Transient Behavior of Polydisperse Emulsions Undergoing Mass

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Ind. Eng. Chem. Res. 2002, 41, 6284-6296

Transient Behavior of Polydisperse Emulsions Undergoing Mass Transfer Alejandro A. Pen ˜ a† and Clarence A. Miller* Chemical Engineering Department, Rice University, 6100 Main Street, Mail Stop 362, Houston, Texas 77005

Numerical simulations of transport in polydisperse oil-in-water emulsions were carried out for three situations in which the aqueous phase contained surfactant micelles: solubilization of a pure oil; Ostwald ripening of a pure oil; and compositional ripening of a mixed emulsion initially containing drops of two pure oils, only one of which was solubilized to a significant extent. Transport was assumed to be controlled by interfacial effects, not by diffusion of micelles in the aqueous phase. The results were compared with published data for changes in drop size distribution in hydrocarbon-in-oil emulsions stabilized by nonionic surfactants for each of the three phenomena. In some cases the published data were supplemented by measurements of interfacial tensions and solubilization rates of single drops of the same oils in solutions of the same nonionic surfactants. Excellent agreement between simulation results and the published data was found for the solubilization and compositional ripening cases with no adjustable parameters. Moreover, accounting for polydispersity was found to be important, i.e., agreement was sometimes not as good when the same transport mechanism was assumed but with a monodisperse distribution corresponding to average drop size. Although results for the Ostwald ripening case showed that interfacial effects and not diffusion controlled transport, confirming that the same mechanism of transport applied in all three cases, numerical agreement between simulations and experiment was not as good, probably because of additional transport between flocculated drops not accounted for in the theory. Introduction An emulsion is a relatively stable collection of droplets of at least one liquid dispersed into another liquid with which it is immiscible. If the droplets are uniformly sized, the emulsion is referred to as monodisperse, but in practice, all emulsions exhibit some degree of polydispersity. The relative stability mentioned above is conferred by the presence of additional agents, usually surfactants or finely divided solids. In any case, emulsions do degrade through several mechanisms such as mass transfer, creaming, flocculation, coagulation, and coalescence, which ultimately leads to the dissolution of the dispersed phase and/or the separation of the liquids.1 Mass transport in emulsions has been the subject of a number of research efforts that might seem unrelated in many cases. First, some investigations2-4 report experiments on the solubilization of oil droplets in micellar aqueous surfactant solutions when the concentration of the transferred oil in the bulk phase is below the saturation limit (hereafter referred to as the equilibrium solubilization capacity, c∞). It might be expected that droplet sizes would diminish because oil is transferred from drops to solution, but these studies indicate that the mean size does not change appreciably in time or even increases in some cases. An explanation for these counterintuitive observations is provided in this paper. Second, several studies5-10 have dealt with so-called Ostwald ripening, a phenomenon in which bigger drops * To whom correspondence should be addressed. Phone: (713) 348-4904. Fax: (713) 348-5478. E-mail: [email protected]. † Permanent address: Laboratorio FIRP, Escuela de Ingenierı´a Quı´mica, Universidad de Los Andes, Me´rida 5101, Venezuela.

grow at the expense of smaller drops through the wellknown Kelvin effect.11 In contrast to the solubilization studies referred to earlier, Ostwald ripening takes place when the bulk phase is slightly supersaturated in the solute that is being transferred. The so-called LSW theory, often used to analyze Ostwald ripening data, was developed by Lifshitz and Slyozov12 and Wagner13 to model precipitation from supersaturated solutions. Finally, when more than one solute is present in the emulsion, differences in concentration among droplets drive mass exchange and induce changes in their sizes.14-16 This phenomenon, which has been referred to as compositional ripening,3 is frequently regarded as a particular case of Ostwald ripening.9,17 However, this generalization is inadequate because, in the case of compositional ripening, mass transfer is not dictated by differences in drop sizes.18 This paper provides the theoretical considerations needed to develop a computational model able to account for mass-transfer kinetics in polydisperse emulsions and emulsion mixtures, based on the following assumptions: (a) transport rates are controlled by resistance to mass transport at the water-oil interfaces, not by diffusion; (b) only one solute is exchanged between the dispersed and continuous phases of the emulsion; (c) mass transfer is the only significant mechanism affecting the transient behavior of the drop size distribution; and (d) interactions among droplets are negligible. The model was applied to simulate experiments reported in the literature for mass transport in emulsions of hydrocarbons in nonionic surfactant solutions, for which the assumptions made above are plausible. Three case studies representative of the processes referred to above, namely, solubilization of a pure oil, Ostwald ripening of a pure oil, and compositional ripening of a mixture

10.1021/ie0109861 CCC: $22.00 © 2002 American Chemical Society Published on Web 05/04/2002

