Transient and persistent Marangoni–Bénard convection in the presence of surfactants

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Abstract

A liquid film subject to a transverse temperature gradient, with cool fluid above and hot fluid below, can erupt into a periodic flow caused by the Marangoni–Bénard instability, a thermocapillary flow driven by the decrease in surface tension with temperature. A unit cell in this periodic flow has an upwelling of hot fluid at its center, and downward flow at the cool edges. There is renewed interest in this flow field, which has been adopted as a means to create periodic structures from particles suspended within evaporating films. Surfactants can alter the onset conditions for the flow through either surfactant-related Marangoni stresses or through increased thermocapillary coupling. We study the effects of an insoluble surfactant on the transients and steady behavior of this flow in a two-dimensional simulation of the non-linearly coupled heat, surfactant transport, and momentum equations in a unit cell of the flow field for the circumstance in which convective effects are strong compared to surface diffusion. This is the usual case in experiment, which cannot be accessed by the perturbative schemes typically adopted in stability analyses. Typically, surface tension reduces with surfactant concentration. For this case, at low surface concentration, surfactant is swept to the outer edges of the unit cell, and the interface can be divided into a surfactant-covered region with zero velocity, and a surfactant-free region of width λfree. Flow persists in the surfactant free region if λfree corresponds to a linearly unstable wavelength. That is, remarkably, the steady response of the non-linear system can be related to classical marginal stability results from a linear stability analysis, provided the wavelength is replaced by λfree. Insoluble surfactants in coexisting liquid expanded and liquid condensed surface states can promote thermocapillary flows, as surface tension is highly coupled to temperature, and decoupled from concentration in two phase coexistence. For such surfactants, linearly unstable conditions are readily established. The flow will rapidly occur and will subsequently self-quench. Surfactants will re-spread, setting conditions for auto-oscillatory behavior.

Graphical abstract

Surfactant–laden interfaces of evaporating liquids can have transient or persistent Marangoni Benard flow.

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Introduction

A film with a vertical temperature gradient with cool fluid above and hot fluid below can erupt into a periodic flow by the Marangoni–Bénard instability, a thermocapillary flow driven by the decrease in surface tension γo with temperature T [1]. The temperature-driven Marangoni stress that drives this flow can be expressed as:sγ=γoTsTwhere γo is the surface tension of the solvent–vapor interface, ∂γo/∂T is the sensitivity of the surface tension to temperature, ∇s is the surface gradient operator, and ∇sT is the surface temperature gradient.

The flow is initiated by perturbations in the temperature at the liquid–gas interface which create cool regions with high surface tension adjacent to warm regions with lower surface tension. The resulting Marangoni stress pulls from warm to cool regions. Liquid drawn from the warm regions is replaced by the up-welling of hot fluid from below, further raising the surface temperature and reinforcing the temperature fluctuation (see Fig. 1a). If this reinforcement occurs rapidly enough compared to the rate at which the initial temperature fluctuation dissipates, the film can become unstable, leading to a periodic steady flow with convective cells characterized by the fastest growing wavelength identified from a linear stability analysis [1], [2]. The ratio of the time-scales for dissipation and reinforcement is expressed via the Marangoni number Ma:Mah02/κh0μ/(|γ/T|(THTC))=γoT(THTC)μh0κwhere (TH  TC) is the vertical temperature difference across the film, h0 is undisturbed film thickness, μ is liquid viscosity, and κ is liquid thermal diffusivity. Instability occurs when Ma is larger than a critical value first identified by Pearson in a stationary state stability analysis [1]. In one manifestation of the instability, a horizontal cross-section of the thin film shows a pair of counter-rotating flows repeating periodically, while a view of the interface from above shows periodic, hexagonal, cellular patterns first reported by Bénard [3], [4] who observed the patterns in heated spermaceti. The hexagonal cellular patterns are shown in Fig. 1b (reproduced from Koschmeider [5]). The streamlines from the center of the hexagonal cells to their edges are sketched in Fig. 1c. Pearson's analysis has been extended by several researchers, most notably Scriven and Sternling [2], who improved on Pearson's assumption of a flat interface by including the effects of interfacial curvature. These authors reported that curvature reduces the critical Marangoni number for the onset of the instability. They also predicted unstable behavior at zero wavenumber, which Smith [6] later showed could be attributed to the absence of gravity in their analysis.

