Ionic Strength and pH as Control Parameters for Spontaneous Surface

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Ionic Strength and pH as Control Parameters for Spontaneous Surface Oscillations N. M. Kovalchuk,†,‡ V. Pimienta,§ R. Tadmouri,∥ R. Miller,‡ and D. Vollhardt*,‡ †

Institute of Biocolloid Chemistry, Vernadsky Avenue 42, 03142 Kiev, Ukraine Max Planck Institute of Colloids and Interfaces, 14424 Potsdam/Golm, Germany § Université de Toulouse, UPS, IMRCP, 118 route de Narbonne, F-31062 Toulouse, France ∥ Laboratory of the Future (LOF), 178 avenue du Docteur Schweitzer, F-33608 Pessac, France ‡

ABSTRACT: A system far from equilibrium, where the surfactant transfer from a small drop located in the aqueous bulk to the air−water interface results in spontaneous nonlinear oscillations of surface tension, is theoretically and experimentally considered. The oscillations in this system are the result of periodically arising and terminating Marangoni instability. The surfactant under consideration is octanoic acid, the dissociated form of which is much less surface-active than the protonated form. Numerical simulations show how the system behavior can be controlled by changes in pH and ionic strength of the aqueous phase. The results of numerical simulations are in good agreement with experimental data.

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INTRODUCTION Systems far from equilibrium are of great scientific and practical interest because they have an impressive ability toward selforganization, a phenomenon bridging artificial systems and living organisms. In systems with liquid interfaces, such selforganization often reveals itself as Marangoni instability caused by heat or solute transfer.1,2 A well-known example is the famous Benard cells. The solutal Marangoni instability can have some specific features compared to the thermal instability because the adsorbed surfactant layer is characterized by its own properties, such as, for example, dilational viscoelasticity,3,4 which can considerably affect the instability development.5 Often, several solutes are transferred to or through the interface, each of them influencing in a specific way the instability patterns. Moreover, chemical reactions are possible at the interface with the formation of new surface active compounds.6−11 This all enormously complicates the analysis of self-organization phenomena caused by surfactant transfer. Certain progress in the understanding of self-organization phenomena as a result of surfactant transfer, concerning the explanation of the mechanism of spontaneous nonlinear oscillations, was achieved by studying a relatively simple system, where such oscillations appear as a result of transfer of a single surfactant from a small drop placed in the bulk of water under the air−water interface.12 According to the simple chemistry of the system, the analysis was performed by numerical simulations in the framework of a common hydrodynamic approach, however, taking into account the dynamic properties of the adsorbed monolayer.13−15 © 2012 American Chemical Society

It was shown in refs 13−15 that, in this system, the oscillations are the result of periodically arising and terminating Marangoni instability. The instability terminates because of the contraction of the surfactant monolayer near the container wall. That means that the dimensions of the interface or, to be more precise, the aspect ratio, i.e., the ratio between the droplet immersion depth and the inner container diameter, is of primary importance. Indeed, it was numerically shown and then experimentally confirmed that either the increase of the container diameter or the decrease of the drop immersion depth results, after a certain threshold, in the change of the dynamic regime. Steady convection is observed instead of spontaneous oscillations. It was also shown that the buoyancy force acting in the bulk solution strongly affects the onset of instability and its termination, complementing the action of surface tension.15 Further studies were aimed at accounting for the effect of chemical interactions on the system dynamics. A simple way to do that is using a fatty acid as a surfactant, whose dissociation degree changes with the pH value in the solution. The first experimental results of this study have been presented in ref 16. The dynamic regime established during the dissolution of fatty acid from a droplet under a free liquid interface can be easily controlled by varying the pH value. At small pH values, spontaneous oscillations of surface tension with rather large amplitudes of 4−6 mN/m have been observed in the system. The amplitude begun to decrease at pH > 5, initially slowly, but Received: February 10, 2012 Revised: March 29, 2012 Published: April 5, 2012 6893

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Figure 1. Geometry of the studied system: (1) vessel with aqueous solution, (2) covering plate, (3) Pasteur pipet, (4) drop of fatty acid, (5) Wilhelmy plate, (6) air, (7) area of high surfactant concentration and low pH, (8) area of low surfactant concentration and high pH, and (9) adsorbed monolayer. The measuring cell is presented in Figure 1. The dynamic surface tension was measured by a homemade tensiometric setup with an electronic balance equipped with a platinum Wilhelmy plate (5). The Wilhelmy plate was annealed before each measurement. The experiments were performed as follows: A total of 40 mL of the aqueous phase with desired pH and ionic strength was poured into a glass vessel (1) with the inner diameter of 44 mm. A cleaned glass Pasteur pipet (3) with an orifice of ∼1 mm was immersed into the aqueous phase to the desired depth. The immersion depth was regulated by a computer-controlled microscrew. The Wilhelmy plate was brought into contact with the aqueous phase, and then the vessel was covered (2) to prevent water evaporation and, in this way, to diminish the thermal convection in the system. An octanoic acid droplet (4) with the diameter of 2−3 mm was formed on the tip of the pipet, and the measurement was started.

