Stratification of a Foam Film Formed from a Nonionic Micellar Solution

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Stratification of a Foam Film Formed from a Nonionic Micellar Solution: Experiments and Modeling Jongju Lee, Alex Nikolov, and Darsh Wasan* Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, Illinois 60616, United States S Supporting Information *

ABSTRACT: Thin liquid films containing surfactant micelles or other nanocolloidal particles are considered to be the key structural elements of foams containing gas and liquid. We report here the experimental results and theoretical modeling for the phenomenon of the stratification (stepwise thinning) of a foam film formed from a nonionic micellar solution. The film stratification phenomenon was experimentally observed by reflected light microinterferometry. We observed that the stepwise layer-by-layer decrease of the film thickness is due to the appearance and growth of a dark spot of one layer less than the film thickness in the film. The dark spot expansion is driven by the diffusion of the dislocation (or vacancy) in the micellar lattice. The vacancies from the meniscus diffuse and condense into the dark spot, leading to its expansion inside the film. We experimentally observed the expansion of the dark spot at various film thicknesses (i.e., the number of micellar layers) and at different film sizes. We also measured the contact angle between the film and the meniscus; we used the data to estimate the structural film interaction energy barrier and the apparent diffusion coefficient. We used the two-dimensional diffusion model to model the dynamics of the dark spot expansion with consideration to the apparent diffusion coefficient and the film size. The model predictions are in good agreement with the experimental observations. On the basis of this model, we carried out a parametric study depicting the effects of the film thickness (or the number of micellar layers) and film area on the rate of the dark spot expansion. We also generalized the model previously proposed by Kralchevsky et al. [Langmuir 1990, 6, 1180−1189], incorporating the effects of the film size, film thickness, and apparent diffusion coefficient to predict the dark spot expansion rate.

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inside the film when the thickness of the foam lamella becomes less than 250−300 nm. The thinner area expands and covers the entire film, which indicates a homogeneous film thickness. After this occurs, the thinner circular area is formed again and expands in the same manner. He suggested the concept of a mica-like sheet-type structure to explain stepwise film thinning. Later, stepwise film thinning (stratification) was discussed by a number of authors with different systems (micelles,5−9 latex particles,10−12 silica particles,13,14 and globular proteins15−17). Nikolov et al.5 investigated the stratification for thin liquid films formed from anionic surfactant micellar solutions of sodium dodecyl sulfate (SDS). They clearly elucidated that the stratification phenomenon can be explained as a layer-by-layer thinning of the ordered structures of micelles or colloidal particles formed inside the film by using reflected light microinterferometry. Moreover, the film changes its thickness with regular stepwise jump transitions and the regular height of each step corresponds to the effective diameter of the micelle (or nanoparticle). Theoretical and/or hydrodynamic models have been proposed in order to obtain a comprehensive understanding of film stratification. Kralchevsky et al.18 suggested that the

INTRODUCTION Foams have a variety of practical applications in industry and in our daily lives. Liquid foams are dispersions of gas in the liquid phase and consist of distorted polyhedral bubbles, Gibbs− Plateau borders, and foam channels.1 Since foams are separated by thin liquid films (lamellae), the properties of foams are affected by the properties of these films. However, the behaviors (e.g., stability) of foams are more complex than those of a single foam film because the area of the foam lamella inside the foam expands due to the drainage of the liquid, resulting in the polydispersity in the foam lamella size; the curvature of the Plateau borders also changes with the capillary pressure.2 For practical purposes, many industries are interested in foams formed from surfactant solutions that have a concentration several times above the critical micellar concentration (CMC). At CMC, it has been observed that the micelles are confined inside the intervening film between the bubbles, indicating that the film is thinning in a stepwise manner (i.e., stratification). The stratification phenomenon occurs when the films (a foam film or emulsion film) contain submicron-sized spherical colloidal particles or micelles at a sufficiently high concentration. In the early 20th century, Johonnott3 and Perrin4 studied the thinning of liquid foam films formed from a surfactant solution at concentrations above CMC. Perrin4 reported that the thinner circular area appears © XXXX American Chemical Society

Received: February 16, 2016 Revised: April 8, 2016

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the domain growth dynamics in stratifying foam films is uncertain. Most recently, Zhang and Sharma23 studied the kinetics of stratification and domain expansion dynamics in thin films formed from anionic surfactant (SDS) solutions. They found that the dynamics of the domain’s expansion changes after the domain makes contact with the meniscus region. An isolated domain shows its radius is proportional to the square root of time, which represents the vacancies diffusion process originally proposed by Kralchevsky et al.18 When the domain coalesces with the meniscus (the surrounding thicker region), the radius of the domain grows linearly with time. However, there is a lack of a theoretical model to predict the dynamics of domain expansion and the change of its kinetics from a constant diffusivity regime to a constant velocity regime. We believe that a reliable model that explains the dark spot formation and its expansion is the vacancies condensation mechanism proposed by Kralchevsky et al.18 According to the vacancies condensation mechanism, the foam film stratification is governed by the self-diffusion of the micelles and/or vacancies. It is worth noting that the vacancy (hole) theory of a liquid has been studied to predict a liquid’s properties, such as its viscosity, diffusion coefficient, and viscoelasticity.24−27 Glasstone et al.28 suggested that any liquid or solid has holes produced from thermal fluctuations. The liquid may be regarded as made up of holes moving about in matter, and the holes are considered to be playing the same role in a liquid. In order to form holes (vacancies), the liquid molecules expend the energy by work done in pushing back other molecules. The jump of the moving molecule from one equilibrium position to the next to produce a hole requires the passage of the system over a potential energy barrier. Frenkel29 studied the hole theory of the liquid state and applied it to estimate the diffusion and viscosity of liquids. The appearance and disappearance of holes are the result of the fluctuations that are related to the heat motion of the liquid. The holes can move from one place to another by closing at some places and opening at neighboring locations when the kinetic energy from the interaction is equal to or larger than the potential energy barrier. Similar ideas have been set forth by Baker et al.,30 who studied the diffusion of vacancies in a one-dimensional colloidal crystal to explain the stratification mechanism of thin liquid films. Because of the nonzero temperatures, a particle’s position is controlled by diffusion. If the particle moves by the diffusion process, the prorogation of the vacancy is viewed in the opposite direction. Later, Chu et al.31 elucidated that the particle density in the direction normal to the film surface increases exponentially as its gets close to the film surfaces by using Monte Carlo numerical simulations. They showed that the thick film containing the middle micellar layers has a large number of vacancies produced from the middle layers with a relatively low particle density. The particles (or micelles) tend to form ordered structures inside the layers as the film thickness (i.e., the number of micellar layers) decreases, thereby reducing the number of vacancies. In the present study, we monitored the expansion of the dark spots inside the film formed from a nonionic micellar solution using reflected light microinterferometry. We did not observe the droplets surrounding the dark spot (the Rayleigh type of instability) for the film formed from the nonionic surfactant solution since nonionic surfactants possess no net charge.32 Moreover, nonionic surfactant micelles are similar to hard spheres, which are more suitable for the modeling system than