Marangoni–Bénard convection is rarely observed in aqueous thin films because of trace surfactant adsorption. The surface tension (i.e. the surface energy) of aqueous–air interfaces is relatively high, so surfactants readily adsorb to lower the surface energy. For this reason, aqueous–air interfaces are highly susceptible to surface contamination even when surfactants are not deliberately added. Berg and Acrivos [7] reported the stabilizing effect of surfactants in a linear stability analysis. When surfactants are present, the surface tension γ depends both on the temperature T and the surface concentration Γ. For this circumstance, the surface tension gradient can be expressed as:sγ=γTsT+γΓsΓ

When the fluid interface experiences a temperature fluctuation as shown in Fig. 1a, the thermocapillary stress causes the interface to contract toward the cold regions which have the elevated surface tension. This motion pulls adsorbed surfactant along with it, causing surfactant to accumulate in the cold region, lowering the surface tension there, as depicted in Fig. 2. This eliminates the driving Marangoni stress. Berg and Acrivos studied the perturbative effects of insoluble surfactants on Marangoni–Bénard flow in terms of this mechanism, determining the dependence of marginal stability states on surface concentration gradients and elasticity parameters describing the coupling between surface tension and surface concentration. Scriven and Sternling [2] included surface viscosities attributable to surfactant monolayers to describe the stabilizing effect of surfactants.

Surfactant surface diffusivities are typically small enough [8] that surface diffusion is weak compared to surface convection. Thus, the surfactant transport, momentum and energy equations are non-linearly coupled for leading order departures of the temperature field and surface concentration field from uniform values, precluding analytical solution. Therefore, we study this flow field numerically by integrating the coupled set of equations for a unit cell for conditions that are predicted to be unstable by the marginal stability analysis of a surfactant-free system. To our knowledge, this is the first study on the role of surfactants in this flow field for non-perturbative departures from uniform surfactant distributions and surface temperatures. We resolve the transients and steady states for two regimes of surfactant behavior, both readily realizable in experiment.

In the first regime we study an insoluble surfactant monolayer in a single surface phase; for this case, regions of elevated surface concentration have reduced surface tension. Since most soluble surfactants at dilute concentrations can be approximated as insoluble (either because of slow bulk diffusion or slow adsorption desorption kinetics [9], [10] compared to the convective flux) the results we uncover will also apply to dilute, soluble, surfactants in this flow.

The second regime concerns the occurrence of surfactant enhanced thermocapillary flows which has been reported for surfactants in two-phase coexistence between the liquid expanded (LE) and the liquid condensed (LC) states. In two phase coexistence, surfactant exists at either the binodal concentration for the LE state or the binodal concentration for the LC state at fixed surface tension. Thus, the usual effect of surface tension decreasing with surfactant accumulation is absent. Furthermore, in LE/LC coexistence, the interface is highly coupled to temperature. Linearly unstable conditions are readily established. The ensuing dynamics are simulated in this work. They will be described concisely in reference to the single surface phase result.

These simulations were motivated by the experimental results of Nguyen and Stebe [11] and Truskett (née Nguyen) and Stebe [12]. In the experiments, an aqueous drop containing micron-sized particles in suspension was allowed to evaporate. The particles were tracers in the flow; the residual pattern of particles after the drop had completely evaporated indicated locations where the streamlines had converged. In the absence of deliberately added surfactant, particles deposited in the expected ‘coffee ring’ pattern created by an outward flow from the drop to the pinned three phase contact line as described by Deegan et al. [13]. However, when the evaporating drop had an insoluble surfactant (n-pentadecanoic acid or PDA) spread in LE/LC states at its interface (as visualized using fluorescence microscopy, see Fig. 3a), the particles were observed to move violently in periodic patterns as the drop evaporated. The residual deposition pattern created by the particles after the liquid in the drop had evaporated was strongly reminiscent of a Bénard convective cell pattern, as shown in Fig. 3b. PDA can form a variety of surface phases at aqueous–air interfaces, depending on its area/molecule at room temperature. The Marangoni–Bénard pattern was observed only for drops covered with monolayers in LE/LC co-existence.

Monolayers in LE/LC coexistence have two key features that allow them to promote Marangoni–Bénard flow. First, in LE/LC coexistence, the surface tension is decoupled from the surface composition, since this surface phase transition is first order. Second, the surface tension is highly coupled to temperature, since there is a pronounced change in the order (and hence the surface entropy) of the monolayer over a small change in area per molecule. This behavior is generic for all surfactants which form these surface phases, (which include all long saturated hydrocarbon surfactants with chains containing between 12–16 carbons and small headgroups, e.g. saturated n-alcohols and n-acids.).

