Pattern formation and mass transfer under stationary solutal Marangoni instability
Graphical abstract
Hierarchical Marangoni cell pattern showing simulated interfacial butanol concentration during mass transfer from cyclohexanol to water
Introduction
The Marangoni effect refers to a mostly small-scale flow driven by local inhomogeneities of interfacial tension σ either caused by temperature (thermocapillary flow) or by concentration gradients (solutocapillary flow). Historically the discovery of this effect goes back to Thomson [2] and Marangoni [3] while the first systematic study of the resulting pattern in plane liquid layers, exposed to a steady temperature gradient, was done in the seminal work by H. Bénard [4]. Pearson's [5] linear stability theory applied to Bénard's experiments revealed that gradients in surface tension are the driving force in thin layers and provided the critical conditions for the onset of what is now called surface-tension-driven Bénard convection.
Motivated by early experimental observations on surprising spontaneous interfacial motions during mass transfer [6], [7], [8], [9], Sternling and Scriven [1] tackled the mass-transfer analog of Pearson's problem in a similarly important paper in 1959 which is detailed later on. Since then, an ongoing research can be noticed in this area which is mainly motivated by two facts. (i) First, there is an enigmatic variety of patterns emerging from the Marangoni effect making such systems to paradigmatic objects of study for pattern formation. The structures found comprise hexagonal [4], [10], polygonal [11], [12], [13], square [14], [15] or round [16] convection cells, eruptions [17], [18] and oscillations [19] as well as different sorts of waves (detailed in 1.3 Oscillatory solutal Marangoni instability, 3.3 Synchronized relaxation oscillation waves). Recently found new structures include circular patterns in volatile binary liquids [20] or the emission of small satellite droplets from pulsating larger droplets [21]. Also chemically reactive systems are affected, e.g. by influencing the shape of propagating autocatalytic reaction fronts [22] or the front velocity [23], see the recent focus issue [24].
The second source for the continuing interest is the seemingly ubiquitous occurrence of this effect both in established techniques, such as extraction, distillation, drying or wetting, and newly emerging technologies such as the self-assembly of patterns of functional materials [25]. To illustrate briefly the relevance of the Marangoni effect we take the extraction process as a first example. Here, the mass transfer enhancement due to the Marangoni effect [12], [26], [27] is an important but critical issue in the correct scale-up of pilot plants designed for the extraction of particular species. Consider next the drying of lacquers or polymer solutions on which many industrial applications such as coating or printing are based. The goal of a smooth surface finish is sometimes not achieved because the solvent evaporation can trigger Marangoni convection which may in turn cause marked surface corrugations [28].
Let us briefly stay at films but turn to newly emerging technologies in which polymer/carbon nanotubes composite films appear as a versatile object for numerous applications. In [29] it was shown that the electrical conductivity and the light transmittance of such films can be improved by controlling the solutal Marangoni instability. Another promising technique is the evaporation of sessile droplets which is attractive for the deposition of rather different types of particles (polymers, DNA, etc.) in self-assembled patterns. Numerous pattern morphologies have been observed since the original work on the single coffee ring stain [30], see e.g. [25]. All of them show a complex interplay between pinning–depinning events of the contact line and Marangoni flows to and away from the contact line arising from solvent evaporation. Criteria for the onset of Marangoni convection in evaporating sessile droplets were established recently [31]. Computer simulations of evaporating pinned sessile water droplets of submicrometer size [32] have shown that the thermal Marangoni flow loses its importance only for very small droplets with diameters L ≤ 10− 5 m. Coalescence of neighboring sessile droplets of different but miscible liquids is suppressed by a Marangoni flow that sucks liquid out of the neck which connects both droplets, thereby counteracting the capillary-driven flow into the neck [33].
Progress in the – by now – large field of research on Marangoni flows was possible thanks to modern experimental and numerical methods. Besides advanced microscopic methods such as wide-field fluorescence microscopy, the growing drop capillary pressure tensiometry [34] has to be mentioned as a tool to measure interfacial tension changes even on short timescales. Furthermore, an extension of the validity limits of the drop profile tensiometry, an originally static technique based on fitting the Gauss–Laplace equation, towards fast growing drops can be expected from the recent combination of computational fluid dynamics and experiments [35]. Novel numerical techniques such as Volume-of-Fluid [36], [37], phase-field methods [38], level-set [39] or pseudospectral formulations [40] in combination with high-performance computing now allow one to perform fully three-dimensional simulations of the mass transfer in liquid–liquid systems.