formation and expansion of the dark spot corresponding to the circular area inside the film with a smaller thickness than its periphery is governed by micellar self-diffusion, which they referred to as the “vacancies condensation mechanism”. According to the vacancies condensation mechanism, the driving force of the layer-by-layer thinning of the film is the chemical potential gradient between the film and the meniscus. The gradient of the chemical potential causes the micelles to diffuse from the film to the meniscus, producing the vacancies as the film thins. When the concentration of the vacancies is sufficient to nucleate, dark spots (i.e., condensed vacancies) appear inside the film. These dark spots have a smaller film thickness than the remaining part of the film, which reflects less light than that found in the periphery. The osmotic pressure of a dark spot is less than that of its periphery, and so it depletes the fluid of the film in the dark spots, which triggers the decreasing film thickness by the effective diameter of micelle, leading to stepwise film thinning. Later, Bergeron et al.19 proposed a hydrodynamic model for the stratification of foam films containing micelles in terms of the disjoining pressure gradient. The main driving force of the dark spot expansion is said to be the equilibrium disjoining pressure that exhibits an unstable thickness region where the disjoining pressure isotherm has a positive slope. The outward fluid flow with the radial pressure gradients by the curvature variation also triggers hole (dark spot) expansion. The dynamics of the dark spot formation and its expansion was predicted by nonlinear hydrodynamic stability analysis with consideration to the oscillatory disjoining pressure. However, their model is valid only for the formation of hole and for a short time of expansion when the foam film thins under constant capillary pressure. It cannot explain all of the dynamics of foam film stratification. The model also does not account for the effect of the film size on the film thinning and the layering structure of the particles (micelles) inside the film. According to Beltrán et al.,20 the dynamics of domain (dark spot) expansion is determined by the film composition. Depending on the film composition, the surfactant charge and concentration affect the local viscosity or structural forces, and these are related to the stratification kinetics. Later, Heinig and Langevin21 applied a hydrodynamic theory to explain the domain’s shape relaxation in terms of the film viscosity. They found that the film viscosity is at least 30 times larger than the bulk viscosity based on the proposed dissipation mechanism. The film is formed by a mixture of an anionic polymer and a cationic surfactant. Unfortunately, the 2-D dissipation mechanism cannot be applied when the domain’s coalescence is governed by the viscoelastic properties of the film or the Saffman length is too close to the length scale of the observed structures. Heinig et al.22 proposed the local-diffusive mechanism to study the expansion dynamics of domains in stratifying foam films without the appearance of droplets near the boundary of the domain. The film tension gradient between the film and the dark spot (domain) controls the material transport, resulting in the domain’s expansion. Using their model, they found that the film viscosity is 60 times larger than that of the bulk viscosity in the case of the film formed by a polyelectrolyte/surfactant mixture. However, the estimated contact angle based on their model is not consistent with the experimental measurements using the Newton-fringe method. Since the prediction of the domain’s growth dynamics is based on the disjoining pressure isotherm (which is related to the contact angle), the validity of the proposed model to explain B

DOI: 10.1021/acs.langmuir.6b00561 Langmuir XXXX, XXX, XXX−XXX

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Figure 1. (a) Photograph depicting the film and meniscus from the top view. (b) Sketch of the film and meniscus from the side view. (c) Meniscus profile with its light intensity profile for the film formed from the 0.052 M Enordet AE 1415-30 surfactant solution. monochromatic light (with a wavelength of 546 nm) through the top of the glass cell was incident on the film surface. The Mach−Zehnder interferometer was combined with an optical microscope; this allowed for the thickness of the foam film changes to produce interference patterns. The interference patterns were used to determine the film’s thickness and meniscus contact angle. A charge coupled device (CCD) camera, in conjunction with a monitor and digital video recorder, was used to record the dynamics of the film’s stratification. Film Thickness and Contact Angle Measurements. Reflected light interferometry showed that the foam film was surrounded by the meniscus, which had consecutive dark and bright Newton interference rings around the periphery of the film. The meniscus profile and the contact angle between the film and the meniscus were obtained from the interference rings (patterns) of the film at the equilibrium state. Figure 1a is a photograph depicting the film and film meniscus from the top view, and Figure 1b is a sketch of the film and meniscus from the side view. Figure 1c shows the light intensity profile of the meniscus region, which represents the thickness change by the successive maxima and minima of the reflected light intensities. The meniscus profile is also shown in Figure 1c, which was obtained using the procedure described in ref 14. In order to determine the film thickness (h), we fitted the data with a second-order polynomial. The value of the film thickness was obtained by the extrapolation of the fitted curve. The contact angle, θeq, was the angle subtended between the film and the meniscus, which was determined from the slope of the intersection point of the meniscus profile with the film. The accuracy of the contact angle determination was ±0.1°. The values for the film thicknesses and the contact angles of the films formed from the 0.052 M Enordet AE 1415-30 micellar solution are tabulated in Table 1.

anionic surfactant micelles containing a double layer (Debye− Huckel length). During the film thinning, we observed that there exists a critical film size at which the film remains with several micellar layers at the equilibrium state and stops the next film thickness transition. This phenomenon only can be interpreted in terms of the vacancies’ diffusion. Thus, we considered the effect of the film size on the stratified foam film based on the vacancies condensation mechanism.18 Our present study presents two models to predict the rate of the dark spot expansion inside the film formed from the nonionic micellar solution. First, we adopted the moving boundary problem of the two-dimensional diffusion equation with cylindrical coordinates to study the dynamics of the dark spot expansion. The interface between the dark spot and the film was regarded as a moving boundary. Because of the discrete structure of the foam film containing micelles, the apparent diffusion coefficient and the film volume are considered in the proposed model. We also propose a model generalizing the work of Kralchevsky et al.18 The predicted rates for the dark spot expansion for different micellar layers using both proposed models are compared with experimental observations.

