Theory of Rate of Solubilization into Surfactant Solutions - American

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Langmuir 2003, 19, 4014-4026

Theory of Rate of Solubilization into Surfactant Solutions D. Sailaja, K. L. Suhasini, Sanjeev Kumar,* and K. S. Gandhi* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India Received November 19, 2002. In Final Form: February 9, 2003 Solubilization of sparingly soluble solutes into aqueous surfactant solutions has attracted considerable attention in view of its application in many areas. Earlier studies differ, both qualitatively and quantitatively, in their conclusions about the beneficial role played by micelles on the rate of solubilization. In the present work, a general model is developed to explain the solubilization kinetics in the presence of micelles. The model considers molecular solubilization and exchange of solute between the micelles and the continuous phase to be the basic steps in the dissolution of solutes. Additionally, in the case of ionic surfactants, contact between the micelles and the solute surface is prohibited to account for ionic repulsion. It is shown that when the partition coefficient of the solute between the micellar phase and the continuous phase is small and the diffusion coefficient of the micelles is comparable to that of the solute in the continuous phase, the micelles and the continuous phase exist in equilibrium. This occurs even if direct contact between the micelles and the solute surface is prevented. Under these circumstances, the solubilization occurs by diffusion, in parallel, through the continuous and the micellar phases. Due to the latter, the solubilization rate can be enhanced in proportion to the magnitude of the partition coefficient and the amount of surfactant added. If the partition coefficient is large and the diffusion coefficient of the micelles is small, the enhancement factor is considerably smaller than that when the two phases are in equilibrium. The enhancement factor in this limit is proportional to the square root of the amount of surfactant added. Under such circumstances, the exclusion of direct contact between the micelles and the solute surface, and surface resistance if present, lower the enhancement factors further. The model quantitatively predicts the observations on the dissolution rates of decane and benzene drops into SDS solutions reported by Todorov et al. (J. Colloid Interface Sci. 2002, 245, 371). For benzene drops, the surface resistance is negligible, the micelles are in equilibrium with the continuous phase, and the enhancement factor is proportional to the surfactant concentration. On the other hand, for decane drops, the micelles are not in equilibrium with the continuous phase, and the enhancement is proportional to the square root of the surfactant concentration. The enhancements in Ostwald ripening rates in the presence of surfactants are also predicted from the model, and these are in good agreement with observations for decane emulsions in sodium dodecyl sulfate. The model is also applied to dissolution of fatty acids into SDS solutions in the rotating disk setup. It is observed that surface resistance plays an important role and that the diffusivities of micelles obtained here are different from the ones obtained in dissolution of benzene drops.

1. Introduction Micellar and microemulsion systems, which are microheterogeneous in nature, are well-known now and have been identified as having enormous potential in many technical applications. They also serve as well-defined prototypes for the more complex biological membrane systems. A few applications where these systems have benefited the processes and enhanced their efficiency are detergency, tertiary oil recovery in the petroleum industry, cleansing of soil from pollutants by the surfactantenhanced solid remediation (SESR) technique, nanosized particle preparation, phase transfer catalysis, wastewater treatment, and drug delivery. One process common and critical to many of the above applications is solubilization of solutes into micellar and microemulsion systems. Apart from the applications just mentioned, the solubilization process is of fundamental importance in understanding destabilization of emulsions caused by Ostwald ripening.1 Consider a pure solute dispersed in a micellar solution. Driven by the difference between the saturation concentration (solubility) of the solute in the continuous phase and the concentration prevailing there, the solute dissolves into the continuous phase by diffusion. This is commonly referred to as molecular solubilization to distinguish it from the following mechanisms in which micelles play a role. Micelles can solubilize solute directly from the solute * To whom correspondence should be addressed. (1) Taylor, P. Adv. Colloid Interface Sci. 1998, 75, 107.

surface and carry it away as they diffuse. This will be referred to as direct solubilization. Alternatively, the micelles take up solute from the continuous phase present around them. This will be referred to as indirect solubilization. The solubilization can be assisted by the presence of micelles through both these mechanisms. The present work is directed toward an understanding of the kinetics of the solubilization process that accounts for the role of micelles. We briefly review the major treatments of the solubilization process to show the different approaches, and we propose a general model that includes all of them. We apply this model to a few systems and discuss various implications of the model. The discussion presented and the model developed are applicable to both micelles and reverse micelles, since the mechanism of solubilization is similar in both systems. 2. Previous Work Usually, a film model is used to calculate the molecular diffusional mass flux, J, of a solute into a continuous phase. When the concentration of the solute in the continuous phase is very small, flux can be calculated from

J)

D C δ s

where D is the diffusion coefficient, Cs is the solubility of the solute in the continuous phase, and δ is the thickness

10.1021/la0268698 CCC: $25.00 © 2003 American Chemical Society Published on Web 04/01/2003

Theory of Rate of Solubilization into Surfactant

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of the diffusion film. Higuchi and Misra,2 in one of the earliest works on solubilization of solutes into micellar media in the context of Ostwald ripening, suggested that the above expression can account for micellar assisted dissolution if Cs is replaced by Ctot, the solubility of the solute in the surfactant solution. A similar expression can also be derived, albeit with a small difference, if it is assumed that the micelles are in contact with the surface of the solute and are saturated with the solute, like the continuous phase in contact with solute is. Such an approach results in the following expression for flux:

J)

Deff C δ s

(1)

The effective diffusion coefficient Deff is given by

Deff ) (1 - φ)D + KeqφDm

(2)

where Keq is the partition coefficient of the solute between the micellar phase and the continuous phase, φ is the volume fraction of the micelles in the solution, and Dm is the diffusivity of the micelles. Expressions similar to the above have been presented by Kabalnov3 and Grimberg et al.4 and validated by Parrott and Sharma5 through their experiments on the dissolution of benzoic acid into solutions of SDS, an anionic surfactant. This model has also been widely validated in the context of extraction of polyaromatic hydrocarbons (PAHs) from soils using reverse micellar solutions of nonionic surfactants, for example, by Grimberg et al.4 and others. The above expression makes a remarkable prediction for the solubilization of sparingly soluble solutes in the continuous phase. The value of Keq for such solutes can be as high as a million while the diffusivity of micelles in which they are solubilized is only about 10 times smaller than that of the solute in the continuous phase. Under such circumstances, the above expression predicts micelles to enhance the mass transfer flux by several orders of magnitude. Several studies6-9 (and the references therein) on the rate of Ostwald ripening in emulsions prepared using nonionic surfactants also seem to agree with the predictions of eq 1 for the enhancement of solubilization due to the presence of micelles. A few dissenting results10 have also been reported. Although a successful prediction of the rate of solubilization of benzoic acid into SDS solutions5 by eq 1 apparently confirms the direct solubilization mechanism, the presence of this mechanism in solutions of ionic surfactants has been questioned. Kabalnov3 observed that enhancements in the rates of Ostwald ripening occurring in emulsions made of SDS were very much smaller than those predicted by eq 1, and this, with some disagreement about the magnitude of the enhancements measured, has been subsequently corroborated by other investigators, for example, Taylor11 and Soma and Papadopolous.12 Kabalnov3 himself provided a valuable insight to resolve this riddle. In the case of ionic surfactant solutions, one (2) Higuchi, W.; Misra, J. J. Pharm. Sci. 1962, 51, 459. (3) Kabalnov, A. S. Langmuir 1994, 10, 680. (4) Grimberg, S. J.; Nagel, J.; Aitken, M. D. Environ. Sci. Technol. 1995, 29, 1480. (5) Parrott, E. L.; Sharma, V. K. J. Pharm. Sci. 1967, 56, 1341. (6) McClements, D. J.; Dugan, R. S. J. Phys. Chem. 1993, 97, 7304. (7) Weiss, J.; McClements, D. J. Langmuir 2000, 16, 5879. (8) Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Rippon, S.; Lubetkin, S. D.; Mulqueen, P. J. Langmuir 1998, 14, 5402. (9) Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Rippon, S.; Lubetkin, S. D.; Mulqueen, P. J. Langmuir 1999, 15, 4495. (10) Hoang, T. K. N.; La, V. B.; Derimaeker, L.; Finsy, R. J. Langmuir 2001, 17, 5166.