Despite of this progress numerous white spots on the Marangoni landscape still exist while new fields become obvious. The complex interaction of different patterns and their transition at high Marangoni driving force, or the delayed onset of Marangoni convection at convective mobile interfaces [41] are examples of the first category. The apparently well understood tears-of-wine phenomenon is a candidate for the second category. As pointed out in [42], the simple explanation in terms of shear force at the air–liquid interface pulling fluid of higher alcohol concentration, hence lower σ, toward the top of the film where σ is higher due to alcohol evaporation might be questionable. However, such debates keep the subject alive and further evolving dynamically.
The present work aims to contribute to this field by reviewing our understanding of a subsection of Marangoni problems, namely the stationary solutal Marangoni instability and its pattern formation in liquid–liquid mass transfer systems. The focus is on plane liquid–liquid interfaces as the generic system while the technically more relevant droplet geometry is also included later on. We start with Sternling and Scriven's theory (Section 1.2) and its predictions of a stationary and an oscillatory primary instability. The oscillatory one is briefly discussed first also to honor Manuel Velarde whose contribution, together with the experimental work of H. Linde and A. Wierschem, has advanced the understanding of this topic considerably. The main part of this work deals with stationary solutal Marangoni instability. To begin with, we present in Section 2.1 a mathematical formulation for the plane liquid–liquid system also used for own numerical simulations presented later. Our view on the typical nonlinear evolution, based on the analysis of numerous experiments from the literature and our own experiments, is summarized in the form of four central theses (Section 2.2). These theses are substantiated in the 3 The building blocks of interfacial convection, 4 Mechanism of hierarchy formation, 5 Time-sequence of structure formation. Section 3 describes the three main types of patterns and their different hierarchy levels. The mechanisms underlying the hierarchy formation for Marangoni roll cells are illustrated in Section 4. In Section 5 we demonstrate that the structures may evolve over time via periodic decays and subsequent re-amplifications. The influence of solutal Marangoni convection on mass transfer between the liquid phases is summarized in Section 6 where droplet geometries are included. Here we also consider the interaction between Marangoni and buoyancy-driven convection. In particular, we focus, by means of numerical simulations, onto the effect of a stabilizing density stratification (Section 6.5) and of geometrical constraints (Section 6.6) not studied before. Finally, a short summary is given in Section 7.
The foundation for a theoretical understanding of the structures arising from solutal Marangoni instability was laid by Sternling & Scriven [1] who studied the linear stability of two semi-infinite layers separated by a planar interface. Fig. 1 summarizes the most relevant cases for liquid–liquid systems on the left-hand side and for liquid–gas systems on the right-hand side. The particular instability regimes in dependence on the mass transfer direction of a conventional surfactant (dσ/dc < 0) and the ratio between the most important material parameters, kinematic viscosity ν(i) and mass diffusivity D(i), are indicated. Furthermore, the value of the f-parameter, the involved structure of which is given in Eq. (1) asplays a certain role. D, ν, ρ, H and Sc(2) refer to the ratios of diffusivities, kinematic viscosities and densities, to the partition coefficient H = ceq(2)/ceq(1) and the Schmidt number of the second phase Sc(2) = ν(2)/D(2). These quantities are summarized in Table 1. μ(2) is the dynamic viscosity of phase 2 and μs refers to the intrinsic surface viscosity which is frequently neglected for estimation of the instability behavior. Finally, αNS stands for the wave number of neutral stability.
The liquid–liquid system is stable when the solute in the donating phase has a higher diffusivity than in the accepting phase provided that viscosity of the latter is higher than that of the donating phase. In the opposite case, stationary instability sets in, if f ≥ 0 holds. Adapting this situation to the much higher diffusivities in gases, a liquid–gas system shows a stationary instability under a viscosity ratio similar to the above one, when the solute transfer is from the liquid to the gas. If the direction of mass transfer is reversed under these conditions, the oscillatory mode of instability occurs. For other combinations not shown in Fig. 1 both oscillatory and stationary modes are possible.
For the transfer of non-standard solutes, which display dσ/dc > 0 as it is the case for inorganic salts, e.g. transferring from water to butanol [43], the same stability criteria are valid for mass transfer in the reverse direction.
The simple rule of Sternling & Scriven can be used as a first estimate to predict the stability behavior of most Marangoni-sensitive systems provided no parasitic buoyancy-driven convection is present which can markedly modify the system's behavior (see Section 6.4). Of course nonlinear effects by established convection or geometrical constraints may lead to other instability effects, which are out of the scope of the linear instability of the diffusive state [1].
