On the Determination of Ostwald Ripening Rates from Dynamic Light

Langmuir 1999, 15, 2327-2332

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On the Determination of Ostwald Ripening Rates from Dynamic Light Scattering Measurements Y. De Smet,* L. Deriemaeker, E. Parloo, and R. Finsy Vrije Universiteit Brussel, Department of Physical and Colloid Chemistry, Pleinlaan 2, B-1050 Brussels, Belgium Received June 24, 1998. In Final Form: December 31, 1998 The Ostwald ripening rate of several alkane in water emulsions stabilized by a nonionic surfactant is determined from dynamic light scattering (DLS) measurements. With the aid of computer simulations, the intensity weighted droplet radii obtained with DLS are converted to number averages, by taking the form of the droplet size distributionswhich evolves continuously toward a stationary distributionsinto account. Thereby the effect of the transition from an initial, log-normal size distribution toward its stationary form is included. Second a model is proposed to account for the effect of the finite size of the surfactant layer (surrounding each oil droplet) on the measured particle size and thus on the ripening rate. It is found that both the effect of the transition from a nonstationary regime toward the stationary Lifshitz-SlyozovWagner regime and the effect of the finite size of the surfactant layer influence the ripening rates significantly.

1. Introduction 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 attained mainly by two mechanisms: first, coagulation, possibly followed by coalescence; second, Ostwald ripening. The former is the one that was most studied (e.g., see 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 owing to the higher solubility of the smaller particles (Gibbs-Thomson or Kelvin effect) and to molecular diffusion through the continuous phase. The Ostwald ripening rate is mainly determined by the solubility of the dispersed phase in the continuous dispersion medium. Since the mutual miscibility of the continuous and dispersed phases is often very low, the effect of Ostwald ripening is sometimes neglected. However, all liquid pairs are mutually miscible to some finite extent and especially in the case of small emulsion droplets this may lead to relatively rapid coarsening. For instance, although the solubility of dodecane in water is extremely low (5.2 × 10-9 mL/mL 2), the average emulsion droplet radius in a dodecane-in-water emulsion stabilized against coagulation, increases from 50 nm for a freshly prepared emulsion to 150 nm in about 10 days.3 A theoretical description of Ostwald ripening in twophase systems has been developed independently by Lifshitz and Slyozov and by Wagner (LSW theory).4-7 It is based on the following assumptions. * To whom correspondence should be addressed. Telephone: +32 2 629 36 89. 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) Hayduk, W.; Laudie, H. AIChE J. 1974, 20, 611. (3) De Smet, Y.; Malfait, J.; De Vos, C.; Deriemaeker, L.; Finsy, R. Bull. Soc. Chim. Belg. 1996, 105, 789. (4) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35. (5) Wagner, C. Z. Elektrochem. 1961, 65, 581. (6) Kabalnov, A. S.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 38, 69. (7) Taylor, P. Adv. Colloid Interface Sci. 1998, 75, 107.

1. Molecular diffusion is the rate-determining step for the transport of material between the particles. 2. There are no particle interactions, i.e., the system is infinitely dilute. 3. The particles are spherical and fixed in space. 4. The concentration of solubilized material is a constant Cm throughout the dispersion medium, except for the direct surroundings of the particles (typically a layer with thickness of the order of the droplet’s radius). The major results of their analysis are as follows. 1. There exists a critical radius ac, given by

ac ) R/ln[Cm/C(∞)]

(1)

where R ) 2γVm/RT is a material dependent constant, called the capillary radius. C(∞) is the bulk solubility of the dispersed phase in the dispersion medium, γ is the interfacial tension, and Vm is the molar volume of the dispersed phase. R and T are the universal gas constant and the absolute temperature, respectively. 2. In the long time limit a stationary regime is reached. (a) In this regime the ripening rate v becomes constant

v≡

daN3 ) 4RDC(∞)/9 dt

(2)

In eq 2 aN denotes the number averaged particle radius and D is the dispersed phase molecular diffusion coefficient. (b) The particle size distribution (PSD) becomes invariant, if scaled to ac

W(u) ≡ W(a/ac) )

81eu2 exp[1/(2u/3 - 1)]

x3 32(u + 3)7/3(3/2 - u)11/13 (u < 3/2)

) 0 (u > 3/2)

(3)

In eq 3 “e” denotes Euler’s number (2.718...). This PSD is referred to as the LSW-PSD and is shown in Figure 1. Note that it is left-skewed (a tail toward the smaller droplet sizes).

