Effect of Foam Films on Gas Diffusion

Journal of Colloid and Interface Science 248, 467–476 (2002) doi:10.1006/jcis.2001.8155, available online at http://www.idealibrary.com on

Effect of Foam Films on Gas Diffusion P. Nguyen Quoc, Pacelli L. J. Zitha,1 and Peter K. Currie Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands E-mail: [email protected] Received April 30, 2001; accepted December 10, 2001; published online March 13, 2002

We report an experimental investigation of the permeability to gas of systems of one or several soap films freely standing in a straight tube, using either reactive gas (NH3 ) or inert gas (argon). The series of soap films appears to be the simplest paradigm of successive lamellae arrangements encountered in foams confined in a porous medium. To conduct the experiments, we devised two novel methods for the determination of gas diffusion fluxes: one based on reactive changes of pH by NH3 and the other on mass spectrometry. The permeability of a single film, stabilized by sodium dodecyl sulfate solution, was found to be 3.50 ± 0.04 10−2 cm /s for argon and 3.18 ± 0.07 10−4 cm /s for NH3 . The permeability value for the inert gas is in good agreement with data obtained by the diminishing-bubble method. When the number of films increases, the permeability decreases considerably as a result of cumulative film resistance effects. We also developed a simple phenomenological model based upon a combination of gas kinetic and energy barrier concepts to interpret our data. This model takes into account gas solubility and the effects of salinity, which have seemingly been ignored in previous models. The predicted film permeability decreases sharply with increase surfactant concentration, indicating the occurrence of higher adsorption and increasingly compact surfactant layers. C 2002 Elsevier Science (USA) Key Words: film permeability; energy barrier; diffusion; freely standing film; surfactant.

I. INTRODUCTION

When a gas and a liquid containing a surfactant flow simultaneously through a granular porous medium, the gas breaks into bubbles that are separated by liquid films. On the macroscopic scale, foams in porous media can be described as a disordered network of bubble trains of which only a fraction is mobile (1–3). The largest part of the remaining fraction represents immobile gas that is, however, connected to the mobile fraction and is called the trapped gas fraction. The partitioning of the gas into mobile and trapped fractions determines to a large extent the dynamic flow resistance of foam in porous media. Because this flow resistance plays an important role in a large number of applications, such as materials engineering, enhanced oil recovery, 1

To whom correspondence should be addressed. 467

and environmental remediation, reliable methods to quantify the trapped gas fraction are needed. Several techniques for determining trapped gas under steady foam flow conditions (e.g., Refs. (4–6)), of which the gas tracer technique has the most potential, have been proposed. With this technique, it is necessary to address the fact that the effluent tracer histories can be substantially influenced by the transfer of the tracer into the surrounding liquid and the trapped gas. To enhance the interpretation of the experimental data, Radke and Gillis (6) developed a tracer mass transfer model for trapped gas prediction, which was verified experimentally by employing a dual gas tracer technique. The model was simplified to the case where the resistance to mass transfer of the tracer in liquid sheaths, separating mobile and trapped gas, and successive trapped lamellae was neglected; i.e., only convective diffusion was responsible for the tracer transfer. Neglecting the presence of foam films in modeling tracer diffusion may be missing the role of surfactant-induced interfacial resistance in particular, or film resistance in general, which has been shown to be significant. Several attempts have been made to explain the mechanisms of gas penetration across the gas–liquid interface in the presence of monolayers of adsorbed hydrocarbon derived materials (7–9). Two mechanisms have been proposed. If the monolayer behaves like a homogeneous phase, gas molecules will encounter a total resistance that is the sum of diffusional resistances in the gas phase, across the monolayer, and through the liquid. This provides an explanation of gas permeability reduction as a phase resistance effect. Alternatively, since the monolayer is not necessarily homogeneous, it may act as an energy barrier to gas molecules striking the interface. The magnitude of this barrier is a measure of physical molecular interaction between the gas and the monolayer. These two conceptual models have been expanded by Princen and Mason (10, 11) to interpret experimental results on surfactant-stabilized film permeability, where a parallel-plane model was first used to describe the two monolayers constituting the film. Nedyalkov et al. (12–14), and Platikanov et al. (15) recently employed the nucleation theory to account for the overlapping adsorption phenomenon (bilayer structure) of the two monolayers. Despite some limitations mentioned later, this theory contributes appreciably to a better understanding of bilayer-structured film. 0021-9797/02 $35.00

 C 2002 Elsevier Science (USA)

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NGUYEN, ZITHA, AND CURRIE

In the above film-permeability studies, the experimental technique based on the shrinkage of a gas bubble owning to gas diffusion (diminishing bubble method) has been developed and improved to investigate the permeability of common or Newtonblack films (NBFs) to gases (10–15). This method, however, requires the data of dynamic bubble shape that has been difficult to measure directly in practice. Moreover, deformation of the bubble due to diffusion may cause the instability of interfacial properties and unsteady film thickness, which determine predominantly the permeability of foam films. We have developed a new method, namely freely standing film diffusion, to investigate the resistance of one or more films to the mass transfer of tracer gases in a stagnant matrix gas. This enables measurements that are difficult, if not impossible, with the diminishing-bubble method. In addition to obtaining a steady film permeability, this method also allows experiments on vapor components, extending the range of investigated gases. A new mechanistic model for relating the effective gas diffusivity to foam film density and permeability has been developed. The latter is also identified further as a complex function of electrolyte concentration, gas solubility, surfactant concentration, and temperature. The paper will first present the theoretical model. This is followed by the description of the two new techniques used to measure the permeability of films to two gases with contrasting solubility: ammonia and argon. The following section is concerned with the agreement of this model with experimental data and the nucleation theory in terms of sensitivity of film permeability to concentration. II. THEORETICAL

