations of state γ (Γ) (e.g.,Eqs. (8)-(12) of Ref. [74]). Other examples of nonlinear equations of state in thin film hydrody-namics are found in Refs. [75–78]. Often, the nonlinearity is incorporated into the Marangoni term24 nd the remaining equation is left unchanged. This may lead to spurious results if the underlyinggradient dynamics structure is broken [79]. If instead, the free energy functional is appropriatelychanged one finds that Marangoni flux, diffusion and adsorption/desorption terms all change in aconsistent manner.In the case without rigidity ( κ = κ s = 0 ) and without evaporation the resulting equations are ∂ t h = −∇ · (cid:20) h η ∇ ( ∇ · ( γ (Γ) ∇ h ) + Π( h )) + h η ∇ γ (Γ) (cid:21) (75) ∂ t Γ = −∇ · (cid:26) h Γ2 η ∇ ( ∇ · ( γ (Γ) ∇ h ) + Π( h )) + (cid:20) h Γ η + D s l kT (cid:21) ∇ γ (Γ) (cid:27) + (cid:101) J ad (Γ , φ ) , (76) ∂ t ψ = −∇ · (cid:26) h ψ η ∇ ( ∇ · ( γ (Γ) ∇ h ) + Π( h )) + h η ψ ∇ γ (Γ) + Dl hkT ∇ p osm ( φ ) (cid:27) − J ad (Γ , φ ) , (77)where we used Γ ∇ f (cid:48) s (Γ) = −∇ γ (Γ) and φ ∇ f (cid:48) ( φ ) = −∇ p osm to express surface and bulk dif-fusion in terms of the surface tension and osmotic pressure, respectively. For a discussion of theadsorption fluxes see section III E 2.Nonlinear equations of state used in the literature are, for instance, the Scheludko equation ofstate [74–76] γ (Γ) = γ [1 + θ Γ] ; (78)the exponential relation γ (Γ) = exp( − α Γ) [77]; and the expression γ (Γ) = γ − RT Γ ∞ ln(1 − Γ / Γ ∞ ) [78]. If diffusion is expressed in the form of Fick’s law j diff = (cid:101) D (Γ) ∇ Γ , the nonlinear‘diffusion constant’ (cid:101) D (Γ) should then be proportional to dγ (Γ) /d Γ – if a constant moleculardiffusivity D s is assumed, cf. Eq. (76). If one does not assume (cid:101) D (Γ) ∼ dγ (Γ) /d Γ ∼ − Γ f (cid:48)(cid:48) s (Γ) , asis the case in all the mentioned works, then it should be realised that implicitly a certain nonlineardependence of the molecular diffusivity on the concentration is being assumed, that may often notbe justified. D. Concentration-dependent wettability
The energy functional F described above in section III A contains well separated bulk con-tributions L b and surface contributions L s , namely Eqs. (12) and (11), respectively. Energeticcouplings (terms that depend on more than one of the independent fields) exist due to the surfacemetric and the introduction of the three independent fields h , (cid:101) Γ and ψ . However, the bulk free25 nergy f ( φ ) , surface free energy f s (Γ) and wetting energy g ( h ) may also depend on the otherfields. First, we discuss a concentration-dependent wetting energy.It has been discussed several times how to incorporate such a dependency into the knownhydrodynamic long-wave equations. One approach is to make the interaction constants withinthe Derjaguin pressure to depend on the surfactant concentration (case of insoluble surfactant)[80–83]. Another is to make the (structural) Derjaguin pressure to depend on the concentrationof nanoparticles to model layering effects [84]. Ref. [85] includes a concentration-dependentdisjoining pressure, and accounts for surfactant layers at the free surface and the solid substrate. Inthe bulk film dissolved surfactant molecules as well as micelles are considered. Similar extensionsare made in Ref. [82] for a two-layer system with surfactant.We argue that incorporating such concentration-dependence of wetting and dewetting phenom-ena has to start with an amended energy functional. Then, a concentration-dependent Derjaguinpressure as introduced in all the papers cited in the previous paragraph, is one natural consequence but is not the only one . We illustrate this by replacing g ( h ) in