Gradient dynamics description for films of mixtures and suspensions - dewetting triggered by coupled film height and concentration fluctuations
A thermodynamically consistent gradient dynamics model for the evolution of thin layers of liquid mixtures, solutions and suspensions on solid substrates is presented which is based on a film height- and mean concentration-dependent free energy functional. It is able to describe a large variety of structuring processes including coupled dewetting and decomposition processes. As an example, the model is employed to investigate the dewetting of thin films of liquid mixtures and suspensions under the influence of effective long-range van der Waals forces that depend on solute concentration. The occurring fluxes are discussed and it is shown that spinodal dewetting may be triggered through the coupling of film height and concentration fluctuations. Fully nonlinear calculations provide the time evolution and resulting steady film height and concentration profiles.
pacs:68.15.+e, 47.50.Cd, 68.08.-p, 47.20.Dr
UT preprint / 1.2
Understanding the behaviour of free surface layers and drops of simple and complex liquids becomes increasingly important because the drive towards further miniaturisation of fluidic systems towards micro- Squires and Quake (2005) and eventually nano-fluidic Mijatovic et al. (2005) devices depends on our ability to gain control of the various interfacial effects on small scales. Liquid layers frequently occur either naturally, e.g., as tear film in the eye or industrially, e.g., as protection or lubrication layers. They are also instrumental in many wet process stages of printing, (nano-)structuring and coating technologies where films or drops of a liquid are applied to a surface with the aim of producing a homogeneous or structured layer of either the liquid or a solute. For reviews see Refs. Oron et al. (1997); Craster and Matar (2009); Bonn et al. (2009); Thiele (2010).
Their omnipresence in natural and industrial processes provides a strong incentive to investigate the creation, instabilities, rupture dynamics, and short- and long-time structure formation of free surface thin liquid films on solid substrates. These processes are well investigated experimentally Reiter (1992); Seemann et al. (2001) and theoretically Thiele et al. (2001); Becker et al. (2003) for films of simple liquids on smooth solid substrates. Continuum models describe the evolution of the film thickness profile as a gradient dynamics for the free energy ttl () that accounts for wettability through the local wetting energy and for capillarity through the local surface energy Thiele (2010). Here, is the long-wave (or small-gradient) approximation of the surface area element in Monge parametrization, is the liquid-gas interface tension, the variational derivative corresponds to the pressure where is the Derjaguin or disjoining pressure de Gennes (1985); Starov and Velarde (2009); Israelachvili (2011), is the mobility function in the case of no-slip at the substrate where is the dynamic viscosity (for the case of slip see, e.g., Münch et al. (2005)), , and . The described model may be derived via a long-wave approximation from the Navier-Stokes and continuity equations with adequate boundary conditions at the free surface and the solid substrate Oron et al. (1997); Thiele (2007); Craster and Matar (2009).
The dynamics of films of simple liquids is rather well understood. However, the situation strongly differs for films of complex liquids as, for instance, colloidal (nano-)particle suspensions, mixtures, polymer and surfactant solutions, polymer blends and liquid crystals. Practically, layers of such complex liquids occur far more widely than films of simple liquid, but a systematic understanding of the possible pathways of their evolution that result from the coupled processes of dewetting, decomposition, evaporation and adsorption has not been reached. Free surface films of such liquids occur, for instance, as tear films Sharma and Ruckenstein (1985), lung lining Grotberg (2011), in the production of organic solar cells Heier et al. (2008) or semiconductor nanoparticle rings Maenosono et al. (1999). Layers of solutions and suspensions with volatile solvent are frequently employed in intermediate stages of the production of homogeneous or structured layers of the solute, e.g., as a non-lithographic technique for covering large areas with regular arrays of small-scale structures. Reviews of experiments, models and applications can be found in Matar and Craster (2009) (surfactant solutions), Frastia et al. (2012); Han and Lin (2012) (deposition processes from solution) and Geoghegan and Krausch (2003) (polymer blends). Although in all these systems the interfacial effects of capillarity and wettability are still main driving forces, they may now interact with the dynamics of inner degrees of freedom as, e.g., the diffusive transport of solutes or surfactants, phase separation and other phase transitions, evaporation/condensation of solvent and concentration-dependent wettability.
