Ostwald Ripening of Alkane Emulsions in the Presence of Surfactant

Langmuir 1999, 15, 6745-6754

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Ostwald Ripening of Alkane Emulsions in the Presence of Surfactant Micelles Y. De Smet, L. Deriemaeker, and R. Finsy* Department of Physical and Colloid Chemistry, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Received February 18, 1999. In Final Form: June 1, 1999 The effect of sodium dodecyl benzenesulfonate (SDBS) surfactant micelles on the Ostwald ripening was investigated by several experiments and by computer simulation. The experimental ripening rates determined by dynamic light scattering were about 2 times higher than the one predicted by the LifshitzSlyozov-Wagner (LSW) theory. This increase is attributed to an increase of the concentration of oil molecules in the continuous phase. The increase in solubility and hence in ripening rates is predicted by Kelvin’s equation, assuming the presence of small oil droplets of the size of the micelles. A study of the solubilization kinetics of emulsion oil droplets into micellar solutions confirms that the main rate-determining mechanism for the exchange of oil between droplets and micelles is molecular diffusion through the continuous phase. Finally the combination of the simulation of Ostwald ripening in the presence of an oil sink with the experimental monitoring of the ripening of emulsions to which continuously a micellar solution is added, confirms the previous model for the transport of oil. There is also evidence that the surfactant micelles are not in local equilibrium with the oil molecules.

1. Introduction

R)

The physical degradation of emulsions is due to the spontaneous trend toward a minimal interfacial area between the dispersed phase and the dispersion medium. Minimizing the interfacial area is achieved mainly by two mechanisms: first coagulation, possibly followed by coalescence, and second Ostwald ripening. The former is the most studied (see, e.g., ref 1). However, if properly stabilized against the coagulation/coalescence process, the latter can cause a substantial breakdown of the emulsion. Ostwald ripening is the process by which larger particles (or, for emulsions, droplets) grow at the expense of smaller ones due to the higher solubility of the smaller particles (Gibbs-Thomson or Kelvin effect) and to molecular diffusion through the continuous phase. A theoretical description of Ostwald ripening in twophase systems has been developed independently by Lifshitz and Slyozov, and Wagner (LSW theory).2-5 One of their major results is that in the long time limit a stationary regime is reached for which the ripening rate v is given by

v)

dan3 4 ) RDmC∞ dt 9

(1)

In eq 1 an denotes the number-average particle radius and Dm is the dispersed-phase molecular diffusion coefficient; R is a material-dependent constant called the capillary radius, defined by * To whom correspondence may be addressed: Tel: +32 2 629 34 85. Fax: + 32 2 629 33 20. E-mail: [email protected]. (1) Tadros, T.; Vincent, B. In Encyclopedia of emulsion technology; Becher, P., Ed.; Marcel Dekker: New York, 1983; p 129. (2) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35. (3) Wagner, C. Z. Elektrochem. 1961, 65, 581. (4) Kabalnov, A. S.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 38, 69. (5) Taylor, P. Adv. Colloid Interface Sci. 1998, 75, 107.

2γVm RT

(2)

Vm stands for the dispersed phase’s molar volume, γ is the interfacial tension, and R and T have their usual meaning of gas constant and absolute temperature. This result predicts that the aging or average droplet size increase is mainly determined by the bulk solubility C∞ of the dispersed phase in the continuous one. This feature of Ostwald ripening has been verified in several experimental studies6-15 of alkane in water emulsions stabilized against coagulation by surfactants. If added in sufficiently large amounts and after complete coverage of the oil-water interface, surfactants spontaneously form micelles in the continuous aqueous phase. The presence of micelles drastically increases the solubility of the oil phase. Therefore, an effect of micelles on the Ostwald ripening may be anticipated. Simplistically it might be expected that replacing the bulk oil solubility C∞ in eq 1 by the concentration of oil solubilized by the micelles and using the micellar diffusion coefficient instead of the molecular one would yield the Ostwald ripening rate in the presence of micelles. This approach predicts an increase of the rates by a factor of about 200 to 1000.5 Experimental studies report hardly any increase. Several, sometimes conflicting, explanations are given.8,11-13 The oil solubilized by micelles however is not dispersed at the molecular level in the continuous phase. Since the (6) De Smet, Y.; Malfait, J.; De Vos, C.; Deriemaeker, L.; Finsy, R. Bull. Soc. Chim. Belg. 1996, 105, 789. (7) Kabalnov, A. S.; Makarov, K. N.; Pertzov, A. V.; Shchukin, E. D. J. Colloid Interface Sci. 1990, 138, 98. (8) Soma, J.; Papadopoulos, K. D. J. Colloid Interface Sci. 1996, 181, 225. (9) De Smet, Y.; Malfait, J.; De Vos, C.; Deriemaeker, L.; Finsy, R. Prog. Colloid Polym. Sci. 1997, 105, 252. (10) McClements, D. J.; Dungan, S. R. Colloids Surf. A 1995, 104, 127. (11) Kabalnov, A. S. Langmuir 1994, 10, 680. (12) Taylor, P.; Ottewill, R. H. Colloids Surf. A 1994, 88, 303. (13) Taylor, P. Colloids Surf. A 1995, 99, 175. (14) Bremer, L.; De Nijs, B.; Deriemaeker, L.; Finsy, R.; Gelade´, E.; Joosten, J. Part. Part. Syst. Charact. 1996, 13, 350. (15) Buscall, R.; Davis, S. S.; Potts, D. C. Colloid Polym. Sci. 1979, 257, 636.