10.1021/la9807513 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/06/1999

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Figure 1. LSW-distribution profile, the PSD attained in the stationary regime of the Ostwald ripening process.

Figure 2. Ripening of the Und-Tween1 emulsion is an example of experimental data, showing nonlinear growth. This deviation from the stationary LSW regime is especially visible in the initial stage.

Nowadays the aging of emulsions is often studied by monitoring the average droplet size using dynamic light scattering (DLS), also referred to as photon correlation spectroscopy (PCS)3,8-15. Ostwald ripening rates v are obtained from plots of aN3 vs time, i.e., from the integrated form of eq 2

aN3(t) ) aN3(t0) + v(t - t0)

(4)

In eq 4 t0 represents the time at which the ripening process starts. Although eq 4 predicts a linear dependence of aN3 with time, several experimental data show a nonlinear initial time dependence.3,8-10,15 See, e.g., Figure 2. Several reasons for this nonlinear behavior can be put forward. First, the results of the LSW theory only describe the long time stationary regime and not the transition from an initial state, i.e., a freshly prepared emulsion with a droplet size distribution different from the limiting LSW-PSD (usually right-skewed, e.g., see Figure 3), to the stationary growth regime. In a previous study22 we showed by a computer simulation that the transition of an initially right skewed size distribution to the left skewed (8) Kabalnov, A. S.; Makarov, K. N.; Pertzov, A. V.; Shchukin, E. D. J. Colloid Interface Sci. 1990, 138, 98. (9) Soma, J.; Papadopoulos, K. D. J. Colloid Interface Sci. 1996, 181, 225. (10) De Smet, Y.; Malfait, J.; De Vos, C.; Deriemaeker, L.; Finsy, R. Prog. Colloid Polym. Sci. 1997, 105, 252. (11) McClements, D. J.; Dungan, S. R. Colloids Surf., A 1995, 104, 127. (12) Kabalnov, A. S. Langmuir 1994, 10, 680. (13) Taylor, P.; Ottewill, R. H. Colloids Surf., A 1994, 88, 303. (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.

Figure 3. Particle size distribution of a freshly prepared squalane in water emulsion, stabilized with sodium dodecyl benzene sulfonate.

LSW distribution results in a nonlinear increase of aN3 with time. Second, in the LSW model, no account is taken of the presence of a surfactant interfacial layer of finite thickness. Third, DLS determines the intensity weighted average droplet size.16-17 Proper conversion to the number averaged size requires more than a constant conversion factorswhich does not transform a nonlinear behavior in a linear onesthroughout the monitored ripening process. Finally, another possible reason is the implicit assumption that the interfacial tension remains constant during the ripening process. This paper presents a scheme to obtain the limiting Ostwald ripening rate v from DLS measurements. The scheme accounts for the initial, nonstationary growth regime, for the fact that DLS measurements yield intensity weighted droplet size averages and not number weighted averages and finally for the effect of an interfacial surfactant layer on the measured particle size. It is assumed that the interfacial tension remains constant. Furthermore the effect of the possible additional diffusional resistance of the adsorption layer is not considered. The problem of the initially nonstationary growth regime and that of the conversion from intensity to number weighted average sizes is tackled with a computer simulation. Finally eq 4 will be extended for the case of the presence of an interfacial surfactant layer with finite thickness δ. 2. Experimental Section 2.1. Materials. n-Undecane and n-dodecane were dispersed in aqueous solutions of polyoxyethyleen-20 sorbitan monolaurate (also known as Tween 20) or nonyl phenol ethoxylate-10 (NPE10). The hydrocarbons were purchased from Sigma (purity more than 99%), the surfactants from Riedel-de-Haen (tech.). The water was distilled before use. 2.2. Emulsion Preparation. A course dispersion of the hydrocarbon in a surfactant solution was obtained with an Ultra Turrax premixer. Further homogenization with a high-pressure homogenizer (Microfluidizer Y110) finally yielded emulsions with droplet radii of about 10-50 nm (intensity weighted averages aI(0), obtained by DLS). The oil volume fraction φ was 1% for all emulsions. In Table 1 an overview of the different prepared emulsions is presented. The amount of surfactant adsorbed at the hydrocarbon-water interface was estimated from the initial particle size distribution (log-normal with width σ ) 0.24, see eq 6), and the specific area As of the surfactants (see section 2.3). The surfactant concentration was in all cases high enough to ensure complete coverage of the oil-water interface. The remaining surfactant concentration in the continuous, aqueous phase was for all emulsions, except for UND-NPE1, well above the critical micellar concen(16) Finsy, R.; De Jaeger, N. Part. Part. Syst. Charact. 1991, 8, 187. (17) Finsy, R. Adv. Colloid Interface Sci. 1994, 52, 79.