The present work provides a consistent framework for the theoretical description of many of the observed dynamical processes in films of liquid mixtures, solutions and suspensions. After introducing the model, we discuss limiting cases and elucidate the physical meaning of the occurring fluxes. As an example we apply the presented general framework to the case of a film of a liquid mixture where the wettability depends on the local concentration. This shows that dewetting may be triggered through the coupling of film height and concentration fluctuations.
We consider a thin non-volatile liquid film of a mixture on a solid substrate (see Fig. 1) that without additional influx of energy relaxes towards some static equilibrium state much as in many of the experiments reviewed in Geoghegan and Krausch (2003). In the case without evaporation the approach to equilibrium for this relaxational system is described by a gradient dynamics of the underlying free energy functional
It is an extension of the above introduced that accounts for (i) a dependence of the wetting energy on local concentration, (ii) the bulk free energy of the mixture per substrate area , and (iii) the energetic cost of strong gradients in the concentration (through where is the interfacial stiffness).
The gradient dynamics has to be written in terms of the conserved fields, film thickness and effective local solute layer thickness . The non-conserved field is the dimensionless height-averaged per volume solute concentration. The general coupled evolution equations for two such conserved order parameter fields in the framework of linear nonequilibrium thermodynamics are
The mobility matrix
is symmetric and positive definite corresponding to Onsager reciprocal relations and the condition for positive entropy production, respectively de Groot and Mazur (1984). is the molecular mobility of the solute.
To perform the variations in Eqs. (2) one has to replace everywhere by . The extended free energy (1) for a film of a mixture ttl () results in convective and diffusive fluxes (for brevity, written in terms of and )
respectively. Employing the fluxes we bring the gradient dynamics equations (2) into the form
Before discussing important limiting cases, we elucidate the physical meaning of the individual flux contributions. In the convective flux [Eq. (4)] the first term is due to Laplace pressure gradients Oron et al. (1997); the second term is the Derjaguin pressure contribution due to wettability; and the final two terms represent the Korteweg flux, i.e., a bulk concentration-gradient driven flux (cf. Anderson et al. (1998) for a discussion of the related bulk model-H). The third term is a flux driven by concentration-gradients within the bulk of the film but only if the film is sufficiently thin such that its two interfaces ’feel’ each other. This novel flux is a direct consequence of the concentration dependence of the wetting energy and has a similar magnitude as the Derjaguin pressure contribution TTL ().
The first term of the diffusive flux [Eq. (5)] is also uncommon in the literature although it is a natural consequence of the gradient dynamics form (2). It represents the influence of the concentration-dependent wettability on diffusion. The second term is the flux due to gradients of the chemical potential in the bulk of the film while the final term is a Korteweg contribution to diffusion that counters steep concentration gradients, e.g., for decomposing solvent-solute films.
The general evolution equations [(6,7) with (4,5)] recover several known models as limiting cases (this is used to determine ). Most importantly: (i) For a constant film height , without wettability contribution () and appropriately defined , Eq. (7) becomes the Cahn-Hilliard equation that describes, e.g., the spinodal decomposition of a binary mixture Cahn (1965); (ii) As in (i) but with and a purely entropic (ideal gas-like)
where is a molecular length scale related to the solute, one recovers the standard diffusion equation with diffusion constant (see, e.g., section IV of Thiele et al. (2012)); (iii) For , and one recovers the conserved part of long-wave equations used, e.g., to study dewetting of and solute deposition from solutions and suspensions Warner et al. (2003); Frastia et al. (2011); ttl (); (iv) Again without wettability, but with Korteweg fluxes (), and employing the double-well potential for the solvent-solute interaction one obtains the thin film limit of model-H Anderson et al. (1998) as derived recently via a long-wave asymptotic expansion Náraigh and Thiffeault (2010); TTL (a).