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contact with the same bulk solution across a flat interface, respectively; v is the molar volume of the solute; and T and R have their usual meaning of absolute temperature and ideal gas constant, respectively. The assumption of local equilibrium at the boundary between the drop interior and the (very thin) interfacial region (see Figure 1) is implicit in eq 1. The concentration of solute in the micellar solution cB can be obtained from an overall balance of dispersed phase in the emulsion. For the case in which drops exhibiting two different initial compositions are present, one obtains Figure 1. Basic configuration of the water-oil interfacial region showing the quasi-steady interface-controlled concentration profile of a solute being transferred from drop to solution.

of two emulsions, are presented and discussed. In so doing, we aim to show that these diverse phenomena can be described using a single theoretical model. Theory Mass-Transfer Model. The following considerations are an extension of a model presented in a previous paper for mass transfer in monodisperse emulsions.18 Here, the more general case of polydisperse emulsions is considered. Figure 1 illustrates the configuration of the interfacial region that is adopted to develop the theory. First, it is assumed that mass-transfer kinetics is dictated by interfacial phenomena, probably by the rates of adsorption and desorption of micelles that act as carriers for the solute, and not by diffusion of micelles in the continuous phase. This assumption is based on the results of numerous experiments on rates of dissolution of single drops of hydrocarbons in solutions of nonionic surfactants at concentrations well above the cmc,19-25 which demonstrate that interfacial phenomena are rate-controlling. Second, ideal solution behavior is assumed for the drop interior. Also, the dispersed phase is either pure or a binary mixture containing only one solute that can be transferred across the water-oil interface. The terms cI,∞ and cI stand for the concentration of solute at either side of the interfacial region, and cB is the corresponding concentration in the bulk solution, far from the interface (see Figure 1). Finally, it is assumed that the molecular solubility of the solute in the bulk phase is negligible when compared to its micellar solubility. The following expression can be used to relate the concentration cI,∞ with the equilibrium solubilization capacity c∞ by a factor that includes the difference in chemical potential because of curvature and composition15

cI,∞ ∆µ ) exp c∞ RT

( )

Φ1,0 + Φ2,0 + vcB,0 )

4π

∆µ ) µ(x,R) - µ0 ) RT ln x +

2γv (1) R

whence

cI,∞ 2γv ) x exp c∞ RTR

(

)

(2)

Here, γ is the water-oil interfacial tension; µ(x,R) and µ0 are the chemical potentials of the solute in a drop of size R and molar composition x and of pure solute in

∑

n2

3

∑

( Ri,1 + Rj,23) + vcB (3) 3V i)1 j)1

where Φm,0 and nm are the initial volume fraction and the initial number of type m drops in a volume V of emulsion. Clearly, emulsions containing one drop type can be treated as a particular case of eq 3. The transient behavior of drop radii can be modeled using the following expression, which has been proposed to describe the micellar solubilization of hydrocarbon droplets in nonionic surfactant solutions18

(

) [

cB cI,∞ cB 2γv dR ) kcS - x exp ) kcS dt c∞ c∞ c∞ RTR

(

)]

(4)

where k is a solubilization rate constant and cS is the surfactant concentration available to form micelles in the bulk solution. This expression holds for the case in which mass-transfer rates are interface-controlled, for which cI ∼ cB as shown in Figure 1. The following expression can be obtained from eq 4 for pure drops by neglecting the effect of interfacial curvature on the surface concentration of solute, i.e., by assuming that cI,∞/c∞ ≈ 1; then multiplying both sides by 4πR2; adding over the entire population of droplets; and further rearranging

dωB A ) k/i (ω∞ - ωB) dt V

()

(5)

Here, k/i is an overall mass-transfer coefficient with units of length × time-1; A/V is the specific surface area of oil drops exposed to the aqueous phase per unit volume of emulsion; and ωB and ω∞ are the mass fractions of oil solubilized in micelles in the bulk at time t and at saturation, respectively. The coefficients k and k/i are related by

k) where

n1

( )

c∞ / v ki cS

(6)

Equation 5 has been used to describe mass transfer in several emulsions where interfacial phenomena seem to determine transport rates by yielding characteristic values of k/i in each case.2-4,26 However, it does not allow one to keep track of the size of individual droplets in time, and therefore, the transient behavior of the drop size distribution and related parameters such as the mean drop size cannot be followed using this approach. LSW Theory for Interface-Controlled Ostwald Ripening in Emulsions. This section concludes by considering the Liftshitz-Slyozov-Wagner theory for Ostwald ripening referred to earlier for the case in

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(c) The third power of R h increases linearly with time, at a rate ω3 given by

ω3 )

dR h3 4 ) RDc∞v dt 9

(11)

Analogous results can be obtained for the case in which mass-transfer kinetics is controlled by interfacial phenomena and not by molecular diffusion. In this case, the rate of change of the radius of a single drop is given by eq 4 instead of eq 7, and the theoretical analysis can be carried on as shown in the work of Lifshitz and Slyozov.12 By doing so, the following results are obtained: (a) The critical radius RC is given by eq 8 as before. (b) At long ripening times, the emulsion adopts a drop size distribution, scaled to u ) R/RC, of the form

{

Figure 2. Plots of the asymptotic solutions for the number-based drop size distribution of emulsions undergoing Ostwald ripening.