then faster, and at pH > 6.5, no oscillations have been detected any more. The last is correct only if the pH value is regulated by a buffer. In the absence of a buffer, oscillations with large amplitude are observed at initial pH > 5 as well. Such a behavior was qualitatively explained on the basis of the mechanism proposed previously in refs 13−15. The main reason for the decrease of the oscillation amplitude is the transition of fatty acid from the more surface active protonated form to the much less surface active dissociated form.16 For the confirmation of this explanation, an inorganic salt is added to increase the surface activity of the dissociated form of fatty acid, which should result in restoring the oscillatory dynamics with a high enough amplitude at high pH. Correspondingly, the model proposed in refs 13−15 was modified to take into account (i) the difference in surface activity of the dissociated and protonated forms of the used fatty acid (octanoic acid), as well as the dependence of the surface activity of the dissociated form on the ionic strength in solution, and (ii) the dependence of the local dissociation degree of the acid on its local concentration. The experimental study was performed under conditions close to those of ref 16 but in the presence of the inorganic salt NaCl. The results of the numerical and experimental studies are presented below.

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MATHEMATICAL MODEL The system geometry used in numerical simulations is similar to the experimental geometry (Figure 1) with a vessel diameter of 40 mm and a height of the aqueous layer of 20 mm. According to refs 13−15, for the considered system geometry, the appearance and characteristics of oscillations depend upon the surfactant solubility, surface activity, and density difference between the saturated surfactant solution and pure solvent (buoyancy force). Buoyancy influences mainly the oscillation period but also the oscillation amplitude, although only a little,15 so that the changes in buoyancy are not taken into consideration in this study. The goal of this study is to find such a formulation that enables one to obtain oscillations with detectable amplitude at high enough pH, where the oscillations have small amplitude or are nondetectable if the only additive present in the solvent is the buffer.16 The two parameters considerably affecting the oscillation amplitude are surfactant solubility and surface activity. The addition of salt changes both of these parameters.

EXPERIMENTAL SECTION

Octanoic acid (for gas chromatography, purity 99.5%, Aldrich) was used as purchased. Sodium chloride (purity 99.5%, Fluka) was annealed for 5 h at a temperature of 800 °C before use to avoid organic contaminations. Phosphate buffer, a mixture of aqueous solutions of disodium hydrogen phosphate dihydrate, Na2HPO4·2H2O (purity >98%, Fluka), and sodium dihydrogen phosphate dihydrate, NaH2PO4·2H2O (purity >99%, Fluka), in a concentration of 0.01 M was used to adjust the desired pH value. The buffer solutions were prepared in ultrapure deionized water produced by a Millipore Milli-Q water purification system. 6894

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⎛ ∂ 2ω ∂(vz ω) ∂(vr ω) 1 ∂ω ∂ω ∂ 2ω + + − Sc ⎜ 2 + + 2 r ∂r ∂t ∂r ∂z ⎝ ∂r ∂z ⎞ ∂(c + c 2) ω − 2 ⎟ + ScRa 1 =0 ∂r r ⎠ (3a)

To account for the effects of pH and ionic strength in the frameworks of the model proposed in refs 13−15, it is convenient to assume that the droplet is composed of a mixture of two different surfactants representing the dissociated form, denoted below as R−, and the protonated form, denoted below as HR acid. Dependent upon the local pH value in the bulk, one form can transform into another form. The system dynamics is described by the set of nonlinear, non-steady-state Navier−Stokes equations, continuity equation, and equations of convective diffusion for each surfactant.13−15 Taking into account cylindrical symmetry of the system under consideration, the equations are rewritten in terms of stream function Ψ and vorticity ω defined by the equations vr =

1 ∂Ψ 1 ∂Ψ , vz = − r ∂z r ∂r

(1)

ω=

∂v ∂vr − z ∂z ∂r

(2)

Equation 3a is used in the simulations discussed below. Two dimensionless parameters governing the system behavior are included in eqs 3−5. The effect of the Schmidt number, Sc, is considered in details in ref 14, whereas the effect of buoyancy is discussed in ref 15. The terms describing dissociation/recombination of acid are not included in eq 5 because it is assumed that those reactions are instantaneous, and therefore, the recalculation of the concentration distribution for the protonated and dissociated forms was performed in a separate step using the total concentration distribution found from eq 5, as described below (eq 14). The approach by Lucassen-Reynders was used for the description of adsorption of an ionic surfactant. This approach discussed in detail in refs 17 and 18 is based on the concept of Gibbs dividing surface being electroneutral by definition. That means that the adsorption of any ion is the sum of its excess in the monolayer and in the electrical double layer; i.e., using this approach, information about the distribution of the ions in the system is unnecessary. According to ref 18, this model enables a correct description of adsorption of ionic surfactants as well as their mixtures, including mixtures with non-ionic surfactants, which is the case in this study. The Langmuir adsorption isotherm is used in the simulations. The Frumkin isotherm, which is more appropriate for describing the adsorption of fatty acids, was not used here because consideration of any exponential term is very timeconsuming for numerical simulations. The surfactant mixture is considered to be ideal,18 and it is supposed that the value of the saturation adsorption Γm is the same for both surfactants, which is a reasonable assumption according to the results obtained in ref 16. In this case, the dimensionless form of the adsorption isotherms is

where r is the radial coordinate, z is the coordinate normal to the interface and directed downward with z = 0 on the interface, and vr and vz are the velocity components in the radial and normal directions to the interface, respectively. Scaling the time, length, velocity, concentration, stream function, and vorticity with H2/D1, H, D1/H, c0, HD1, D1/H2, respectively, where H is the characteristic length scale, the height of the liquid layer, D1 is the bulk diffusion coefficient of one of the surfactants, c0 is the total surfactant solubility being the sum of two individual solubility values c0 = c01 + c02 = c0HR + c0R−, the dimensionless form of the governing equations is ⎛ ∂ 2ω ∂(vz ω) ∂(vr ω) 1 ∂ω ∂ω ∂ 2ω + + − Sc ⎜ 2 + + 2 r ∂r ∂t ∂r ∂z ⎝ ∂r ∂z ⎞ ⎛ ∂c ∂c ⎞ ω − 2 ⎟ + Sc ⎜Ra1 1 + Ra 2 2 ⎟ = 0 ⎝ ∂r ∂r ⎠ r ⎠