In our previous study (16), we formulated a one-dimensional diffusion problem in which a gas, under a constant driving force, is allowed to diffuse through a straight cylindrical tube in the absence or presence of one or more freely standing foam films. The effective gas permeability was found to be dependent on background (or film-free) gas diffusivity as well as film permeability and density. In the present work, we expand this functional relationship using a more detailed mechanistic model for the foam film penetration process, taking into account the surface dynamics of surfactant in interaction with gas molecules. 1. Multiple-Films Diffusion Let us first briefly recall the analytical solution of Fick’s second law describing the unsteady diffusion flux of a certain tracer gas in a gaseous mixture contained in a cylindrical tube of length L:  n=∞   ∂C D  −(2n L+x)2 /(4Dt) N (x, t) = −D = (C1 − C0 ) e ∂x πt n=0  n=∞  −(2(n+1)L−x)2 /(4Dt) . [1] + e n=0

Equation [1] can be reduced to the following simpler form, also known as Fick’s first law, under steady state conditions, which are attained for t > L 2 /Dπ , N=D

(C1 − C0 ) = kg (C1 − C0 ), L

[2]

where D is the background gas diffusivity in the gaseous mixture. The ratio of D/L defines the mass transfer coefficient, kg . C1 and C0 are the time-independent inlet and outlet concentrations, respectively. In the presence of n f intervening foam films of equal thickness δ, the gas flux N (x, t) is reduced as a result of film resistance. It is not uncommon to treat the films as an additional phase separating 1D-gas space into (n f + 1) sections. This recalls the classical problem of mass transfer across a phase boundary in which the mass transfer coefficient is frequently defined for the overall process. Following this line, we introduce the effective diffusivity De for the whole system. The difference between De and D is therefore a measure of the film resistance. For the steady state, using the usual rule of additivity of resistances in series, the effective mass transfer coefficient of the investigated gas, ke (or De /L), through n f foam films of thickness δ, taking into account the film resistance, 1/kf , can be written as   1 (L − n f δ) n f H 1 2 1 , = , = + + ke D kf kf km kl

[3]

where kl is background gas mass transfer coefficients in bulk liquid, km is monolayer permeability, and H is Henry’s coefficient. Equation [3] is derived from Eq. [2] written for (n f + 1) gas sections and n f foam films of equal thickness δ. The above relations between the various coefficients are valid provided that the transfer rate is linearly related to the driving force and that the equilibrium relationship is a straight line. Equation [3] can be reduced further to a simpler form in the case of NBF—lacking the central liquid layer between its two interfaces (the bilayer structure),   1 1 1 = + nf H ke kg ks

for δ
L and

1 2
, kl km

[4]

where ks is the permeability to gas of the bilayer film. In agreement with Ref. (8), we identify ks simply as the product of the collision rate of gas molecules and the fraction of molecules entering the film, α, ks =

Hα H e(−E/RT ) = , 1/2 (2π M RT ) (2π M RT )1/2

[5]

where M is the gas molecule weight, R the ideal gas constant, and T the temperature. This relation allows an approximation of the penetration rate of gas molecules across a homogeneous interface characterized by the penetration activation energy, E. In consequence, Eq. [5] should be applied for clean interfaces

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EFFECT OF FOAM FILMS ON GAS DIFFUSION

or close-packed monolayers. In the present work, our principal concern lies in the fact that the state of unsaturated monolayers varies considerably with the dynamic adsorption behavior of the surfactant, which depends in turn on the presence of electrolytes in the solution. As a result the permeability of a foam film to gas is dependent on the surface coverage of the surfactant. For greater generality, consider one interface of a foam film normal to the diffusion direction, comprising sites occupied by the adsorbed surfactant molecules. In the classic Langmuir adsorption theory, the effective fractional area of the occupied sites is given by θ (= n e /n o ), the ratio of equilibrium (n e ) to close-packed (n o ) adsorption density of surfactant, assuming a constant area of the adsorbed molecules. However, it is well known that the effective area per molecule of surfactant in the interface varies substantially with monolayer pressure, and decreases with the increasing θ (18). For SDS surfactant used in the present work, the practical equilibrium effective area decreases approximately by a factor of 2 at CMC (25), while a hypothetical maximum factor of 4 can be assumed representing the order of the polar group area (19). Therefore, we propose the following approximate form of the effective fraction of the occupied sites: θ raised to power 1/(λ − 2θ ) − the reciprocal of the number of surfactant molecules per occupied site, where λ (= 4) denotes the maximum change in the effective area per surfactant molecule with varying surface concentrations of surfactant. This relation is approximately valid for insoluble surfactants with experimentally determined λ and close-packed effective area. From kinetic theory the overall penetration rate of gas molecules across this interface can be written   1 1 2 ks = 1 − θ λ−2θ Fe(−Ew /RT ) + θ λ−2θ Fe(−Ef /RT ) ,

[6]

where F=

H (2π M RT )1/2

[7]

with H = H0 e

1 1 ( − H R ( T − T )) 0

.