does not expect any direct contact between the micelles and the solute drop due to the repulsion between the coions adsorbed on the solute surface and the similarly charged micelles. Hence, if micelles can assist solubilization, they would have to enter the diffusion film from the bulk without coming in direct contact with the surface of the solute, solubilize solute from the continuous phase only, and carry it back. Kabalnov3 estimated the time required for a micelle to solubilize the solute to its full capacity and the time required for a micelle to traverse the thickness of the diffusion film. He found the latter to be very small compared to the former, which indicates that micelles do not spend enough time to pick up solute to their full capacity. Thus, the solubilization rates would be much smaller than those predicted by eq 1. Kabalnov and Weers13 captured the new picture by an intuitive model. Since micelles may extract solute to only a very small fraction of their full capacity, it is assumed in this model that the micelles act like a reactant with which the solute reacts irreversibly. This model, when applied to Ostwald ripening, showed qualitative agreement between the rates predicted by the model and the observations. Information regarding the diffusivity and solubilization capacity of the micelles does not enter the model at all. The model therefore represents a limiting case of a more general process of solubilization and does not have the ability to assess the relative effects when parameters of the system change. Consider solubilization of drops of decane of size 100 nm into SDS solution. The time required for a micelle of size 3 nm to be saturated is estimated3 to be 0.1 s while the diffusion time is calculated3 to be about 0.001 s. If the drop size is increased to 30 µm, the latter increases to about 90 s while the former remains unaltered, and one concludes that micelles have enough time to get fully saturated. Recently, Todorov et al.14 reported carefully carried out measurements on dissolution of a single drop of decane of size 30 µm into SDS solutions. They found that the observed dissolution rates were far less than those predicted by eq 1 even though, following the argument of Kabalnov,3 there was enough time for the micelles to get saturated. Todorov et al.14 used the model of Kabalnov and Weers13 and had to additionally incorporate surface resistance to secure agreement between the model and the observations. It is also to be noted that they used the same model for predicting observations on solubilization of benzene into SDS solutions even though the capacity of micelles to solubilize benzene is not high. Thus, criteria to assess the effects of micelles on solubilization into surfactant solutions are still uncertain. It is important to point out one more issue. A feature relevant to reverse micellar systems, which is absent in micellar systems, is the exchange of solute between reverse micelles by fusion of two micelles and fission of the dimer. It will be difficult to apply the criteria developed by Kabalnov3 for such systems. Cussler and associates15-17 studied detergency and dealt with ionic surfactants and solutes with low solubility in (11) Taylor, P. Colloids Surf., A 1995, 99, 175. (12) Soma, J.; Papadopolous, K. D. J. Colloid Interface. Sci. 1996, 181, 225. (13) Kabalnov, A. S.; Weers, J. Langmuir 1996, 12, 3442. (14) Todorov, P. D.; Kralchevsky, P. A.; Denkov, N. D.; Broze, G.; Mehreteb, A. J. Colloid Interface Sci. 2002, 245, 371. (15) Chan, A. F.; Evans, D. F.; Cussler, E. L. AIChE J. 1976, 22, 1006. (16) Chan, A. Solubilization kinetics in detergency. Ph.D. Thesis, Carnegie-Mellon University, 1977. (17) Shaeiwitz, J. A.; Chan, A. F.-C.; Cussler, E. L.; Evans, D. F. J. Colloid Interface. Sci. 1981, 84, 47.

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the continuous phase. Assuming that the solutes were insoluble, they proposed a five-step mechanism: (i) diffusion of surfactant molecules toward the solid surface, (ii) adsorption of the surfactant at the surface, (iii) surface “reaction” of the solute and the adsorbed surfactant to produce micelles with solute, (iv) desorption of the micelles containing the solute, and (v) diffusion of the desorbed micelles into the bulk. Solute exchange between micelles and the continuous phase, and direct contact between micelles and the solute surface, both are not permitted in this model. Instead, surfactant molecules form micelles containing solute on the surface, and these desorb. From studies conducted on dissolution of lauric, palmitic, and stearic acid into SDS solutions in a rotating disk configuration, they15,17 found desorption of micelles and their diffusion to be the rate-limiting steps. The model proposed by them reduces to the one represented by eq 1 if all steps other than the diffusion of micelles are fast. It would therefore be interesting to see if their model is in some ways equivalent to the model proposed by Kabalnov and Weers.13 There is clearly a need for a theory which is able to capture the role of micelles in the solubilization process in a comprehensive way. This model should allow prediction of conditions which lead to different effects of micelles on mass transfer. In the following sections, such a model is proposed. It focuses on ionic surfactant systems where direct contact between micelles and solute surface is not permitted. We also briefly touch upon the consequences of allowing such a contact which might be applicable to solubilization into solutions of nonionic surfactants. 3. Present Model In the present work, we model the surfactant solution containing micelles as a micro-heterogeneous medium. We treat micelles as the dispersed phase and the rest of the solution as the continuous phase. Solubilization occurs when a micellar solution is brought into contact with the solute, present as either solid or another immiscible liquid. We refer to this for short as the solute phase. The following mechanism of solubilization is proposed. The solute can dissolve into the continuous phase to the extent of its solubility there. The solute molecules enter directly into the solvent due to the difference in the chemical potential by the usual molecular solubilization process. We, however, allow for the presence of surface resistance. Thus, unless the surface resistance is negligible, the solute phase and the continuous phase will not be in equilibrium at the interface. Micelles present in the continuous phase can augment the rate of solubilization by picking up solute from the continuous phase (indirect mechanism) and diffusing to regions away from the solute surface. Such solute exchange occurs by the process of diffusion from the continuous phase to the surface of the micelle, and then into its interior or vice versa, and is referred to as interphase mass transport, well described by Bird et al.18 Independent of these processes, micelles can also solubilize solute directly from the solute surface provided they can approach/adsorb onto it. The adsorbed micelles may desorb from the surface, either partially or fully saturated with the solute, and then diffuse away, carrying solute with them. Micelles, irrespective of how they pick up solute, can fuse and redisperse in the bulk, redistributing their contents among the daughter micelles randomly. In the present work, we do not include the (18) Bird, R. B.; Stewart, W. E.; Lightfoot, N. E. Tranport Phenomena; Wiley: 1961.