(3)

∂ 2Ψ ∂ 2Ψ 1 ∂Ψ + − − ωr = 0 2 r ∂r ∂z 2 ∂r

(4)

KLi Γi csi = KL1 1 − c0KL1Γ1 − c0KL1Γ2

∂ci ∂(vrci) ∂(vzci) ∂ 2c vc D ⎛ ∂ 2c + + + r i − i ⎜ 2i + 2i ∂t ∂r ∂z r D1 ⎝ ∂r ∂z +

1 ∂ci ⎞ ⎟=0 r ∂r ⎠

(6)

where KLi is the Langmuir constant for the ith surfactant, csi = ci(z = 0) is its subsurface concentration, and Γi is its adsorption scaled by c0KL1Γm. For the adsorption of the ionic surfactant in the presence of an electrolyte, the mean ionic product (5)

c* = f ((cel + c)c)1/2

3

where t is time, Sc = v/D1 is the Schmidt number, Rai = (gc0H / ρ0vD1)(∂ρ/∂ci) is the Rayleigh number for the ith surfactant (i = 1 and 2), v is the kinematic viscosity of the liquid, ρ = ρ0 − (c1/c01)(ρ0 − ρs1) − (c2/c02)(ρ0 − ρs2) is the solution density, ρ0 is the pure solvent density, ρsi is the density of the saturated solution of the surfactant i, and g is the acceleration due to gravity. Taking into account the explicit form of the equation for the solution density, the Rayleigh number can be rewritten as Rai = (gc0H3/ρ0vD1) ((ρ0 − ρsi)/c0i). For the special case when the dissociated and non-dissociated forms of the surfactant have the same density and the same effect on the solution density, we obtain Ra1 = Ra2 = Ra = (gH3/vD1)(1 − (ρs/ρ0)), where ρs is the average density of the saturated solution. For this particular case, eq 3 can be rewritten as

(7)

should be used instead of concentration c.18 In eq 7, cel is the bulk concentration of the electrolyte and f is the average activity coefficient of the ions in the bulk. The value of f can be calculated according to the Debye−Hückel equation, which for the 1:1 electrolyte reads18 log(f ) = −

0.5115 I + 0.055I 1 + 1.316 I

(8)

where I = (1/2)∑i n= 1cizi2 is the ionic strength of the solution, ci is the molar concentration of the ith ion expressed in mol/L, and zi is the charge number of the ith ion. The no-slip boundary conditions are used for the bottom and side wall of the vessel, the capillary, and the droplet surface. 6895

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KHR [R−] = c KHR + [H+]

The air−water interface is considered to be non-deformable in normal direction (vz = 0 at the interface) in accordance with the experimental observations with the flow visualization. The equation of state corresponding to the adsorption isotherm (eq 6) written for dimensional variables is ⎛ Γ̂ Γ̂ ⎞ Π̂ = −RT Γm ln⎜1 − 1 − 2 ⎟ Γm Γm ⎠ ⎝

The solution of eqs 13 and 14 enables the calculation of the concentration of the dissociated ([R−]) and non-dissociated ([HR]) forms from the known total concentration c. It is assumed that the concentration at the droplet interface is equal to the solubility limit and that the dissociated and protonated forms are in equilibrium at the interface. Also assuming that the solubility of the non-dissociated form of the weak acid is independent of pH and taking c0HR = 4.8 mM for octanoic acid according to eqs 13 and 14 in the solution at the interface of the octanoic acid droplet dissolving in pure water, one obtains pH 3.6, c0R− = 0.25 mM, and c0 = c0HR + c0R− = 5.05 mM. Those are the boundary conditions for the concentrations at the droplet interface for the system with pure water. In the case when the fatty acid is added to the buffer solution, the equilibrium pH is determined by the concentration and composition of the buffer and concentration of the acid. It is easy to find that, for the mixture of Na2HPO4·2H2O and NaH2PO4·2H2O used in the presented experimental study and in ref 16, the resulting pH obeys the equation

(9)

Substituting eq 9 in the tangential stress balance (eq 10)

∂vr̂ ∂z ̂

= z=0

1 ∂Π̂ ρv ∂r ̂

(10)

one obtains the dimensionless boundary condition for the vorticity at the interface ω = Ma