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Table 1. Main Characteristics of the Studied Emulsions emulsion code

oil

Und-NPE1 Und-NPE2 Und-Tween1 Und-Tween2 Und-Tween3 Dod-NPE Dod-Tween

n-undecane n-undecane n-undecane n-undecane n-undecane n-dodecane n-dodecane

surfactant

aI(0) (nm)

surfactant concn (M)

NPE-10 NPE-10 Tween20 Tween20 Tween20 NPE-10 Tween20

26 24 49 34 31 13 26

2.3 × 10-5 6.2 × 10-3 1 × 10-3 5 × 10-2 1 × 10-2 6 × 10-3 1 × 10-2

a 1000 times, to rule out droplet interactions and multiple scattering effects.10 In a previous study it was shown that this procedure, i.e., diluting a sample taken from the undiluted emulsions just before particle size measurement, did not affect the Ostwald ripening rate.10 To ensure that the surfactant molecules are not desorbed from the interface, dilution was performed with a surfactant solution with about the estimated surfactant concentration in the aqueous phase of the undiluted emulsion. Intensity averaged droplet radii aI were computed from the intensity autocorrelation data with the cumulants method.18-19 The polydispersity index (typically about 0.1), computed from the second cumulant, indicates no more than that the size distribution is narrow for the considered experimental technique. No detailed information about the size distribution can be obtained from it. Therefore only the average radii aI are retained.

3. Effect of the Nonstationary Ripening and Conversion of Intensity to Number Averaged Sizes

Figure 4. Plot of the water-air surface tension (γ) vs ln(Cs) for Tween 20. Table 2. Critical Micellar Concentration and Specific Surface Area of NPE-10 and Tween 20 surfactant

cmc (M)

specific area (nm2)

NPE-10 Tween 20

4.7 × 10-5 7.1 × 10-5

2.0 2.7

tration (cmc). For UND-NPE1 the concentration in the aqueous phase was at about the cmc. Since the interfacial area decreases during the ripening process and since it was initially completely covered, surfactant will be released to the aqueous phase during the ripening process. Thereby the oil-water interface remains completely covered. Therefore, it is assumed that the interfacial tension remains constant during the ripening process. 2.3. Critical Micellar Concentration (cmc) and Specific Area (As) of the Surfactants. 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 vs the natural logarithm of Cs one can calculate the cmc and the specific area As using the Gibbs adsorption equation, assuming an ideal dilute behavior of the surfactant solution

dγ 1 Γ)RT d(ln(Cs))

(5)

where Γ ) NA/As is the adsorption of surfactant molecules at the droplet’s surface, γ is the surface tension and NA is Avogadro’s number. The cmc of the surfactants 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 Tween 20 solutions. Table 2 summarizes the results for NPE-10 and Tween 20. The oil-water interfacial tension (γ ) 10 mN/m) was estimated as about the ones reported by Kabalnov et al.8 for several alkane emulsions stabilized by sodium dodecyl sulfate (SDS). 2.4. DLS Measurements. The experimental setup consists of an Ar+ laser (wavelength λ ) 488 nm), a thermostated sample holder allowing to control the temperature (with accuracy (0.1 °C) 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 166 MHz processor. The measurements took place at 25 °C. Before each measurement a sample was prepared by diluting the original emulsion

The conversion of intensity to number averaged droplet radii and the effect of the initially nonstationary, i.e., nonLSW growth regime, is undertaken by the following procedure. The final aim of this procedure is to compare the experimental average particle sizes, determined with DLS and thus intensity weighted, with average values computed from a simulation procedure of the ripening process. This simulation procedure is started up with a priori determined initial particle size distributions. The evolution of the particle size distribution in time is then computed. At each time step the number and intensity weighted average particle sizes are computed from the actual form of the distributions obtained from the simulation. The ratio of both average sizes yields then at each time step the conversion factor between number and intensity weighted average radii and vice versa the conversion factor of intensity to number average. 3.1. Effect of the Nonstationary Ripening. The amplitude of this effect is evaluated by simulation. Thereby the results of the growth regimes starting with two different initial PSD’s are compared. The first growth regime is the stationary one predicted by the LSW theory, i.e., for an initial (stationary) LSW size distribution given by eq 3. Since the size distribution of a freshly prepared emulsion is right skewed,20 see, e.g., Figure 3, unlike the left skewed limiting LSW distribution, the growth starting with an initial log-normal distribution (eq 6) is also simulated.