Next we present as an example the practically relevant case of a solute-dependent wettability, i.e., . For clarity we only include entropic bulk terms for the solute-solvent interaction, i.e. [Eq. (8)] and , implying absolute stability against bulk solute-solvent decomposition. For the wetting energy we use the combination of long-range van der Waals interactions and an always stabilising () short-range contribution Israelachvili (2011); Pismen (2002):
Note that we combine a concentration-dependent Hamaker ’constant’ and a constant . One may as well introduce a concentration-dependent short-range contribution or use a different form for the short-range contribution TTL (c), however, these choices do not affect the main results. The Derjaguin pressure is while could be called a Derjaguin chemical potential. is determined employing homogenization techniques. For many experimentally employed mixtures as e.g., PMMA/PS, toluene/acetone or PS/toluene on Si or SiO a linear dependence is an excellent approximation over the entire concentration range Todorova (2013). Selecting the case where the pure solvent is wetting , we write where the nondimensional number quantifies the strength of the concentration-dependence of wettability. Experimentally, may be changed by choosing a different solute. For the materials we are interested in, varies in the range Nm and lies in the range . For example, a mixture of polystyrene (PS) and poly(methyl methacrylate) (PMMA) on a silicon (Si) substrate (used, e.g., in Heriot and Jones (2005)) yields and and for a solution of PS in toluene on silicon oxide (SiO) one obtains and , while a mixture of toluene and acetone on SiO gives .
Note that for and , both, a film of pure solvent and a film of pure solute, are absolutely stable. With the bulk solute-solvent mixture is stable as well. A film of mixture might then be expected to be stable for all and to become unstable for when because then . This expectation, however, assumes that the mixture in the film remains homogeneous, i.e., that concentration fluctuations are always damped. However, a linear stability analysis of flat homogeneous films with respect to fluctuations and shows that the fluctuations in film height and concentration couple, rendering the system more unstable. Fig. 2(a) shows that even for where all decoupled subsystems are stable, the film of a mixture can be linearly unstable in an extended experimentally accessible range of the parameter space.
Here the dimensionless number is the ratio of entropic and wetting influences TTL (b). In other words a film of stable solvent can be destabilized by a stable solute if the diffusion of the solute is sufficiently weak, i.e., if is sufficiently small. For common mixtures, solutions and nanoparticle suspensions can range from to . The estimate is based on a typical precursor thickness of nm Popescu et al. (2012) and a solute length scale between nm and nm (this corresponds e.g. to the size of molecules or (nano-)particles diffusing in the film) Todorova (2013). Also for the film becomes unstable at smaller than expected under the assumption that the mixture stays homogeneous (dashed line in Fig. 2(a)). Because and are both conserved, the instability is of long wavelength, i.e., at onset (at critical or ) it has zero wavenumber (cf. Thiele (2010)). Therefore, for finite domains the stability borders in Fig. 2(a) are slightly shifted.
Starting from a homogeneous flat film, we illustrate in Fig. 2(b,c) the resulting spontaneous structure formation TTL (d). During the shown linear and nonlinear stages of the short-time evolution, the steady state shown in Fig. 3(b) is approached TTL (d). In a large domain many such small droplets will undergo a long-time coarsening process (not shown) to reach pancake-like drops as shown in Fig. 3(b) for . Inspecting the and profiles and the energy in Fig. 3 the physical mechanism that drives the structuring becomes clear: Although the film can not reduce its energy by modulating its thickness profile at homogeneous concentration, it is still able to do so by simulaneously modulating its thickness and concentration profiles. In the present example the solute is enriched [depleted] in the thicker [thinner] part of the profile. The characteristics of the coexisting flat parts visible in Fig. 3(b) for may also be obtained through an analysis of the binodals of the system, i.e., of the film height and concentration values at coexistence Todorova (2013). Note that the structuring results in extended flat regions of different heights that are still much larger than the vertical lengthscale, i.e., all regions may still accommodate a diffusing solute with . Furthermore, one may include steric effects due to the solute size into the free energy.
The presented example illustrates that the above introduced thermodynamically consistent long-wave model allows one to predict a novel interface instability for thin films (below about 100nm) of liquid mixtures and suspensions under the influence of long-range van der Waals forces that are concentration dependent. The resulting coupling of film height and concentration fluctuations always renders such films more unstable than the decoupled subsystems. The chosen numerical example shows that the destabilization can even occur if all decoupled subsystem are unconditionally stable. However, the presented gradient dynamics formulation has further far reaching implications for the description of thin film of complex fluids: For so-called nanofluids (nano-particle suspensions) often a structural Derjaguin pressure Deryagin and Churaev (1974) is included into the hydrodynamic description Wasan and Nikolov (2003); Craster et al. (2009). However, Eqs. (5) shows that this is incomplete. Instead, a structural wetting energy has to be introduced what results, in consequence, in additional contributions to the convective, diffusive (and evaporative) flux. An accounting for attractive solvent-solute interactions (beyond the entropic term considered in the example) allows one to investigate how the various decomposition and dewetting instability modes couple, resulting in a number of different instability types and evolution pathways somewhat similar to the ones described for two-layer films of immiscible liquids Pototsky et al. (2005).