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EXPERIMENTAL SECTION

Materials and Preparation of Solutions. A commercial nonionic surfactant, Shell’s ethoxylated alcohol, was used; Enordet AE 1415-30 contains 30 ethoxy groups per surfactant molecule with 14- and 15-carbon atom straight paraffinic chains. The concentration of the surfactant solution is 0.052 M, which is about 1200 times above the critical micellar concentration (CMC). The surfactant solutions were prepared by dissolving the Enordet AE 1415-30 surfactant in deionized water using a “Milli-Q” water system deionizer from Millipore Corporation. Monitoring of the Stratification Phenomena in a Foam Film. The dynamics of the foam film stratification (stepwise thinning) that occurred with the 0.052 M Enordet AE 1415-30 micellar solution was monitored using reflected light microinterferometry. Microscopic horizontal films were formed in a glass cell combined with a thickwalled cylindrical capillary; the hydrophilic inner wall had a radius of 1.35 mm. The horizontal microscopic films were formed by suctioning out the liquid from a biconcave drop inside the capillary through an orifice in the wall.5,6,14 An optical bench was equipped with common interferometry to monitor the dynamics of the film thinning as a function of time. In the reflected light mode of the microscope, the

Table 1. Values of the Film Thickness (hn), Contact Angle (θeq,n), Dimensionless Structural Energy Barrier (ΔWnd2/ kT), and Apparent Diffusion Coefficient (Dn) for the 0.052 M Enordet AE 1415-30 Surfactant Solution

n 1 2 3 4 5 C

film thickness, hn (nm) 23.4 33.1 43.0 52.4

± ± ± ±

2 2 2 2

contact angle, θeq,n (deg)

dimensionless structural energy barrier, ΔWnd2/kT

apparent diffusion coefficient, Dn × 108 (cm2/s)

± ± ± ±

2.17 0.80 0.29 0.11 0.04

0.9 3.5 5.9 7.1 7.6

0.92 0.67 0.56 0.42

0.1 0.1 0.1 0.1

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Figure 2. Evolution of film thinning for the 0.052 M Enordet AE 1415-30 solution (the film is undergoing the stratification thickness transitions).

ness.34 When the micellar concentration and film thickness are fixed, the concentration of the vacancies is proportional to the film size. The total amount of the vacancies inside the film is insufficient to condense into a dark spot when the film size is small enough (smaller than the critical film size). As a result, a dark spot cannot form and the film thickness transition stops. In order to make the film move into the fourth film thickness transition, the radius of the film formed from the Enordet AE 1415-30 micellar solution was gradually increased from 110 to 350 μm. As a result, a dark spot containing one micellar layer appears inside the film containing two micellar layers (i.e., the fourth film thickness transition initiates). Since the film size increases, more vacancies are supplied from the meniscus region. When the film reaches a sufficient vacancies concentration, a dark spot is formed and expands. After the dark spot containing one micellar layer covers the whole film area, we reduced the film radius from 350 to 110 μm to measure the film thickness and the contact angle at the same size. It is worth noting that the capillary pressure increases when the film size increases. Thus, the stepwise film thinning is affected by both the film size and the capillary pressure simultaneously. Although the capillary pressure affects the film thinning process, the stepwise film thinning (stratification) depends more on the film size than on the capillary pressure.34 If the film size decrease is smaller than the critical film size, the dark spot begins to shrink and vacancies from the spot diffuse into the film area to restore the equilibrium between the condensed vacancies in the dark spot and vacancies inside the film.35 When the 0.052 M Enordet AE 1415-30 film radius decreased from 350 to 110 μm, the vacancies practically stopped diffusing from the film meniscus to the film. However, the reverse process was not observed in this case, since the film size is not small enough to drive the reverse process. Here, we verified that a critical film size exists in the case of the film formed from the 0.052 M Enordet AE 1415-30 surfactant solution; if the film size is smaller than the critical film size, the next thickness transition never occurs due to the insufficient vacancies inside the film to form a dark spot. The vacancies concentration inside the film controls the formation of the dark spots in stratifying foam films. The effective micellar volume concentration was estimated to be 40 vol % based on the number of film thickness transitions.36 The effective micellar diameter was 9.7 nm for the 0.052 M Enordet AE 1415-30 micellar solution, which corresponds to the mean amplitude of the stepwise film thickness transitions. The effective diameter of the micelle was confirmed by dynamic

Micellar Size and Distribution Measurements. The Enordet AE 1415-30 surfactant solutions were characterized by the dynamic light scattering (DLS) method. DLS measurements were carried out using a ZETASIZER 3000HSA (Malvern Instruments, Worcestershire, UK), which uses a helium−neon laser and integrated analysis software. The temperature was adjusted to 25 °C, and the scattering angle was set to 90° before the measurements were taken. The data were expressed as the z-average (the hydrodynamic diameter of the particle) and polydispersity index (the width of the size distribution).