Eq. (12) by the general expression g ( h, Γ , φ ) for the case without rigidities ( κ = κ s = 0 ) but keep f = f ( φ ) and f s = f s (Γ) . Thenthe variations in long-wave approximation are p = δFδh = f − φ∂ φ f + ∂ h g − φh ∂ φ g − ∇ · (˜ ω ∇ h ) (79) µ s = δFδ (cid:101) Γ = ∂ Γ g + ∂ Γ f s (80) µ = δFδψ = 1 h ∂ φ g + ∂ φ f (81)with the generalised surface tension ˜ ω = f s − Γ ∂ Γ f s − Γ ∂ Γ g. (82)Note the new contributions that depend on ∂ φ g or ∂ Γ g which appear in p , µ s , µ and ˜ ω . They areoften missing in the literature. The full expressions for κ (cid:54) = 0 , κ s (cid:54) = 0 and general f and f s aregiven in Appendix B.With Eqs. (79) to (81) the general gradient dynamics form (42)-(45) of the evolution equations26 ecomes ∂ t h = ∇ · (cid:18) h η (cid:20) ∇ ( ∂ h g − ∇ · (˜ ω ∇ h )) − ∇ φh ∂ φ g (cid:21) + h Γ2 η ∇ [ ∂ Γ g + ∂ Γ f s ] (cid:19) − J ev ( h, Γ , φ ) , (83) ∂ t Γ = ∇ · (cid:18) h Γ2 η (cid:20) ∇ ( ∂ h g − ∇ · (˜ ω ∇ h )) − ∇ φh ∂ φ g (cid:21) + (cid:18) h Γ η + D s l Γ kT (cid:19) ∇ [ ∂ Γ g + ∂ Γ f s ] (cid:19) + (cid:101) J ad ( h, Γ , φ ) , (84) ∂ t ψ = ∇ · (cid:18) h ψ η (cid:20) ∇ ( ∂ h g − ∇ · (˜ ω ∇ h )) − ∇ φh ∂ φ g (cid:21) + hψ Γ2 η ∇ [ ∂ Γ g + ∂ Γ f s ] + Dl ψkT ∇ (cid:20) h ∂ φ g + ∂ φ f (cid:21)(cid:19) − l s (cid:101) J ad ( h, Γ , φ ) . (85)The non-conserved terms are only written in summary form, but can be easily obtained withEqs. (79) to (81) from Eqs. (49) and (50).Inspecting Eqs. (83)-(85), one notices that the above mentioned cross-coupling terms depend-ing on ∂ φ g or ∂ Γ g contribute to all conserved and-non-conserved fluxes. These terms are importantfor very thin films and in contact line regions where the free liquid-gas interface approaches thesolid-liquid interface. There they contribute to diffusion, act as Marangoni-like driving terms of theconvective flux and influence adsorption and evaporation. For drops of mixtures, a concentration-dependent wettability might, e.g., result in a local phase decomposition in the contact line region orin a single-component wetting layer (precursor films) as, e.g., observed in experiments with poly-mer solutions [86, 87]. Note that Derjaguin pressure isotherms for binary mixtures have alreadybeen discussed in Ref. [88].It is our impression that the cross-coupling terms are often missing in the literature. This is alsoimportant on general grounds since without them the gradient dynamics structure of the dynamicequations is broken. We believe that this is the reason why Ref. [82] reports traveling and standing“dewetting waves” that are clearly unphysical in a relaxational setting. It seems also likely that thecusps in the dispersion curves obtained in [81] result from transitions between real and complexeigenvalues. The latter could again result from a broken gradient dynamics structure. However,the character of the eigenmodes is not explicitly mentioned in Ref. [81], here we only deduce thispossibility from the appearance of the dispersion curves.27 . Surfactant phase transitions and mixture decomposition - bulk and surface rigidity In sections IV B and IV C we have discussed concentration-dependent bulk energies f ( φ ) andsurface energies f s ( φ ) . If these are nonlinear and exhibit negative second derivatives, then thesystem is thermodynamically unstable over the corresponding concentration range. In such a casea phase decomposition in the bulk film [89] or a surfactant phase transition [90, 91] may occur.Then a theoretical description needs to include rigidity effects, i.e., κ (cid:54) = 0 and/or κ s (cid:54) = 0 toassign an energetic cost to strong concentration gradients. Long-wave models that include theseterms were already developed for non-surface