10.1021/la9901736 CCC: $15.00 © 1999 American Chemical Society Published on Web 08/05/1999

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main rate-determining step in the Ostwald ripening process is the oil transport by diffusion of single oil molecules through the continuous phase, the approach of replacing the bulk solubility C∞ by the total concentration of oil solubilized by micelles is misleading. Another picture is to consider the solubilized oil as small oil droplets dispersed in a continuous, aqueous phase. The molecular oil solubility Cm in the aqueous phase in equilibrium with small oil droplets is then given by the Kelvin equation

Cm/C∞ ) exp(R/a)

(3)

in which a is the oil droplet radius. Inserting, e.g., the value for R for an undecane in water emulsion (R ) 1.72 nm) in eq 3 and assuming that the radius a is that of the micelles (typically 2-5 nm), one obtains Cm/C∞ ≈ 1.4-2.4. Hence on the basis of this approach, only a slight increase is expected in better agreement with the observed effects.8,11-13 Another possible and not considered effect of surfactants and surfactant micelles is that they just withdraw oil from the emulsion droplets and hence that the oil in the micelles does not contribute to the Ostwald ripening process. When the emulsion droplets grow, the interfacial area decreases, and thus surfactant molecules are released to the continuous phase. For surfactant concentrations above the critical micellar concentration, this means that the number of micelles increases and that during the ripening process oil is also transferred continuously from the emulsion droplets to the micelles. This process may even slow the ripening instead of enhancing it. To account for this effect the problem of Ostwald ripening in the presence of an oil sink is presented. Results from both computer simulations and the experimental monitoring of the aging of some model emulsions are reported. 2. Ostwald Ripening in the Presence of a Sink 2.1. The Simulation Model. The growth law corresponding to Ostwald ripening is usually derived with the aid of Fick’s first diffusion law and Kelvin’s equation (assuming R , a), yielding16

dn ) 4πDmC∞R(a/ac - 1) ≡ P(a) dt

(4)

in which n is the number of oil molecules in a droplet with radius a; ac is the critical radius, i.e., the radius of a droplet that neither grows nor shrinks, thus P(ac) ) 0. This critical radius can be calculated from the mass-transfer balance of the dispersed oil phase. For a particle size distribution (PSD) described by a (discrete) set of N droplet sizes {ai; i ) 1, ..., N}, the mass-transfer balance can be written as N

P(ai) ) 0 ∑ i)1

(5)

Combining eqs 4 and 5 yields

C h ) C∞ exp(R/ac)

(7)

Note that in all the above equations both the individual oil droplet radii and the critical radii are time-dependent variables. The presence of an oil sink can now be introduced by stating that per unit time a number Ns of oil moleculess which needs not be a constant in timesis withdrawn from the oil emulsion droplets (e.g., due to solubilization by micelles). Taking this sink into account in the masstransfer balance, the critical radius is now given by N

ac ) K

ai/(KN - Ns) ∑ i)1

(8)

where K stands for the constant 4πDmC∞R. Note that eq 8 predicts that the critical radius is larger than the number-average radius. Hence the average concentration C h of oil molecules in the continuous phase is smaller in the presence of a sink than without a sink, and one can anticipate that the presence of a sink will slow the ripening process. The oil released by the small particles is only partly transported to the larger particles, so it can be argued that the average droplet size will increase less rapidly. To verify this on a quantitative basis, a computer simulation of the ripening in the presence of a sink was performed. The simulation of the growth with and without sink is performed with the procedure described in ref 17. The main steps are the following: 1. An initial set of droplets with radii {ai; i ) 1, ..., N} was generated, according to an a priori model of the particle size distribution (PSD). Two models are considered. First a LSW-PSD, that is the limiting PSD (see eq 11 in ref 17) in the long time limit as predicted by the LSW theory. Since such a LSW-PSD is not so realistic for the size distribution of freshly prepared emulsions, an initially log-normal distribution was also used.

W(a) )

1

x2πσ

[

exp -

]

[ln(a) - ln(ap)]2 2σ2

(9)

A typical value for ap, the radius at which the distribution has its peak, of 25 nm was chosen. The width σ ) 0.24 of this distribution was the one obtained from a cryo-TEM (transmission electron microscopy) study of a virtually nongrowing squalane in water emulsion.18 2. Transport of dispersed-phase (oil) molecules from one droplet to another was carried out according to the growth rule corresponding to the ripening without sink (eqs 4 and 6) and with sink (eqs 4 and 8). The following two sink models are investigated: (a) The sink Ns,1 is nothing more than the number of oil molecules solubilized per time unit into the micelles, created by the release of surfactant from the decreasing interfacial area. Ns,1 is calculated as follows. At every time step tj the total interfacial area Atot of the droplets is calculated as

N

ac )

ai/N ≡ an ∑ i)1

(6)

i.e., the critical radius equals the number-average radius. The average concentration C h of oil molecules in the continuous phase is related to the critical radius by Kelvin’s equation: (16) Kahlweit, M. Adv. Colloid Interface Sci. 1975, 5, 1.

N

Atot (tj) ) 4π

ai2(tj) ∑ i)1

(10)

The number of micelles released in the time interval (17) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1997, 13, 6884. (18) De Smet, Y.; Danino, D.; Deriemaeker, L.; Talmon, Y.; Finsy, R. Langmuir, submitted.

Ostwald Ripening in the Presence of Surfactant Micelles

Figure 1. Simulation results for evolution of fraction of molecules transported by Ostwald ripening, fOR (two upper curves) and of number of molecules withdrawn from the ripening process by solubilization into micelles created by the release of surfactant from decreasing interfacial area, fs,1 (two lower curves). The full lines correspond to the ripening of a droplet set with initially a log-normal profile, the dashed lines to one with initially an LSW profile.