which mass transport is limited by interfacial phenomena. Wagner13 and Marqusee and Ross27 have considered this case for the last stage of interface-kineticslimited precipitation of solids, and we extend it here to the case of ripening emulsions. This theory will be used to assess the rate-controlling mechanism for a particular experiment and also as a reference for testing the numerical algorithm used in the simulations. When applied to emulsions, the LSW theory is based on the following assumptions: (a) The rate controlling step for mass transfer is the molecular diffusion of the solute across the bulk phase. If so, the following equation applies to describe the rate of change of the radius in a single droplet28

dR Dv ) (c - cI,∞) dt R B

(7)

where D is the corresponding diffusion coefficient. (b) The drops are spherical and fixed in space. (c) The emulsion is dilute; that is, interactions among droplets are negligible. (d) The bulk concentration cB is constant everywhere, except at the surface of the droplets, where cI,∞ is given by eq 2. The most important results of this theory are the following: (a) There is a critical radius RC given by

RC )

2γv R where R ) RT ln(cB/c∞)

(8)

24u -3u/(2-u) e pInt(u) ) (2 - u)5 0

u 200 h. Using such linear fits, correlation coefficients of 0.988 were obtained in both cases, and therefore it is not feasible to determine the limiting mass-transfer mechanism based on the quality of the linear fit. If diffusion dictates the Ostwald ripening rate, eq 11 holds at long ripening times and the diffusion coefficient is given by

D)

Figure 6. Case study 2: Experimental and numerical results for the transient behavior of the mean drop size. Top, plot of R h 2 vs time; bottom, plot of R h 3 vs time. Points, experimental data; solid lines, linear fits at long ripening times; dashed lines, results from numerical simulations.

sumption ensures that the initial collection of droplets in the monodisperse emulsion has the same surface-tovolume ratio as the drops in the initial polydisperse emulsion. It is seen that the calculated trend departs significantly from the experimental data and predicts a fast decay of the concentration of oil present as droplets in the emulsion. This significant result indicates that, by neglecting the polydispersity of the initial emulsion, it is possible to obtain results inconsistent with experimental evidence, even though a model that is indeed representative of the underlying mass-transfer mechanism is used to perform the calculations. Case Study 2: Ostwald Ripening. Figure 6 shows (points) the time-dependent behavior of the mean drop size for an emulsion of n-tetradecane (5 wt %) dispersed in an aqueous solution of Tween 20 (5 wt %).10 Data h 3 vs time because a are represented either as R h 2 or R linear behavior should be observed in either case at long ripening times according to the LSW theory, depending on whether the mass-transfer process is controlled by interfacial phenomena or by diffusion, respectively. A common practice when processing Ostwald ripening data is to find the best linear fit by considering all experimental points. However, the LSW theory provides asymptotic solutions for the Ostwald ripening problem. Unless the initial drop size distribution is the same as the asymptotic solution, there will be a transient as the

9 ω3 4 Rc∞v

(18)

With the value of ω3 obtained from Figure 6 (bottom) (ω3 ) 3.16 × 104 nm3/h) and the physical properties listed in Table 2, a diffusion coefficient of 1.54 × 10-14 m2/s is calculated from eq 18. Weiss et al.10 report that the diffusion coefficient of Tween 20 micelles is approximately 4.2 × 10-11 m2/s. With this value, an Ostwald ripening rate 1000 times higher [(ω3)Micelles ) 8.61 × 107 nm3/h] is predicted. Therefore, the diffusion of micelles acting as carriers for the solute was not ratelimiting in this case. In addition, the molecular diffusion coefficient of n-tetradecane in water is ca. 5 × 10-10 m2/s. This value was obtained from the Wilke-Chang equation, which is recommended by Taylor9 for estimating molecular diffusion coefficients for Ostwald ripening studies. With this coefficient, the interfacial tension reported in Table 2, and the molecular solubility of n-tetradecane in water30 (1.1 × 10-5 mol/m3), an Ostwald ripening rate approximately 18 times lower than the experimental one [(ω3)Molecules ) 1.74 × 103 nm3/h] is obtained. Hence, molecular diffusion cannot be primarily responsible for the observed ripening. Moreover, it seems unlikely that micelles would fail to play a major role in Ostwald ripening in view of their dominant role for solubilization and compositional ripening in similar systems, as demonstrated elsewhere in this paper. On the other hand, if interfacial phenomena limit mass-transport kinetics, the mass-transfer coefficient k can be calculated from eq 15 as

k)

81 ω2 32 RcS

(19)

With the experimental Ostwald ripening rate of 101 nm2/h reported in Figure 6 (top) and the parameters given in Table 2, an effective mass-transfer rate keff ) 2.05 × 10-12 m4/(mol s) is obtained. An experimental value of k was calculated from solubilization tests of single drops of n-tetradecane in oil-free aqueous solution containing Tween 20 (5 wt %). In these experiments,