To identify key parameters associated with the LE/LC coexistence state, consider the isotherm of surface pressure π vs. area per molecule A for PDA on an acidified aqueous subphase in Fig. 4a, reproduced from Hifeda and Rayfield [14]. The surface pressure π = γo  γ(A), where γo is the surface tension of the solvent–vapor interface and A = 1/Γ. For A  40 Å2/molecule the surfactant is in the LE phase. As A is decreased quasi-statically, the surface pressure π rises until A = 1/Γ1, where Γ1 is the binodal concentration for forming the LE phase. Thereafter, the interface undergoes a first-order phase transition to the LC state [15], [16], [17]. Compression of the monolayer in LE/LC coexistence causes islands of LC phase at the binodal concentration Γ2 to grow at constant π. When the area per molecule is reduced to 1/Γ2, the entire surface is in an LC state. Further reduction of A causes π to increase. These data support the first important feature of these surfactants; the surface tension is decoupled from changes in the mean surface concentration for monolayers in LE/LC coexistence, so Marangoni stresses associated with surfactant distributions are zero. Fig. 4b shows πA isotherms for PDA obtained at different temperatures (reproduced from Ref. [15]). The inset to Fig. 4b is a plot of surface tension γ vs. temperature T in the LE/LC coexistence regime; the data for air-water interfaces are also presented as a reference. −∂γ/∂T is pronounced for monolayers in LE/LC coexistence. A Marangoni number calculated for PDA based on −∂γ/∂T in LE/LC coexistence on an aqueous thin film (with a nominal vertical temperature difference of 2 K across a thin aqueous film with h0 = 0.01 cm, κ = 1.5 × 10−3 cm2/s, μ = 1.0 cp, and −∂γ/∂T = 1.042 mN/(m K)):MaLE/LCγT(THTC)μh0κ=1389is well above the critical value for onset of convective instability, so the conditions for initiating Marangoni–Bénard instability are in place. Thus, linearly unstable conditions indeed prevailed in the experiments reported in Refs. [11], [12]. Furthermore, the lengthscale of the polygonal domains of deposited particles in those experiments corresponded closely to the most unstable wavelength according to Pearson's stability analysis.

However, a consideration of the streamlines sketched in Fig. 1c indicates that the surfactant can be swept to the edges of the hexagonal domains, where it can accumulate, and away from the center of the domains, where it can be depleted. Thus, the question arises: can the flow field drive the surfactant out of two phase coexistence? And if so, can the flow persist?

The results of this study will be of particular interest in applications which exploit evaporating liquid films, for example, to deposit particles in patterns from suspensions [11], [12], [13], [18] or to enhance heat or mass transport from liquid films [19], [20].

Section snippets

Governing equations

Consider a two-dimensional unit cell consisting of a thin liquid film of width λ and undisturbed film thickness h0. The width λ represents a wavelength of the Marangoni–Bénard flow. A Cartesian coordinate system is defined in this cell with 0 < x < λ and 0 < y < h(x,t), where y = h(x,t) locates the time evolving location of the air–liquid interface. A temperature difference TH  TC is present across a thin liquid film of thickness h0 and viscosity μ. When the temperature is perturbed at the air–liquid

Numerical solution of the governing equations

The fully-coupled system of equations was solved using Galerkin's method of weighted residuals on a finite element mesh [23]. Biquadratic basis functions were used to parametrize velocity, temperature and concentration fields in the Galerkin/finite element framework; linear, discontinuous basis functions were used to discretize pressure; and a backward difference approximation was used for the time derivatives. The kinematic condition at the air–liquid interface (13) was used to provide

Results and discussion

The dimensionless width λ was fixed arbitrarily at 2, and Pes was arbitrarily held constant at an elevated value to study regimes of weak surface diffusion. Pes = 103 for single phase surfactant. The remaining values of dimensionless parameters are based on typical values of the physical quantities of which they are comprised for aqueous films near 293 K. In addition to parameters in the surface equation of state cited in (28), the following quantities are specified. The Prandtl number Pr = μ/ρ/κ is

Conclusions

In this work, we have investigated two surfactant effects on Marangoni–Bénard convection. First, we studied the typical effect of surfactant in quenching thermocapillary flows. At extremely dilute surface concentrations, surfactants are seen to be effective at quenching any thermocapillary flow even under conditions when the surfactant-free film is susceptible to the Marangoni–Bénard instability, i.e., when the Marangoni number is supercritical. Further, we demonstrate the existence of the

Acknowledgements

Partial support of this research to the Donors of the American Chemical Society Petroleum Research Fund through grant number ACS PRF #41231-AC9 and to NSF grant CTS 0244592.

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    Present address: Solyndra, Inc., 47700 Kato Rd., Fremont, CA 94538, USA.

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