∂(Γ1 + Γ2) 1 at z = 0 (1 − KL1c0 Γ1 − KL1c0 Γ2) ∂r (11)

where Ma = (RTc0KL1ΓmH/ρvD1) is the global Marangoni number for the system,14 R is the gas constant, and T is the temperature. The effect of the Marangoni number on the oscillations produced by dissolution of a surfactant droplet under the air−liquid interface is discussed in detail in ref 14. The mass balance of each of the surfactants on the air−liquid interface is described by the equation

cb2 =

where cb is the total buffer concentration, cb2 is the concentration of Na2HPO4·2H2O, Kb1 = 10−6.865 is the dissociation constant for H2PO4−, and Kb2 = 10−12.319 is the dissociation constant for HPO42−. Because the minimum pH considered below is about pH 5, the reaction of recombination of H2PO4− to H3PO4 is neglected because the dissociation constant of phosphoric acid is K = 10−2.148. When a weak acid with the dissociation constant KHR and concentration c is added to the buffer solution, the resulting pH obeys the equation

(12)

where Dsi is the surface diffusion coefficient for the ith surfactant. Octanoic acid is a weak acid with the dissociation constant pKa of 4.89. At low pH (below pH 3), octanoic acid is mainly in the non-dissociated (protonated) form (HR). This form has a rather low solubility in water c0HR = 4.8 mM19 and a high surface activity (parameters of Langmuir isotherm are Γm = 8.3 × 10−6 mol/m2 and KL = 1.35 m3/mol20). At high pH (above pH 7), octanoic acid is mainly in the dissociated form (R−). This form has much higher solubility in water in comparison to the non-dissociated form [for sodium octanoate, critical micelle concentration (cmc) ≈ 0.4 M21] and much lower surface activity (KL ≈ 0.018 m3/mol16). When octanoic acid, HR, is dissolved in pure water, having pH 7, it dissociates according to the reaction HR = H+ + R−, lowering in this way the pH value. The relationship between the equilibrium pH value and the total concentration c of the acid, being the sum of the concentrations of protonated and dissociated forms, c = c1 + c2 = [HR] + [R−], can be easily found from the mass balance and electroneutrality condition as ⎛ K ⎞⎛ [H+] ⎞ c = ⎜[H+] − W+ ⎟⎜1 + ⎟ ⎝ [H ] ⎠⎝ KHR ⎠

2Kb1Kb2 + Kb1[H+] KW + − [H ] + c b [H+] Kb1Kb2 + Kb1[H+] + [H+]2 (15)

∂Γi ∂(Γivr) Γv D ⎛ ∂ 2Γ 1 ∂Γi ⎞ H ∂ci ⎟− + + i r − si ⎜ 2i + ∂t ∂r r D1 ⎝ ∂r r ∂r ⎠ KL1Γm ∂z = 0 at z = 0

(14)

⎛ K [H+] ⎞⎛ c = ⎜1 + ⎟⎜cb2 + [H+] − W+ [H ] KHR ⎠⎝ ⎝ − cb

⎞ ⎟ + Kb1[H+] + [H+]2 ⎠

2Kb1Kb2 + Kb1[H+] Kb1Kb2

(16)

or in terms of the concentration of the non-dissociated acid, [HR] [HR] =

K [H+] ⎛ ⎜cb2 + [H+] − W+ [H ] KHR ⎝ − cb

⎞ ⎟ + Kb1[H ] + [H ] ⎠

2Kb1Kb2 + Kb1[H+] Kb1Kb2

+

+ 2

(16a)

−

K [R ] = HR [HR] [H+] (13)

(14a)

Equation 16 coincides with eq 13 when cb2 = cb = 0. Equations 16 and 14 are used for the calculation of the concentration of the dissociated and non-dissociated forms from the known total concentration c of the fatty acid in buffer solution. Assuming that the solubility of the non-dissociated form of the weak acid is independent of pH at small enough buffer concentrations, for octanoic acid with c0HR = 4.8 mM according to eqs 14a and 16a at the interface of the droplet dissolving in 0.01 M buffer

where [H+] = 10−pH is the concentration of hydrogen ions, KHR = (f 2[H+][R−]/[HR]) = 10−4.89 M is the dissociation constant of octanoic acid, KW = [H+][OH−] = 10−14 M2 at 25 °C is the ionic product of water, f is the average activity coefficient of the ions in the bulk phase defined by eq 8, and square brackets designate the concentrations in mol/L = M. In the absence of salt, f is close to unity. The part of the dissociated acid is then 6896

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solution (pH 7) we obtain pH 5.0, c0R− = 5.64 mM, and c0 = c0HR + c0R− = 10.44 mM. Those are the boundary conditions for concentrations at the droplet interface for the system with buffer solution. For simulations in the presence of salt, it was assumed that the pH of the buffer is independent of the salt concentration. However, it is well-known that the solubility of organics in water decreases with the increase of the salt concentration. The solubility of the non-dissociated form of octanoic acid in the presence of salt can be found from the Setchenow equation22 log(c0HRs/c0HR ) = −K saltcsalt

dissociated and protonated forms of octanoic acid in buffer solution and pure water and the increase of the mean ionic activity for the dissociated acid in the presence of salt. After the formation of a droplet, the fatty acid begins to dissolve in water. Because of diffusion- and buoyancy-driven convection, it is redistributed over the bulk. Because the solution density is lower than that of the solvent, buoyancy convection appears in the aqueous bulk with the streamline pattern shown in Figure 1 (right-hand side). An example for the distribution of the total acid concentration (i.e., sum of [HR] and [R−] in notations of eqs 13−16) in the initial period of time can be seen in Figure 2a. To be precise, Figure 2