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the drop size decayed linearly in time, and the value of k was determined from the slope as explained in a previous paper.18 It is worth noting that the linear decay of drop sizes in solubilization tests is indicative of interface-controlled mass-transport kinetics.18-25 An average value of k ) 4 × 10-13 m4/(mol s) was calculated from these experiments, whence an Ostwald ripening rate of 20 nm2/h is obtained from eq 15. Also, Weiss et al.10 report the mass-transfer coefficient k/i for an emulsion of n-tetradecane suspended in a 2 wt % Tween 20 solution (k/i ) 1.9 × 10-8 m s-1). A value of k ) 7.0 × 10-13 m4/(mol s) is obtained using eq 6, and a corresponding Ostwald ripening rate of 35 nm2/h is predicted by eq 15. Therefore, the ripening rates that are calculated from data reported independently in refs 10 and 18 are comparable but consistently smaller than that of the ripening experiment depicted in Figure 6. Whereas coalescence is not expected to play a significant role in this ripening rate as noted by Weiss et al., 10 flocculation might well be responsible for the observed difference between the experimental and theoretical rates for interface-controlled mass transfer. Flocculation by a depletion mechanism would be favored by the high concentration of nonionic surfactant present in the system (5 wt %). For droplets in contact, it is plausible to consider other mechanisms in addition to micellar transport. Kumacheva et al.31 used electron microphotography to show the formation of bridges between the adsorption layers at the interfaces of flocculated drops in a ripening oilin-water emulsion stabilized with a nonionic surfactant (Proxanol-268) and claimed that such “bridging” intensified the process of Ostwald ripening and caused deviations of transient behavior of R h 3 from linearity. In this case, it is plausible to think that mass transfer is aided by the transient opening of holes smaller than the critical size for coalescence across the films between flocculated droplets. Also, migration of oil associated with the tails of Tween 20 molecules adsorbing at the water-oil interfaces could take place. This mechanism would be aided by the fact that the aqueous thin films in contact with flocculated drops for which R > RC exhibit local supersaturation in n-tetradecane. The latter mechanism is analogous to that suggested by Wen and Papadopoulos32 to explain transport rates of water in W1/O/W2 emulsions across a thin oil film via hydration of the headgroups of an oil-soluble nonionic surfactant (SPAN 80). In the latter study, mass-transfer rates between W1/O and O/W2 interfaces in contact were 20 times those found between separated interfaces. The coefficient k/i noted above was obtained from tests with an emulsion in which some flocculation might be expected. The fact that the corresponding rate k was larger but still comparable to that obtained from solubilization tests of individual droplets suggests that micellar transport contributes significantly to the overall ripening rate and that mass transfer was enhanced by concomitant mechanisms such as those suggested above. The dashed lines in Figure 6 represent numerical simulation results with the model presented in the theory section using the given initial drop size distribution, keff ) 2.05 × 10-12 m4/(mol s) obtained from the slope shown in Figure 6 (top), and the constants specified in Table 2. As expected, the slopes at long ripening times in the plot of R h 2 vs time for the solid and dashed lines are practically identical, and the difference

Figure 7. Case study 2: Validation of the extension of the LSW theory for interface-controlled Ostwald ripening via numerical simulations. Top, transient behavior of the drop size distribution; bottom, correlation between the mean size R h and the critical radius RC.

between the two is explained by the approximation exp(R/R) ≈ 1 + R/R that holds for eq 15 but was not needed for the numerical simulations. The most relevant feature of the numerical results is that the model correctly accounts for the time frame of the transition between the initial state and the asymptotic regime at long ripening times. Therefore, the mass-transfer model proposed in this work based on interfacial control is adequate for simulating the ripening of the emulsion. Finally, the Ostwald ripening theory for interfacecontrolled mass-transfer kinetics was useful for validating the numerical algorithm and vice versa. Figure 7 shows the time-dependent behavior of the cumulative form of the drop size distribution and of the mean radius R h with respect to the critical radius RC using keff ) 2.05 × 10-12 m4/(mol s). The theoretical expression derived for the cumulative form of the limiting drop size distribution of the emulsion (eq 14) is also plotted as a solid line for comparison. Results show that the simulated drop size distribution does adopt the theoretical limiting curve at ripening times beyond approximately 300 h. Also, the companion chart illustrates that the relationship R h ) 8/9RC is indeed correct. These findings

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Figure 8. Case study 3: Drop size distributions of the initial emulsions before mixing. Solid bars and lines, experimental data; dashed lines, numerical fit using the log-normal pdf (eq 16). Table 3. Parameters and Physical Properties Used in Mass-Transfer Simulations for the Decane (7 vol %)/ Squalane (1 vol %)/Water/C12E8 (2.5 wt %) Mixed Emulsion parameter (units)

value

parameter (units)

value

(dgV)1,0 (m) σ10 (dgV)2,0 (m) σ2,0 n* Φ1,0 Φ2,0 x1,0 x2,0

7.80 × 10-7 0.4417 1.72 × 10-5 0.4023 15000 0.01 0.07 0 1

θ0 k [m4/(mol s)] c∞ (mol/m3) cS (mol/m3) T (K) γ (N/m) v (m3/mol) vI (m3/mol) ∆t (s)