[tj-1, tj] is then given by

[Atot(tj-1) - Atot(tj)]/(NaggAs)

(11)

Nagg is the aggregation number of the surfactant micelles and As is the specific surface area of a surfactant molecule at the oil droplet’s interface. Finally, if we multiply this with the solubilization capacity, that is, the number Nsol of oil molecules solubilized by a single micelle, and divide by the time interval, we find Ns,1:

Ns,1 )

[Atot(tj-1) - Atot(tj)]Nsol NaggAs(tj - tj-1)

(12)

(b) To mimic the solubilization of oil by micelles added to the system from outside, an additional constant number of oil molecules Ns,2 is withdrawn per time unit on top of the previous sink Ns,1. This situation corresponds to, e.g., an oil in water emulsion to which an aqueous micellar solution is added continuously at a constant rate. 3. Computation of the number-average radii an at every time step tj of the simulation. The simulation was carried out for sets of N ) 5000 particles, containing 7.47 × 108 and 5.04 × 108 oil molecules (Noil) for respectively the initial LSW and the log-normal distributions, and ended when the number-average radius reached 100 nm (i.e., 4 times the initial number-average radius). 4. Finally, the growth rate is estimated as the (average) slope of an3 as a function of time. 2.2. Results. The parameter values for the simulation procedure were those typically for an undecane in water emulsion (R ) 1.72 nm, Dm ) 4.31 × 10-6 cm2 s-1, Vm ) 211.23 cm3 mol-1, γ ) 10-2 Nm-1, T ) 298 K, and C∞ ) 2.0 × 10-8 mL mL-1 6). In a first step the growth rate without sink (Ns,1 ) Ns,2 ) 0) was estimated from the average slope of an3 as a function of time yielding 6.60 and 8.25 nm3 s-1 starting with the LSW and log-normal distributions, respectively. In Figure 1 the fraction fOR of the number of molecules transported from the smaller to the larger droplets N |P(ai)|/Noil is plotted as a estimated as fOR ) (1/2)K∑i)1 function of time. Initially this fraction is relatively large due to the presence of a relative large number of small

Langmuir, Vol. 15, No. 20, 1999 6747

Figure 2. Simulation results for the evolution of an3 with time in the presence of a sink (both Ns,1 and Ns,2 * 0). The different curves correspond to different values of Ns,2: (top to bottom) 0, 1.6, 4, 5, 12, 20 (all in 10-6 s-1).

particles. It drops to an almost constant value of about 1.0 × 10-5 s-1 and 1.1 × 10-5 s-1 respectively in simulations started with a log-normal and an LSW-PSD, for the considered increase in average particle size by a factor of 4. The initial relative number of molecules transported by Ostwald ripening, i.e., the fraction fOR, is larger when the simulation is started with a more realistic log-normal distribution. The main reason for this is that, if an amount of oil molecules Noil is to be partitioned over a set of oil droplets that is log-normally distributed, one ends up with many more droplets than when it is LSW-distributed (with the same average size). Therefore, there are more terms N |P(ai)|, yielding a larger value of fOR at the in the sum ∑i)1 beginning. In the considered time interval the log-normal PSD does not attain the limiting LSW profile,17 so it can be expected that the two values of Ns,1 do not coincide at the end of the same time interval. In a second step a simulation is performed accounting for the rate of solubilization Ns,1 into the micelles, created by the release of surfactant from the decreasing interfacial area. Hereto we used the ratio Nsol/Nagg of the number of oil molecules Nsol solubilized by one single micelle to the number Nagg of surfactant molecules per micelle. The ratio was estimated as 0.118 from the solubility of undecane in a 0.1 M SDS solution (2.5 × 10-3 mL/mL) reported by Kabalnov.11 For As we take the value of 0.62 nm2 (i.e., the value for the surfactant used throughout this study; see section 3.3 for its determination). In Figure 1 the evolution of the fraction of oil molecules withdrawn by the sink Ns,1, i.e., fs,1 ) Ns,1/Noil, during the ripening process is shown in comparison with fOR. The fraction fs,1 decreases from 1.5 × 10-6 to about 8.1 × 10-9 s-1 when the simulation is started with an LSW-PSD; for a log-normal PSD it decreases from 3.3 × 10-6 to 6.9 × 10-9 s-1, illustrating that the number of oil molecules withdrawn from the oil droplets is more than a factor 103 smaller (except at the beginning) than the number of molecules transported by the Ostwald ripening process. As expected, this effect hardly affects the average growth of the oil droplets. The average slopes hardly differ: the ratio of the slopes with sink to the corresponding ones without sink are 0.9996 and 0.9994 when the simulation starts with a LSW and a log-normal PSD, respectively. Finally simulationssstarting with a log-normal PSDs with the additional sink Ns,2 were performed for different constant values of Ns,2. In Figure 2 the evolution of an3 during the ripening in the presence of both sink terms Ns,1 and Ns,2 is displayed for several values of Ns,2. For values of the fraction of oil molecules fs,2 ) Ns,2/Noil absorbed

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Figure 3. Effect of the sink Ns,2, i.e., withdrawal of molecules from a ripening system, on the relative ripening rate. The relative rate (ratio of the rate with a sink to the one without sink) is plotted as a function of fs,2, the fraction of molecules absorbed in the sink. Diamonds/squares: initial ripening rate and ripening rate for the set with initially an LSW-PSD. Triangles/crosses: initial ripening rate and ripening rate for the set with initially a log-normal PSD.