(17)

where c0HRs is the solubility in the presence of salt, csalt is the concentration of salt in mol/L, and Ksalt is the Setchenow constant. The value of the Setchenow constant for octanoic acid in the presence of NaCl was estimated as Ksalt = 0.262 by extrapolation of the data for butanoic, hexanoic, and heptanoic acids given in ref 23. For csalt = 1 M, we obtain c0HRs = 2.6 mM. According to eqs 14a and 16a at the interface of the droplet dissolving in 0.01 M buffer solution with pH 7 in the presence of 1 M NaCl, it results in pH 4.8, c0R− = 5.7 mM, and c0 = c0HR + c0R− = 8.3 mM. For the sake of simplicity, simulations have been performed for the same value of the diffusion coefficient D = 6.1 × 10−6 cm2/s for the dissociated and protonated forms of octanoic acid in water; the surface diffusion coefficient Ds was also chosen as Ds = D = 6.1 × 10−6 cm2/s. It is noteworthy that the changes in the Ds values in the range of several orders of magnitude have only a small effect on the system dynamics.14 The density difference between pure water and saturated surfactant solution was kept constant Δρ = 10−5 g/cm3 for all systems. The two values KL1 = 1.5 m3/mol and KL2 = 0.015 m3/mol, close to those reported in the literature, were used for the parameter of the Langmuir isotherm of protonated and dissociated acids, respectively, whereas Γm = 8.3 × 10−6 mol/m2 was used for both forms. According to the system symmetry, the computation domain for the numerical simulations in the radial direction was limited to half of the vessel, as shown in Figure 1. The governing equations with corresponding boundary conditions are solved by the explicit finite difference method on a regular grid with the resolution of 120 × 120 mesh points. Equation 4 was solved by the Gauss−Seidel iteration method. For eqs 3 and 5, the two-point forward difference approximation is used for the time derivatives, the three-point centered differences are used for the diffusion terms, and the modified upwind differences are used for the convective terms.24 After the concentrations of protonated and dissociated acids were found by solving eq 5, the total concentration was calculated by the summation of these two values. The pH value was found using linear interpolation to the previously tabulated values obtained from eq 13 or 16. The tabulation table contained about 300 pairs of c versus pH values. Finally, the dimensionless concentration of dissociated acid [R−]/c0 was found from eq 14, and the dimensionless concentration of protonated acid was [RH]/c0 = c − ([R−]/c0).

Figure 2. Distribution of pH in the liquid bulk at dissolution of a octanoic acid droplet in pure water, with a capillary immersion depth h = 10 mm, (a) before the instability onset (t = 10 min) and (b) after the instability onset (t = 20 min): (1) pH 4, (2) pH 4.9, (3) pH 6.99, and (4, 5, and 6) pH 7.

represents the bulk distribution of the pH value. However, because the pH value is fully determined by the total acid concentration, the pH contours simultaneously represent also the concentration contours. It is seen from Figure 2a that the concentration profiles deviate from the spherically symmetrical shape, especially in the vicinity of the capillary as a result of buoyancy convection. Near the droplet, the acid concentration is relatively high (region 7 in the left-hand part of Figure 1). Dissociation of the acid supplies protons H+ to the solution, lowering the pH value, especially in the absence of buffer. The pH value for pure water is ∼7, which means that the concentration of H+ in it is ∼10−7 M. The solubility limit of octanoic acid in pure water is ∼5 × 10−3 M. In comparison to these two values, one can see that the dissociation of the only relatively small part of the acid dissolving near the droplet is

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RESULTS AND DISCUSSION The performed numerical simulations have shown considerable differences between the dynamic regimes developing during the dissolution of a droplet of fatty acid in pure water, buffer solution, and buffer solution in the presence of salt. These differences are the result of different ratios between the 6897

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large enough to decrease the pH value and to prevent further dissociation. Therefore, the acid near the droplet is mainly in the protonated form (left, region 7 of Figure 1). The total acid concentration decreases very quickly with the increase of the distance to the droplet, becoming comparable and even less than the proton concentration in pure water. Therefore, in this region, dissociation of even the whole acid has a much smaller impact on the pH value and the acid is mainly dissociated (left, region 8 of Figure 1). The distribution of pH presented in Figure 2a shows that, after 10 min of the droplet dissolution in pure water, only in a relatively small part of the liquid bulk (inside curve 2) is the concentration of protonated acid larger than that of dissociated acid. In the region outside curve 3 in Figure 2a, the pH value is practically equal to that of pure water and all of the acid is dissociated. When some detectable amounts of surfactant reach the surface, an additional mechanism of transfer, namely, Marangoni convection, appears in the system. The driving force of Marangoni convection is the non-uniform surfactant distribution at the interface. Resulting from the system geometry, it is an obvious fact that both diffusion- and buoyancy-driven convection bring more surfactant to the capillary region than to the more distant regions; i.e., the surfactant concentration at the interface decreases in the direction from the capillary to the wall (left, region 9 of Figure 1). Marangoni convection enhances the buoyancy convection because it has the same direction: from the capillary to the wall at the interface, downward at the wall, from the wall to the vessel axis at the bottom, and upward at the axis (right, Figure 1). Moreover, because this flow brings more concentrated solution from the droplet to the interface in the capillary region, Marangoni convection enhances itself; i.e., there is a feedback in the system enabling the development of Marangoni instability. For details of the instability development, see refs 13−15. The intensity of Marangoni convection depends upon the surface tension gradient, which means the surfactant concentration at the interface. In the case of octanoic acid, two surfactants are transferred in the system, with one of them, the protonated acid, being 2 orders of magnitude more surface active than the dissociated acid. At the droplet interface, the amounts of protonated and dissociated acids as well as the pH value always remain the same solely depending upon the solubility limit of the protonated form. In the rest of the bulk, with the decrease of the total concentration, the protonated form is partially transformed into the dissociated form, i.e., the less surface active form. Figures 3 and 4 demonstrate the changes in the pH value and in the fraction of the protonated form depending upon the total concentration of octanoic acid, c, calculated by eqs 16 and 14, respectively. When the droplet of octanoic acid dissolves in pure water, the pH value as well as the fraction of protonated acid changes essentially with the change of the total concentration. Using the buffer solution instead of pure water, the capacity of buffer is insufficient to keep the desired pH 7 at high total concentrations of acid near the droplet. However, already at concentrations below 10% of the solubility limit, the pH is high enough and practically all acid is in the dissociated form. Note that the instability develops in these systems at subsurface concentrations of the order of 0.1% of the solubility limit and, during the oscillation, the total concentration does not exceed 5% of the solubility limit. According to Figure 4, under these conditions, about half of the acid is in the