active mixtures [28, 71] and non-soluble surfactants[29, 92, 93]. In the case of constant rigidities κ s and κ , a model for soluble surfactants essentiallycombines the rigidity-related expressions developed in [28] and [29]. Therefore, here we do notexplicitly write the bulky expressions. However, the variations of the energy functional in thegeneral case are given as Eqs. (B24) to (B26) in Appendix B, so the dynamic equations can beeasily obtained by introducing them into the general gradient dynamics form (42)-(45). The case ofconcentration dependent rigidities may also be treated and these result in additional contributionsto the variations. Finally, note that the effect of substrate-mediated condensation described in[90, 91] naturally results in a free energy f ( φ, h ) that depends on both φ and h , that is also coveredin Appendix B.This section ends the discussion of the special cases of the presented general model. The fol-lowing conclusion includes a discussion of possible further extensions and open questions. Note,that there are two appendices: Appendix A clarifies an issue in the comparison of hydrodynamiclong-wave approach and the present variational approach and Appendix B gives the variations ofthe energy functional in the most general case covered by the present work. V. CONCLUSIONS
We have shown that a thin film (or long-wave) model for the dynamics of liquid films on solidsubstrates with a free liquid-gas interface that is covered by soluble surfactants can be broughtinto a gradient dynamics form. Note that we always consider regimes where inertia does notenter (small Reynold number). The gradient dynamics form is fully consistent with linear non-equilibrium thermodynamics including Onsager’s reciprocity relations [3]. In the dilute limit, themodel reduces to the well-known hydrodynamic form that includes Marangoni fluxes due to a28 inear equation of state relating surface tension and surfactant concentration at the free surface[5]. In this case the free energy functional incorporates wetting energy (resulting in a Derjaguinor disjoining pressure), surface energy of the free interface (constant contribution plus entropicterm, resulting in capillarity - Laplace pressure - and Marangoni flux) and bulk mixing free en-ergy consisting solely of an (ideal-gas) entropic term that results in a dependence of evaporationon osmotic pressure but does not influence the convective flux. The entropic contributions alsodetermine surfactant diffusion within and on the film and adsorption/desorption fluxes.The advantage of the gradient dynamics form is that one may amend the energy functional (in-corporating non-entropic mixing and surface energies, bulk and surface rigidities, concentration-dependent wetting energies, etc.) and so one automatically obtains a thermodynamically consis-tent set of updated expressions for the Laplace and Derjaguin pressures, Marangoni, Korteweg anddiffusion fluxes, and evaporation as well as adsorption/desorption terms. There are also new cross-coupling terms, e.g., in the case of a concentration-dependent wettability. The general model wehave presented contains as limits the case of films of non-surface active mixtures [27, 28] and insol-uble surfactants [29]. Such models with specific energies are furthermore found in Refs. [71, 94]and [92, 93], respectively. However, our work has also shown that many models existing in theliterature are incomplete because they directly modify the hydrodynamic long-wave equations byincorporating, e.g., concentration-dependent Derjaguin pressures or nonlinear equations of state(for examples see section IV, but also the discussions in [27–29]). Such ad-hoc changes shouldbe avoided