Sailaja et al.

micellization-demicellization19 mechanism, as it has been considered unlikely3 in ionic surfactant solutions. We further assume that the presence of solute in the system does not change the critical micellar concentration, micelle size and shape, and their diffusivities. Spatial dependence on only one dimension is assumed to simplify presentation. We cast the model in the spherical coordinates to facilitate comparison with the results obtained on dissolution of single drops by Todorov et al.14 Thus, the space of interest is contained in the range R e r < ∞, where R is the radius of the drop and r is the radial coordinate. Since convection is not present in most of the experimental results considered later, it is not included in the development. No new principles are involved in removing these two restrictions to further generalize the model, and we present the final equations for transfer from a rotating disk that include convection, and compare predictions with the experiments of Chan.16 3.1. Conservation Equations. Description of Micelles. The micelles present in the same neighborhood may not contain solute at the same level of concentration due to their random diffusive movement, change in solute content due to exchange with the continuous phase, and fusion and fission processes which are possible for reverse micelles. The most appropriate way of describing such populations is through population balances.20 Such an approach has been used by us21 successfully to model nanoparticle formation in reverse micellar systems. In this approach, the micelles are characterized by their location r, time t, and solute content. Thus, let n(Cm, r, t) be the number density of micelles present at location r at time t; n(Cm, r, t) dCm represents the number of micelles with solute concentration in them in the range Cm to Cm + dCm at location r at time t per unit volume of the surfactant solution. Though we have chosen continuous description for the solute concentration, the model can also be developed equivalently in terms of number of molecules present in the micelle, as long as the average occupancy is not very much smaller than unity. Solute Exchange between the Micelles and the Surroundings. If the partition coefficient between the micellar and the continuous phase is large, as is expected in all cases of practical interest, the rate of solute transfer from the continuous phase into the micelles is controlled by diffusion from the continuous phase to the surface of the micelles.18 In view of the smallness of the Peclet numbers, the diffusive flux of solute normal to the micelle surface, Jsm, is then given by18

Jsm )

(

)

Cm 2D′ Cdm Keq

(3)

C is the concentration of solute in the continuous phase, and Keq is the partition coefficient and is defined as the ratio of concentration in the micelles per unit volume of micelle to that in the continuous phase per unit volume of the continuous phase under conditions of equilibrium. D′ is the diffusion coefficient of solute in the micellar solution, and it can be different from D, the diffusivity in the pure continuous phase, since the viscosities of the micellar solution and solvent can be different. Micelles are assumed to be spherical and of diameter dm. The rate of the total local solute exchange (M ˙ ) between the micelles and the solvent surrounding it, per unit volume of the (19) Carroll, B. J. Colloid Interface Sci. 1981, 79, 126. (20) Ramkrishna, D. Population balances; Academic Press: 2001. (21) Bandyopadhyaya, R.; Kumar, R.; Gandhi, K. S.; Ramkrishna, D. Langmuir 1997, 13, 3610.


entire system, is then given by

M ˙ )

(

C

)

∫0C 2πD′dm C - Keqm n(Cm, r, t) dCm T

Here, CT is the maximum possible concentration of solute in a micelle and corresponds to the maximum amount of solute that can be solubilized in a micelle. Suppose a surfactant solution with surfactant concentration S solubilizes solute up to a concentration of m. If Vm is the volume of a single micelle and Nag is the aggregation number, then

(m - Cs)Nag

(m - Cs) CT ) ) ≡ KeqCs φ (S - cmc)NAVm where cmc is the critical micellar concentration and NA is Avogadro’s number. If (m - Cs) increases linearly with (S - cmc), both CT and Keq are independent of surfactant concentration. In the case of solubilization into surfactant solution, it is expected that Cs g C and CT ) KeqCs g KeqC > Cm and Jsm approaches zero as Cm approaches KeqC. Mass Balance of Solute in the Continuous Phase. The processes involving the solute dissolved in the continuous phase are diffusion of the solute through the continuous phase and its exchange with that in the micelles. The equation for the solute mass balance in the continuous phase for these processes can be written as

(1 - φ)

( )

1 ∂ 2 ∂C ∂C ) (1 - φ)D′ 2 r ∂t ∂r r ∂r Cm CT 2πD′dm C n(Cm, r, t) dCm (4) 0 Keq

(

∫

)

We assume that the solution is initially uniform. The boundary condition at r ) R should account for surface resistance, while that at r f ∞ should specify that the concentration remains undisturbed at the initial value:

For all r: C ) C∞; t ) 0 For all t: ∂C ) ks(Cs - C), at r ) R; C ) C∞ as r f ∞ (5) -D′ ∂r ks characterizes surface resistance. To solve eq 4, the concentration in the micelles has to be determined. Population Balance Equation for Micelles in the Continuous Phase. The number of micelles can change in some solubilization processes when the surfactant concentration changes or when the size of micelles changes due to the uptake of solute, for example, in Ostwald ripening. If a change in their total number is small, it can be assumed to remain constant. The population balance equation for diffusion of micelles, exchange of solute with the continuous phase, and their fusion and fission is given by (see eqs 2.10.3 and 3.3.5 of ref 20)

∂ ∂ (n(Cm, r, t)C˙ m) ) n(Cm, r, t) + ∂t ∂Cm CT 1 ∂ ∂ Dm 2 r2 n(Cm, r, t) + qn(Cm, r, t) 0 n ∂r ∂r r CT CT (Cm, r, t) dCm - q 0 0 n(C′m, r, t) n(C′′m, r, t)

∫

( )

∫ ∫

P(Cm:C′m, C′′m) dC′m dC′′m (6)

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C˙ m is the rate of change of concentration in a micelle due to exchange with the continuous phase, q is the coalescence frequency, and P is the probability that a micelle with solute concentration Cm is produced when two micelles with concentrations C′m and C′′m fuse and dissociate. According to the mass transfer model described in the previous section, C˙ m is given by

C˙ m )

(

Cm 12D′ C2 Keq dm

)

(7)

In writing the population balance equation, we have assumed that a dimer formed upon coalescence dissociates into two micelles immediately and the contents of the dimer are redistributed randomly. This assumption can be relaxed by allowing for breakage frequency. The coalescence frequency q can be taken to be zero for micellar solution and a suitably modified Brownian collision frequency for reverse micellar systems. The initial condition needed to solve the above equation is uniform concentration in micelles, in equilibrium with the continuous phase. Thus,

n(Cm, r, 0) ) N0δ(Cm - KeqC∞)

Rer 0 (12)

Coupling between the Two Populations of Micelles. The coupling between the conservation equations for micelles in the continuous phase and the equations for those adsorbed on the surface is provided by the balance of the number of micelles at the interface. Hence, the micellar number flux at the solute surface equals the net rate of adsorption onto the surface, that is,

-Dm

∂ n(Cm, r, t) ) Kdnˆ (Cm, t) ∂r ˆ 0) t > 0, r ) R (13) Kan(Cm, r, t)(1 - aˆ pN

3.2. Moments of Balances. The models of solubilization proposed earlier in the literature deal with overall concentrations and mass transfer rates. The present model can be connected to those by calculating the moments of the population balance equations of the previous section. Zeroth Moment. The zeroth moments of the balance equations and boundary conditions are obtained by integrating the respective equations from Cm ) 0 to CT. These equations, though not given here, show that the total micelle population N0(r, t) is a constant, independent of position and time. This is consistent with the assumption that micelles neither leave the system nor are created or consumed. First Moment. The first moment of the population balance equation in the micellar phase gives an equation for C h m, the average concentration of solute in the micellar phase, defined as

C h m ) KeqC∞ at all times as r f ∞

dC ˆm ) Kd(C h m(R, t) - C ˆ m) + kˆ map(KeqCs - C ˆ m) dt

(16)

(17)

where C h m(R, t) is the average concentration of solute in the micelles at the interface. The first moment of the initial condition yields C ˆ m ) 0. (iii) Coupling Condition. The first moment of the coupling condition (eq 13) is given by