The oscillatory mode of the solutal Marangoni instability was first detected by Linde and coworkers in the 1960s [44], [45], [46]. Thanks to the strong support by Manuel G. Velarde, new experimental initiatives were undertaken between 1990 and 2005 in close collaboration between Linde, Wierschem and Velarde and the theoretical modeling was advanced [47], [48], [49]. As a result, the complex patterns observed could be traced back essentially to the interaction of three dominant wave types. This has led to a rather comprehensive understanding of the oscillatory Marangoni patterns as summarized in the book of Nepomnyashchy, Velarde and Colinet [50] or by Linde et al. [51].
In line with Fig. 1, the experiments employed to a large part the absorption of a gaseous surfactant, e.g. pentane, into an organic solvent, e.g. toluene, while to a smaller part also heat transfer experiments were included. Shadowgraphy was used to visualize the wave pattern in connection with laser beam deflection to measure the time-dependent surface deformations associated with the waves. After the start of the experiment, the instability sets in via surface waves which are driven by the Marangoni shear stress. These waves show an anomalous dispersion in the sense that their phase velocity v decreases with increasing wavelength λ, i.e. dv/dλ < 0. With growing time, hence decreasing forcing, the waves experience first a regularization towards a periodic wave train. Later on, a second transformation of the wave types occurs. Due to the built-up of a stabilizing density stratification with the pentane absorption in toluene, the decaying surface Marangoni waves excite internal gravity waves which now dominate the dynamics. This goes along with a change toward normal dispersion since the internal waves obey dv/dλ > 0. Single wavetrains as well as two counterrotating wavetrains were observed in cylindrical or annular containers. These wavetrains show properties of dissipative solitons which became apparent after studying the behavior after head-on and oblique collisions of two solitary waves, see Fig. 2, or after the reflection on a wall, which turned out to be of Mach–Russell type [53].
Section snippets
Stationary solutal Marangoni instability
According to Section 1.2, the stationary mode of the Marangoni instability in liquid–liquid systems, which forms the focus of the present work, sets in when the solute transfer is out of the phase with lower mass diffusivity but higher viscosity, thus D(2)/D(1) < 1 and ν(2)/ν(1) > 1 provided f ≥ 0 (Fig. 1). This condition can easily be understood when assuming ν(2) = ν(1) for simplicity. Consider a fluctuation leading to a local increase of solute concentration at the interface. Since interfacial
Roll cells
The quasi-stationary Marangoni roll cells are the basic structure. This cellular convection is initiated by small fluctuations creating a concentration increment at the interface. A higher solute concentration implies a lower interfacial tension σ (normal surfactant). The resulting Marangoni shear stress, dσ/dc ⋅ ∂c/∂x, drives a convective motion of the interface directed to regions with high interfacial tension. When the Marangoni number exceeds the critical one, the convection is reinforced by
Mechanism of hierarchy formation
To point out the main mechanisms underlying the formation of hierarchical patterns, this section uses the example of RC-II and discusses the temporal evolution shown in Fig. 7 in more detail. A conspicuous feature of this evolution, shown by two zoomed snapshots in Fig. 19a,c, is the coarsening of the higher-order structures.
Since the size of the substructure does not change notably, the degree of hierarchy increases, i.e. the RC-II host a larger number of RC-I in Fig. 19c. From the
Time-sequence of structure formation
The characteristic feature of the hierarchical structures is their spatial periodicity on multiple scales. However, as already indicated for the RORC structures in Section 3.2, the Marangoni pattern can also exhibit a temporal periodicity. This mechanism is not limited to the RORC as a specific type of structure [69]. Also the more stable patterns, RC and ROW can likewise show a temporal periodicity with more and less intensive convection states, but taking place on a larger time scale. This is
Basic aspects
Mass transfer across a liquid–liquid interface is just what occurs in extraction processes. However, this unit operation of chemical engineering is implemented in various scales and extractor types providing different geometries and flow conditions for the liquid phases. Depending on the chemical system employed, Marangoni convection has the potential to significantly augment mass transfer rates [96]. With regard to the diversity of structures described in the previous sections, it can be
Summary and outlook
More than six decades of research on both the oscillatory and the stationary mode of solutal Marangoni instabilities has led to an impressive accumulation of knowledge on the particular patterns, their nonlinear evolution and their capability of transferring mass. While a rather comprehensive understanding of the dominant wave patterns could be already established for the oscillatory Marangoni instability, a unified picture on pattern evolution for the stationary counterpart was achieved only
Acknowledgments
Financial support by the Deutsche Forschungsgemeinschaft in the form of the Priority Program 1506 and grant nos. Bo1668/6 (T.B.) and Ec201/2 (K.E.) is gratefully acknowledged. Furthermore, we thank the computing center (UniRZ) of TU Ilmenau and the computing center of FZ Jülich (NIC) for access to its parallel computing resources.
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These authors contributed equally to this work.