by the sink Ns,2 less than about 10-6 hardly any effect is observed. Only when this fraction becomes comparable to or greater than the average fraction fOR of oil molecules transported by the Ostwald ripening process is a slowdown of the growth of an3 observed. The largest values of fs,2 are still smaller than the initial fraction transported by Ostwald ripening fOR. However, during the ripening process fOR drops to values below fs,2. Therefore, some of the plots of an3 show an initial increase, due to Ostwald ripening, followed by a decrease. The decrease is due to the extreme withdrawal of oil molecules from the droplets by the sink Ns,2, which models, e.g., the solubilization of oil under a continuous and constant addition of micelles. Since the plots of an3 are nonlinear with time, the effect of the sink Ns,2 on the ripening rate was estimated as follows. The initial rate was estimated from the initial slope, and a reduced rate was estimated at the point where an3 had increased to 60 times its initial value. Both rates relative to the one without sink (determined in the first simulation) are plotted in Figure 3 as a function of fs,2. The initial ripening rate is not or hardly affected by the presence of the sink. The rates at the point where an3 reaches 60 times an3(t ) 0) do start to drop when fs,2 approaches fOR. 3. Experimental Section 3.1. Materials and Emulsion Preparation. Several oil-inwater emulsions were prepared with distilled water, each with oil (alkane) volume fraction of 0.01. The alkanes used were decane, undecane, dodecane, and tridecane; all were 99+% pure (Aldrich). Sodium dodecyl benzenesulfonate (SDBS) (tech., Aldrich, purity 96%) was added as a surfactant. The oil component was added to the aqueous SDBS surfactant solution (0.1 M). After 10 min of premixing with an Ultra-Turrax T25 with rotor S25-18G, the coarse emulsion was further homogenized using an Y-110 Microfluidizer for another 5 min (external pressure 4 bar). 3.2. Particle Size Measurements. Droplet sizes were determined with dynamic light scattering (DLS). The experimental setup consists of an Ar+ laser (wavelength λ ) 488 nm), a thermostated sample holder allowing control of the temperature, and a photomultiplier (EMI 9863A) mounted at a detection angle of 90°. A Brookhaven BI9000 correlator was used. Data analysis was performed with a PC with Pentium 233 MHz processor. The measurements took place at (25 ( 1) °C. Just before each measurement a sample was prepared by diluting the original emulsion 1000 times with 5 × 10-3 M SDBS, in order to rule out droplet interactions and multiple scattering effects.9 The 5 ×

Figure 4. Surface tension of aqueous SDBS solution versus SDBS concentration. 10-3 M SDBS concentration was chosen close to the SDBS concentration in the aqueous phase of the undiluted emulsion in order to minimize possible changes of the distribution of surfactant molecules between the droplet surface and the aqueous phase. For each sample, the intensity-averaged particle size aI was computed with the cumulants method.19 3.3. Characteristics of the Surfactant SDBS: Critical Micellar Concentration and Specific Surface Area. A drop volume tensiometer (Lauda TVT1) was used to measure the surface tension of surfactant solutions in air as a function of surfactant concentration Cs. From the slope of the plot of the surface tension γ versus the natural logarithm of Cs, the specific area As was calculated using the Gibbs adsorption equation, assuming an ideal dilute behavior of the surfactant solution,

Γ)-

dγ 1 2RT d(ln Cs)

(13)

where Γ ) NA/As is the adsorption of surfactant molecules at the droplet’s surface (NA is Avogadro’s number). The critical micellar concentration of the surfactant is determined as the concentration at which the linear part of the γ vs ln(Cs) plot equals the limiting constant value of γ. This is illustrated in Figure 4, showing γ as a function of ln(Cs) for the SDBS solutions. This analysis resulted in values of 5 × 10-3 M for the critical micellar concentration and 0.62 nm2 for the specific surface area A s. (19) Koppel, D. J. Chem. Phys. 1972, 57, 4814.


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Table 1. Ostwald Ripening Rates and Parameters Estimated from Enhanced Ripening Rates as a Function of Surfactant Concentration for Several Undecane Emulsions

in water of m ) 1.0826, the wavelength λ ) 488 nm, and the scattering angle θ ) 90°. The intensity-averaged particle size aI, for a given value of ap, is then computed as24

SDBS overall concn av SDBS after ripening av rate micellar emulsion concn coverage rate enhancement radius Nsol/ no. (M) (M) (nm3/s) Nagg factor (nm) 1 2 3 4 5

0.033 0.069 0.110 0.130 0.150

0.030 0.066 0.107 0.127 0.147

13.4 11.1 10.4 10.2 9.5

2.05 1.70 1.59 1.56 1.45

2.4 3.2 3.7 3.9 4.6

1.42 1.88 2.18 2.30 2.70

4. Ripening of Emulsions in the Presence of Micelles 4.1. Ostwald Ripening Rates of Alkane Emulsions Stabilized by Sodium Dodecyl Sulfate (SDS). Several studies on the effect of the presence of micelles have been reported in the literature. Recently Taylor5 gave a review. There is some controversy about the effect of the surfactant concentration on the ripening rate. Kabalnov11 claims that there is hardly any effect. The measurements of Taylor13 and Soma and Papadopoulos8 on the other hand point toward a small increase, though significant enough to notice in an experiment, with surfactant concentration Cs. Although the surfactant SDS is probably the most widely used in scientific studies, it has the disadvantage that in aqueous solution it may hydrolyze to dodecanol.20-22 Since Kabalnov uses purified SDS, whereas Taylor and Soma and Papadopoulos use commercially available SDS without purification, it cannot be excluded that the difference in results and conclusions is due to the difference in purity of the surfactant used. To circumvent the problem of hydrolysis of SDS, another similar surfactant, sodium dodecyl benzenesulfonate (SDBS), which does not hydrolyze, was used in this study. 4.2. Ripening Rate of Undecane-Water Emulsions at Several Surfactant Concentrations above the Critical Micellar Concentration. A mother emulsion with undecane volume fraction φ of 0.05 and overall SDBS surfactant concentration of 0.15 M was prepared as described in section 3.1. This emulsion was subsequently diluted 5 times with five different aqueous SDBS solutions, so as to obtain emulsions with different surfactant concentrations, but each with φ ) 0.01. The overall surfactant concentration Cs of each emulsion is reported in Table 1. 4.2.1. Surfactant Concentration in the Continuous Phase after Complete Coverage of the Oil-Water Interface. To find out the surfactant concentration after complete coverage of the oil-water interface, the following procedure is followed. We assume that the freshly prepared emulsion’s initial PSD has a log-normal profile given by eq 9 with σ ) 0.24. The parameter ap was determined from the initial intensity-averaged aI,0sin its turn determined by DLSs as follows. For different values of ap and σ ) 0.24 lognormal distributions were generated. Each distribution was discretized as a set of 5000 particles with different radii {ai; i ) 1, ..., 5000}. For a droplet with given radius ai the scattering power Si(ai,m,λ,θ) was calculated with the aid of the Mie scattering coefficients for homogeneous spheres,23 using the relative refractive index of undecane (20) Fang, J. P.; Joos, P. Colloids Surf. 1992, 65, 113. (21) Miles, G. D. J. Phys. Chem. 1945, 49, 71. (22) Miles, G. D.; Shedlovski, L. J. Phys. Chem. 1944, 48, 57. (23) Bohren, C. F.; Huffman, D. R. Absorption and scattering of light by small particles; Wiley: New York, 1983; p 83.