[8]

E f and E w are respectively the penetration activation energies across the occupied and unoccupied sites. H is the enthalpy of solution. The squared fractional area, (1 − θ 1/(λ−2θ ) )2 , of the unoccupied sites accounts for the surfactant monolayer in the other interface. However, a power other than 2 can also be taken, depending on the overlapping structure of the two monolayers. It should be lower for sufficient thick foam films, and vice versa. Therefore, this particular form of Eq. [6], or other film permeability models based on the concept of bilayer films in general, cannot describe properly foam film permeability at far below CMC, where bilayer structure may not characterize thick films (26). Also E f and E w should be dependent of θ on account of intermolecule interactions. An explicit expression of these relations is however out of the scope of this work.

Bringing in surfactant solution properties, an estimate of θ can be obtained from the kinetic theory of adsorption from solution (19) θ=

(B/n o )Cs e(Ea /kT ) , 1 + (B/n o )Cs e(Ea /kT )

[9]

where Cs is concentration of surfactant, E a is the desorption activation energy, B is the adsorption–desorption equilibrium constant, taking the value of 1.6 × 1014 for sodium dodecyl sulfate (SDS) adsorbed at air–water interface (19). The closepacked adsorption density n o for SDS equals 3 × 1014 given by the radioactive-counts method (19). Equation [9] has the form of the Langmuir adsorption isotherm, although the coefficient in the Langmuir expression is no longer constant due to θ -dependence of E a . Taking this into account, we use the given relation of E a with respect to θ , surfactant formulation, and salinity, as follows (see detail of its derivation in Refs. (18, 19)), Ea =

521m + 12 × 10−6 m θ 0.5 n 0.5 o − ze ψ0 NA

[10]

with   2365(T Ce )−1/2 ψ0 = 0.17T sinh−1 , σ

[11]

where m is the number of CH2 group in the hydrophobic part of the surfactant, and z and e are the valance number and electric charge, respectively. The first and second terms in Eq. [10] account for the net adsorption energy of the hydrophobic part taking into account the interchain cohesion effect, while the last term indicates reduced adsorption due to electrolyte effects. The Gouy potential (18), ψ0 , is expressed through Eq. [11] as a function of electrolyte concentration, Ce , and effective area per surfactant molecule, σ . For consistency, the latter parameter should vary with the surface coverage of the surfactant through the relation, σ = (σ0 /θ )θ 1/(λ−2θ ) , where σ0 represents σ at the lowest ˚ 2 for surfactant concentration, taken approximately to be 80 (A) the practical stability of SDS stabilized films (25). The incorporation of ψ0 (Ce ) and E a (θ) into the prediction of the film permeability is proposed here for the first time. In summary, the above relations allow the prediction of the effective gas diffusivity as a function of foam film density (Eqs. [3], [4]) with the foam film permeability predicted from Eqs. [6]–[11] for the given surfactant system and environmental conditions. As discussed later, if E w is presumed to be constant for a clean air–water interface, this model as a whole has just one single parameter to fit to experimental data, E f . III. EXPERIMENTAL

Due to the solubility dependence of the foam film permeability in the above relations, we selected two solubility-contrast

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NGUYEN, ZITHA, AND CURRIE

gases, ammonia vapour (NH3 ) and argon gas (Ar). In our previous work (16), only NH3 was investigated, using a commercial aqueous surfactant. In this work, in addition to repeating the previous measurements with analytical reagent graded surfactants, we also developed a new experimental setup that allows the investigation of a wide range of gases. 1. Principle and Apparatus Reactive gas diffusion setup. The schematic of the apparatus is presented in Fig. 1. The left vessel contains 1000 ml of extra pure ammonium solution, while the right vessel contains 200 ml of sulfuric acid solution of varying concentrations. During the experiment, ammonia vapor is allowed to diffuse through a glass tube of 8 mm inner diameter, and is absorbed into the acid solution. Foam films can be placed in the tube. The tube ends are fabricated from Pyrex cone-and-socket joints fitting into the two vessels. Because the diffusion length L influences the magnitude of diffusion flux and hence the resolution of pH change of the absorbing solution (Eq. [14]), two different lengths, 33.6 and 47.7 cm, are used to clarify this. Also placement of more than one foam film was found easier with the long diffusing tube. Conversion of ammonia flux N (L , t) from Eq. [1] into the pH change of the absorbing solution is based on the equations − 2NH3 + H2 SO4 → 2NH+ 4 + SO4

2NH+ 4 (aq)