as they alter only one ‘transport channel’ (e.g. Marangoni flux or pressure gradient)while the underlying change of the energy functional affects all transport channels. So does, e.g.,a change in the concentration-dependence of the surface free energy. This not only changes thesurface equation of state and the Marangoni flux, but also affects surfactant diffusion and adsorp-tion/desorption. A concentration-dependent wettability results in a concentration-dependent Der-jaguin pressure and furthermore it gives a new Marangoni-type flux, affects diffusion, evaporation,and adsorption/desorption. We expect that our general model with appropriately adapted energiescan describe the film dynamics and incorporate the effects of, e.g., the spreading of patches ofhigh-concentration surfactants on a liquid layer, that exhibit a local concentration maximum at theadvancing surfactant front [95, 96], or the adsorption/desorption dynamics of nanoparticles thatact as surfactant [97, 98].Besides the amendments to the energy functional that we have discussed at length, an im-portant element of a thermodynamically-consistent gradient dynamics structure are the mobilities29 hat form a positive-definite (positive entropy production) and symmetric (Onsager’s reciprocityrelations) matrix. Whenever a similar model for a relaxational situation is derived by makinga long-wave approximation, a transformation into the gradient dynamics form should result insuch a mobility matrix - thereby providing a valuable check that not all models in the literaturepass. Here, we have not changed the convective mobilities, but allowed for general diffusive ones, M ( φ ) and M s (Γ) . A further discussion of the former [ M ( φ ) ] is found in [36], where a solvent-solute symmetric model is developed (without surface activity) that is valid also for high soluteconcentrations. However, the convective mobilities may also be amended: for instance, one canincorporate slip at the substrate or solvent diffusion along the substrate as discussed in Refs. [45]and [99] for films of simple liquids and layers of organic molecules, respectively. Less is knownabout the mobility coefficients of the non-conserved fluxes, so they are often approximated as aconstant. A discussion of different mobility functions in the evaporation term is found, e.g., in[15], although there also a constant is often used [46]. The influence of the mobilities should befurther studied – in the present three-field case we expect a larger influence than in the one-fieldcase of a film of simple liquid. There, the various convective mobilities mainly change the relativetiming of the different stages of the time evolution without much change to the pathway itself[99]. Another important factor that we have not discussed here, is the dependence of the liquidviscosity on solute concentration. This is easy to incorporate, as long as the liquid is Newtonian.A further future task is the incorporation of surface viscosity [100] that should results in changesto the mobility matrix.The gradient dynamics approach that we have presented may also be applied to situations wheremore than the three fields considered here (effective bulk solute height, projected surface concen-tration, film height) matter. For example, systems with surfactant adsorption at the solid substratehave relevance, e.g, for chemically-driven running droplets [101, 102] where the transfer of a sur-factant between different media and a solid substrate plays an important role. To model such sys-tems one needs to account for adsorption at the substrate and diffusion of the adsorbate along thesubstrate. This can be achieved through the incorporation of a fourth field (adsorbate concentra-tion) into the gradient dynamics structure and an appropriate amendment of the energy functional.This leads to a fourth evolution equation that couples through additional adsorption/desorptionfluxes with the dynamics of the other fields. Such considerations are also important if one is seek-ing to model the dependence of the fluid dynamics in the contact line region on the concentration,including the concentration-dependence of all the involved interfacial tensions and of the equilib-30 ium contact angle. Such a model would allow one to describe the dynamics of effects like, e.g.,surfactant-induced autophobing [103].Another important extension is the incorporation of micelle dynamics [104, 105]. This playsan important role, e.g., for super-spreading, as does adsorption at the substrate [106–108]. To dothis, one must again incorporate additional fields into the gradient dynamics approach. One couldemploy the free energy approach of Ref. [109] and combine it with the present ideas to obtaincoupled equations for the film height, effective solute height, effective micellar height and surfaceconcentrations. This is straightforward if the micelles are monodisperse in size. However, thenumber of equations will proliferate if the number of molecules per micelle is considered in detail.In hydrodynamic long-wave models only one size is normally considered [5, 105, 110].Since the adsorption at the substrate may be physisorption or chemisorption, the question ariseswhether, in general, chemical reactions may be incorporated into a gradient dynamics. Ref. [68]provides such a formulation for reaction-diffusion systems that may be coupled to the presentformulation of thin film hydrodynamics. Preliminary considerations show that this is possible andresults, e.g., in cross-couplings between chemical reactions and wettability. However, as brieflydiscussed in Section III E 2, what the correct way to construct the mobilities such that they agreewith the ones obtained via kinetic considerations is still an open question.Throughout the present work we have nearly exclusively referred to relaxational situations, i.e.,experimental settings without any imposed influxes or through-flows of energy or mass, where theinitial state relaxes towards a minimum of the underlying energy functional. However, the resultinggradient dynamics formulation for the time evolution can now be supplemented by well-defined(normally non-variational) terms to describe systems that are permanently out of equilibrium.Example of this are film flows and drop dynamics on inclined planes where a gradient dynamicsmodel is obtained by incorporating the potential energy of the liquid into the energy functional[10].Other examples include models for dip-coating and Langmuir-Blodgett transfer processeswhere a film of solution or suspension is transfered from a bath onto a moving plate [31]. Thenthe relaxational gradient dynamics is supplemented by a dragging or comoving frame term that to-gether with lateral boundary conditions representing the bath and the deposited layer, respectively,effectively transforms the model into a non-relaxational out-of-equilibrium model that often showsmultistability or self-organised pattern formation [31, 93, 111, 112]. It is similar for dragged filmsof simple liquids (aka the Landau-Levich problem) [113, 114], films and drops on/in rotating31 ylinders [115, 116] and also for evaporative dewetting of suspensions (in the comoving frame ofa planar evaporation front) [15, 117]. Furthermore, one may impose certain in- and/or out-fluxesof material that break the gradient dynamics structure (e.g., caused by heating) [118].Finally, we point out that such an approach