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RESULTS AND DISCUSSION Film Stratification (Stepwise Thinning) Phenomena. Using reflected light microinterferometry, we monitored the drainage characteristics of the microscopic horizontal foam films formed from the Enordet AE 1415-30 (Shell Co.) micellar solution at a concentration of 0.052 M. The evolution of film thinning is depicted in the movie clip in the Supporting Information. After the film is formed, it immediately starts to decrease its thickness. When the film thickness becomes less than 102 nm (corresponding to the last light intensity maximum), the film thins in a stepwise manner (stratification). The film rests for a few seconds; it is homogeneous in color, indicating the uniform film thickness. When the film reaches a sufficient concentration of vacancies, the vacancies tend to condense and a dark dot is formed inside the film. The dark dot contains one micellar layer less than its periphery inside the film. Therefore, it reflects less incident light than that in the remaining part of the film. It moves randomly in the film area and slowly expands in size as the vacancies condense. The expanding dark dot becomes a dark spot. Once the dark spot is formed, it radially expands and covers the film area. When the dark spot covers the entire area of the film, the film rests and is a homogeneous color (thickness) for several seconds. In the same manner, the film thickness deceases in regular steps with the formation and expansion of the dark spot. Figure 2 shows the evolution of film thinning for the 0.052 M Enordet AE 1415-30 micellar solution. The first thickness transition occurs around 40 s after the film formed. The second and third film thickness transitions follow. After the third film thickness transition, the film remains stable at a thickness of ∼33 nm with two micellar layers inside it. According to Nikolov and Wasan,33 a critical film size exists in which the film contains several micellar layers at the equilibrium state, and the next film thickness transition does not occur. The concentration of the vacancies inside the film depends on its size, micellar concentration, and film thickD

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spot and the diffusion coefficient are Pd and Dd, respectively. The number of vacancies per unit area in the film region and the diffusion coefficient are denoted by Pf and Df, respectively, in the region of r > R(t). Here, R(t) is the interface between the dark spot and the film (i.e., the moving boundary). The moving boundary problems have been studied by a number of authors.37−42 In cylindrical coordinates, the solution of the two-dimensional diffusion equation is given by the exponential integral, −Ei(−r2/4Dft).43 At the moving boundary (r = R(t)), the number of vacancies per unit area for the dark spot and that for the film (Pd and Pf) have to be conserved, so

light scattering measurements. The size of the Enordet AE 1415-30 micelle was 9.8 nm (determined from the dynamic light scattering (DLS) measurements).

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MODELING In order to get a better understanding of the stratifying foam film formed from the micellar solution, the dynamics of the dark spot expansion was modeled by considering the micellar layer structure as well as the film size (area). Our proposed models only take into account the dynamics of the dark spot expansion before the dark spot coalesced with the meniscus. And, we assumed the dark spot area is much smaller than the film area. The sketch in Figure 3 illustrates the film with a circular dark spot (the area of one micellar layer less than that of the

Pd = Pf = ρd ,

at r = R(t )

(2)

Since the vacancies are supplied from the film region to the expanding dark spot (−Df(∂Pf/∂r)), the moving boundary (r = R(t)) advances in the opposite direction of the influx vacancies (−ρd(dR/dt)). Therefore Df

∂Pf dR = ρd , ∂r dt

at r = R(t )

(3)

The additional boundary conditions are the following: Pd(0 < r ≤ R(t )) = ρd

(4)

Pf (r → ∞) = ρf

(5)

Equation 4 indicates that the number of vacancies per unit area in the dark spot is constant as ρd. Equation 5 represents the number of vacancies per unit area in the film, ρf, at large distances. The following solutions were obtained by the Neumann method, which is described elsewhere.38 From the formal solution of the governing equation (i.e., −Ei(−r2/4Dft)) with the boundary conditions (eqs 4 and 5), the solutions of eq 1 are Pd = ρd

⎛ r2 ⎞ Pf = ρf − C Ei⎜ − ⎟ ⎝ 4Df t ⎠

Figure 3. Sketch of the foam film with a dark spot’s (a) top view and (b) side view. R(t) is the moving boundary between the dark spot and film, P(r,t) is the number of vacancies per unit area of film (or dark spot), D is the diffusion coefficient, subscript “d” represents the dark spot, and subscript “f” represents the film.

(6)

for (r ≥ R ) (7)

where C is a constant. Then, eq 2 becomes ⎛ R2 ⎞ ρd = ρf − C Ei⎜ − ⎟ ⎝ 4Df t ⎠

surrounding film) expanding at the center of film. In the case of a thicker film (e.g., 4 and 5 micellar layers), several dark spots appear simultaneously, resulting from the relatively high vacancies concentration inside the film since the vacancies concentration is proportional to the film thickness. However, the total area of several dark spots is regarded as the area of a single circular dark spot for the purpose of modeling. Governing Equations. We adopted the two-dimensional Fickian diffusion equation of cylindrical coordinates for the thin liquid film because the film thickness is much smaller than the film size (area):37 ⎛ ∂ 2P ∂P 1 ∂P ⎞ = D⎜ 2 + ⎟ ∂t r ∂r ⎠ ⎝ ∂r

for (0 < r ≤ R )

(8)

2

For all of the values of t, R should be proportional to t:

R(t )2 = 4λ 2Df t

(9)

where λ is a numerical constant. Substituting eqs 8 and 9 into eq 7, we obtain 2

Pf = ρf −

⎛ r2 ⎞ Ei⎜ − ⎟ Ei( −λ ) ⎝ 4Df t ⎠ ρf − ρd 2

for (r ≥ R ) (10)

Scaling Up the Model. Equation 9 predicts that a circular dark spot area (R2) increases linearly with time, which agrees with our experimental measurements. Equation 9 does not consider the role of the film thickness in the dynamics of the dark spot expansion and is thereby valid only for a single layer. Since the film formed from the micellar solution contains multiple micellar layers depending on the micellar volume fraction of the solution, eq 9 needs to consider the discrete structure (e.g., micellar layering) of the foam film formed from micellar solutions. When the film thickness is less than ∼100

(1)

where r is the radial distance from the center of the film, t is the time from the initial formation of the dark spot, D is the diffusion coefficient, and P(r,t) is the number of vacancies per unit area of film or dark spot depending on r. In the region of 0 < r < R(t), the number of vacancies per unit area in the dark E

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Langmuir nm, the micelles self-organize in the confined film region during film thinning.5 Because of the layered structure of micelles, the film viscosity (μf) increases and the apparent diffusion coefficient (Df, Dn) decreases with the decreasing film thickness. Moreover, the vacancies supply depends on the film volume, which takes into account the effects of both the film thickness and film area. From the experimental measurements of the rate of the dark spot expansion, the rate of the dark spot expansion increases as the film thickness increases for a fixed film size (3, 4, and 5 micellar layers formed from the 0.052 M Enordet AE 1415-30 solution with a film radius (rf) of 110 μm). The rate of the dark spot expansion also increases when the film size (or radius) increases from 110 to 350 μm, as in the case of the two micellar layers formed from the 0.052 M Enordet AE 1415-30 solution. Therefore, we took into consideration the effects of the micellar layering, film thickness, and film size on the dynamics of the dark spot expansion in terms of the apparent diffusion coefficient (Dn) as well as the film volume (Vn). Therefore, the following relation is proposed: ⎛ D ⎞⎛ V ⎞ R(t )n 2 = 4λ 2(Df )⎜ n ⎟⎜ n ⎟t ⎝ Df ⎠⎝ Vf ⎠

Figure 4. Oscillatory structural film interaction energy isotherm as a function of the film thickness for the 0.052 M Enordet AE 1415-30 micellar solution.