[12] −

2NH3 (aq) + H2 O ↔ + OH (aq)  

t V C 0 (H+ ) − A 0 N (L , t) dt pH = −log10 , V

[13] [14]

where C 0 (H+ ) is the initial concentration of H+ , V is the volume of the absorbing solution, and A is the diffusion area. It is of importance to note that at very dilute acid concentrations (pH > 6), a small percentage ammonia also reacts incompletely with water to make some OH− according to Eq. [13]. This reaction implies that small amounts of ammonia trapped in the solution will not be detected by the measurement of the pH change. A computerized meter is used to read pH values (to the second decimal place) and temperature histories. Obtaining the homogeneity of the solutions requires sufficient agitation in both vessels. Inert gas diffusion setup. The schematic of the apparatus used for the diffusion of inert gases is presented in Fig. 2. Unlike the ammonia setup, this setup allows a direct measurement of the diffusion flux of the investigated gas along the straight diffusing tube of 2.5 mm inner radius. Two ends of the diffusing tube are permanently connected perpendicularly to two identical cylindrical straight tubes of 8 mm (called the Ar and N2 tube). The investigated and carrying gases (Ar and N2 , respectively) flow respectively in these tubes at constant rates monitored by two gas mass flow controllers. Foam films can be placed in the diffusing tube. In film-free diffusion experiments, undesired convection in the diffusing tube is minimized by using the same flow rate Q for both gases, but a complete elimination of convection is difficult due to unavoidable asymmetries of the hand-made tube connections and differences in gas properties. When foam films are present, convection is not a problem, since flow rates can be adjusted to give zero-movement of the films. A possible mixing vortex at the ends of the diffusing tube is minimized by choosing a high radius ratio for the Ar and diffusing tubes. The insignificance of this effect on the diffusion process was confirmed by the Winkelmann method (31).

FIG. 1. Experimental setup for the measurement of (effective) diffusivity of ammonia (NH3 ) with and without foam films: NH3 diffusion flux at the gasabsorbing solution (dilute H2 SO4 solution) interface is determined through the pH change rate, assuming negligible resistance at the interface, and homogenous pH in the absorbing solution due to sufficient agitation.

471

EFFECT OF FOAM FILMS ON GAS DIFFUSION

FIG. 2. Experimental setup for the measurement of effective diffusivity of argon (Ar) in the presence of one or several foam films: Ar diffusion flux at the outlet of the diffusion tube is determined through the measurement of Ar concentration in the carrying gas N2 , using the same flow rates of Ar and (N2 ) in two identical vertical tubes.

Driving force for the diffusion is a difference in partial pressures of the diffusing gas over the length of the diffusing tube L (6.8 cm). The background and effective gas diffusivity can thus be determined from Eqs. [1] and [2] with the diffusion flux calculated from the expression N (L , t) = ADe

 ∂C(x, t)  = Qxw (t), ∂ x x=L

[15]

where xw (t) is the mole fraction of the diffusing gas measured in the N2 flow less its native fraction in the N2 supply. This quantity is determined by a mass spectrometer. Temperature needs to be well controlled by a thermal controller. All experiments including reactive and inert gases, were conducted at 25◦ C. The film drainage is monitored with a Leica Orthoplan transmitted light microscope for higher magnification associated with a stereomicroscope with transmitted light for overview.

Vapor and gases. NH3 vapor is supplied from 29% solution (analytical reagent grade, purchased from Merck) with vapor pressure of 450 mbar under standard conditions. For the inert gas diffusion experiments, the quality of the investigated (argon) and carrying (nitrogen) gases was analyzed by the mass spectrometer as presented in Table 1. 3. Procedures Absorbing solution. Procedures for preparing the absorbing solution followed the standard method for the determination of ammonia in the atmosphere (20). Film-free diffusion. An experiment provides the baseline for comparison of diffusion rates. The diffusing tube is cleaned with distilled water, and then put in the oven at 80◦ C for 30 min. The dried tube is cooled for 30 min. in the thermal controller to the experimental temperature before connecting to the vessels.

2. Materials Absorbing solution. In ammonia diffusion experiments, aqueous solution of sulfuric acid 0.05 M (analytical reagent grade, purchased from Merck) is diluted to various concentrations. Film-making solution. Foam films are made from sodium dodecyl sulfate solution (M 288) with varying concentrations from 1.5 × 10−3 M (minimum concentration for film stability) to 4 × 10−3 M in the presence of 0.5 M NaCl in all experiments.

TABLE 1 Quality of the Investigated and Carrying Gases Quality (%)

Investigated gas (Ar supply)

Carrying gas (N2 supply)

N2 Ar O2 CO2

4.03 95.24 0.72 0.008

99.09 0.18 0.70 0.04

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NGUYEN, ZITHA, AND CURRIE

IV. RESULTS AND DISCUSSIONS

The film-free diffusion rates (base-case experiments) were first determined to confirm the theoretical background diffusivity of NH3 vapor and Ar gas, as well as to calibrate the system. Several repeated measurements of the single-film diffusion rate followed to obtain the value of the maximum film resistance, 1/kf . Note that this is an estimate of film resistance 1/ks , according to Eqs. [3] and [4]. Fitting Eq. [6], with the help of equations [7]–[11], to the experimentally determined ks allows the energy of penetration, E f , to be calculated with the known value of E w . This is followed by the prediction of experimental data on the ke –n f relationship through Eq. [4]. 1. Film-Free Diffusion Ar gas. The use of a low flow rate for carrying gas N2 may reduce the reproducibility of experimental results due to poor mixing in the N2 tube, as well as degrade the quality of the supplied Ar and N2 due to admixture with air. High flow rate may also cause mechanical disturbance to the film-making process,