In summary, we have presented a general gradient dynamics model and a particular underlying free energy ttl () which is able to describe a wide range of dynamical processes in thin films of liquid mixtures, solutions and suspensions on solid substrates including the dynamics of coupled dewetting and decomposition. We have argued that on the one hand the model recovers known limiting cases including the long-wave limit of model-H. On the other hand we have discussed the physical meaning of important contributions that are missing in the hydrodynamic literature, and have shown that they are needed for a thermodynamically consistent description of, e.g., evolution pathways controlled by concentration-dependent wettability. As an example, we have investigated the dewetting of thin films of liquid mixtures and suspensions under the influence of long-range van der Waals forces that are concentration dependent.
The presented gradient dynamics form will allow for systematic future developments. Most importantly, the here presented model for a film of a mixture without enrichment or depletion boundary layers at the interfaces may be combined with models for films with an insoluble surfactant Köpf et al. (2010); Thiele et al. (2012) to also describe systems where enrichment or depletion layers form at the interfaces, including instabilities and structuring processes as observed in Thomas et al. (2010).
Acknowledgements.This work was supported by the European Union under grant PITN-GA-2008-214919 (MULTIFLOW).
- T. M. Squires and S. R. Quake, Rev. Mod. Phys. 77, 977 (2005).
- D. Mijatovic, J. C. T. Eijkel, and A. van den Berg, Lab Chip 5, 492 (2005).
- A. Oron, S. H. Davis, and S. G. Bankoff, Rev. Mod. Phys. 69, 931 (1997).
- R. V. Craster and O. K. Matar, Rev. Mod. Phys. 81, 1131 (2009).
- D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Rev. Mod. Phys. 81, 739 (2009).
- U. Thiele, J. Phys.: Condens. Matter 22, 084019 (2010).
- G. Reiter, Phys. Rev. Lett. 68, 75 (1992).
- R. Seemann, S. Herminghaus, and K. Jacobs, J. Phys.: Condens. Matter 13, 4925 (2001).
- U. Thiele, M. G. Velarde, and K. Neuffer, Phys. Rev. Lett. 87, 016104 (2001).
- J. Becker, G. Grün, R. Seemann, H. Mantz, K. Jacobs, K. R. Mecke, and R. Blossey, Nat. Mater. 2, 59 (2003).
- Note, that is sometimes called an ”effective interface Hamiltonian” (see Bonn et al. (2009)), but could equally be called a ”Lagrangian” (based on an analogue to point mechanics).
- P.-G. de Gennes, Rev. Mod. Phys. 57, 827 (1985).
- V. M. Starov and M. G. Velarde, J. Phys.-Condes. Matter 21, 464121 (2009).
- J. N. Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 2011), 3rd ed.
- A. Münch, B. Wagner, and T. P. Witelski, J. Eng. Math. 53, 359 (2005).
- U. Thiele, in Thin Films of Soft Matter, edited by S. Kalliadasis and U. Thiele (Springer, Wien, 2007), pp. 25–93.
- A. Sharma and E. Ruckenstein, Amer. J. Optom. Physiol. Opt. 62, 246 (1985).
- J. Grotberg, Phys. Fluids 23, 021301 (2011).
- J. Heier, J. Groenewold, F. A. Castro, F. Nüesch, and R. Hany, in Organic Optoelectronics and Photonics III, Proc. of SPIE Vol. 6999 (2008), p. 69991J.
- S. Maenosono, C. D. Dushkin, S. Saita, and Y. Yamaguchi, Langmuir 15, 957 (1999).
- O. K. Matar and R. V. Craster, Soft Matter 5, 3801 (2009).