(11)

2

where t is the time, R(t)n is the squared dark spot radius, Dn is the apparent diffusion coefficient, and Vn is the film volume for the “nth” micellar layer. Df and Vf are the apparent diffusion coefficient and the film volume for the reference micellar layer, respectively. λ2 is a constant obtained from the experimental measurements of the squared dark spot radius versus time with the apparent diffusion coefficient. The apparent diffusion coefficient (Df, Dn) is calculated by the structural film interaction energy barrier (discussed below). The film volume (Vf, Vn) is the product of the measured film thickness and film area. Estimation of the Apparent Diffusion Coefficient. In order to obtain the apparent diffusion coefficient, the structural film interaction energy isotherm was employed. The following relation was used in order to calculate the structural film interaction energy isotherm:14,44 ⎛ 2πh ⎞ ⎛ h⎞ ⎟ exp⎜ − ⎟ = 2σ (cos θ − 1) A cos⎜ a/l eq ⎝ d ⎠ ⎝ d⎠

minimum. The values for the dimensionless structural energy barriers (ΔWd2/kT) corresponding to the respective micellar layers are tabulated in Table 1. It is worth noting that the structural energy barrier increases when the film thickness decreases. Since the contribution of the structural film interaction energy with micellar layering to the film thinning is more pronounced as the film thickness decreases, the stepwise thinning transitions occur within a longer period of time than those transitions that occur in the thicker micellar layers.45 The dimensionless structural energy barriers are used first to calculate the film viscosity and then the apparent diffusion coefficient for each micellar layer. On the basis of the vacancy theory,24−27,29 Frenkel proposed a relation to estimate the viscosity of liquids in terms of the energy barrier. According to the Frenkel,29 the vacancies can move from one place to another by closing at some places and opening at neighboring locations when the kinetic energy from the interaction is equal to or larger than the potential energy barrier. In the case of foam film containing micelles, micelles tend to form ordered structures inside the layers as the film thickness decreases. Therefore, the film viscosity should depend on the micellar layering phenomenon in terms of the film interaction energy barrier. Therefore, the film viscosity (μf) was obtained from the equation

(12)

where A is the amplitude of the oscillation, θeq is the filmmeniscus contact angle, h is the film thickness, and d is the effective diameter of the micelle corresponding to the mean step size of the film thickness transitions. σa/l is the surface tension, which was determined with the drop shape analysis method, and the apex radius of the curvature of the drop was determined by fitting the drop profile with the Laplace equation (σa/l = 46.9 mN/m). Parameter A was determined by choosing the film thickness and film-meniscus contact angle corresponding to one micellar layer. The film interaction energy per unit area (W) is given by14 ⎛ 2πh ⎞ ⎛ h⎞ ⎟ exp⎜ − ⎟ W = A cos⎜ ⎝ d ⎠ ⎝ d⎠

⎛ ΔWd 2 ⎞ μf = B exp⎜ ⎟ ⎝ kT ⎠

(14)

where B is a constant coefficient, ΔWd 2 /kT is the dimensionless film interaction energy barrier, W is the film interaction energy per unit area, k is the Boltzmann constant, T is the temperature, and d is the effective diameter of the micelle. The pre-exponential constant B refers to the bulk viscosity (μb) when the energy barrier is small (B corresponds with μb as ΔWd2/kT → 0). The bulk viscosity for the 0.052 M Enordet AE 1415-30 solution was determined (5.7 cP) based on its effective micellar volume fraction (ϕ = 0.4).46 Substituting B into the corresponding bulk viscosity (μb) in eq 14 leads to the expression of the film viscosity:

(13)

The oscillatory structural interaction film energy isotherm for the 0.052 M Enordet AE 1415-30 micellar solution is presented in Figure 4. The film interaction energy per unit area (W) was nondimensionalized using the effective diameter of the micelle (d) and the kinetic energy of the micelles (kT). The structural energy barriers were obtained for each corresponding micellar layer (or film thickness) from one maximum to the next F

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Figure 5. Squared dark spot radius versus time for the film formed from the 0.052 M Enordet AE 1415-30 micellar solution. The film with 3 micellar layers was selected to obtain the experimental value of λ2, and then the rates of the dark spot expansion for the film with 2, 4, and 5 micellar layers were predicted. (Experimental data are marked with symbols, and the solid lines are the model predictions from eq 18.)