W(a) ) C exp{-[ln(a) - ln(a)]2/2σ2}

(6)

In eq 6, ln(a) is the average of ln(a), and σ characterizes the width of the log-normal PSD. The idea is that the latter simulates a real, initially nonstationary growth process, while the first simulation corresponds to growth under stationary LSW conditions. 3.2. Conversion of Intensity to Number Averaged Radii. Given the actual form of the PSD as a set of N droplets with radii {ai; i )1, ..., N}. Note that in the description of the PSD each different radius occurs only once. Hence the number and intensity averaged radii (respectively aN and aI) can be calculated using eqs 7 and (18) Koppel, D. J. Chem. Phys. 1972, 57, 4814. (19) Finsy, R. Particle size analysis-Photon Correlation Spectroscopy; ISO 13321: 1996(E), International Organization for Standardization: Gene`ve, Switzerland, 1996. (20) Vold, R. D.; Vold M. J. Colloid and Interface Chemistry; AddisonWesley: London, 1983; p 403. (21) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley: New York, 1983; p 83. (22) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1997, 13, 6884.

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8.16 In eq 7 Si ) Si(λ, ai, m, θ) is the scattering power at N

aN )

ai/N ∑ i)1

N

aI )

(7)

N

Si/ai) ∑ ∑ i)1 i)1 Si/(

(8)

angle θ (90° in this study) of a droplet with radius ai and relative refractive index m (1.0826 and 1.0689 were used for undecane and dodecane emulsions respectively), illuminated with light with wavelength λ (488 nm in this study). Explicit formulas for Si can be found in the work of Bohren and Hufmann.21 For a given PSD (i.e., a given set of radii) the ratio aI/aN yields the required conversion factor. Note that since the shape of the PSD as well as the different droplet sizes change during the ripening process and since droplet scattering power is a nonlinear function of droplet size, the ratio aI/aN is not a constant during the ripening process. 3.3. Simulation. The simulation of both growth regimes is performed with the procedure described elsewhere.22 The main steps are the following. 1. Generation of an initial set of droplets with radii {ai; i ) 1, ..., N}, according to an a priori model of the PSD. Two initial PSD’s are considered: (i) the LSW-PSD given by eq 3 and (ii) a log-normal PSD (eq 6). The parameters of the latter are determined as follows. The width σ ) 0.24 was determined by cryo-TEM measurements on a virtually nongrowing emulsion of squalane-in-water stabilized with sodium dodecyl benzene sulfonate (SDBS),23 prepared in the same way as the studied emulsions. Since the shape of the droplet size distribution is determined in the first place by the preparation procedure, we assume that the shape of the droplet size distribution of the studied emulsions is the same as that of the squalane/SDBS/water emulsion. The parameter ln(a) is determined so that the initial intensity averaged radius equals the experimentally determined initial intensity weighted average radius. 2. Transport of dispersed phase (oil) molecules from one droplet to another according to the growth rule corresponding to Ostwald ripening (eqs 1 and 8 in ref 22). 3. Computation of the number and intensity weighted average radii using eqs 7 and 8 at every time step tj of the simulation. The results for the number and intensity averaged radii of the simulation with initially the LSWPSD are denoted respectively as aN,LSW (tj) and aI,LSW (tj). The number and intensity weighted average radii of the simulation with initially the log-normal PSD are referred to as aN,LN (tj) and aI,LN (tj). 3.4. Amplitude of Both Effects. In Figure 5 the results of both simulations are displayed for a typical example. Note the nonlinear time dependence of aN,LN3 and aI,LN3 as a function of time, in contrast to the linear dependence of aN,LSW.3 Figure 5 also illustrates that both effects (nonstationary growth and conversion of the intensity to number weighted averages) are significant and have to be taken into account in the interpretation of experimental data. This is done as follows. The amplitude of both effects together is estimated at each time during the growth as the ratio aI,LN3/aN,LSW.3 Hence the experimental DLS data for aI3 are converted to number averaged data for a stationary growth by multiplying by the inverse of the previous ratio. (23) De Smet, Y.; Danino, D.; Deriemaeker, L.; Talmon, Y.; Finsy, R. To be published.