to interface-dominated out-of-equilibrium pro-cesses may also be applied to the modelling of (bio-)active soft matter. For instance, Ref. [119]presents a model for the osmotic spreading dynamics of bacterial biofilms where a relaxationalmodel for a mixture of aqueous solvent and biomass is supplemented by growth terms that modelthe proliferation of biomass. Another example considers a dilute carpet of insoluble self-propelledmicro-swimmers on a liquid film and describes it using an extension of models developed for insol-uble non-self-propelling surfactant particles [120, 121]. To describe higher concentrations of themicro-swimmers one could employ the present model of soluble surfactants and add contributionsresulting from the self-propulsion. Acknowledgments
We acknowledge discussions with many colleagues about the concept of gradient dynamicsin the context of long-wave hydrodynamic models, for instance, Richard Craster, Oliver Jensen,Michael Shearer and Tiezheng Qian. We thank the Center of Nonlinear Science (CeNoS) and theInternationalisation Funds of the Westf¨alische Wilhelms Universit¨at M¨unster for their support ofour collaborative meetings and an extensive stay of LMP at M¨unster, respectively. Further wewould like to thank the Isaac Newton Institute for Mathematical Sciences at the University ofCambridge for the Research Program “Mathematical Modelling and Analysis of Complex Fluidsand Active Media in Evolving Domains” (2013) where where many discussion with colleaguestook place and the first part of this work was perceived. We are thankful to Sarah Trinschek andWalter Tewes for triple-checking part of our calculations.32 ppendix A: Asymptotic long-wave expansion vs. variational approach
There is an interesting issue in the variational form of the evolution equations for an insolublelayer of surfactant on a liquid layer as presented in Ref. [29]. There, in Eq. (15) the Laplacepressure takes the form − ∂ x ( γ∂ x h ) , where γ = γ (Γ) is the surfactant concentration-dependentsurface tension that emerges as the local grand potential [122].Consider the curve representing the surface of a fluid in two dimensions with surface tension γ = γ ( s ) as a function of arclength s . On mechanical grounds one should expect that the force ona curve element to be the derivative w.r.t. arclength of γ ( s ) t , i.e., dds ( γ ( s ) t ) = dγ ( s ) ds t + γ ( s ) d t ds = dγ ( s ) ds t + γ ( s ) K n (A1)where n = 1 ξ ( − ∂ x h, T , t = 1 ξ (1 , ∂ x h ) T , K = ∂ xx hξ are the normal vector, tangent vector and curvature of the surface, respectively, and ξ = (cid:0) ∂ x h ) (cid:1) / .This seems to indicate that the Laplace pressure term in a long-wave model should be − γ∂ xx h since Eq. (A1) gives the r.h.s. of the classical hydrodynamic force boundary condition (BC) at afree surface while the left hand side is ( τ in − τ out ) · n .We show next that the form − ∂ x ( γ∂ x h ) in Ref. [29] that also appears in all the models presentedhere naturally arises when projecting the force BC not onto n and t (as done for general interfaces),but onto the cartesian unit vectors e x = (1 , T and e z = (0 , T , as appropriate when performinga long-wave approximation.The stress tensor is τ = − p I + η ( ∇ v + ( ∇ v ) T ) . (A2)where p ( x, z ) stands for the pressure field and I is the identity tensor. The force equilibrium is ( τ − τ air ) · n = γK n + ( ∂ s γ ) t (A3)where the surface derivative is defined by ∂ s = t · ∇ and we assume that the ambient air does nottransmit any shear stress ( τ air = p gas I ) and introduce p = p liq − p gas .The boundary condition (A3) is of vectorial character, i.e. one can derive two scalar conditionsby projecting it onto two