⎛ ΔWd 2 ⎞ μf = 5.7 exp⎜ ⎟ ⎝ kT ⎠

were used to calculate the rate of the dark spot expansion using the proposed model (eq 11). Effect of the Film Thickness on the Dynamics of the Dark Spot Expansion. The vacancy supply depends on the film thickness. For a fixed film size at a radius of 110 μm (in the cases of films with 3, 4, and 5 micellar layers), the rate of the dark spot expansion increases with the film thickness since the vacancies supply (i.e., total influx of vacancies) increases as the film becomes thicker. When the film thickness is reduced, the rate of the dark spot expansion decreases because the micelles form a better layered structure inside the film. Once the micelles form a layered structure in the confined region of the film, the film viscosity increases and the apparent diffusion coefficient decreases. As a result, the rate of the vacancies condensation into the dark spot (or the rate of the dark spot expansion) becomes slower. For example, in the case of the films formed from the 0.052 M Enordet AE 1415-30 solution, we chose the film with 3 micellar layers as a reference in order to predict the rates of the dark spot expansion for films with 4 and 5 micellar layers. The apparent diffusion coefficient (D3) and the film volume (V3) for 3 micellar layers were substituted in for the apparent diffusion coefficient (Df) and the film volume (Vf) in eq 11. From the experimental measurements of the squared dark spot radius versus time for 3 micellar layers (shown in Figure 5) with the corresponding apparent diffusion coefficient (D3), the experimental value of λ2 was obtained using eq 11 (λ2 = 8.1 ± 0.5). Therefore, eq 11 becomes

(15)

In the case of the film containing one micellar layer, the film viscosity is 49.8 cP (based on eq 15), which is 8.7 times larger than the bulk viscosity. Since the micelles tend to form a layered structure inside the confined surfaces of the film as the film becomes thinner, the film viscosity increases, resulting in an increasing drainage time for the liquid film. And then, we used the relation between the viscosity and diffusion coefficient (D) which is the Stokes−Einstein equation:47,48 D=

kT 3πμd

(16)

where k is the Boltzmann constant, T is the temperature, μ is the viscosity, and d is the effective diameter of the micelle. However, eq 16 is only valid for the dilute system (i.e., the particle moves independently of all the other particles). We adopted the Stokes−Einstein equation taking into consideration the film with different micellar structure for each micellar layer. Thus, the viscosity (μ) term in eq 16 was substituted with the film viscosity (μf), and we have Dn =

kT 3πμf d

(17)

Using the eq 17 which considers the micellar layering inside the film in terms of the film interaction energy barrier, the apparent diffusion coefficients (Dn) were obtained for the film with different number of micellar layers. We found that the apparent diffusion coefficient decreases as the film becomes thinner (see Table 1). It is also related to the micellar structuring inside of the film. When the film thickness decreases, the micelles form a better micellar structure,5 thereby taking a longer time for the micelles to diffuse from the film to the meniscus. The apparent diffusion coefficients versus the film thickness (micellar layer)

⎛V ⎞ R(t )n 2 = 32.38Dn⎜ n ⎟t ⎝ V3 ⎠

(18)

This equation states that the rate of the dark spot expansion corresponding to the “nth” micellar layer can be predicted with the combination of the apparent diffusion coefficient (Dn) and film volume (Vn). Figure 5 shows the experimental measurements of the squared dark spot radius and the model predictions from eq 18 in the films with 3, 4, and 5 micellar G

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Langmuir layers made with the 0.052 M Enordet AE 1415-30 solution when we set the reference for the films as the film with 3 micellar layers. From the experimental measurements and model predictions, we found that the rate of the dark spot expansion is constant with time for certain film thicknesses and film sizes, indicating that the total influx of vacancies is constant. Moreover, the rate of the dark spot expansion increases with the increased film thickness for the fixed film size. Effect of the Film Size (Area) on the Dynamics of the Dark Spot Expansion. Nikolov and Wasan33 reported that the rate of the dark spot expansion at a given micellar concentration and film thickness depends on the film size. When the film size increases, the stepwise thickness transition occurs more rapidly due to the high vacancies concentration in the film. We observed the effect of the film size on the rate of the dark spot expansion inside the film formed from the 0.052 M Enordet AE 1415-30 solution. During the stepwise thinning of the film, the film remains with two micellar layers and the film radius (rf) increases from 110 to 350 μm to make the film containing 2 micellar layers initiate the next film thickness transitions. Since the increased film area is able to supply more vacancies, the film containing 2 micellar layers reaches the sufficient vacancies concentration to form a dark spot. We measured the squared dark spot radius as a function of time for 2 micellar layers (see Figure 5). It was observed that the rate of the dark spot expansion for the film containing 2 micellar layers is faster than that of the film containing thicker micellar layers (3, 4, and 5 micellar layers) despite the higher film viscosity and lower apparent diffusion coefficient than those of the thicker layers. This is because the film area for a film with 2 micellar layers is 8.5 times larger than that of films with 3, 4, and 5 micellar layers, thereby increasing the rate of the dark spot expansion. It indicates that the film area also plays an important role in controlling the dynamics of the dark spot expansion. The rate of the dark spot expansion for the film with 2 micellar layers was predicted using eq 18, as presented in Figure 5. Since eq 18 takes into consideration the film volume (i.e., the film thickness and the film area) with respect to the vacancies supply, the proposed model eq 18 is able to predict the rate of the dark spot expansion when the film size (area) and/or the film thickness is varied. Parametric Study. In eq 18, the rate of the dark spot expansion (dR2/dt) is related to the film volume (Vn) and the apparent diffusion coefficient (Dn). The film volume (Vn) is the product of the film thickness (h) and the film area, i.e., the squared film radius (rf2). Since the apparent diffusion coefficient (Dn) depends on the number of micellar layers (n), the apparent diffusion coefficients with the corresponding film thicknesses of the micellar layers were used to perform the calculations. Figure 6a shows results of our parametric study depicting the effective film size on the rate of the dark spot expansion. The film radius was varied from 50 to 350 μm. The larger film size (or radius) resulted in a faster rate for the dark spot expansion. For a fixed film thickness (or micellar layer), the rate of the dark spot expansion increased with the increasing film radius. This indicates that the film area (rf2) is directly proportional to the vacancy supply and thereby the rate of the dark spot expansion (rf2 ∝ dR2/dt). The variation in the rate of the dark spot expansion as a function of the film thickness (or micellar layer) with varying film radii is presented in Figure 6b. The rate

Figure 6. (a) Rate of the dark spot expansion as a function of the squared film radius with various thicknesses (or micellar layers). (b) Rate of the dark spot expansion as a function of the film thickness (or micellar layer) with various film radii.

of the dark spot expansion increases as the film thickness (micellar layers) increases for a fixed film radius. When the film thickness (or micellar layer) increases, the apparent diffusion coefficient also increases (see Table 1). As a result, the vacancies diffusion occurs more rapidly at the thicker micellar layer. In addition, the vacancy supply is proportional to the film thickness. The increased vacancy supply causes the increasing rate of the dark spot expansion. Generalization of Kralchevsky et al.’s Model.18 On the basis of micellar self-diffusion, Kralchevsky et al. reported that the total influx of the vacancies (J) can be calculated from the relation18