0.15 8.9 × 10-11 10.2 46.5 293 1.1 × 10-3 1.95 × 10-4 5.22 × 10-4 1

are independent of the mass-transfer coefficient that is used, as found in additional simulations with other values of k. Again, the match between the numerical results at long times and the theory is very close in all cases, but it is not exact because of the approximation exp(R/R) ≈ 1 + R/R referred to earlier. Finally, Ratke and Thieringer33 studied the effect of particle motion (Stokes and Marangoni flow) on Ostwald ripening and derived alternative expressions for the asymptotic drop size distribution and ripening rates. Whereas these considerations might be relevant when diffusion is rate-controlling because particle motion alters the shape of the isoconcentration contours surrounding the droplets, convective effects should not significantly affect the mass-transfer kinetics for isothermal systems where interfacial phenomena are limiting. Case Study 3: Compositional Ripening. In this section, the experiment of Binks et al.15 in which two emulsions, one containing n-decane dispersed in a 2.5 wt % solution of the nonionic surfactant C12E8 and another in which squalane was dispersed in a similar surfactant solution, were blended to make a mixed emulsion containing 7 vol % of n-decane and 1 vol % of squalane. Figure 8 shows the volume-weighted drop size distributions of the emulsions before mixing (solid bars) and the fit of such distributions with the log-normal distribution (eq 16) (dashed lines). The parameters of the initial distribution, along with the constants that were used in the simulations referred to later in this section, are reported in Table 3. It has been shown that squalane is practically insoluble in the surfactant solution whereas n-decane exhibits limited but measurable solubility.15,18 There-

fore, it can be assumed that only n-decane is transferred across the continuous phase by the micellar aggregates. As time proceeds, the drops containing squalane (type 1) grow at the expense of the solubilization of the n-decane (type 2) droplets. Figure 9 illustrates the time-dependent behavior of the volume-weighted drop size distributions for the mixed emulsion. The peak on the left-hand side of each frame corresponds to the drops containing initially pure squalane (type 1), whereas the peak on right-hand side corresponds to n-decane (type 2) droplets. Clearly, the experiment was devised in a way in which significant overlapping of the two peaks was avoided to facilitate interpretation of the results and characterization of the transient distribution of each drop type. The experimental data are given as solid bars for the drop size distribution and as solid lines for the corresponding cumulative function. The continued solubilization of n-decane becomes evident, because the peak at the right-hand side vanishes as time proceeds. The amount of decane initially present in the emulsion is significantly greater than the equilibrium solubilization capacity of decane in the surfactant solution (see Table 3), and therefore n-decane is transferred into the squalane drops until a single peak is obtained at the end of the experiment. The numerical results are presented as dashed lines for the drop size distribution and as dot-dashed lines for the corresponding cumulative distribution function. It is seen that the agreement between the experiments and the numerical results is remarkable, even at t ) 60 s when differences are more noticeable possibly because of the adjustment of the bulk concentration of n-decane at the beginning of the test. It is worth emphasizing that no adjustable parameters were used in the simulations. Importantly, because the algorithm kept track of the transient behavior of every drop within the population n*, it was feasible to assess the behavior of the distribution of sizes for each drop type individually. When this was done, some overlapping was observed at long times, but up to 8460 s, the overlap was always small and not likely to affect the characterization of each peak individually from the experimental data. It is natural to wonder how it is that the two peaks do not overlap significantly, if the n-decane (type 2) drops, which are larger than the type 1 drops at the beginning of the experiment, are solubilized and their sizes reduced. This apparent paradox can be explained

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Figure 9. Case study 3: Transient behavior of the drop size distribution for the mixed emulsion. Solid bars and lines, experimental data; dashed lines, results from numerical simulations.

by considering two facts: (a) From the data provided in Table 3, one can calculate that the population of type 1 droplets is greater than the population of type 2 droplets by a factor of 1000. Therefore, when drops of the two types are present in the same class of the histogram, the fraction of type 2 drops present is negligible. (b) The distributions reported in Figure 9 are volume-weighted and not number-based. This is why regardless of the fact that the number of type 2 drops is much smaller, the corresponding peaks of the type 2 droplets contribute significantly to the overall distribution at large drop sizes but not at the small drop sizes, for which the contribution of type 1 drops is clearly more significant.