-Dm

∂C hm φˆ ) Kd (C ˆ -C h m(0, t)) t > 0, r ) R (18) ∂r φ m

where φ and φˆ , the volume fraction of micelles in the bulk and the area fraction occupied by them on the surface, are ˆ 0aˆ p, respectively. Equation 18 provides the N0vp and N second boundary condition needed to solve eq 15. Solute Concentration in the Continuous Phase. Using the definition of the first moment and the expression derived for the solute exchange between micelles and the continuous phase, eq 4 is rewritten as

(1 - φ)

( )

(

)

C hm 1 ∂ 2 ∂C 12D′φ ∂C ) (1 - φ)D′ 2 r C2 ∂t ∂r ∂r K r dm eq (19)

(15)

The initial and boundary conditions are still given by eq 5. 3.3. Special Cases and Analytical Solutions. Two special cases, representing extremes of micellar solubilization behavior, are considered in this section. One allows for direct pickup of solute by the randomly diffusing micelles while the other disallows any contact between the micelles and the solute surface but accounts for surface resistance. Mass Flux with Direct Solubilization. Here we consider direct pickup of solute by the continuous phase and the micelles under steady-state conditions with no surface resistance to all surface processes. The balance equations developed in the previous section (eqs 15 and 19) can therefore be solved directly after dropping the un-steadystate terms. The boundary conditions, however, need to be considered in detail. The solute is assumed to dissolve into the continuous phase without experiencing any surface resistance. Likewise, the micelles adsorbed on the solute surface are assumed to rapidly saturate with solute without experiencing any surface limitations. In this situation, ks and kˆ m approach infinity. Consequently, the driving force for transport into the continuous phase and the micelles on the solute surface (r ) R) can be equated to zero. The boundary conditions therefore simplify and are stated as

The initial condition and one of the boundary conditions for this equation, obtained by multiplying eqs 8 and 9 by

h m ) KeqCs at r ) R, and C f C ) Cs, C h m ) KeqC∞ as r f ∞ C∞, C

C h m(r, t) )

∫0C Cmn(Cm, r, t) dCm

1 N0

T

(14)

(i) Average Solute Concentration in the Micellar Phase. An equation for C h m is obtained by multiplying eq 6 with Cm and integrating from Cm ) 0 to CT. Thus,

φ

( )

(

)

hm C hm ∂C hm 1 ∂ 2 ∂C 12D′φ ) φDm 2 r + C2 ∂t ∂r Keq r ∂r dm


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The micelles on the surface are thus in equilibrium with the solute and the coupling between the micelle population on the surface and that in the bulk becomes redundant. If eqs 15 and 19 are added up after multiplying with Keq/[(1 - φ)D′] and -1/(φDm), respectively, the following equation results for the steady state:

( )

1 d 2 d r (KeqC - Cm) r2 dr dr φ D′ 12 (KeqC - C h m) ) 0 + dm2 (1 - φ) KeqDm

(

)

The boundary conditions for this new differential equation, derived from those stated earlier, are

h m ) 0 both at r ) R and as r f ∞ KeqC - C

diffusion equation, given below, is valid.

( )

∂Cf 1 ∂ 2 ∂Cf r )D 2 ∂t ∂r r ∂r

Jd ) -φDm

dC hm dC - (1 - φ)D′ r)R dr dr

(20)

which simplifies to

Jd )

D′(1 - φ) + φDmKeq (Cs - C∞) R

-

Cf ) C;

The total flux of solute into the system, Jd, is the sum of fluxes through the continuous phase and the micellar phase. Thus,

(21)

Mass Flux with Only Indirect Solubilization. (i) Exclusion Layer. Due to the repulsive forces exerted by an ionic surfactant adsorbed on the solute surface, the micelles are excluded from direct contact with it. An exact theory will have to treat micelles as macroions and calculate their concentration in the vicinity of the charged solute surface. The concentration of micelles, similar to what is observed in the double layer, is expected to be nearly zero at the solute surface and attain the bulk value after some distance away from the surface. An exact calculation of the concentration profile is very complex, and to avoid it, we model the effect by postulating that, in the immediate vicinity of the solute surface, a thin layer of only continuous phase exists. Thus, only molecular solubilization and diffusion occur in the region R < r < R + κ, where κ is the thickness of the “exclusion” layer. As micelles cannot come into this layer, the flux of micelles is zero at the boundary of this thin layer and the micellar solution. However, the solute that diffuses through this layer and enters the domain r > R + κ occupied by the micelles can be picked up by them by the indirect solubilization mechanism. Mathematically, the exclusion layer is incorporated in the general model presented earlier by merely displacing the surface at r ) R to r ) R + κ and simplifying or modifying the boundary conditions, as the case may be. In the region R e r < R + κ, the solute diffuses only through the continuous phase, and hence the usual

(22)

Here, Cf represents the concentration of the solute in the micelle-free zone. Equations 15 and 19 for R + κ e r < ∞ and eq 22 for R e r < R + κ thus form a complete system of equations. (ii) Solution of the Equations. In the experiments of Todorov et al.14 as well as in Ostwald ripening experiments, the diffusion layer around the drop is in pseudo-steadystate with respect to the bulk conditions. The convective velocities are small. The steady-state version of eqs 15, 19, and 22 thus constitutes the complete set of equations that needs to be solved to predict the dissolution rates. The boundary conditions are given by

Hence, C h m ) KeqC is the only solution for all values of r. Substituting this into the steady-state versions of eqs 15 and 19 and solving them, the concentration profiles are determined to be

C - C∞ h m - KeqC∞ R 1 C ) ) Cs - C∞ Keq Cs - C∞ r

RereR+κ

dCf ksR ) (C - Cf) at r ) R dr D s

hm dCf (1 - φ)D′ dC dC ) ; ) 0 at r ) R + κ dr D dr dr C)

C hm ) C∞ r f ∞ Keq

(23)

The first boundary condition represents surface resistance for solute transport into the continuous phase. The second and third boundary conditions represent continuity of concentration and flux at the interface of the micelle free zone and the micelle occupied zones. The fourth boundary condition represents no entry (zero flux) of micelles into the micelle free zone (and can also be obtained from eq 18). The last boundary condition specifies the concentrations far away from the solute interface to be the same as those taken initially. Similar equations are also obtained in solving problems of diffusion accompanied by reversible chemical reaction, and their solutions are well documented.22 We thus give the solution directly in terms of nondimensional quantities defined below:

r* )

C - C∞ / r / Cf - C∞ ; Cf ) ; C* ) ;C h ) R Cs - C∞ Cs - C∞ m C h m - KeqC∞ Keq(Cs - C∞)

(

1 - C/f (r*) ) J* 1 -

C*(r*) C/f (λ)

)

)

D 1 + r* ksR

(24)

(25)

1 λ [Dam(Rλ + 1) + DaeR(λ-r*)] r* R2 + DamRλ (26)

C/m(r*) C/f (λ)

)

Dam 2

λ (27) [(Rλ + 1) - eR(λ-r*)] r*

R + DamRλ

where R ) xDa+Dam and λ ) 1 + κ/R. Da and Dam are two Damkohler numbers for the continuous and micellar (22) Crank, J. The mathematics of diffusion; Oxford, 1957.