which results in low film stability. From numerous film-free experiments for a wide range of flow rates, these effects were minimized in the range 0.1–0.18 cm3 /s (standard conditions). For an optimum, we selected a constant flow rate of 0.15 cm3 /s for all experiments with this setup. Since Ar is a nonpolar gas, the relation of Fuller, Schettler, and Giddings (FSG) (22) can be used to accurately estimate the background diffusivity, giving D Ar = 0.21 ± 0.02 cm2 /s. Although one single flow rate was applied for both Ar and N2 , convection was still present in the absence of a film. Indeed without a film, the experimentally determined background diffusity of Ar, 1.62 ± 0.08 cm2 /s, is about seven times higher than that estimated from the FSG relation. We believe this discrepancy is due to convection. However, the presence of foam films helps preclude the effect of convection, giving a more accurate estimate of the background diffusivity, as shown later. NH3 vapor. The NH3 setup is subject to the following constraints for valid results. The maximum time for each measurement should not be longer than 5 h, and the initial pH used should be higher than 3 for better resolution of the measured pH profiles. Figure 3 shows a good agreement between measured and predicted film-free diffusion rates for two different diffusion distances, L, using the known background diffusivity of NH3 in air (DNH3 = 0.28 cm2 /s) (17). This implies that the diffusion lengths used give good resolution of the pH histories. 2. Single-Film Diffusion Ar gas. The effective transfer coefficient of Ar with a single film, keAr , was measured in the presence of 0.5 M NaCl for a range of five SDS surfactant concentrations. The lowest surfactant concentration, 1.5 × 10−3 M, corresponds to the critical stability of the foam film obtained by this setup. The experimental data shows a constant value of keAr to be 1.5 ± 0.4 × 10−2 cm/s, independent of surfactant concentration. Taking D Ar of 0.21 ± 0.02 cm2 /s estimated from the FSG relation, this results in a value

5.5 5.0 4.5

L = 33.6 cm

4.0

pH

Across-films diffusion. The film-making process involves three steps. First, a film is created at the end of the diffusing tube and then displaced to a marked position in the middle of the tube by lightly blowing. Second, to remove as much as possible the gravity effect that causes an irregular film thickness, the film must be drained out to an equilibrium lamella. This can be done by closing one end of the tube and centrifuging at sufficiently high velocity until a minimum and uniform width of plateau border next to the glass wall is obtained. To ensure this, the width of the plateau border is further monitored with varying directions of the light source and camera for 1 h before the measurement of gas diffusion. Note that the thickness of the film plateau border is larger than that of the central part of the film, although its effective area for gas diffusion is negligible relative to the overall area of the foam film (21). Elimination of the plateau border is impossible by any methods developed so far. Notwithstanding that the visual technique cannot assess the actual film thickness, with the above type and concentration of surfactant and electrolyte, the equilibrium films will all be NBFs (28, 29), and as reported by Radke and co-workers (30), of very nearly constant thickness to all possible film disjoining pressures. The next step is only applied for the reactive gas diffusion experiments. To avoid lateral absorption, the wet inner wall of the diffusing tube must be dried out manually using a cotton stick. This is one of the most delicate parts of the experiment because the thinnest films will have less chance to survive in the presence of mechanical disturbances. To clean the wall as closely as possible to the edges of the film plateau borders, the microscope is again employed to direct the cotton stick. The cleaning is carried out after attaining an equilibrium film. Note that this procedure is only possible for the measurement with a single film.

3.5

L = 47.7 cm

3.0 2.5

X

2.0

Experimental Theoretical

1.5 1.0 0

20

40

60

80

Time (min) FIG. 3. Experimental and theoretical pH variation with time of NH3 for zero-film with two diffusion lengths (SDS surfactant solution of 2 × 10−3 M).

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EFFECT OF FOAM FILMS ON GAS DIFFUSION

film is the sum of penetration through its holes-free film area and penetration through the holes of different sizes, expressed as K = Ko +

 i

FIG. 4. Experimental and theoretical foam film permeabilities for singlefilm diffusion of Ar (SDS surfactant solution), compared to those from Ref. (13).

3.5 ± 0.4 10−2 cm/s for the film permeability ksAr , on the same order as determined experimentally by Goodridge and Bricknell (8) for the close-packed monolayer of SDS. As presented in Fig. 4, Nedyalkov and co-workers, using the diminishing-bubble method, found the asymptotic mean permeability of NBF (called “background permeability” in Ref. 13) to be slightly lower than that in the present work under similar experimental conditions (13). If the central part of the investigated foam film should account for the diffusion flux of Ar, this slight discrepancy results most likely from using commercial preparations of surfactants, and differences in the accuracy of the two methods and in the gases used. It is, however, difficult to separate the magnitudes of these effects. The minimum surfactant concentration used in this work is above CMC (CMC of SDS in the presence of 0.5 M NaCl is about 6 × 10−4 M (19)). This restricted our observations to saturation films as shown by circle points on the predicted solid curve in Fig. 4. Nevertheless our model predicts well the experimental data of Nedyalkov et al. (13) for film permeability at low surfactant concentration, determined by the diminishing-bubble method. In fact, the permeability of a shrinking bubble might vary slightly with time (10), although the same mechanism of gas penetration through a foam film is dominant in both the diminishing bubble and our method. In this sense, our model gives an alternative mechanistic explanation on the regimes of gas penetration across a film. At very low surfactant concentrations, the film permeability decreases rapidly with slightly increasing concentration, caused by a significant change of the effective area per surfactant molecule, σ . This is followed by a more gradual decrease ascribed to the increase in area of the occupied sites, partially counterbalanced by the compression of σ . Approximately near CMC, a constant value is reached due primarily to the formation of a close-packed monolayer. In fact, the present model is in good agreement with nucleation theory where the overall gas penetration rate across the foam