Figure 3. Solution pH versus concentration of octanoic acid normalized by the solubility limit (c0 = 5.05 mM for pure water, and c0 = 10.44 mM for buffer solution): (1) in pure water and (2) in 0.01 M phosphate buffer.

Figure 4. Fraction of protonated acid versus concentration of octanoic acid normalized by the solubility limit (c0 = 5.05 mM for pure water, and c0 = 10.44 mM for buffer solution): (1) in pure water and (2) in 0.01 M phosphate buffer.

protonated form when using pure water as the solvent, whereas practically all acid is dissociated in buffer solution. Figure 2b shows that, during the instability development, the pH value near the capillary, where the influx of the surfactant to the interface takes place, as well as near the surface, is close to 4.9, confirming the conclusion that nearly half of the surfactant here is in the protonated form. Therefore, the adsorption of highly surface active protonated acid present in pure water results in oscillations with an amplitude of ∼3.7 mN/m (curve 1 of Figure 5). In contrast, in the buffer solution, where the only acting surfactant is the dissociated acid with 100 times lower surface activity, oscillations are practically not detectable, having an amplitude of ∼0.05 mN/m (curve 2 of Figure 5). The results in Figure 5 as well as in Figures 6−10 are calculated for the distance r = 10 mm from the vessel axis. The results are rather similar to each other for r = 5−15 mm. The radial distributions of the surface velocity and surface concentration are discussed in detail in refs 13−15. The oscillations of surface tension are the result of Marangoni instability periodically arising and terminating,13−15 and they are accompanied by oscillations of the surface and bulk velocities. An example is given in Figure 6 for oscillations 6898

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Figure 7. Oscillations of pH at the interface by dissolution of a droplet of octanoic acid: (1) in pure water and (2) in 0.01 M phosphate buffer (numerical results; pipet immersion depth of h = 10 mm).

Figure 5. Oscillations of surface tension produced by dissolution of a droplet of octanoic acid: (1) in pure water, (2) in 0.01 M phosphate buffer, and (3) in 0.01 M phosphate buffer + 0.5 M NaCl (numerical results; pipet immersion depth of h = 10 mm).

Figure 6. Oscillations of surface velocity accompanying oscillations of surface tension by dissolution of a droplet of octanoic acid in pure water (numerical results; pipet immersion depth of h = 10 mm).

produced by the dissolution of an octanoic acid droplet in pure water. It is seen from Figure 6 that the sharp decrease of surface tension corresponds to the considerable growth of the surface velocity, i.e., to the development of convective Marangoni instability, whereas the gradual increase of surface tension occurs after the instability termination, when the Marangoni convection becomes negligible and the surfactant transfer is due to diffusion- and slow buoyancy-driven convection only. Figure 7 demonstrates the changes in pH near the air−water interface taking place during the drop dissolution. In the very beginning of the dissolution process, we have pH 7 in both systems, with pure water and with buffer. However, when the instability starts to develop, the pH in the system with pure water begins to decrease very quickly because of the increase of the acid concentration near the air−water interface and decreases to a value of about pH 4.5, where more than half of the acid is in the protonated form (curve 1 of Figure 7). The synchronized changes in the subsurface concentration of dissociated and protonated acids, surface tension, pH, and surface velocity at a smaller time scale during the instability development are presented in Figure 8. From Figure 8, it is seen that, shortly after the instability onset, the concentration of dissociated acid stops to grow (curve 1) and all acid supplied at

Figure 8. Synchronous changes in the (1) subsurface concentration of dissociated acid [R−], (2) subsurface concentration of protonated acid [HR], (3) surface tension, (4) pH, and (5) surface velocity at instability onset during the dissolution of a droplet of octanoic acid in pure water (pipet immersion depth of h = 10 mm).