different directions. In Refs. [4, 5, 123] projections onto n and t are33 sed, resulting in t : η [( u z + w x )(1 − h x ) + 2( w z − u x ) h x ] = ∂ s γ (1 + h x ) (A4) n : p + 2 η h x (cid:2) − u x h x − w z + h x ( u z + w x ) (cid:3) = − γK (A5)Note that to highest order in long-wave scaling (see below) this results in BC (when keeping allthe surface tension terms) p = − ε γh xx and ηu z = ε∂ x γ .Here, instead, we project onto e x and e z obtaining e x : − h x (2 ηu x − p ) + η ( u z + w x ) = − h x γK + ∂ s γ (A6) e z : − ηh x ( w x + u z ) + 2 ηw z − p = γK + h x ∂ s γ (A7)Next we introduce the long-wave scaling with length scale ratio ε = H/L . Note, that we do notnon-dimensionalize. We also replace K ≈ h xx and ∂ s γ ≈ ∂ x γ - formally introducing scaled(long-wave) variables x (cid:48) = εx and w (cid:48) = w/ε . After dropping the dashes we have e x : − εh x (2 ηεu x − p ) + η ( u z + ε w x ) = − ε γh x h xx + ε∂ x γ (A8) e z : − εηh x ( ε w x + u z ) + 2 εηw z − p = ε γh xx + ε h x ∂ x γ (A9)In the usual way [123] one takes into account that all velocities are small, introducing u (cid:48) = u/ε , w (cid:48) = w/ε ; dropping small terms with the exception of surface tension related terms. After drop-ping the dashes one has e x : εh x p + εηu z = − ε γh x h xx + ε∂ x γ (A10) e z : − p = ε γh xx + ε h x ∂ x γ (A11)Introducing Eq. (A11) into Eq. (A10) one has εh x ( − ε γh xx − ε h x ∂ x γ ) + εηu z = − ε γh x h xx + ε∂ x γ (A12)i.e. ηu z = (1 + ε h x ) ∂ x γ ≈ ∂ x γ. (A13)The second condition (A11) is identical to p = − ε ∂ x ( γ∂ x h ) . (A14)As the previous two equations give the BC for the bulk equations u zz = p x and p z = 0 , theinvolved quantities have to scale as O ( ε γ ) = O ( ∂ x γ ) = O ( p ) = O ( u ) = O (1) , i.e., in other34 ords ∂ x ( γ∂ x h ) ≈ γ∂ xx h . The difference is of higher order in ε . Our consideration poses theinteresting question whether an asymptotic expansion should in general be done in such a way thatit does not break deeper principles. Here the deeper principle is the thermodynamically consistentgradient dynamics formulation required for the description of a relaxational process. Therefore ∂ x ( γ∂ x h ) should be preferred over γ∂ xx h . Appendix B: Variations in the general case
The free energy F [ h, Γ , φ ] for the thin liquid film covered with soluble surfactant (aka film ofa mixture with surface active components) is F (cid:34) h, (cid:101) Γ ξ , ψh (cid:35) = (cid:90) hf (cid:18) h, ψh (cid:19) + g (cid:32) h, (cid:101) Γ ξ , ψh (cid:33) + ξf s (cid:32) h, (cid:101) Γ ξ (cid:33) + h κ (cid:18) ∇ ψh (cid:19) + κ s ξ (cid:32) ∇ (cid:101) Γ ξ (cid:33) dA. (B1)We define F (cid:34) h, (cid:101) Γ ξ , ψh (cid:35) = F bulk + F wet + F surf + F gradbulk + F gradsurf (B2)and separately calculate the variations of the five terms in the free energy. For simplicity, weonly consider the one-dimensional case. An extension to the general two-dimensional case isstraightforward. Initially, we keep the full expression ξ = (cid:112) ∂ x h ) and introduce the long-wave approximation for ξ later on. This implies ∂∂h ξ = 0 , ∂ξ∂ ( ∂ x h ) = 1 ξ ∂ x h, ∂ x ξ = 1 ξ ( ∂ x h )( ∂ xx h ) and ∂∂ ( ∂ x h ) 1 ξ = − ξ ∂ x h. (B3)