Ki = Ko +



ai Cs−i ,

[16]

i

where i denotes the number of vacancies of half-surfactant molecule area in hole i, K o is the holes-free foam film permeability, and the constant ai is a function of the surfactant molecule area, diffusivity of i-sized hole molecule, intermolecule interaction energy, and the film thickness (see details in Refs. 12–14, 26, 27). Nedyalkov et al. (13) suggested the best fit form of Eq. [16], the dashed curve in Fig. 4, to their experimental data that gave K o = 0.032 cm /s, i = 3, and ai = 2.7 × 10−13 cm mol/s dm3 at 25◦ C. The two theoretical curves in Fig. 4 (from Eqs. [6] and [16]) have almost the same shape with somewhat different asymptotic permeabilities. Moreover, closer examination of the plateau region (second regime) shows that ks is more sensitive to concentration in the case of the nucleation theory. Whether this is the result of ignoring all constants ai other than a3 for a good fitting of Eq. [16] or whether the fractional effective area of the unoccupied sites in the first term of Eq. [6] should be raised to a power higher than 2 for the reasons discussed in the theoretical section is still an open question. Note that our model could include a relationship between the fractional effective area of the occupied sites and its corresponding energy of penetration, E f . The quantitative estimation of this relation is beyond the scope of this work. For simplicity the fact that the activation energy for the diffusion process across a clean gas–liquid interface, E w , has approximately the same value for all non- and poorly ionized gases suggests E w has a constant value of 5.5 kcal/mol determined experimentally from Ref. (23). Unlike E f , the fractional-effective-area independence of E w can be confidently assumed as a good approximation provided that the significance of the constant a3 from Eq. [16] still holds. Indeed, the constant a3 responsible for the first regime implies the dominance of holes sized 1.5 times the surfactant molecule area. ˚ 2 (taking 30 (A) ˚ 2 as the This implies that the area is about 45 (A) effective SDS molecule area in a close-packed monolayer), still significantly larger than triatomic gas molecules on the order of ˚ 2 . In other words, it is reasonable to assume that E w is 20 (A) insensitive to the dynamic fractional effective area in the first regime. The foregoing arguments strengthen confidence in the fitted value of E f of 9.43 kcal/mol. NH3 vapor. For a single film, Fig. 5 shows a significant delay of diffusion rate at the initial pH of 3, corresponding to a reduction of the NH3 diffusivity by 11.8% (i.e., De = 0.882D). Consequently, ksNH3 can be calculated from Eq. [3], giving a value of 3.18 ± 0.07 10−4 cm/s. In the present work we neglect the effect of the additional cation NH+ 4 (Eq. [6]) on the film disjoining pressure which is sensitive to film thickness. Nevertheless, the value of the equilibrium constant of Eq. [13], pKb , about 4.8

474

NGUYEN, ZITHA, AND CURRIE

5.5 5.0 Zero film

4.5

pH

4.0 One film

3.5 3.0 X

2.5

Experimental Theoretical

2.0 1.5 1.0 0

20

40 Time (min)

60

80

FIG. 5. Experimental and theoretical pH variation with time of NH3 for zero and one film (L = 33.6 cm, SDS surfactant solution of 2 × 10−3 M).

(24), indicates that there will be insignificant amount of cation NH+ 4 in the very small volume of the foam film. Prediction of foam film permeability to NH3 for the wide range of surfactant concentration is also presented in Fig. 6. The dashed curve was plotted from Eqs. [6]–[11], taking E w = 5.5 kcal/mol, the same as in the Ar diffusion, and assuming the influence of cation NH+ 4 on the surface potential ψ0 to be negligible. A best fit to the experimental data gives E f an approximate value of 7.24 kcal/mol. Using this value, the theoretical curve predicts that ksNH3 decreases first to a minimum and then increases slightly to its asymptotic value, rather than decreasing monotonicaly with concentration. Lack of data in the low-concentration range make it impossible to confirm this prediction. If the prediction is indeed correct, the minimum may be due to traces of impurity in the surfactant solutions, since it is very difficult in practice to prepare pure water, surfactant, and electrolytes. For instance, if the regions of the foam film interface occupied by impurity have the resistance of a clean surface,