J=

1 d(πR2) δ l 2 dt

(19)

where πR is the area of the dark spot, t is the time, and δl2 is the lateral area occupied by a vacancy in the film. It was assumed that the size of a vacancy corresponds with that of a micelle. Equation 19 shows that the total influx of vacancies leads to an expansion of the dark spot. From the experimental measurements of the squared dark spot radius versus time, it 2

H

DOI: 10.1021/acs.langmuir.6b00561 Langmuir XXXX, XXX, XXX−XXX

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area on the rate of the dark spot expansion in terms of the film volume. A parametric study showed the dependence of the rate of the dark spot expansion on the film thickness (or micellar layer) and the film size. Kralchevsky et al.’s model for the rate of the dark spot expansion was based on the vacancies condensation mechanism in terms of the total influx of vacancies.18 Their model did not consider the effects of the film thickness and its size. However, our generalized model in this study is able to predict the total influx of vacancies versus micellar layers by taking into consideration the effects of the micellar structuring and film size on the rate of the dark spot expansion. It is based on the proportionality between the total influx of the vacancies, the apparent diffusion coefficient, and the film volume. The proposed model makes good predictions for the rate of the dark spot expansion with various film thicknesses and film areas.

was found that the rate of the dark spot expansion is faster when the film is larger and/or thicker (see Figure 5). In order to predict the rate of the dark spot expansion for various film thicknesses and/or areas, we should consider the film volume for the supply of vacancies and the apparent diffusion coefficient because of the micellar layering phenomenon. Since the rate of the dark spot expansion is proportional to the film volume and the apparent diffusion coefficient, the total influx of vacancies is also proportional to them. Therefore, we generalized eq 19 by scaling the total influx of vacancies for different micellar layers with the film volume and the apparent diffusion coefficient as follows:

Jn JR

=

Dn Vn DR VR

(20)

where Jn is the total influx of vacancies, Dn is the apparent diffusion coefficient, and Vn is the film volume for the “nth” micellar layer. JR, DR, and VR are for the reference micellar layer. In order to predict the total influx of vacancies, the reference micellar layer has to be selected first. For instance, we selected the film with 3 micellar layers as a reference and then predicted the total influx of vacancies for the films with 2, 4, and 5 micellar layers. Therefore, J3, D3, and V3 can be substituted with JR, DR, and VR in eq 20. From the experimental measurements of the squared dark spot radius versus time for the film with 3 micellar layers and eq 19, we obtained the total influx of the vacancies for the film with 3 micellar layers (J3 = 6.36 × 106 s−1). We find

Jn = J3

Dn Vn D3 V3

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CONCLUSIONS The goal of this study is to elucidate the multiple stepwise thinning (stratification) of foam films made from nonionic micellar solutions by observing the decrease in the local stepwise thicknesses driven by the dark spot formation− expansion and to propose an analytical expression for the rate of the dark spot expansion. The following is a brief summary of our conclusions: (1) Reflected monochromatic light interferometry was used to study aqueous foam films formed from the Enordet AE 1415-30 (a nonionic surfactant with 14- and 15-carbon chains and 30 alcohol ethoxylates) surfactant solution at a concentration higher than the CMC. Below the ∼100 nm film thickness, the film thins with multiple stepwise thickness transitions. The mean amplitude of the film thickness transition is independent of the film thickness and film size, which corresponds to the diameter of a nonionic micelle; this was verified by both reflected light interferometry and dynamic light scattering measurements. (2) We observed that the 0.052 M Enordet AE 1415-30 film with a film radius of 110 μm stopped thinning and remained with 2 micellar layers inside it; this shows that the critical film size exists. When the film size is smaller than the critical film size, the stepwise film thickness transitions do not occur and the film exists at an equilibrium with several micellar layers inside it. However, when the film size increases, it becomes larger than the critical film size; this initiates the next film thickness transition, since the vacancies concentration inside the film increases with the film size, which is sufficient to form a dark spot. Therefore, the stepwise film thinning is governed by the vacancies condensation mechanism proposed by Kralchevsky et al.18 (3) The two-dimensional Fickian diffusion approach was applied to model the expansion of the dark spot. The analytical expression for the dynamics of the dark spot expansion takes into consideration both the apparent diffusion coefficient and the volume of the film. On the basis of the relation between the rate of the dark spot expansion (or total influx of vacancies) and the apparent diffusion coefficient and the film volume, we generalized the model of Kralchevsky et al.18 The model predictions for the rate of the dark spot expansion versus micellar layers (various film thicknesses and/or film sizes) are in good agreement with the experimental observations.

(21)

Equation 21 allows us to predict the total influx of the vacancies for different micellar layers (Jn) with the apparent diffusion coefficient (Dn) and the film volume (Vn). Knowing the total influx of the vacancies allows us to calculate the rate of the dark spot expansion using eq 19. It is important to note that the model predictions using eq 11 should be identical with the model predictions based on eqs 19 and 20. After selecting the reference micellar layer from the experimental measurements, the scaling relations with apparent diffusion coefficients and film volumes in both models are the same. Therefore, the generalized model’s predictions for the rate of the dark spot expansion are the same as the model predictions in Figure 5. We verified that the rate of the dark spot expansion decreases with the decreasing film thickness in the case of a film of a fixed size (a film radius of 110 μm). Since the film size is proportional to the vacancies supply, the rate of the dark spot expansion for a film with 2 micellar layers (film radius of 350 μm) is faster than that for the rest of the films with more micellar layers (3, 4, and 5 micellar layers) despite having a low apparent diffusion coefficient. In summary, our proposed models in this study predict the rate of the dark spot expansion for different micellar layers by considering the apparent diffusion coefficient as well as the film volume. We adopted the two-dimensional diffusion equation for the thin liquid film containing micelles because the film thickness was much smaller than the film size (area). In order to consider the effect of micellar layering for different micellar layers on the rate of the dark spot expansion, we explored the structural film interaction energy isotherm to obtain the apparent diffusion coefficient (or the film viscosity). Moreover, the proposed model takes into account the effect of the film I