4020


Sailaja et al.

phases, respectively, and are given by

Da )

12φ R2 R2 D′ ; Dam ) 12 2 2 1-φd KeqDm d m

(28)

m

Further, J* is the nondimensional flux of the solute,

J* )

dCf JR at r* ) 1 )dr* D(Cs - C∞)

(29)

Dam, approaches zero. Kabalnov3 has interpreted Dam as the ratio of the residence time of a micelle in the diffusion zone (of length dimension equal to drop radius) to the time required by a micelle to get saturated. Kabalnov3 therefore considers the limit of Dam , 1 to be adequate for the micelles to remain empty as they go away from a drop. On these grounds, Dam . 1 is expected to be adequate for the micelles to become fully loaded or equilibrated. Kabalnov and Weers13 subsequently made use of the limiting behavior for Dam , 1 and solved the following mass balance equation for the continuous phase

and is given by

1 D 1 ) + J* p ksR

0) (30)

Ω)

(1 - φ)γRλ(Rλ + 1) (31) p) (1 - φ)γR(Rλ + 1)(λ - 1) - Daλ + R(Rλ + 1)

4. Results and Discussion It is interesting to note that eqs 15 and 19 with the boundary conditions given by eqs 5 and 16 are the usual diffusion equations for a two-phase system. These could have been written directly also if it is assumed that all the micelles at a given location have the same solute concentration in them.23 The general framework used here shows how the conclusions drawn with the above assumptions hold even when the solute concentration in micelles is not the same due to fusion and fission of micelles and the random amount of solute solubilized by them from the solute surface. 4.1. Direct Solubilization by Micelles. Equation 21 gives the steady-state flux when direct solubilization is permitted. It shows that as long as the micelles are saturated with the solute at the solute-surfactant solution interface, C h m is always equal to KeqC everywhere, and the two phases are in local equilibrium at all points. Under these conditions, solute is not exchanged between the micelles and the continuous phase. The solubilization of the solute by the continuous phase and the micelles has become parallel and independent. This conclusion, a direct consequence of the micelles being saturated at the solute surface, is independent of the values of the diffusion coefficients of the solute and micelles and the interfacial area between the two phases. Equation 21 becomes identical to eq 1 when the bulk is devoid of solute, that is, C∞ ) 0 and the diffusivity of the solute in the surfactant solution is the same as that in the continuous phase. Thus, as long as the micelles can equilibrate with the solute at the interface, the model makes the interesting finding that they remain in local equilibrium with the external phase at all points irrespective of the other conditions, and the enhancements predicted by eq 1 are valid. 4.2. Comparison with the Theory of Kabalnov and Weers. The Damkohler number Dam represents the ratio of the rate at which a micelle receives solute to equilibrate itself to the rate at which it is carried away by diffusion. If the capacity of micelles to absorb solute is large, that is, Keq f ∞, the Damkohler number in the micellar phase, (23) Mehra, A. Chem. Eng. Sci. 1988, 43, 899.

(

)

(32)

to obtain a new criterion for the micelles to be empty.

where p is

and γ is D′/D. The steady-state mass flux expected in the absence of micelles is simply equal to D(Cs - C∞)/R. Thus, J* is the factor by which mass flux is enhanced due to the presence of micelles.

1 d dC* r*2 - DaC* 2 dr* dr* r*

[

][ ][ ][ ]

R2 1 D′ CS 4πD′dmCSNA , 1 (33) Dm Nag Dm Cmic

The present model, in the limit of Dam f 0 and with no direct pickup of solute allowed from the solute surface, yields C h m ) 0 and an equation identical to eq 32 for mass balance in the continuous phase. The criterion expressed through eq 33, when expressed in terms of the notation used in the present work, yields

Dam ,

[

]

Da (1 - φ)NS Dam 4Nag

-1

(34)

NS is the number of solute molecules in fully saturated micelles. This is a much less severe constraint than the one (Dam , 1) considered by Kabalnov,3 but it is not yet complete, as we will see later. The theory of Kabalnov and Weers also does not make any predictions for the other end of the spectrum represented by fully saturated micelles. We will shortly show, while considering various limiting cases, that this picture is incomplete and needs modification. 4.3. Limit of Low Absorption Capacity in Micelles. Let us first consider the special case where the thickness of the exclusion layer is zero and surface resistance is absent, that is, κ ) 0 or λ ) 1, and D/(ksR) ) 0. The flux is then given by

J)

(1 - φ)D′(Cs - C∞) R2 + R R R + Dam

(35)

The micelles have low absorption capacity if Keq is not large. In this limit, as long as the surfactant concentration is not very low, both Damkohler numbers should be large and comparable in magnitude. Hence, R2 . R and Dam . R, which, after they are combined, yield Dam . 1 + (Da/ Dam). Under these conditions, the expression for flux simplifies to

J)

(

)

(1 - φ)D′(Cs - C∞) Da 1+ ) R Dam Cs - C∞ (D′(1 - φ) + KeqφDm) (36) R

This is identical to the prediction made by eq 1 despite prohibiting direct solubilization by the micelles. This result can be understood as follows. If both Damkohler numbers


Langmuir, Vol. 19, No. 9, 2003 4021

are large, we expect that C h m ) KeqC, that is, the micellar phase is in equilibrium with the continuous phase. However, this cannot be valid at r ) R, since there the boundary conditions for the two concentrations require

C ) Cs

J)

dC hm )0 while dr

Thus, no matter how large the Damkohler numbers are, there is a very thin layer near the surface where the two phases are not in equilibrium. Equations 26 and 27 show that C* and C/m/Keq differ for (r* - 1)R log(Da/RDam) ∼ (r* - 1)R log R ∼ 1, or over a layer of thickness R/R ln R. In this layer, to be called an exchange boundary layer, micelles act as sinks for solute and enhance the mass transfer rate. In the regions outside the exchange boundary layer, the micelles are in equilibrium with the local environment and no exchange occurs. The larger the value of the Damkohler number Dam, the thinner is the exchange boundary layer, and the closer is the flux to that obtained by assuming local equilibrium everywhere. Three features of the enhancement with equilibrated micelles (eq 36) are worth noting. The first is that the enhancement caused by micelles, J*, is proportional to φ. As the volume fraction of micelles is proportional to the surfactant concentration, the enhancements are expected to be proportional to the surfactant concentration as well. The second feature is that the enhancement factor is independent of the size of the drop, or the flux is inversely proportional to the size of the drop. The linear dependence on surfactant concentration and inverse proportionality to drop size are features characteristic of this limit and will be referred to later. The third feature is the inadequacy of the criterion that when Dam . 1, micelles reach equilibrium with the continuous phase. A comparison between Da and Dam is needed to arrive at a condition for local equilibrium to hold, which results in a substantially more accurate constraint, that is, Dam . 1 + Da/Dam. Let us now consider the additional effect of having a thin layer (κ/R , 1) that excludes the micelles. Equation 30 under these conditions simplifies to

Dam + R κ 1 D 1 ) + + J* R (1 - φ)γ R2 + R ksR

amounts of solute or when the solubility in the continuous phase is low. In this limit, we expect Dam f 0, which results in the following expression for mass flux:

(37)

The right-hand side can be interpreted as nondimensional resistances to mass transfer. The first term is due to the exclusion layer, the second is due to the presence of micelles, and the last is due to the surface resistance. The effects of the exclusion layer are expected to be unimportant when

κ R2 + R ,1 (1 - φ)γ R Dam + R For commonly employed conditions, the above criterion can be simplified to

R2 κ ,1 R Dam + R 4.4. Limit of High Absorption Capacity in Micelles. Here also we begin with the special case of κ ) 0 or λ ) 1 and D/(ksR) ) 0 of the previous section (eq 35). Keq values tend to be very large when micelles can absorb large

( x ( ))

(1 - φ)D(Cs - C∞) 1+ R

12φ R 1 - φ dm

2

(38)