- L. Frastia, A. J. Archer, and U. Thiele, Soft Matter 8, 11363 (2012).
- W. Han and Z. Lin, Angew. Chem. Int. Ed. 51, 1534 (2012).
- M. Geoghegan and G. Krausch, Prog. Polym. Sci. 28, 261 (2003).
- S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (Dover publications, New York, 1984), ISBN 0-486-64741-2.
- L. Frastia, A. J. Archer, and U. Thiele, Phys. Rev. Lett. 106, 077801 (2011).
- D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Ann. Rev. Fluid Mech. 30, 139 (1998).
- Most literature includes a concentration-dependent wettability only through the Derjaguin pressure and therefore breaks the overall gradient dynamics form (2). An exception is Ref. Clarke (2005) which proposes a gradient dynamics model for the coupled decomposition and dewetting of a thin film of a binary polymer blend in terms of constrained functional derivatives w.r.t. the conserved field and the non-conserved field . The resulting system of integro-differential evolution equations does not correspond to a limit of our Eqs. (2)-(1). The model of Clarke (2005) is used in Thomas et al. (2010) to discus the lateral instability of an enrichment layer on a film of a polymer blend.
- J. W. Cahn, J. Chem. Phys. 42, 93 (1965).
- U. Thiele, A. J. Archer, and M. Plapp, Phys. Fluids 24, 102107 (2012).
- M. R. E. Warner, R. V. Craster, and O. K. Matar, J. Colloid Interface Sci. 267, 92 (2003).
- Note, that adding a non-conserved term to the first equation of (2) accounts for the evaporation of a volatile solvent including the osmotic pressure dependence missed in Warner et al. (2003); Frastia et al. (2011).
- L. Ó. Náraigh and J. L. Thiffeault, Nonlinearity 23, 1559 (2010).
- A small discrepancy in the mobilities arises because the free energy and the mobility, on which the Cahn-Hilliard Equation is based, are approximated independently of each other.
- L. M. Pismen, Colloid Surf. A-Physicochem. Eng. Asp. 206, 11 (2002).
- For the analysis, Eqs. (2)-(1) are nondimensionalised employing scales , , and for vertical and horizontal length scales and time, respectively, where the energy scale is . In the case of a partially wetting solvent, corresponds to the precursor film height.
- Note that other combinations of long-range destabilizing and short-range stabilizing terms in the wetting energy may be used to describe partially wetting liquids on a solid substrate (see, e.g., Refs. Seemann et al. (2001); Thiele (2007); Israelachvili (2011)). Here, we employ a combination that arises as sharp interface limit of an underlying density functional theory based on Lenard-Jones molecular interactions with an attraction between molecules and hard-core repulsion Pismen (2002). Note, however, that the system behaviour is very similar if wetting energies of a similar form are chosen that result in identical precursor film heights and equilibrium contact angles.
- D. Todorova, Ph.D. thesis, Loughborough University (2013).
- S. Y. Heriot and R. A. L. Jones, Nat. Mater. 4, 782 (2005).
- M. Popescu, G. Oshanin, S. Dietrich, and A. Cazabat, J. Phys.-Condes. Matter 24, 243102 (2012).
- For the time simulations we use a second order central finite difference scheme in space and a variable-order and variable-step backward differentiation algorithm in time. Periodic boundary conditions are used and the simulation is initiated with flat and profiles perturbed by a small uniform noise. Steady profiles and bifurcation diagrams are numerically obtained employing pseudo-arclength continuation.
- B. V. Deryagin and N. V. Churaev, J. Colloid Interf. Sci. 49, 249 (1974).
- D. T. Wasan and A. D. Nikolov, Nature 423, 156 (2003).
- R. V. Craster, O. K. Matar, and K. Sefiane, Langmuir 25, 3601 (2009).
- A. Pototsky, M. Bestehorn, D. Merkt, and U. Thiele, J. Chem. Phys. 122, 224711 (2005).
- M. H. Köpf, S. V. Gurevich, R. Friedrich, and L. F. Chi, Langmuir 26, 10444 (2010).
- K. R. Thomas, N. Clarke, R. Poetes, M. Morariu, and U. Steiner, Soft Matter 6, 3517 (2010).
- N. Clarke, Macromolecules 38, 6775 (2005).