DOI: 10.1021/acs.langmuir.6b00561 Langmuir XXXX, XXX, XXX−XXX

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(8) Langevin, D.; Sonin, A. Thinning of soap films. Adv. Colloid Interface Sci. 1994, 51, 1−27. (9) Krichevsky, O.; Stavans, J. Micellar stratification in soap films: a light scattering study. Phys. Rev. Lett. 1995, 74 (14), 2752. (10) Dushkin, C.; Nagayama, K.; Miwa, T.; Kralchevsky, P. Colored multilayers from transparent submicrometer spheres. Langmuir 1993, 9 (12), 3695−3701. (11) Denkov, N.; Yoshimura, H.; Nagayama, K.; Kouyama, T. Nanoparticle arrays in freely suspended vitrified films. Phys. Rev. Lett. 1996, 76 (13), 2354. (12) Basheva, E.; Danov, K.; Kralchevsky, P. Experimental study of particle structuring in vertical stratifying films from latex suspensions. Langmuir 1997, 13 (16), 4342−4348. (13) Tan, S.; Yang, Y.; Horn, R. Thinning of a vertical free-draining aqueous film incorporating colloidal particles. Langmuir 2009, 26 (1), 63−73. (14) Nikolov, A.; Kondiparty, K.; Wasan, D. Nanoparticle selfstructuring in a nanofluid film spreading on a solid surface. Langmuir 2010, 26 (11), 7665−7670. (15) Koczo, K.; Nikolov, A.; Wasan, D.; Borwankar, R.; Gonsalves, A. Layering of sodium caseinate submicelles in thin liquid filmsa new stability mechanism for food dispersions. J. Colloid Interface Sci. 1996, 178 (2), 694−702. (16) Husband, F.; Wilde, P. The effects of caseinate submicelles and lecithin on the thin film drainage and behavior of commercial caseinate. J. Colloid Interface Sci. 1998, 205 (2), 316−322. (17) Dimitrova, T. D.; Leal-Calderon, F.; Gurkov, T. D.; Campbell, B. Surface forces in model oil-in-water emulsions stabilized by proteins. Adv. Colloid Interface Sci. 2004, 108, 73−86. (18) Kralchevski, P.; Nikolov, A.; Wasan, D.; Ivanov, I. Formation and expansion of dark spots in stratifying foam films. Langmuir 1990, 6 (6), 1180−1189. (19) Bergeron, V.; Jimenez-Laguna, A.; Radke, C. Hole formation and sheeting in the drainage of thin liquid films. Langmuir 1992, 8 (12), 3027−3032. (20) Beltrán, C.; Guillot, S.; Langevin, D. Stratification phenomena in thin liquid films containing polyelectrolytes and stabilized by ionic surfactants. Macromolecules 2003, 36 (22), 8506−8512. (21) Heinig, P.; Langevin, D. Domain shape relaxation and local viscosity in stratifying foam films. Eur. Phys. J. E: Soft Matter Biol. Phys. 2005, 18 (4), 483−488. (22) Heinig, P.; Beltrán, C.; Langevin, D. Domain growth dynamics and local viscosity in stratifying foam films. Phys. Rev. E 2006, 73 (5), 051607. (23) Zhang, Y.; Sharma, V. Domain expansion dynamics in stratifying foam films: experiments. Soft Matter 2015, 11 (22), 4408−4417. (24) Powell, R.; Roseveare, W.; Eyring, H. Diffusion, thermal conductivity, and viscous flow of liquids. Ind. Eng. Chem. 1941, 33 (4), 430−435. (25) McLaughlin, E. Viscosity and self-diffusion in liquids. Trans. Faraday Soc. 1959, 55, 28−38. (26) Eyring, H.; Ree, T. Significant liquid structures, VI. The vacancy theory of liquids. Proc. Natl. Acad. Sci. U. S. A. 1961, 47 (4), 526. (27) Abbott, A. P. Application of hole theory to the viscosity of ionic and molecular liquids. ChemPhysChem 2004, 5 (8), 1242−1246. (28) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena; McGraw-Hill Book Company, Inc.: 1941. (29) Frenkel, J. Kinetic Theory of Liquids; Dover: 1955. (30) Barker, G. C.; Grimson, M. J.; Richmond, P. Vacancy diffusion in colloidal crystals. J. Chem. Soc., Faraday Trans. 1991, 87 (3), 391− 394. (31) Chu, X.; Nikolov, A.; Wasan, D. Thin liquid film structure and stability: The role of depletion and surface-induced structural forces. J. Chem. Phys. 1995, 103 (15), 6653−6661. (32) Grossman, P.; Colburn, J. Capillary Electrophoresis: Theory and Practice; Academic Press: 2012.

(4) The results of our parametric study are reported, showing the dependence of the dark spot expansion on both the film thickness and the film size. An additional study is warranted to elucidate the dynamics of the dark spot expansion after it makes contact with the meniscus.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b00561. The movie clip depicts the foam film thinning process of 0.052 M Enordet AE 1415-30 under reflected light microinterferometry. After 40 s from when the film is initially formed, the first film thickness transition is observed. The second and third film thickness transitions occur in succession. The film remains with 2 micellar layers and the film thickness transitions stop; this indicates that the film size (film radius is 110 μm) is smaller than the critical film size, so there are not enough vacancies to nucleate into a dark spot to initiate the next film thickness transition. In order to observe the fourth film thickness transition, we increase the film radius from 110 to 350 μm gradually. After 5 min from when the film reaches the equilibrium state with 2 micellar layers, a dark spot containing 1 micellar layer appears inside the film region with a relatively faster rate of its dark spot expansion than that of the films with 3, 4, and 5 micellar layers. The film volume of 2 micellar layers is larger than the film volumes of 3, 4, and 5 micellar layers due to the larger size of the film, in spite of it being thinner than these other films. The concentration of the vacancies is proportional to the film volume, thereby increasing the rate of the dark spot expansion in the case of the film containing 2 micellar layers. (AVI)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (D.W.). Notes

The authors declare no competing financial interest.

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REFERENCES

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