The above limiting flux is, in fact, valid in a much larger range of Dam , 1 + Da/Dam if there is an enhancement in transport rate due to micelles. Except for the constants appearing in eq 38, it is the same result as that obtained by Kabalnov and Weers13 in the limit Ω , 1 (eqs 33 and 34), which corresponds to the presence of empty micelles in the diffusion zone. We thus see that there are three criteria proposed so far for the presence of an empty micelle: Kabalnov3 proposed Dam , 1, followed by Dam , Da/Dam by Kabalnov and Weers13 and Dam , 1 + Da/ Dam in the present work. Dam , 1 indicates that micelles are empty if they stay for shorter time in the diffusion zone than that required for their saturation. The second criterion (although not interpreted physically by Kabalnov and Weers) brings in the rate at which solute can dissolve from the surface. Thus, if micelles stay for a long enough time in the diffusion zone, they still remain empty if the rate at which solute can dissolve from the surface is much less than that at which it can be removed by the micelles (indicated by the ratio Da/Dam). The third criterion, Dam , 1 + Da/Dam, of the present work brings out the complete picture. Micelles can remain empty under two conditions: when mass transfer into them is slow or the dissolution of solute from the surface itself is slow or both. Equation 38 predicts that the enhancement of mass flux by micelles in this limit, given by the second term in the parentheses, is only proportional to the square root of the surfactant concentration. It is not therefore necessary that when micelles play a role, their effect should be directly proportional to their numbers. Thus, this limit is characterized by flux being proportional to the square root of surfactant concentration and independent of the drop size, in contrast to the previous limit of low Keq. 4.5. Other Cases. Low Diffusivity of Micelles. If the diffusivity of micelles is very small, it is possible that Dam > Da. In the limit of Dam f ∞, the micelles will be in local equilibrium at steady state but will not carry any solute. Thus, the flux will correspond only to transport through the continuous phase. Exclusion Layer Dominates. If resistance due to the exclusion layer dominates all others, it can be seen from eq 37 that flux is independent of radius and depends on surfactant concentration just as κ does. It is likely that this limit is reached at high surfactant concentrations. 4.6. Phase Space. The discussion presented in the previous sections shows that solubilization of solutes in micelles gives rise to many interesting situations depending on the values of the two Damkohler numbers Da and Dam. The behavior predicted by the present model is summarized in Figure 1. The curves going from the extreme left to the right (dashed lines) in Figure 1 divide the phase space into three regions. In the uppermost region, the condition Dam . 1 + Da/Dm is satisfied, which ensures equilibrium between the micelle and the external phase. Under these conditions, the nature of the surface boundary condition (direct versus indirect solubilization) does not play any role. The lower region, bounded by the second curve going from the extreme left to the right represents conditions

4022


Sailaja et al. Table 1. Parameters Used in the Model for Benzene and Decane benzene

Figure 1. Phase space delineating different behaviors expected for solubilization in micelles with no surface resistance, zero exclusion layer thickness, but no direct pickup.

needed for nearly empty micelles in the diffusion zone. Between these two extreme behaviors lie many other features. For example, the entire region of phase space on the upper left side represents the condition of no enhancement by the micelles, due to either their very small diffusivity or their extremely low hold up. Thus, there are regions in the phase space where J * f(φ), J ∝ xφ, and J ∝ φ, and there are regions which mark the transition from one simple limiting description to another. 4.7. Comparison of Predictions with Experiments. The predictions of the theory are compared with the observations of Todorov et al.14 They monitored the radii of single drops of benzene and decane as a function of time during their dissolution into solutions of SDS. The initial radius of the drop and the concentration of the surfactant were varied in these experiments. The radius of a dissolving drop can be predicted from the expression for flux derived earlier. From a mass balance, we have

-

DCs F dR )J) J* ln 2 M dt R

(39)

Equation 39 can be integrated to find the radius as a function of time after substituting for J* from eq 30 and using the known initial size of a drop. It should be noted that J* is a function of the radius itself. In the experiments of Todorov et al.,14 a single drop was located on the wall of a cylinder. The radius of the cylinder is very large compared to the radius of the drop. Hence, the geometry can be treated as a drop located on an infinite plane. Diffusion from a sphere in this configuration is not radially symmetric. The rate of dissolution of such a drop is given by [2πDR(Cs - C∞)] ln 2. The factor ln 2 is the correction factor for asymmetric mass transfer and can be derived from the solution for viscous flows given by O’Neill.24 This correction factor has been used in the above balance assuming that it is valid even when the micelles are participating in diffusion. Thickness of the Exclusion Layer. A value for the thickness of the exclusion layer is needed before the predictions of the model can be compared with observations. As the surfactant concentration in the system is increased, micelles form when the concentration exceeds the cmc. A double layer also forms around the benzene drops as a result of adsorption of the surfactant. Very few micelles will, however, be present for surfactant concentration just above the cmc, and hence, it can be assumed (24) O’Neill, M. E. Chem. Eng. Sci. 1968, 23, 1293.

parameter

value

mol wt solubility in water (mol/cm3) Keq diffusivity in water (cm2/s) diffusivity of micelle (cm2/s)

72 2.275 × 10-5 37.04 8.1 × 10-6 2 × 10-6

decane ref

value

ref

142.68 14 3.65 × 10-10 26 25 1.075 × 106 27 14 7.8 × 10-6 14 fitted 2 × 10-6

that the electrical potential, ψ, calculated corresponding to the concentration of the cmc will still be valid. As the size of the drops is large, the surface can be treated as planar. If the charge density on the drop is assumed to be the same as that on a micelle; it turns out to be large, and hence the electrical potential is given by

eψ ) 4 exp(-lx) kBT where l is the double layer thickness calculated at the concentration of the cmc. The concentration of micelles around the drop can then be calculated by using the Boltzmann distribution:

(

)

RsNageψ N ) exp N0 kBT

where Rs is the fraction of the total micellar charge that is ionized, ψ is the electrical potential, kB is the Boltzmann constant, T is the temperature, and e is the electronic charge. The exclusion layer thickness can be estimated as the distance from the drop surface where the micellar concentration reaches 95% of the bulk value. The exclusion layer thickness calculated this way, assuming that Rs ) 0.3, turns out to be about 25 nm and should be smaller than this value for higher surfactant concentrations. Results for Benzene. To start with, the surface resistance was assumed to be zero, that is, ks f ∞. The solubilities of benzene in water and SDS solutions, and the diffusivities of benzene and micelles are needed to calculate the various parameters. The solubility and diffusivity of benzene in water reported by Todorov et al.14 were used. The solubility of benzene in SDS solutions reported by Liu et al.25 was used. D was assumed to be equal to D′. The diffusivity of micelles is not known. The diameter of the micelle has been given by Todorov et al.14 as 4.8 nm. The diffusivity estimated using this size and the Stokes-Einstein relation turns out to be ∼4 × 10-6 cm2/s. We have used a value of 2 × 10-6 cm2/s, and it fits the data well, as will be shown a little later. The values of the parameters used are listed in Table 1. The Damkohler numbers can be calculated using these values and the initial radius. Consider a drop of initial size of 34.2 µm dissolving in a surfactant solution of 0.2 M concentration. The Damkohler numbers are estimated as Da ) 6.8 × 107 and Dam ) 6.7 × 107. The thickness of the exclusion layer was estimated to be of the order of 25 nm. Using this as an estimate of κ, we find that κ/R in eq 37 is much less than unity, which indicates that the exclusion layer does not play any role. As R2 . R and Dam . R, this system then corresponds to the limit where the enhancement of mass transfer by micelles is proportional to the surfactant concentration and the flux is inversely proportional to the size of the drop, and can be predicted by eq 21, even though micelles are prohibited from direct (25) Liu, G.; Roy, D.; Rosen, M. Langmuir 2000, 16, 3595.