1. Variations with respect to h δF bulk δh = f + h∂ h f − φ∂ φ f (B4) δF wet δh = ∂ h g − φh ∂ φ g + ddx (cid:20) Γ ξ ( ∂ Γ g ) ∂ x h (cid:21) (B5)Note that the final term was missed in Eq. (A4) of Ref. [29]. This then also results in amendmentsin their Eq. (23), namely there is an additional − Γ ∂ Γ g in the surface tension γ in their Eq. (23) andthe Marangoni force is ∇ γ − ( ∂ Γ g ) ∇ Γ (Note that our g is their f ).35 ext, we have δF surf δh = ξ∂ h f s − ddx (cid:20) ξ f s ∂ x h − ξ ( ∂ Γ f s ) (cid:101) Γ ∂ x h (cid:21) (B6) = ξ∂ h f s − ddx (cid:20) ξ ( f s − Γ ∂ Γ f s ) ∂ x h (cid:21) . (B7)For the next variation we need to use δ ( (cid:82) (cid:63)dx ) δh = ∂(cid:63)∂h − ddx ∂(cid:63)∂ ( ∂ x h ) + d dx ∂(cid:63)∂ ( ∂ xx h ) . (B8)We also need ∂ x (cid:101) Γ ξ = ∂ x (cid:101) Γ ξ − (cid:101) Γ ξ ∂ x ξ (B9) = ∂ x (cid:101) Γ ξ − (cid:101) Γ ξ ( ∂ x h )( ∂ xx h ) (B10)The variations of the gradient terms are then δF gradbulk δh = κ (cid:18) ∂ x ψh (cid:19) + κ (cid:18) ∂ x ψh (cid:19) (cid:20) − ∂ x ψh + 2 ψh ∂ x h (cid:21) + ddx (cid:20) κ ψh (cid:18) ∂ x ψh (cid:19)(cid:21) = κ ∂ x φ ) + κ φh ( ∂ x h ) ( ∂ x φ ) + κφ∂ xx φ (B11)and δF gradsurf δh = − ddx − κ s (cid:32) ∂ x (cid:101) Γ ξ (cid:33) ∂ x hξ − κ s ξ (cid:32) ∂ x (cid:101) Γ ∂ x h + (cid:101) Γ ∂ xx h − (cid:101) Γ ξ ( ∂ x h ) ∂ xx h (cid:33) ∂ x (cid:101) Γ ξ − d dx (cid:34) κ s ξ (cid:32) ∂ x (cid:101) Γ ξ (cid:33) (cid:101) Γ ∂ x h (cid:35) = − ddx (cid:26) κ s ξ (cid:20) −
12 ( ∂ x Γ) ∂ x h − (cid:18) ∂ x Γ ∂ x h + Γ ∂ xx h − ξ ( ∂ x h ) ∂ xx h (cid:19) ∂ x Γ − (cid:18) ξ ( ∂ x h ) ∂ xx h − ∂ x Γ ∂ x h − Γ ∂ xx h (cid:19) ∂ x Γ + Γ ∂ x h∂ xx Γ (cid:21)(cid:27) = ddx (cid:26) κ s ξ (cid:20)
12 ( ∂ x Γ) ∂ x h + Γ ξ ( ∂ x h ) ( ∂ xx h ) ∂ x Γ − Γ ∂ x h∂ xx Γ (cid:21)(cid:27) (B12)
2. Variations with respect to (cid:101) Γ δF bulk δ (cid:101) Γ = 0 and δF gradbulk δ (cid:101) Γ = 0 (B13)36 F wet δ (cid:101) Γ = 1 ξ ∂ Γ g (B14) δF surf δ (cid:101) Γ = ∂ Γ f s (B15) δF gradsurf δ (cid:101) Γ = − κ s ξ ( ∂ x Γ)( ∂ x h )( ∂ xx h ) − κ s ddx (cid:20) ξ ∂ x Γ (cid:21) = κ s ξ ( ∂ x Γ)( ∂ x h )( ∂ xx h ) − κ s ξ ∂ xx Γ (B16)
3. Variations with respect to ψ δF surf δψ = 0 and δF gradsurf δψ = 0 (B17) δF wet δψ = 1 h ∂ φ g (B18) δF bulk δψ = ∂ φ f (B19) δF gradbulk δψ = − κ h ( ∂ x φ )( ∂ x h ) − κ∂ xx φ (B20)
4. Collecting the terms
The resulting expressions for the variations are p = δFδh = f + h∂ h f − φ∂ φ f + ∂ h g − φh ∂ φ g + ξ∂ h f s + κ ∂ x φ ) + κ φh ( ∂ x h ) ( ∂ x φ ) + κφ∂ xx φ (B21) − ∂ x (cid:20) ξ (cid:18) f s − Γ ∂ Γ f s − Γ ξ ∂ Γ g − κ s ξ ( ∂ x Γ) + κ s ξ Γ ∂ x (cid:18) ξ ∂ x Γ (cid:19)(cid:19) ∂ x h (cid:21) µ s = δFδ (cid:101) Γ = 1 ξ ∂ Γ g + ∂ Γ f s − κ s ξ ∂ x (cid:18) ξ ∂ x Γ (cid:19) (B22) µ = δFδψ = 1 h ∂ φ g + ∂ φ f − κh ∂ x ( h∂ x φ ) (B23)37 his seems the appropriate stage in the derivation to apply the long-wave approximation, i.e., touse ( ∂ x h ) ∼ ε (cid:28) . Therefore ξ ≈ O ( ε ) and one obtains to highest order p = δFδh = f + h∂ h f − φ∂ φ f + ∂ h g − φh ∂ φ g + ∂ h f s + κ ∂ x φ ) + κ φh ( ∂ x h ) ( ∂ x φ ) + κφ∂ xx φ (B24) − ∂ x [˜ ω∂ x h ] µ s = δFδ (cid:101) Γ = ∂ Γ ( f s + g ) − κ s ∂ xx Γ (B25) µ = δFδψ = ∂ φ f + 1 h ∂ φ g − κh ∂ x ( h∂ x φ ) (B26)where we have introduced ˜ γ = ˜ ω = f s − Γ ∂ Γ f s − Γ ∂ Γ g − κ s ∂ x Γ) + κ s Γ ∂ xx Γ (B27)corresponding to the surface grand potential density for the nonlocal case. Note that ∇ ˜ γ = − Γ ∇ µ s − ∂ Γ ∇ Γ . The free energy in the general case (B1) may be simplified by assuming thatcross-couplings between composition and film height are all contained in g ( h, Γ , φ ) and do notappear in the bulk and surface energy. The latter are then f ( φ ) and f s (Γ) , respectively. In conse-quence, ∂ h f = 0 and ∂ h f s = 0 Eqs. (B24)-(B26) simplify accordingly. The general expressionsfor the variations, i.e., Eqs. (B24) to (B26) are then introduced into the general gradient dynamicsform (42)-(45). With specific simplifying assumptions for the individual terms of the energy func-tional, one obtains several models in the literature and all the models introduced above as specialcases. [1] L. Onsager. Reciprocal relations in irreversible processes. II.
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