Threshold of Bénard-Marangoni instability in drying liquid films

We here show how evaporation/condensation processes lead to efficient heat spreading along a liquid/gas interface, thereby damping thermal fluctuations and hindering thermocapillary flows. This mechanism acts as an effective thermal conductivity of the gas phase, which is shown to diverge when the latter is made of pure vapor. Our simple (fitting-parameter–free) theory nicely agrees with measurements of critical conditions for Bénard-Marangoni instability in drying liquid films. Heat spreading is also shown to strongly affect wavelength selection in the nonlinear regime. In addition to providing a quantitative framework for analyzing transitions between complex evaporation-driven patterns, this also opens new perspectives for better controlling deposition techniques based on drying. Evaporation and condensation are widespread processes in nature and technology, both at large scales (e.g., water cycle, salt lake drying, . . .) and at small scales (e.g., heat exchangers, deposition and coating techniques, . . .). In the latter case, such phase change phenomena are generally coupled with so-called Marangoni flows, resulting from surface tension gradients along the interface. For pure liquids evaporating into air, the latter are due to temperature gradients generated by the consumption of latent heat, often resulting in Bénard-type patterns [1,2]. In contrast, when the gas phase contains only vapor (e.g., in boiling applications), such flows are typically absent since the interface is bound to remain close to the saturation temperature (see, e.g., [3], showing the predominance of buoyancy in that case). However, despite the numerous applications, the nature of such interfacial temperature homogenization process remains unclear, and has never been accurately quantified as a function of the vapor content of the gas phase. Intuitively, it can be expected that temperature gets uniformized by the transport of energy (in the form of latent heat) from hot/evaporating to cold/condensing regions along the surface. Our goal here is therefore to assess the efficiency of this “heat spreading” (a)E-mail: [email protected] (b)E-mail: [email protected] mechanism as a function of fluid properties and ambient conditions. After having described a simple modeling of this effect in quite general conditions, we analyze its impact on Bénard-Marangoni (BM) patterns in liquid films drying into ambient air, for liquids of different volatilities (the most volatile ones eventually approaching the limiting case of pure vapor). It is well known in that respect that the critical conditions for BM instability somehow depend on thermoconvective processes in the gas phase, generally lumped into an empirically determined heat transfer coefficient [4]. Even though various theories have been proposed to generalize these one-sided approaches and to calculate the (effective) heat transfer coefficients (see, e.g., [5,6]), none of them was ever validated by direct comparison with accurate experiments. Actually, the critical Marangoni number Mac for the onset of patterns was experimentally checked only for non-volatile liquids, in which case the value is about 80 [4,7]. In view of the heat spreading mechanism discussed above, one therefore expects a strong damping of the instability, hence much larger values of Mac, when volatility increases. In the present work, this is investigated in detail by detecting the transition from the convective to the conductive state occurring when the drying-film thickness decreases below some threshold value. In addition to allowing an accurate validation of our new theory, it is worth noting that results presented in this letter could open interesting perspectives to better control techniques using drying films such as polymer coating [8] or nanoparticle deposition [9]. In order to understand and quantify the heat spreading mechanism as a function of the volatility, let us consider a flat interface (at z = 0, with z pointing to the gas) at temperature TΣ, where evaporation occurs at a mass flux density J (in kg/ms). Hereafter, a subscript Σ will denote a quantity evaluated at the interface, the gas mixture is taken to be perfect, its total pressure pg is supposed to be constant and uniform (small dynamic viscosity), and the inert gas (say, air) is not absorbed into the liquid. This implies (see, e.g., [5]) J =− DMv RTΣ ∂zpv 1−ω ∣∣∣∣ z=0 , (1) where D is the vapor-air diffusion coefficient, Mv is the molar mass of the vapor, pv is the partial pressure of vapor in the gas phase, ω= pv/pg its mole fraction, and R is the perfect gas constant. In addition, the energy balance at the interface reads ∂Tl ∂z ∣∣∣∣ z=0 =− JL λl + λg λl ∂Tg ∂z ∣∣∣∣ z=0 , (2) where Tl and Tg are, respectively, the liquid and gas temperatures, L is the latent heat of vaporization, while λl and λg are, respectively, the liquid and gas thermal conductivities (with λg≪ λl in general). We now consider fluctuations (denoted by tilded quantities) around a particular steady (or quasi-steady) state distinguished by a superscript 0. The determination of this particular state needs not be detailed for the moment, and in principle, the following reasoning applies to both evaporation (J > 0) and condensation (J < 0). The interface temperature is written TΣ = T 0 Σ + T̃Σ, and assuming local chemical equilibrium at the interface (generally valid except at very small scales [1,5]), the corresponding fluctuation of vapor partial pressure there reads p̃vΣ = p ′ sat(T 0 Σ) T̃Σ, where psat(T ) is the coexistence (i.e., Clausius-Clapeyron) curve and a prime denotes its derivative. As fluctuations satisfy ∇p̃v = 0 in the limit of a small Péclet number (defined on a typical length scale of the fluctuations, assumed to be much smaller than the typical size H of the gas phase) and in the quasi-static hypothesis, we have p̃vq = p ′ sat(T 0 Σ) T̃Σq e , where a subscript q indicates the Fourier component with wave vector q (in the horizontal plane). Similarly, one also has ∇T̃g = 0, because the Lewis number Le=D/κg is O(1) in the gas (κg is the gas thermal diffusivity). Hence, assuming Tg = Tl (= TΣ) at z = 0, we have T̃gq = T̃Σq e . Finally, linearizing eq. (1), we can calculate (the Fourier transform of) the phase change rate fluctuation J̃q = |q| DMv RT 0 Σ psat′(T 0 Σ) 1−ω0 Σ T̃Σq, (3) where fluctuations of the denominator have been neglected (this is rigorously valid for |q| ≪H, as will be shown elsewhere). Then, Fourier-transforming the interfacial energy balance (2) and grouping terms, we get ∂T̃lq ∂z ∣∣∣∣ z=0 +α|q| T̃Σq = 0, (4)