5000

aI )

∑ i)1

5000

Si/(

Si/ai) ∑ i)1

(14)

In this way, a table of values of aI and ap is established. The value of ap corresponding to the experimentally measured intensity average aI,0 is then looked up. For the measured value of aI,0 ) 35 nm this yields ap ) 24.5 nm. Knowing the emulsion’s droplet size distribution, the total interfacial area Atot can be calculated with eq 10. Hence the interfacial area per unit volume of oil phase is given by

AS,V ) 3(

∑i ai2)/(∑i ai3)

(15)

and the number of moles of surfactant ns required for complete coverage of the interfacial area of a unit volume of oil dispersed in water is given by

ns ) AS,V/(NAAs)

(16)

For an emulsion with ap ) 24.5 nm and σ ) 0.24 and with oil volume fraction of 0.01, this leads to a concentration of SDBS of 2.5 × 10-3 M. The remaining surfactant concentration in the continuous aqueous phase is then estimated, as the difference between the total surfactant concentration and the surfactant concentration needed for complete coverage of the oil-water interface. In Table 1 the initial surfactant concentration in the continuous phase is reported for the five emulsions. Clearly all these concentrations are above the cmc. 4.2.2. Determination of the Ostwald Ripening Rate. For each emulsion the particle size growth was monitored by DLS, until the average particle radius aI reached about 100 nm (nearly 3 times the initial intensity average radius). From the data of aI(t) as a function of time, the Ostwald ripening rates were determined by the procedure reported elsewhere.25 In this procedure intensity-averaged radii aI are converted into number-average radii an and the effect of the transition of an initially log-normal size distribution toward the limiting LSW distribution is taken into account. 4.2.3. Results. In Figure 5 the raw experimental data for aI3 as a function of time are shown. To assess a rough estimate of the reproducibility of the experimentally determined ripening rates, every experiment (preparation of the mother emulsion, dilution to obtain emulsions with five different surfactant concentrations, monitoring of the particle growth, and determination of the ripening rates) was repeated twice. In Figure 6 the resulting ripening rates are given as a function of the surfactant concentration (in Table 1 the average of the two series of experiments is reported). In both experiments a slight decrease, rather than an increase, of the ripening rate with surfactant concentration is observed. The observed rates are a factor of 1.4-2.1 higher than the rate (6.6 nm3 s-1) predicted by eq 1, with the values of Dm, Vm, γ, T, and C∞ given in section 2.2. These enhanced values compare well with the value of 1.4-2.4 estimated with Kelvin’s equation (eq 3) and are (24) Finsy, R.; De Jaeger, N. Part. Part. Syst. Charact. 1991, 8, 187. (25) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1999, 15, 2327.

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Figure 5. Evolution of aI3 with time for the five experiments described in section 4.2. For clarity the different series have been translated along the ordinate over 105 nm3. The series in this figure are arranged as in Table 1: the upper curve corresponds to the emulsion with the highest surfactant concentration, the lower to the one with the lowest surfactant concentration.

Figure 6. Experimental ripening rates (circles and triangles, two independent series of experiments) of undecane emulsions for several values of surfactant concentration (in the aqueous phase, after coverage of the undecane-water interface).

consistent with the model of growing droplets in the presence of a micellar solution of oil molecules, presented in the Introduction. From the enhanced values the micellar radii can be estimated with eq 3 yielding values ranging from 2.4 to 4.6 nm (see Table 1). The observed, slight but systematic decrease of the enhancement factors can be explained by the fact that more oil molecules are solubilized per micelle as the surfactant concentration increases, as, e.g., observed by Kabalnov11 for the solubility of undecane in SDS solutions. The ratio of oil molecules per micelle Nsol to the number Nagg of surfactant molecules per micelle is, for a spherical micelle with oil core radius a, given by

Nsol NAAsa ) Nagg 3Vm

(17)

The results for Nsol/Nagg are presented in Table 1. These estimates are about an order of magnitude larger than the one that can be estimated from the solubilities of undecane in SDS solutions11 (Nsol/Nagg ranging from 0.09 to 0.24 in the SDS concentration range of 0.033-0.3 M).