this time to the surface remains in the protonated form (curve 2) because of the low pH value (curve 4). An additional feedback mechanism occurs in this system. A decrease in pH results in an increase of the fraction of protonated acid with a higher surface activity, which, in turn, causes an increase of the surface and bulk velocity. Correspondingly, the increasing flow brings more acid to the air−water interface and, in this way, lowers the pH value here. This feedback also works during the instability termination, providing a faster stabilization. Thus, here, one can see an example of additional chemical contribution supporting and enhancing the development of solutal Marangoni instability and its termination. This synergism may be of considerable importance for the explanation of the oscillation mechanism in other systems of interest. Note that, in the system with buffer 6899

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solution, this additional mechanism does not work, because the pH value during the oscillations remains in close vicinity of pH 7 (curve 2 of Figure 7). When a salt is added to the buffer solution, the pH changes remain in the range of curve 2 in Figure 7; i.e., in this case, the concentration of protonated acid is negligible and the instability development is determined solely by the dissociated acid. However, if the concentration of salt is large enough, the activity of the dissociated acid increases considerably, resulting in well-detectable oscillations of surface tension, as presented by curve 3 in Figure 5. In this case, the oscillation period is essentially larger than that presented by curves 1 and 2, but one can see a very small oscillation nearly at half of the period. Such merging of two smaller oscillation periods into one larger period was also observed in experiments discussed in ref 25. The merging effect is seen in more detail in Figure 9. Here,

Figure 10. Quasi-stationary regime after first oscillation produced by dissolution of a droplet of octanoic acid in pure water at a pipet immersion depth of h = 6 mm.

example is given in curve 2 of Figure 11, where the oscillation amplitude is practically not detectable by the tensiometric setup

Figure 9. Oscillations of surface tension produced by dissolution of a droplet of octanoic acid in 0.01 M phosphate buffer with salt: (1) 0.5 M NaCl and (2) 1 M NaCl (numerical results; pipet immersion depth of h = 6 mm). Figure 11. Oscillations of surface tension produced by dissolution of a droplet of octanoic acid: (1) in pure water, (2) in 0.01 M phosphate buffer at pH 7, and (3) in 0.01 M phosphate buffer at pH 7 with 1 M NaCl (experimental results; pipet immersion depth of h = 10 mm).

curve 2 represents the same salt concentration as curve 3 in Figure 5 but for a different immersion depth of the droplet. Thus, this phenomenon is most likely geometry-dependent. Obviously, the increase of the immersion depth causes merging of oscillations of a smaller period into a larger period. For a larger salt concentration, the merging occurs already at the immersion depth of h = 6 mm (curve 1 in Figure 9). It is noteworthy that, in the absence of salt at the immersion depth of h = 6 mm, the oscillatory regime is replaced by a quasi-stationary regime, as shown in Figure 10, where after the first oscillation, the surface tension decreases continuously because of quasi-stationary Marangoni convection. This quasistationary regime appears by either a decrease of the drop immersion depth below a certain threshold value, keeping the vessel radius constant, or an increase of the vessel radius, keeping the capillary immersion depth constant.13−15 The reason for such a transition from an oscillatory to a quasistationary regime is that the retarding force arising as a result of compression of the adsorbed monolayer near the wall and the change in buoyancy in the wall region become insufficient to suppress completely the Marangoni instability. In the presence of salt, this transition occurs at a smaller immersion depth than in pure water. The experimental study performed in ref 16 has shown that, indeed, in accordance with the results of numerical simulations, the increase in pH of the aqueous phase using phosphate buffer causes a considerable decrease of the oscillation amplitude. An

used, being on the order of 0.1 mN/m. Curve 1 in Figure 11 represents the oscillations in pure water having an essentially larger amplitude. The addition of salt to the buffer solution results in an increase of the oscillation amplitude in full accordance with the results of numerical simulations (curve 3 of Figure 11). It is seen that the addition of salt also causes a considerable decrease of the oscillation period. This can be the result of the increase of the density difference between the solvent and the solution of octanoic acid. The experiments were performed with 0.5 and 1 M NaCl. The average oscillation amplitude was smaller in the presence of 0.5 M NaCl (about 0.5 mN/m) than that in the presence of 1 M NaCl (about 2 mN/m). The effect of salt on the transition from the oscillatory to quasi-stationary regime was observed in experiments as well. In the presence of salt, the oscillatory regime was observed even at the immersion depth of h = 2 mm (Figure 12), whereas in the case of pure water, the transition to a quasi-stationary regime was observed at h = 4−6 mm.

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CONCLUSION Numerical simulations based on the first principles, namely, on the Navier−Stokes and convective diffusion equations with 6900