Figure 10 shows (points) the transient behavior of the geometric mean size of the volume-weighted distribution for each drop type as reported by Binks et al.15 It is observed that, whereas the average size of the drops containing squalane (type 1) increases steadily, the behavior of the mean size of the type 2 droplets is rather erratic but exhibits a slight decrease. This same experiment was analyzed in a previous paper in which it was shown that mass-transfer kinetics are limited by interfacial phenomena and not by diffusion of the solute across the bulk phase either as individual molecules or dissolved into micellar aggregates for this system.18 The dashed lines in Figure 10 represent calculations from an interface-controlled

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Figure 10. Case study 3: Transient behavior of the volumeweighted mean drop sizes of the two drop types present in the mixed emulsion. Solid bars and lines, experimental data; dashed lines, results from numerical simulations.

mass-transfer model analogous to the one presented here but simplified for the case of a mixture of two monodisperse emulsions. It is seen that the agreement between the calculated profile for the type 1 drops and the experimental data is excellent, whereas for the type 2 drops, the model predicts a decay of the mean size somewhat more pronounced than that observed experimentally. The solid lines in Figure 10 represent the predictions from the mass-transfer model presented in this work, considering the polydispersity of the original emulsions before mixing them. It is seen that the model reproduces very well the experimental results both for the type 1 and the type 2 droplets. Therefore, the lack of agreement between theory and experiments for the type 2 drops that had been reported by Binks et al.15 when considering diffusion-controlled kinetics and by Pen˜a and Miller18 when assuming interface-controlled kinetics can be attributed to the finite polydispersity of drop sizes in the emulsion, as suggestedsbut not provedsearlier by these authors. On the other hand, the relative insensitivity of the prediction for type 1 drops to polydispersity suggests that the number-based drop size distribution changes in a different fashion depending on whether the drops are shrinking or growing. Figure 11 shows numerical results for the number-based drop size distributions for type 1 and type 2 drops separately. It illustrates that,

Figure 11. Case study 3: Numerical results for the timedependent behavior of the number-based drop size distributions of the two drop types present in the mixed emulsion.

when the drops are shrinking, the distribution becomes broader and tailed toward small drop sizes, exactly as found in the solubilization example discussed earlier. If the distribution is volume-weighted, the relative contribution of the smaller drops becomes negligible and the volume-weighted mean size exhibits a relatively small change. On the other hand, if the drops increase their sizes, the shape and width of the distribution remain practically unchanged, and the geometric mean grows at the same rate at which the distribution is displaced toward larger drop sizes. This assessment is, in this case, practically independent of the base in which the distribution is reported because, as mentioned earlier in the discussion of eq 16, corresponding numberbased and volume-weighted log-normal distributions have the same width σ and their geometric means differ by a constant factor if σ remains unaltered. Finally, the numerical results presented in this section for the model that considers the polydispersity of the emulsions were recalculated neglecting the contribution of interfacial curvature to the surface concentration of n-decane by setting exp(2γv/RTR) ≈ 1. These calculations did not differ significantly from those reported above, thus confirming that mass transfer was driven by concentration gradients of decane between the drops and the bulk phase and not by differences in drop sizes. Therefore, compositional ripening alone degraded this emulsion.

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Conclusions A model was proposed to evaluate mass transfer in polydisperse emulsions, limited to the case in which one solute is transferred from drop to bulk solution and vice versa. This model includes polydispersity and assumes transport rates dictated by interfacial phenomena and not by molecular or micellar diffusion in the surfactant solution. It was satisfactorily used to simulate results from three experiments reported in the literature for mass transfer in emulsions of hydrocarbons dispersed in aqueous nonionic surfactant solutions, namely, the solubilization of drops in an undersaturated surfactant solution,2 the Ostwald ripening of a single emulsion,10 and the compositional ripening of a mixed emulsion.15 Simulations for tests in which mass transport took place at a bulk concentration below the solubility limit (solubilization, compositional ripening) showed that considering the polydispersity in drop sizes significantly improved the agreement between predicted and experimental transient characteristics such as the mean drop size or the concentration of solute in the surfactant solution. Also, it was demonstrated that the trends that are observed for the transient drop size distribution or the mean size of drops undergoing solubilization are heavily influenced by the basis (number, area, or volume of droplets) upon which these properties are reported. Calculations for the Ostwald ripening experiment indicated that mass transfer was not limited by diffusion. Our simulations strongly suggest that mass transfer was controlled by events taking place at the interfaces and that the experimental ripening rate was somewhat higher than its corresponding theoretical value because of phenomena not included in the model, such as enhanced transport between drops that have flocculated and are nearly in contact. Finally, the extension of the LSW theory presented in this work for interface-controlled Ostwald ripening of emulsions was validated via numerical simulations.

χ ) R1,0/R2,0. R1,0 and R2,0 are reference radii for type 1 and type 2 drops, respectively. These reference radii are arbitrary, but it is advisable to choose plausible values such as a characteristic mean of the initial drop size distributions. When θ is calculated in successive iterations u and u + 1 from eq 20, one obtains u+1

θ

u

rθ -

n1

κ

{χ

c∞v

n2

u+1 3 u 3 [(rj,2 ) - (rj,2 ) ]} ∑ j)1

θu+1 r θu -

κ

Dimensionless Finite-Difference Formulation of the Mass-Transfer Model. In this appendix, an algorithm that is able to account for mass transfer in polydisperse emulsions with drops having two different initial compositions and one mobile solute is developed. Emulsions with drops of the same composition can be treated as particular cases of the derivation. The dimensionless concentration of solute in the bulk solution θ ) cB/c∞ can be readily obtained from eq 3 as

c∞v

(Φ1,0 + Φ2,0) -

∑{n1χ3[(ri,1u+1)3 - (ri,1u )3] +

n*c∞vi)1

u+1 3 u 3 ) - (ri,2 ) ]} (22) n2[(ri,2

Either eq 21 or eq 22 requires knowledge of the drop radii at time steps u and u + 1. Equation 4 can be written for the type 1 droplets as