Figure 7. Comparison of relative ripening rates determined in this study, to the work of Taylor,13 Kabalnov,11 and Soma and Papadopoulos.8

The presented model of ripening in the presence of a micellar solution of oil molecules implies that there is no direct exchange of oil molecules between micelles and oil droplets. Exchange of oil is achieved only by diffusion of individual oil molecules through the continuous aqueous phase. The fact that the observed ratios Nsol/Nagg are an order of magnitude different from the ones estimated from the equilibrium solubilities indicates that, in the exchange of oil between emulsion droplets and surfactant micelles, more oil is involved than expected from the equilibrium solubility of the oil in a micellar surfactant solution, or in other words there is no local equilibrium between surfactant (micelles) and oil. This observation will be confirmed further by the experiments described in section 4.4. Note that this interpretation is consistent with the one presented by Kabalnov:11 micelles do not exchange oil directly with the bulk oil droplets and the micelles are not in local equilibrium with the medium. Kabalnov11 also reports results for the ripening rate of undecane in water emulsions stabilized with SDS as a function of surfactant concentration. To account for the effect of the oil volume fraction, an enhancement factor of the ripening rate of 1.75 is taken into account.7,26 The ratio of the experimental to the theoretically predicted rates ranges from 1.8 to 2.4. Assuming that the initial PSDs in Kabalnov’s work can be modeled by a log-normal with ap ) 107 nm and σ ) 0.24, and a value of As ) 0.5 nm2 for SDS, we estimate that about 2 × 10-2 M surfactant concentration is needed to cover the oil-water interface completely. Hence in all cases the continuous phase surfactant concentration is above the cmc ()0.0082 M). In Figure 7 Kabalnov’s relative ripening rates (ratio of the experimental rate to the theoretical one) are plotted as a function of the estimated surfactant concentration in the continuous phase. The enhanced rates are again also consistent with the same picture of growing droplets in the presence of micelles. In an earlier paper Kabalnov7 mentions that one of the possible reasons for the higher values of the experimental rates might be the Brownian motion of the emulsion droplets, not taken into account in the LSW theory, which assumes immobile particles. Accounting for the Brownian motion of the droplets would correspond to replacing the molecular oil diffusion coefficient Dm with a relative diffusion coefficient Dr ) Dm + Dp ,27 whereby Dp is the (26) Voorhees, P. W. J. Stat. Phys. 1985, 38, 231. (27) See e.g. Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1991; Vol. 1, p 441.


droplet’s diffusion coefficient. However, since Dp , Dm, hardly any increase of the ripening rate can be explained this way. Taylor13 reports results for decane in water emulsions stabilized by SDS (BDH AnalaR reagent grade, used without further purification). The relative ripening rate for emulsions with oil volume fraction φ ) 0.025 increases from about 2 at SDS concentrations just below the cmc to about 4-6 at higher SDS concentrations. This means that the presence of micelles enhances the ripening rates by a factor of about 2-3, which agrees with the picture of growing droplets in the presence of a micellar solution of oil molecules. The surfactant concentration in the continuous phase is addressed again with the aid of an estimated initially log-normal distribution with ap ) 110 nm and σ ) 0.24. In this case 2 × 10-3 M surfactant is required to cover the oil-water interface completely. The results are plotted in Figure 7 as a function of the estimated surfactant concentration in the continuous phase. All surfactant concentrations are above the cmc, except perhaps the two lowest concentrations. Recently Soma and Papadopoulos8 also reported results for the effect of SDS concentration on the ripening rate of decane in water emulsions. The studied emulsions were very dilute (decane volume fraction of 2 × 10-3). For an initial average droplet radius of 100 nm, we estimate that 1 × 10-4 M SDS is needed to cover the oil-water interface. The relative ripening rates increase from 3-4 below the cmc to 7-13 above the cmc. Note that again the presence of micelles increases the ripening rate by a factor of about 2-3, in agreement with the picture of growing droplets in the presence of a micellar solution. The authors explain the increased rates by an increase of the effective concentration of the oil in the bulk phase. They propose another model to explain the higher concentration of oil monomers in the bulk phase: micellar dissociation, monomer adsorption at the oil-water interface, micelle formation and solubilization at the interface, desorption of the swollen micelle, and finally diffusion of the swollen micelle into the bulk phase. However, no quantitative treatment of this model is presented. The reasons for the difference in absolute rates between the four studies (Kabalnov, Taylor, Soma and Papadopoulos, and this study) are unclear. The main difference between the four studies is the nature and the purity of the surfactants used. The lowest values of the relative rates are observed with the SDBS emulsions, i.e., with the surfactant that does not hydrolyze, and emulsions prepared with freshly purified SDS. In the latter case hydrolysis into dodecanol probably is less significant than in the studies where SDS from a commercial batch was used without further purification. 4.3. Solubilization Kinetics of Some Alkanes in SDBS Micelles. Another kind of experiments whereby oil molecules are exchanged between emulsion droplets and micelles is the solubilization of oil by micellar solutions. Therefore, the solubilization of the oil phase of alkane in water emulsions into micellar SDBS solutions was studied by monitoring the scattered light intensity (at a scattering angle 90°) as a function of time, in parallel with the determination of the average particle size by dynamic light scattering. Four alkane (decane, undecane, dodecane, and tridecane) in water emulsions were studied (all with oil concentration of 0.01 by volume and SDBS surfactant concentration of 0.1 M). For each the average droplet size was monitored in time with DLS, until the (intensity weighted) average size reached 100 nm. At that moment,

Langmuir, Vol. 15, No. 20, 1999 6751

Figure 8. Evolution of scattered light intensity I with time during the solubilization of tridecane emulsion droplets in an aqueous micellar SDBS solution.