say 0.002 s/cm, one calculates that even 0.01% of these in the monolayer may increase the foam film permeability by 20%. Alternatively, the minimum may arise from an imperfect choice for E w . NH3 is highly soluble and partially reacts with water, as partially implied by its value of E f (= 7.24 kcal/mol) being lower than that of Ar ( = 9.43 kcal/mol). Consequently, E w for NH3 may take a lower value, giving monotonic variation of ks with Cs . Monotonic behavior is found if E w ≤ 5 kcal/mol. The solid curve in Fig. 6 shows the ks –Cs relationship corresponding to upper limit of E w = 5 kcal/mol. Furthermore, to achieve a good fit to the experimental data, E w cannot be smaller than 4 kcal/mol. Thus, while lacking experimental data for ammonia in the low-concentration regime, the experimental results are consistent with monotonic decrease of ks with Cs only if E w is in the narrow range of 4 to 5 kcal/mol. Despite this uncertainty, the interaction between the properties of diffusing gases and dynamics of the interfacial properties in governing foam film permeability is described fairly well by our model. 3. Multiple-Film Diffusion Prediction of the ke –n f relationship is essential for the determination of trapped gas in porous medium using the tracer gas technique. The validity of the newly developed foam film permeability model can be tested by comparing ke estimated from Eq. [4] with experimental data of multiple-film diffusion. Ar gas. Placement of more than one foam film makes drying out the inner wall of the segment between two adjacent films impossible. To check the effect of the lateral diffusion that may cause transfer of gas into the wetting film, several film-free diffusion experiments were carried out with the wet inner wall. The results showed the same value of gas diffusivity as in the case of dry inner wall. The lateral diffusion is thus negligible for Ar gas, which is reasonable due to its low solubility in water. Figure 7 shows that the measured effective film resistance 1/ke varies linearly with n f , in good agreement with Eq. [4]. 240 200

1/ke (s/cm)

160 120 80 40 0 0 FIG. 6. Experimental and theoretical foam film permeabilities for singlefilm NH3 diffusion with E f = 7.24 kcal/mol (L = 33.6 cm, SDS surfactant solution).

1

2

3 n_ film

4

5

6

FIG. 7. Predicted and experimental reciprocal of the effective film permeability to Ar as a linear function of the number of films (SDS surfactant solution of 2 × 10−3 M).

475

EFFECT OF FOAM FILMS ON GAS DIFFUSION

5.5 5.0 4.5 One film

pH

4.0 3.5

Two films

3.0 2.5

X

2.0

Experimental Theoretical

1.5 1.0 0

20

40

60

80

Time (min) FIG. 8. Predicted and experimental film permeability reduction factor for Ar gas as a function of the number of films (SDS surfactant solution of 2 × 10−3 M).

The intersection between the straight line fit and the y axis gives the background diffusivity DAr to be 0.22 ± 0.03 cm2 /s, in good agreement with DAr of 0.21 ± 0.02 cm2 /s estimated from the FSG relation (22). This corrects the erroneously high value found in Section 1 above, caused by convection. Moreover, from the slope, the value of ks = 3.8 ± 0.3 10−2 cm2 /s, is almost the same as the above experimentally determined value for the single-film diffusion. For a high population of trapped films during steady foam flow in a porous medium, it is also interesting to plot the theoretical diffusivity reduction factor, the ratio of effective diffusivity to film-free diffusivity, versus a large number of n f . Figure 8 predicts a drastic decrease of ke for few films. For increasing number of films, ke further reduces only slowly, since the resistance in the bulk gas phase becomes marginal relative to the total resistance of the films. This has an important implication

FIG. 10. Experimental and theoretical pH variation with time of NH3 for one and two films (L = 47.7 cm, SDS surfactant solution of 2 × 10−3 M).

for the diffusional transfer of tracer gas into trapped foams. Particularly, the coarsening of trapped foam after its formation in steady foam flow will not increase substantially the transfer rate of the tracer gas. Rather foam films located near the boundary between flowing and trapped foam will control the diffusion process. Finally, Fig. 9 shows the effect of the number of films on the Ar diffusion flux in a start-up diffusion process, predicted using data from Fig. 7 and Eq. [1]. The potential of the tracer gas technique is once again confirmed with this figure, which emphasizes the important role of the time scale of the tracing process. NH3 vapor. Figure 10 compares the predicted and experimental diffusion rates of NH3 in the presence of one and two films. The pH histories show a perfect agreement for the case of one film. However, the experimental curve in the case of two films shows a delay in time compared with the theoretical prediction, indicating that the theory overestimates the actual effective gas diffusivity. An explanation of this discrepancy lies in the fact that the wet inner wall of the tube confined between two films (3 mm in distance) results in substantial lateral absorption. Note that Henry’s coefficient cancels in Eq. [4] when substituting ks (H ) from Eqs. [6]–[8], although gas solubility still has a significant effect on ke due to the gas-solubility dependence of the energy of penetration. Finally, a comparison of effective film resistance for NH3 and Ar demonstrates the effect of the higher solubility of NH3 . This corrects our previous work (16) in which we assumed that the energies of penetration E w and E f are approximately the same for most gases. V. CONCLUSIONS

FIG. 9. Theoretical dimensionless diffusion flux of Ar using data from Fig. 7 (SDS surfactant solution of 2 × 10−3 M). Dimensionless flux is defined as Nd (xd , td ) = N (x, t)/(DC1 /L), with td = t D/l 2 and xd = x/L.