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(5) Kovalchuk, N. M.; Vollhardt, D. Marangoni instability and spontaneous non-linear oscillations produced at liquid interfaces by surfactant transfer. Adv. Colloid Interface Sci. 2006, 120, 1−31. (6) Arai, K.; Kusu, F. Electrical potential oscillations across a water− oil−water liquid membrane in the presence of drugs. In Liquid Interfaces in Chemical, Biological and Pharmaceutical Applications; Volkov, A. G., Ed.; Marcel Dekker, Inc.: New York, 2001; Surfactant Science Series, Vol. 95, pp 699−727. (7) Nakache, E.; Dupeyrat, M.; Vignes-Adler, M. Experimental and theoretical study of an interfacial instability at some oil−water interfaces involving a surface-active agent. 1. Physicochemical description and outlines for a theoretical approach. J. Colloid Interface Sci. 1983, 94, 187−200. (8) Yoshikawa, K.; Matsubara, Y. Chemoreception by an excitable liquid membraneCharacteristic effects of alcohols on the frequency of electrical oscillation. J. Am. Chem. Soc. 1984, 106, 4423−4427. (9) Shioi, A.; Sugiura, Y.; Nagaoka, R. Oscillation of interfacial tension at a liquid interface composed of de(2-ethylhexyl) phosphoric acid and calcium chloride. Langmuir 2000, 16, 8383−8389. (10) Pradines, V.; Tadmouri, R.; Lavabre, D.; Micheau, J. C.; Pimienta, V. Association, partition and surface activity in biphasic systems displaying relaxation oscillations. Langmuir 2007, 23, 11664− 11672. (11) Sczech, R.; Eckert, K.; Acker, M. Convective instability in a liquid−liquid system due to complexation with a crown ether. J. Phys. Chem. A 2008, 112, 7357−7364. (12) Kovalchuk, V. I.; Kamusewitz, H.; Vollhardt, D.; Kovalchuk, N. M. Auto-oscillation of surface tension. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 60, 2029−2036. (13) Kovalchuk, N. M.; Vollhardt, D. Theoretical description of repeated surface-tension auto-oscillations. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 66, 026302. (14) Kovalchuk, N. M.; Vollhardt, D. Effect of substance properties on the appearance and characteristics of repeated surface tension autooscillation driven by Marangoni force. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2004, 69, 016307. (15) Kovalchuk, N. M.; Vollhardt, D. Effect of buoyancy on appearance and characteristics of surface tension repeated autooscillations. J. Phys. Chem. B 2005, 109, 15037−15047. (16) Kovalchuk, N. M.; Vollhardt, D. Auto-oscillations of surface tension: effect of pH on fatty acid systems. Langmuir 2010, 26, 14624−14627. (17) Fainerman, V. B.; Lucassen-Reynders, E. H. Adsorption of single and mixed ionic surfactants at fluid interfaces. Adv. Colloid Interface Sci. 2002, 96, 295−323. (18) SurfactantsChemistry, Interfacial Properties, Applications. Studies in Interface Science; Fainerman, V. B., Möbius, D., Miller, R., Eds.; Elsevier: Amsterdam, The Netherlands, 2001. (19) Small, D. M. The Physical Chemistry of Lipids, Handbook of Lipid Research; Plenum Press: New York, 1986, Vol. 4. (20) Malysa, K.; Miller, R.; Lunkenheimer, K. Relationship between foam stability and surface elasticity forces: Fatty acid solutions. Colloids Surf. 1991, 53, 47−62. (21) Zemb, T.; Drifford, M.; Hayoun, M.; Jehanno, A. Light scattering of solutions of sodium octanoate micelles. J. Phys. Chem. 1983, 87, 4524−4528. (22) Setchenow, J. Z. Uber die konstitution der salzlosungen auf grund ihres verhaltens zu kohlensaure. Z. Phys. Chem. 1889, 4, 117− 125. (23) Ni, N.; Yalkowsky, S. H. Prediction of Setchenow constants. Int. J. Pharm. 2003, 254, 167−172. (24) Roache, P. J. Computational Fluid Dynamics; Hermosa: Albuquerque, NM, 1976. (25) Kovalchuk, N. M.; Vollhardt, D. Autooscillations of surface tension in water−alcohol systems. J. Phys. Chem. B 2000, 104, 7987− 7992.

Figure 12. Oscillations of surface tension produced by dissolution of a droplet of octanoic acid in 0.01 M phosphate buffer at pH 7 with 1 M NaCl (experimental results; pipet immersion depth of h = 2 mm).

appropriate boundary conditions, have shown that there is a simple but efficient way, namely, the change of ionic strength and the pH, to control the oscillations caused by adsorption of a surfactant transferred from the source situated in the bulk of solution, for instance, from a droplet, to the free liquid interface. This method is applicable to surfactants being a weak base or acid, such as fatty acids. An increase in pH and correspondingly in the degree of dissociation of fatty acid causes a decrease of the oscillation amplitude. In the presence of buffer at pH 7, when practically all acid is in the less surface active dissociated form, the oscillations become undetectable. The addition of salt increases the activity of the dissociated acid, and the system reveals the oscillation with large amplitude again. The presence of salt shifts the transition from an oscillatory to a quasi-stationary regime at smaller immersion depths of the droplet. Note that the ionic strength influences the dynamics of the system with any ionic surfactant. The results of numerical simulation are in good agreement with the experimental data obtained for octanoic acid.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work was supported by a project from the DFG SPP 1506 (Mi418/18-1) and joined project BMBF and the Ukrainian Ministry of Education and Science (UKR 10/039). We acknowledge CALMIP Toulouse for the presented time on cluster computer (Project P0908).

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REFERENCES

(1) Colinet, P.; Legros, J. C.; Velarde, M. G. Nonlinear Dynamics of Surface-Tension-Driven Instabilities; Wiley-VCH: Berlin, Germany, 2001. (2) Convective Transport and Instability Phenomena; Zieper, J., Oertel, H., Eds.; Braun-Verlag: Karlsruhe, Germany, 1982. (3) Joos, P. Dynamic Surface Phenomena; VSP: Dordrecht, The Netherlands, 1999. (4) Interfacial rheology. In Progress in Colloid and Interface Science; Miller, R., Liggieri, L., Eds.; Brill: Leiden, The Netherlands, 2009; Vol. 1. 6901

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