a sample of the original emulsions was diluted by a factor of 4 with 0.1 M aqueous SDBS solution. Finally the scattered light intensity I of this diluted sample was followed until it reached that of a 0.1 M SDBS solution. The emulsion preparation conditions (equipment, duration, surfactant concentration and type, oil concentration) were the same for all four emulsions; only the oil type was different. Therefore it is plausible to argue that the four emulsions ripened in the same way (starting with the same size distribution), until the (intensity weighted) average droplet radius of 100 nm was reached. Hence, subsequent differences in the time evolution of the scattering behavior of these emulsions are attributed in the first place to the difference in oil type. Figure 8 shows the evolution of the scattered light intensity I with time for the case of tridecane: as more and more oil from the emulsion droplets is solubilized by the micelles, both the number and the sizes of oil droplets decrease, which is reflected in a decreasing scattered light intensity. In the case of oil molecules diffusing through the continuous phase from the oil droplets to the surfactant micelles, Kabalnov and Weers28 argue that the mass flux Joil through the oil-water interface is proportional to DmC h. It is reasonable to neglect the differences in Joil for the four alkanes due to diffusion (Dm) and to attribute them to the differences in C h , for which C∞ is a fair estimate. Plotting the scattered light intensity as a function of t′ ) t/C∞, whereby the values for C∞ were taken from ref 29, should then yield almost identical decay rates for the different alkanes. This is shown in Figure 9. Clearly the experimental decays of I as a function of t′ are roughly equal, illustrating that the characteristic solubilization times of the different alkanes are in a first approximation inversely proportional to the molecular solubility of the oil in the continuous phase. This confirms the absence of a direct exchange of oil between emulsion droplets and micelles: otherwise the solubilization rates would be proportional to the ratio of the number of oil molecules to that of surfactant molecules (a constant in this study) and independent of the chemical nature of the oil molecules. It confirms once more that the main ratedetermining factor in the exchange of oil between emulsion droplets and micelles is the molecular diffusion of single oil molecules through the continuous phase. (28) Kabalnov, A.; Weers, J. Langmuir 1996, 12, 3442. (29) McAuliffe, C. J. Phys. Chem. 1966, 70, 1267.

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Figure 9. Evolution of scattered light intensity I with time (this time scaled with the alkane’s aqueous bulk solubility) for the four alkanes decane, undecane, dodecane, and tridecane (from bottom to top). The ordinates of the different curves are translated by a value of 50 × 103 counts s-1.

The transport rate (number of oil molecules transported per time unit) of the solubilization process can be estimated as the number of oil molecules present initially, divided by the time it takes to solubilize them into the micellar solution. Initially there is 0.005 mL of undecane present in the form of emulsion droplets, which are solubilized into the micelles in a time span of about 600 s. For undecane this comes down to about 2.4 × 1016 molecules per second. The average transport rate in the corresponding Ostwald ripening process was estimated as fORNoil, where Noil is the number of oil molecules in 0.005 mL and the value of fOR is taken from the computer simulations of section 2.2, yielding a transport rate which is at least a factor of 100 slower than the solubilization rate. 4.4. Ripening of an Undecane in Water Emulsion in the Presence of an Oil Sink. To assess the possible effect of micelles on the Ostwald ripening process in an enhanced way, we determined the ripening rates of several undecane emulsions, prepared in such a way that the continuous-phase surfactant concentration is above the cmc. An aqueous micellar surfactant solution is added at a constant rate. Since the solubilization rate of the oil in the added micellar solution is fast compared to the Ostwald ripening rate, the studied system can in a first approximation be considered as a model system of ripening in the presence of an instantaneous oil sink. Mother emulsions of undecane in a 0.1 M SDBS aqueous solution were prepared. The oil volume fraction was 0.01. From the experiments in the previous section, we learned that the oil is solubilized very quickly by the SDBS micelles, if the latter are present in large amounts. To prevent the solubilization from masking the actual ripening by too fast a destruction of the oil droplets, the micellar SDBS solutions were added slowly and continuously, using a computer-controlled syringe. The mixture was kept homogeneous by continuously stirring it magnetically. In the first couple of experiments, 0.05 M SDBS aqueous solutions were added to samples of 10 mL of the mother emulsions at the rates of 1 and 4 mL/h. In a second series of three experiments, 0.15 M SDBS solutions were added to 10 mL samples of the mother emulsions at rates of 1, 2, and 4 mL/h. Every 10-15 min the intensity-weighted average droplet size aI was determined. The monitoring of the particle size was pursued until the scattering power became too small to reliably determine the average droplet size. During all size experiments the growth of the average

De Smet et al.

Figure 10. Comparison of evolution of aI3 for undiluted, original emulsions (triangles) to that of emulsions continuously diluted with SDBS solution (circles). The ordinates of the different curves are translated by a value of 2 × 105 nm3. The curves are arranged from bottom to top as in Table 2: the lower curve corresponds to emulsion 6 in Table 2, the upper curve to emulsion 10.

Figure 11. Relative ripening rate as a function of SDBS addition rate.

particle size of samples of the mother emulsions was also monitored in parallel. The evolution of aI3 in these experiments is shown in Figure 10, together with the evolution of aI3 determined in parallel for the mother emulsion. Both ripening curves coincide almost completely. The scatter of the aI3 data at the end of the monitoring of the emulsion to which the surfactant solution was added is attributed to its very weak scattering power, making the particle size determination less reliable. In Figure 11 the ratio of the ripening rates of the emulsions to which the micellar solutions were added to those of the mother emulsions, which ripened without addition of surfactant, are plotted as a function of the addition rate. Clearly the values of this ratio of rates scatter around 1, showing that there is no significant effect on the growth rate due to the addition of micelles. The average ripening rate of the five mother samples (no addition of SDBS solution) was 10.5 nm3 s-1, i.e., a factor of 1.59 larger than the theoretical rate of 6.60 nm3 s-1. The evolution of the average scattered light intensity of the continuously diluted emulsion, however, shows a decrease after an initial increase (Figure 12). The initial increase is attributed to the average droplet size growth, while the subsequent decrease is due to solubilization.