The relationship between the effective diffusivity and foam films has been predicted well by a new film penetration model. In good agreement with the nucleation theory, the model suggests that: (a) the overlapping structure of the two monolayers constituting the foam film imposes a significant resistance to gas

476

NGUYEN, ZITHA, AND CURRIE

mass transfer; and (b) the dynamics of the surface, particularly the effective area of adsorbates, should be taken into account. This theory gives a good estimation of the quantities of interest in this study, but further work is needed on the expression of the penetration energy-effective area relationship. The effect of surfactant type, particularly hydrophobic chain length, is expressed partially through the expression of the surface potential. To elucidate this effect requires an explicit expression of the energy penetration across the occupied sites. It may be surmized that this relation should take the same form as the above expression of E a . Our newly developed freely standing film diffusion method can be used to measure the diffusivity of gases and vapors and the permeability of foam film to these gases. This method is also designed for the investigation of tracer diffusion across a population of trapped foam films in porous media, where these foam films are frequently assumed to stay flat in pore throats. The lateral absorption observed in the NH3 experiments has an important implication for resistance to tracer diffusion. On one hand, it will cause significant delay of the tracer transfer into trapped gas in addition to the resistance effect of foam films. More tracer may also be lost into the liquid phase due to its high solubility on the other hand. ACKNOWLEDGMENTS We thank Michiel H. G. Thissen who conducted most of the experiments with the inert gas reported here. This work was conducted in the framework of the foam project sponsored by Shell and Halliburton European Research Centre.

REFERENCES 1. Kovscek, A. R., and Radke, C. J., in “Fundamentals and Applications in the Petroleum Industry,” ACS Advances in Chemistry Series 242. American Chemical Society, Washington, DC, 1994. 2. Rossen, W. R., and Gauglitz, P. A., Am. Inst. Chem. Eng. J. 36, 1176 (1990). 3. Quoc, P. N., Alexandrov, V. A., Zitha, P. L. J., and Currie, P. K., paper SPE 58799, in “Proc. SPE International Symposium on Formation Damage Control,” Lafayette, Lousiana, 23–24 Febuary 2000, p. 735. 4. Bernard, G. G., Holm, L. W., and Jacobs, L. W., SPE J. 5, 295–300 (1965). 5. Friedmann, F., Chen, W. H., and Gauglitz, P. A., SPE Reservoir Eng. 6, 37–45 (1991).

6. Radke, C. J., and Gillis, J. V., paper SPE 20519, in “Proc. 65th Annual Technical Conference and Exhibition of the SPE,” New Orleans, 23– 26 September 1990, p. 475. 7. Molder, E., Tenno, T., and Nigu, P., Crit. Rev. Anal. Chem. 28, 75–80 (1998). 8. Goodridge, F., and Bricknell, D. J., Trans. Inst. Chem. Engrs. 40, 54–60 (1962). 9. Harvey, E. A., and Smith, W., Chem. Engrg. Sci. 10, 274 (1959). 10. Princen, H. M., and Mason, S. G., J. Colloid Interface Sci. 20, 353–357 (1965). 11. Princen, H. M., and Mason, S. G., J. Colloid Interface Sci. 24, 125–130 (1967). 12. Nedyalkov, M., Krustev, R., Kashchiev, D., Platikanov, D., and Exerowa, D., J. Colloid Polym. Sci. 266, 291–296 (1988). 13. Nedyalkov, M., Krustev, R., Platikanov, D., and Stankova, A., Langmuir 8, 3142–3144 (1992). 14. Nedyalkov, M., Krustev, R., Platikanov, D., and Stankova, A., J. Disp. Sci. Technol. 18, 789–800 (1997). 15. Platikanov, D., Nedyalkov, M., Krustev, R., and Kashchiev, D., Langmuir 12, 1688–1689 (1996). 16. Quoc, P. N., Alexandrov, V. A., Zitha, P. L. J., and Currie, P. K., in “Proc. 3rd Euroconference on Foams, Emulsions, and Applications,” 4–8 June 2000, p. 215. 17. Coulson, J. W., and Richardson, J. F., in “Chemical Engineering,” Vol. 1, Chap. 10. Pergamon, Oxford, 1990. 18. Paul, C. H., in “Principles of Colloid and Surface Chemistry,” Vol. 9, Chap. 7. Marcel Dekker, New York, 1986. 19. Davies, J. T., and Rideal, E. K., in “Interfacial Phenomena,” Chap. 4. Academic Press, New York, 1961. 20. Lodge, J. P., in “Methods for Air Sampling and Analysis.” Lewis, Chelsea, 1995. 21. Malhotra, A. K., and Wasan, D. T., AIChE J. 9, 1533–1540 (1987). 22. Robert, H. P., and Don, W. G., in “Perry’s Chemical Engineers Handbook,” 50th ed. Chemical Engineers Series, 3-285. McGraw-Hill, New York, 1985. 23. Cullen, E. J., and Davidson, J. F., Trans. Inst. Chem. Engrs. 35, 51–59 (1957). 24. Holtzclaw, H. F., Robinson, W. R., and Odom, J. D., in “General Chemistry,” 7th ed., Chap. 4. Health, Lexington, 1991. 25. Cockbain, Trans. Faraday Soc. 50, 874 (1954). 26. Exerova, D., and Kashchiev, D., Contemp. Phys. 27, 429–442 (1986). 27. Exerova, D., Kashchiev, D., and Platikanov, D., Adv. Colloid Interface Sci. 40, 201–212 (1992). 28. Jones, M., Mysels, K. L., and Scholten, P., Trans. Faraday Soc. 62, 1336– 1342 (1966). 29. Exerova, D., and Kruglyakov, P. M., Foam and Foam Films. Elsevier, Amsterdam, 1998. 30. Aronson, A. S., Bergeron, V., Fagan, M. E., and Radke, C. J., Colloids Surf. A: Physicochem. Eng. Aspects 83, 109–120 (1994). 31. Winkelmann, A., Ann. Physik. 22, 152 (1884).