Langmuir, Vol. 19, No. 9, 2003 4023

Figure 2. Comparison of model predictions with observations for benzene. S ) 0.2 M, R0 ) 34.24 and 45.7 µm; S ) 0.25 M, R0 ) 21.83 and 25.77 µm.

contact with the surface of benzene drops. Thus, the rate of change of the drop radius is given by

Cs F dR ) J ) (D(1 - φ) + KeqφDm)(ln 2) M dt R Figure 2 shows a comparison between the temporal dependence of the radius of the drop predicted by the model using eq 39 and the observations. The agreement for all cases is very good. The radius decreases nonlinearly with time. The dependence is such that R02 - R2 ∼ t, which is to be expected, as the flux is inversely proportional to the instantaneous value of R for this case. Another consequence of the success of the model is that the micelles should be in equilibrium with the continuous phase everywhere. This implies that C* ∼ C/m for all r*. Though not shown, a plot of the radial concentration profiles in the continuous phase and the micellar phase confirms that C* ∼ C/m. In view of the good agreement, it can be concluded that surface resistance is absent. Todorov et al.14 used the model of Kabalnov;3 that is, eq 32 was used along with the boundary condition given by C ) Cs at r ) R. However, as we discussed earlier, eq 32 can be used only if Dam f 0. They also confirm this conclusion indirectly, as they found that the micelles do pick up a significant amount of benzene, contrary to the assumptions made. Results for Decane. The solubility of decane in SDS solutions has been reported by Kabalnov et al.27 To be consistent, we have also used the solubility of decane in water reported by the same authors, which is 3.65 × 10-7 mol/dm3. It should be noted that this widely used value has been originally reported by McCauliffe26 and is about (26) McCauliffe, C. A. Science 1969, 163, 478. (27) Kabalnov, A. S.; Makarov, K. N.; Pertzov, A. V.; Shuchikin, E. D. J. Colloid Interface Sci. 1990, 138, 98.

Table 2. Values of K for Different Surfactant Concentrations surfactant conc (M)

φ

κ (nm)

0.025 0.05 0.1 0.15 0.25

0.0089 0.022 0.048 0.074 0.126

85 33 22 19 18.5

three times smaller than the value reported by Todorov et al.14 The value of the diffusivity of micelles used for benzene was used here as well. The values of the parameters used are listed in Table 1. Surface resistance was assumed to be absent here as well. Consider a drop of size 30 µm and a surfactant concentration of 0.1 M. The Damkohler numbers for this system based on the initial value of the radius are Da ) 2.44 × 107 and Dam ) 1757. It can be seen that Dam , 1 + Da/Dam. We therefore expect that the enhancement of dissolution rate should be proportional to the square root of the surfactant concentration. We also see that as Rκ/R > 1, the exclusion layer plays an important role in determining the mass flux. It was however observed that predictions made using the estimated value of κ, that is, 25 nm, or, for that matter, any single value of κ did not fit all the data accurately. This also rules out surface resistance as being the sole cause for the discrepancy between theory and predictions. κ was then fitted to obtain agreement with observations. Table 2 shows the values of κ used for making predictions. The value of κ fitted to secure agreement at the lowest surfactant concentration, though similar in order of magnitude to that estimated, is much larger than expected. The values of κ at other surfactant concentrations are not very different from the estimated value. It can be observed that there is a systematic decrease in the value of the thickness with increasing surfactant concentration. The

4024


Sailaja et al.

Figure 3. Comparison of model predictions with observations for decane. S ) 0.025, 0.05, 0.1, and 0.25 M.

decrease might be expected, as the double layer thickness decreases with increasing ionic strength. The exact nature of the decrease is hard to predict, as the solution contains charged drops, charged micelles, co-ions of surfactant monomers, and counterions. Complications arise, since a micelle is a macroion, and the methods to treat them are not clearly established. The complications have been discussed by Hansen and Lowen28 and by Vlachy.29 The temporal dependence of the radius of the drop is predicted using eq 39. Figure 3 shows the comparison between the model predictions and observations. Once again, the agreement is very good. It is observed that the radius is a linearly decreasing function of time. However, it should be pointed out that whenever dissolution rates are small, drop radius initially decreases linearly with time, as demonstrated below. Equation 39 when integrated (after neglecting surface resistance and the small contributions made by logarithmic terms) yields

( )

D′(1 - φ)t CsM 1 + ξ Dam ln 2 ) (1 - ξ) + 2 F 2 R0 R2 (1 - ξ)

(

)

Da κ κ Da κ + 3 - φγ - (1 - γ) R0 R0 R2 R0 R

The Damkohler numbers are based on the initial radius, R0, and ξ ) R/R0. Thus, the initial decrease of the radius will be linear with time. However, the independence of flux on radius for all times is valid only if the coefficient of the first term of the right-hand side of the above equation is small compared to that of the other side. This is found to be valid for the conditions of experiments of Todorov et al.14 (28) Hansen, J.-P.; Lowen, H. Annu. Rev. Phys. Chem. 2000, 51, 209. (29) Vlachy, V. Adv. Chem. Phys. 1999, 50, 145.

Table 3. Enhancements in Rates of Ostwald Ripening of Decane in SDS Solutionsa enhancement in rate surfactant conc (M)

exp

model with κ ) 25 nm

model with fitted κ

0.07 0.0375 0.02 0.009

5-6.75 3.75-5.5 3.5-4.75 2.25-2.75

4.35 4.03 3.58 2.13

4.35 2.58 2.07 1.17

a

Comparison of the experimental and the model predictions.

4.8. Comparison of the Model Predictions with Data on Ostwald Ripening. Ostwald ripening is a complex phenomenon. Taylor11 as well as Soma and Papadopolous12 have reported observations on Ostwald ripening of decane drops in SDS solutions. Both these investigators studied the effect of surfactant concentration and presented their results for enhancement of ripening rates due to micelles. The enhancement factor is obtained by dividing the ripening rate by the rate expected if micelles were not present. Thus, it is equal to J* in the present model. In Table 3 the enhancements predicted by the model and the experimental observations are reported. The predictions were made in two ways. In the first procedure, the values of κ fitted to the data of Todorov et al.14 were interpolated for the surfactant concentrations employed by Taylor11 and used to make predictions of the enhancement in the ripening rates. In the second procedure, the estimated value of 25 µm was used for κ and the enhancement in the ripening rates was predicted. We find that the model predictions of the enhancements are in reasonable agreement with the range of experimental values predicted for all surfactant concentrations for a single value of κ. The agreement is not as good when the values of κ required to fit the data of Todorov et al.14 are used. The maximum surfactant concentration in Table 3


Langmuir, Vol. 19, No. 9, 2003 4025

is 0.07 M, and hence the interpolated values of κ are dominated by the large fitted value obtained at the lowest surfactant concentration used by Todorov et al.14 It is fair to conclude that evaluation of the thickness of the exclusion layer needs more investigation. Beyond a surfactant concentration of 0.07 M, Taylor11 observes a decrease in the ripening rates, and this feature is not predicted by the model. Taylor11 speculated about the effect of surfactant concentration on diffusivity. We have tried to incorporate this effect. But the model is still unable to make a prediction of a maximum. In his experiments, Taylor11 found that R3 ≈ t, and going by the conventional theory of Ostwald ripening, this implies that the dissolution flux in his experiments depends inversely on drop radius, whereas the model predicts that the flux is independent of R. It should be mentioned here that there have been reports30 that R3 ≈ t even when the flux is independent of R. Furthermore, scatter in the experimental data in some situations3 allows both behaviors to be fitted reasonably well. 4.9. Dissolution from a Rotating Disk. The rotating disk system was used for studying the solubilization kinetics of fatty acids by Cussler and co-workers.15-17 They studied dissolution of lauric, palmitic, and stearic acids into solutions of SDS in water. Their model, reviewed briefly in a previous section, explains their observations with desorption of micelles followed by their diffusion into the bulk as the rate-limiting steps. The model proposed in the present work is also applied to their data. The model has to deal with the convective system. The equations can be derived following the procedures presented for the previous case. We omit the details and provide only the final mass balance equations:

Exclusion Layer 0 < z < κ ∂2Cf ∂Cf )D 2 vz(z) ∂z ∂z

(40)

Micellar Solution z > κ vz(z)(1 - φ)

(

) )

C hm ∂2C 12φD′ ∂C ) (1 - φ)D′ 2 C2 ∂z K ∂z d eq m

vz(z)φ

(

∂C hm C hm ∂2C h m 12φD′ ) φDm + C2 2 ∂z Keq ∂z d m

(41)

Cf(z) ) C, D

∂Cf ) ks(Cs - Cf), at z ) 0 ∂z

(43)

hm ∂Cf ∂C ∂C ) (1 - φ)D′ , )0 ∂z ∂z ∂z all at z ) κ (44) C ) Cm f 0 as z f ∞

and is valid only when Da . 1. The expression for flux is given by

[

(45)

The solution to these equations can be worked out following the procedure given by Pleskov and Filinovskii,31 and only the result for the flux is presented here. The solution is obtained by a singular perturbation technique (30) Neogi, P.; Narasimhan, G. Chem. Eng. Sci. 2001, 56, 4225. (31) Pleskov, Y.; Filinovskii, V. The rotating disc electrode; Consultants Bureau: New York, 1976.

]

Cs (1 - φ)δrdp1 1 κ ) + + J Drd[(1 - φ) + φKeq] D ks Drd ) dm ) µc

[

]

(1 - φ)D′ + φDmKeq KeqφDmµc , p1 ) 1 + (1 - φ) + φKeq (1 - φ)D′δrd

( )()

x

Drd 12D′ 12φ + , δrd ) 1.61 ν (1 - φ) DmKeq

1/3

ν ω

1/2

(46)

where ν is the kinematic viscosity of the surfactant solution and ω is the angular velocity of the rotating disk. Comparison with Experiments. The expression for flux can then be treated as the driving force divided by three resistances:

flux )

[

Cs (1 - φ)δrdp1

Drd[(1 - φ) + φKeq]

(42)

The boundary conditions are given by

-D

Figure 4. Cs/observed flux vs R1 + R2 for all the fatty acids.

+

κ 1 + D ks

]

(47)

The first term in the denominator is the resistance due to the diffusional film, the second is due to the exclusion layer, and the third is due to surface resistance. Thus, R2 ) κ/D, R1 ) (1 - φ)δrd[Drd((1 - φ) + φKeq)], and R3 ) 1/ks. Hence, the intercept in a plot of Cs/observed flux versus (R1 + R2) should give an estimate of the surface resistance. All the data, except the solubilities and the diffusion coefficients of the fatty acids in water, were taken from ref 16. Diffusivities of the solute in the continuous phase were calculated from the Wilke-Chang equation. The solubilities of fatty acids in water reported by Singleton32 have been used. With these parameter values, the resistance due to the exclusion layer, even if it is taken to be 85 nm, turns out to be very small here. Plots of R1 + R2 versus Cs/observed flux for lauric, palmitic, and stearic acids for several surfactant concentrations and speeds of rotation are shown in Figure 4. Here the surfactant concentrations ranged from 0.01 to 0.2 M and the Reynolds ranged numbers from 900 to 6400. All the curves shown in the figure represent a linear (32) Singleton, W. Fatty acids: their chemistry, properties, production and uses: Part I; Interscience: 1960.

4026


dependence of Cs/observed flux on R1 + R2, but those with nonzero intercept on the y-axis appear curved on a loglog plot and clearly emphasize the intercept. The figure shows that the data for lauric and palmitic acids superimpose, nearly fall on 45° line, and are predicted by the model very well. It is to be noted that the present model does not allow for micelles to be in direct contact with the solute surface while the model of Chan et al.15 proposes formation of micelles on the surface of the solute and their desorption as the rate-limiting steps. Surface resistance for stearic acid dissolution, however, is seen to be significant and is estimated to be 8.5 s/cm from the plot. Chan et al.15 explain the differences in the dissolution behavior of stearic acid from those of others on the basis of the lack of penetration of water into fatty acids. Lawrence et al.33 found that water does not penetrate into fatty acid substances at low temperatures. However, when a critical temperature is crossed, liquid crystalline phases33 between stearic acid and SDS solutions form, and the rate of dissolution is enhanced. Chan et al.15 attribute rate-limiting desorption behavior to the lack of penetration of water. It also seems feasible to interpret lack of penetration of water as surface resistance. Chan et al.15 found that the resistance due to desorption step becomes negligible when the temperature is raised above 40 °C for palmitic acid, since at that temperature water penetration into the acid increases substantially. If the present interpretation is valid, surface resistance should vanish at high temperatures. Though not shown, a plot of Cs/observed flux versus R1 + R2 for palmitic acid indeed indicated that surface resistance disappears. One can only conclude that both models are mathematically equivalent and give identical results. The diffusivities of micelles containing fatty acids were found to be significantly different from those containing benzene. They were 4.44 × 10-7, 2.67 × 10-7, and 1.04 × 10-7 cm2/s for micelles containing lauric, palmitic, and stearic acid, respectively. That these values are lower than the corresponding value for benzene is not unexpected. However, the decrease in diffusivity values, from lauric (33) Lawrence, A.; Bingham, A.; Capper, C.; Hume, K. J. Phys. Chem. 1964, 68, 3470.

Sailaja et al.

to stearic, is unexpected, since the chain lengths of the acids are not so very different and moreover the number of acid molecules solubilized in micelles also decreases in the same order. This might indicate that, apart from diffusion, an additional resistance to the entry of these long surface active molecules into the micelle is present. 5. Conclusions Widely varied results have been reported on the extent of enhancement of dissolution into solvents caused by micelles. A model to describe the dissolution of solutes into micellar solutions has been developed to rationalize the observations. It accounts for the existence of surface resistance, repulsive forces between co-ions that prevent direct contact between micelles and the solute surface, and the exchange of solute between the micelles and the continuous phase. Two Damkohler numbers, Da and Dam, are found to play an important role in the dissolution process. Da represents the ratio of the rate at which the solute is absorbed by micelles to the rate at which it diffuses away through the continuous phase. Dam represents the ratio of the rates at which the micelles receive solute to the rate at which it is carried away by micelles by diffusion. When Dam . 1 + Da/Dam, which occurs when the partition coefficient is not large, the mass transfer is enhanced by micelles, as if they can dissolve solute directly. On the other hand, if Dam , 1 + Da/Dam, the enhancement is considerably less. These predictions have been confirmed against observations of dissolution of drops of benzene and decane, as well as fatty acid solids, into solutions of SDS. Comparison of the predictions with observations suggests that modeling of prevention of direct contact between micelles and solute surfaces through a layer where micelles are excluded needs more investigation. Further, results obtained for fatty acids suggest that there may be some resistance, in addition to diffusion, for the entry of these solutes into micelles. Acknowledgment. The authors wish to acknowledge discussions with Prof. D. Ramkrishna on the population balance approach to adsorption. LA0268698

Theory of Rate of Solubilization into Surfactant Solutions - American

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