Langmuir, Vol. 15, No. 20, 1999 6753 Table 2. Ostwald Ripening in the Presence of a Sink: Experimental Data and Estimated Parameters

emulsion no.

surfactant concn of added solution (M)

addn rate of surfactant (mL/h)

time at which scattered intensity peaks (103 s)

Nsol/Nagg

6 7 8 9 10

0.05 0.05 0.15 0.15 0.15

1 4 1 2 4

18.7 3.9 6.5 2.9 1.8

0.52 0.59 0.56 0.58 0.60

a

b

Figure 12. Evolution of scattered light intensity with time for emulsions that were continuously diluted with (a) 0.05 M SDBS solution and (b) 0.15 M SDBS solution (addition rates 1, 2 and 4 mL/h: labeled with “1”, “2”, and “4”).

The maximum in the intensity is related to the moment at which solubilization takes over from Ostwald ripening, that is when the fraction fOR of oil molecules exchanged between the droplets has decreased to the value of the fraction of oil swallowed by the sink (fs,1 + fs,2). To compare the experimental results to predictions from the model of Ostwald ripening with a sink, the time evolution of the scattered light intensity was also computed. To this end the initially log-normal distribution (eq 9) with σ ) 0.24 and ap as determined from the experimental initial intensity average size as explained in section 4.2.1 was chosen. The scattered intensity corresponding to the distribution, sampled as before by 5000 droplets with different sizes, is computed in arbitrary units with 5000

I(t) )

Si(ai,t) ∑ i)1

(18)

The number of surfactant molecules Ns added per second in an experiment is related to the sink term Ns,2 in the simulation by

Ns ) Ns,2(Nagg/Nsol)

(19)

The ratio Nsol/Nagg is left as a parameter in the simulation. Its value is determined by the condition that the times at which the maximum in I(t) occurs coincide

Figure 13. Simulation of evolution of scattered light intensity for emulsions that were continuously diluted with (a) 0.05 M SDBS solution and (b) 0.15 M SDBS solution (addition rates 1, 2 and 4 mL/h: labeled with “1”, “2”, and “4”).

with the maxima in the experimental curves. The results for the values of Nsol/Nagg obtained this way are reported in Table 2. In Figure 13 the corresponding simulated curves for I(t) are shown. The ratio Nsol/Nagg is fairly constant, with an average of 0.57. Note that this is about a factor of 5 larger than the value predicted from the equilibrium solubility of undecane,11 and that the largest value of 0.60 is obtained for the highest addition rate, while the smallest value of 0.52 (closer to the equilibrium value of 0.12) occurs at the slowest addition rate. This shows that in the exchange of oil between emulsion droplets and surfactant micelles more oil is involved than expected from the equilibrium solubility of the oil in a micellar surfactant solution. It again supports the model of surfactant micelles not in local equilibrium with oil molecules, as already pointed out in section 4.2.3.

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5. Conclusions The experimental data for the Ostwald ripening in the presence of micelles and for the solubilization of oil in micellar solutions can be explained by a model of exchange of oil molecules between emulsion oil droplets and the oil phase solubilized by surfactant micelles. The transport goes by diffusion of individual oil molecules through the continuous phase, and the surfactant micelles are not in local equilibrium with the oil molecules. The slight enhancement by a factor of about 2 of the Ostwald ripening rates in emulsions of undecane in aqueous SDBS micellar solutions can be attributed to the slightly higher molecular oil concentrations in the continuous phase in overall equilibrium with small micellar oil droplets. The enhancement factors are predicted by Kelvin’s equation assuming that the solubilized oil phase consists of spherical droplets with radii in the range of 2-5 nm. The slight decrease of the enhancement factor of the ripening rates with the surfactant or micellar concentration can be explained by an increase of the oil solubilization capacity of the SDBS micelles. The solubilization rate of several alkane emulsions in 0.1 M SDBS appears to be proportional to the oil solubility in water, confirming the hypothesis that there is no direct exchange of oil molecules between emulsion droplets and the oil phase solubilized by the micelles. The ratedetermining factor for the transport of oil is the molecular diffusion of oil through the continuous aqueous phase. Simulations indicate that at surfactant concentrations in the continuous phase above the critical micellar

De Smet et al.

concentration a negligibly small fraction of oil molecules is solubilized during the ripening process by surfactant released from the decreasing interfacial area. It was also shown, both by the simulation of Ostwald ripening in the presence of an oil sink and by an experimental study of the ripening of emulsions to which micellar solutions were added continuously, that the Ostwald ripening rate is not affected significantly, as long as the rate of withdrawal or solubilization of oil molecules is smaller than the rate of transport of oil molecules from smaller to larger droplets. The experimental data also confirm the hypothesis that the surfactant micelles are not in local equilibrium with the oil molecules. It should be noted that in all experiments an anionic surfactant was used. Since electrostatic repulsive interactions are important in such systems, it may well be that the observations and conclusions made in this study do not apply to systems with nonionic surfactants, such as in the studies by McClements and Dungan.30-32 Acknowledgment. The Fonds voor Wetenschappelijk Onderzoek (FWOsBelgium) is acknowledged for partial support of this research. LA9901736 (30) McClements, D. J.; Dungan, S. R.; German, J. B.; Kinsella, J. E. Food Hydrocolloids 1992, 6, 415. (31) McClements, D. J.; Dungan, S. R. J. Phys. Chem. 1993, 97, 7304. (32) McClements, D. J.; Dungan, S. R. Colloids Surfaces A 1995, 104, 127.

Ostwald Ripening of Alkane Emulsions in the Presence of Surfactant

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