[

( )]

dri,1 1 σ ) θ - xi,1 exp dτ χ χri,1

n1

κ

n2

ri,13 + ∑rj,23) ∑ i)1 j)1

c∞v

(χ3

(20) where κ ) 4πR2,03/3V, ri,1 ) Ri,1/R1,0, ri,2 ) Ri,2/R2,0, and

1 e i e n*

(23)

where τ ) (kcS/R2,0)t and σ ) (2γv)/(RTR2,0). For the type 2 drops, one obtains

1 e i e n*

(24)

The finite-difference formulation arises when the rate of change of the dimensionless radii is replaced by the forward difference approximation u+1 u dri,m ri,m - ri,m ≈ dτ ∆τ

Appendix A

θ ) θ0 +

n*

( )

A.P. is a J.W. Fulbright scholar at Rice University. The Rice University Consortium for Processes in Porous Media, OndeoNalco Energy Services, and the Chemical Engineering Department at Universidad de Los Andes (Venezuela) also supported this research project.

(21)

Equation 21 provides a way to calculate the bulk concentration at time step u + 1. However, it is computationally inefficient if used as shown since either number of droplets n1 or n2 could be excessively large. A way to resolve this issue is to choose a number of droplets n* large enough to be representative of the drop size distributions of both drop types but small enough to perform calculations within a reasonable computational time. If so, the following approximation can be made

dri,2 σ ) θ - xi,2 exp dτ ri,2

Acknowledgment

1

u+1 3 u 3 [(ri,1 ) - (ri,1 ) ]+ ∑ i)1

3

(25)

with ∆τ ) (kcS/R2,0)∆t. The combination of eqs 23-25) yields the following expressions for the radii of the two drop types at the iteration u + 1 u+1 u r ri,1 + ri,1

[

( )] [

σ ∆τ u u θ - xi,1 exp u ; χ χri,1 u+1 u u r ri,2 + ∆τ θu - xi,2 exp ri,2

( )] σ u ri,2

(26)

A numerical solution for the transient behavior of each drop size within the sample of n* type 1 and n* type 2 drops and for the transient behavior of the concentration u+1 of solute in the bulk can be obtained by calculating ri,1 u+1 u+1 and ri,2 using eq 26 and then determining θ from eq 22. These results are further assigned as initial values for the next time step. The drop population

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6295

Figure 12. Effect of the selection of the time step ∆τ on the numerical response of the proposed algorithm for case study 1.

balance can be accounted for by assigning zero radius u+1 to drops for which ri,m < 0 and subtracting the number of drops of either type for which this assignment is made from the corresponding sample population. Finally, the molar fraction of mobile solute in each drop type, xi,m, that is needed in eq 26 is determined from a mass balance in the drop, from which the following expression is obtained

xi,m )

ri,m3 - φmηi ri,m3 - φmηi(1 - δ)

where

ηi )

( ) Ri,0 Rm,0

3

and δ )

v (27) vI

or otherwise xi,m ) 1 if pure drops of solute are considered. In eq 27, v and vI are the molar volumes of the mobile and immobile species, respectively, and φm is the initial volume fraction of immobile compound present in the type m drops. Note that in the definition of ηi, Ri,0 stands for the initial radius of drop i, whereas Rm,0 is the reference size for the type m drops referred to earlier. Equations 22 and 26 define an explicit numerical scheme, because the variables at the time step u + 1 can be calculated from their values at time step u. The robustness of such scheme depends on the selection of ∆τ, because large values of this variable lead to numerical errors and instabilities. For each simulation, a small ∆τ was set for each case by successive trials, with the aim of ensuring numerical stability, negligible error due to the size of the time step, and a reasonable computational time. Figure 12 shows the influence of ∆τ on the calculated behavior of the dimensionless concentration of oil in the bulk phase θ for case study 1. It is seen that the value of ∆τ that was chosen for these calculations (∆τ ) 2.35 × 10-4 or ∆t ) 30 s) was 3 orders of magnitude smaller than the one at which divergence of the numerical response starts to become noticeable. A similar criterion was applied to determine ∆τ for case studies 2 and 3. Literature Cited (1) Walstra, P. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1990; Vol. 4.

(2) McClements, D. J.; Dungan, S. R. Light Scattering Study of Solubilization of Emulsion Droplets by Nonionic Surfactant Solutions. Colloids Surf. A 1995, 104, 127. (3) Weiss, J.; McClements, D. J. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Solubilization. Langmuir 2000, 16, 5879. (4) Weiss, J.; Coupland, J.; McClements, D. J. Solubilization of Hydrocarbon Emulsion Droplets Suspended in Nonionic Surfactant Micelle Solutions. J. Phys. Chem. 1996, 100, 1066. (5) Kabalnov, A. S.; Pertzov, A. V.; Shchukin, E. D. Ostwald Ripening in Emulsions. J. Colloid Interface Sci. 1987, 118 (2), 590. (6) Kabalnov, A. S.; Shchukin, E. D. Ostwald Ripening Theory: Applications to Fluorocarbon Emulsion Stability. Adv. Colloid Interface Sci. 1992, 38, 69. (7) Kabalnov, A. S. Can Micelles Mediate a Mass Transfer between Oil Droplets? Langmuir 1994, 10, 680. (8) Soma, J.; Papadopoulos, K. D. Ostwald Ripening in Sodium Dodecyl Sulfate-Stabilized Decane-in-Water Emulsions. J. Colloid Interface Sci. 1996, 181, 225. (9) Taylor, P. Ostwald Ripening in Emulsions. Adv. Colloid Interface Sci. 1998, 75, 107. (10) Weiss, J.; Canceliere, C.; McClements, D. J. Mass Transport Phenomena in Oil-in-Water Emulsions Containing Surfactant Micelles: Ostwald Ripening. Langmuir 2000, 16, 6833. (11) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1986. (12) Lifshitz, I. M.; Slyozov, V. V. The Kinetics of Precipitation from Supersaturated Solutions. J. Phys. Chem. Solids 1961, 19, 35. (13) Wagner, C. Theorie der Alterung von Niederschla¨gen durch Umlo¨sen (Ostwald-Reifung). Z. Elektrochem. 1961, 65, 581. (14) Higuchi, W. I.; Misra, J. Physical Degradation of Emulsions via the Molecular Diffusion Route and the Possible Prevention Thereof. J. Pharmacol. Sci. 1962, 51 (5), 459. (15) Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Rippon, S.; Lubetkin, S. D.; Mulqueen, P. J. Kinetics of Swelling of Oil-inWater Emulsions. Langmuir 1998, 14, 5402. (16) Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Rippon, S.; Lubetkin, S. D.; Mulqueen, P. J. Kinetics of Swelling of Oil-inWater Emulsions by Different Surfactants. Langmuir 1999, 15, 4495. (17) Kabalnov, A. Ostwald ripening and related phenomena. J. Dispersion Sci. Technol. 2001, 22 (1), 1. (18) Pen˜a, A. A.; Miller, C. A. Kinetics of Compositional Ripening in Emulsions Stabilized with Nonionic Surfactants. J. Colloid Interface Sci. 2001, 244, 154. (19) Carroll, B. J. The Kinetics of Solubilization of Nonpolar Oils by Nonionic Surfactant Solutions. J. Colloid Interface Sci. 1981, 79, 126. (20) Carroll, B. J.; O’Rourke, B. G. C.; Ward, A. J. I. The Kinetics of Solubilization of Single Component Nonpolar Oils by a Nonionic Surfactant. J. Pharm. Pharmacol. 1982, 34, 287. (21) Carroll, B. J. Solubilisation of Two-component Oil Mixtures by Micellar Surfactant Solutions. J. Chem. Soc., Faraday Trans. 1 1986, 82, 3205. (22) Donegan, A. C.; Ward, A. J. I. Solubilization Kinetics of n-Alkanes by a Non-ionic Surfactant. J. Pharm. Pharmacol. 1987, 39, 45. (23) O’Rourke, B. G. C.; Ward, A. J. I.; Carroll, B. J. Investigation of the Solubilization Kinetics of Binary Mixtures of Non-polar Oils by Non-ionic Surfactants. J. Pharm. Pharmacol. 1987, 39, 865. (24) Chen, B. H.; Miller, C. A.; Garrett, P. R. Rates of Solubilization of Triolein into Nonionic Surfactant Solutions. Colloids Surf. A 1997, 128, 129. (25) Chen, B. H.; Miller, C. A.; Garrett, P. R. Rates of Solubilization of Triolein/Fatty Acid Mixtures by Nonionic Surfactant Solutions. Langmuir 1998, 14, 31. (26) Tai, B. H. The Kinetics of Solubilization and Transport by Micelles in Oil-in-Water Emulsions. M.S. Thesis, University of California, Davis, CA, 1999. (27) Marqusee, J. A.; Ross, J. Kinetics of Phase Transition: Theory of Ostwald Ripening. J. Chem. Phys. 1983, 79 (1), 373. (28) Smoluchowski, M. V. Versuch einer Mathematischen Theorie der Koagulationskinetik Kolloider Lo¨sungen. Z. Phys. Chem. 1918, 92, 129. (29) Lim, J. C.; Miller, C. A. Dynamic behavior and detergency in systems containing nonionic surfactants and mixtures of polar and nonpolar oils. Langmuir 1991, 7, 2021.

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Received for review December 6, 2001 Revised manuscript received March 25, 2002 Accepted March 28, 2002 IE0109861