Figure 5. Simulated behavior of the cube of the number averaged (aN,LN3) and intensity weighted (aI,LN3) particle radius in case of an initially log-normal distribution profile, compared to the cube of the number averaged particle radius (aN,LSW3) in the case of an initially limiting LSW-distribution profile. Note the linear evolution of aN,LSW3 as opposed to the nonlinear evolution of aI,LN3 and aN,LN3.

Figure 6. Conversion of the cubed, intensity weighted particle radii aI3 (circles) into cubed, number averaged radii aN3 (triangles) by multiplication by [aN,LSW/aI,LN]3, applied to the data of the Und-Tween1 emulsion.

Figure 7. Sketch of the oil droplet and its surrounding surfactant layer with thickness δ.

In Figure 6 the results are shown for a typical example. Note that the conversion procedure straightens to some extent, though not completely, the nonlinear time dependence. To account for the remaining nonlinearity a third effect, i.e., the effect of the presence of a surfactant layer of finite thickness, is now considered. 4. Effect of the Finite Size of the Interfacial Surfactant Layer The following model (see Figure 7) is proposed to include the effect of the surfactant monolayer around the oil droplets. It is assumed that the oil core undergoes Ostwald ripening as described in the LSW theory and that the thickness of the surfactant layer, referred to as δ, remains constant.

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Hence the increase of the number averaged droplet radius with time can be written as

a(t) ) acore(t) + δ

(9)

The growth of the core is now given by eq 4

acore3(t) - acore3(t0) ) v(t - t0)

(10)

Combining eqs 9 and 10 yields

a3(t) - a3(t0) ) v(t - t0) - 3δ[a2(t) - a2(t0)] + 3δ2[a(t) - a(t0)] (11) Note that hereby a stationary (LSW) growth regime is assumed. In Figure 8, a3(t) predicted by eq 10 (for a0 ) 30 nm, δ ) 5 nm and v ) 8 nm3/s) as well as the linear increase of acore3 with time is plotted. Clearly the presence of a surfactant layer has a significant effect. The final step of the determination of the limiting Ostwald ripening rate v makes use of eq 11. To this end the experimental data for aI3, converted to aN,LSW3 for the stationary regime by the procedure explained in the previous section, are fitted to eq 11 with a priori determined values of the surfactant layer thickness δ. An upper limit of δ was estimated as the length of a surfactant molecule in full extension. These values are computed using the Unichem24 chemical software package. Table 3 summarizes the estimates of the layer thickness δ for the surfactant molecules. Hence the only remaining unknown in the fitting procedure is the Ostwald ripening rate v in eq 11.

Figure 8. Comparison between a3 as predicted by eq 10 (for a0 ) 30 nm, δ ) 5 nm, and v ) 8 nm3/s; full line) and acore3 (dotted line). Table 3. Molecular Length in Full Extension of the Surfactants Used surfactant

δ (nm)

Tween 20 NPE 10

4.7 4.6

5. Results and Discussion To extract the Ostwald ripening rates from the raw experimental data, i.e., aI(t), the following four methods were considered. (1) The conversion of intensity to number averaged radii aN(t) was performed by division of aI3(t) by a constant factor K1 ) [aI /aN]3 of 1.64 as proposed in ref 8. The converted data for aN(t) were fitted to eq 4 with a linear least squares procedure. This method assumes implicitly a stationary (24) Unichem 4.1, Oxford Molecular Ltd., 1998.

LSW distribution throughout the whole ripening process. The finite thickness of the surfactant layer is neglected. (2) The second method follows the same procedure as the previous one, except that a nonconstant conversion factor K2 ) [aI, LSW/aN,LSW]3 was used. Again a stationary LSW distribution is assumed throughout the ripening process, and the surfactant layer thickness is neglected. (3) The same procedure as in method 2 is used, except that now a nonconstant conversion factor K3 ) [aI,LN/ aN,LSW]3 is used. In this method, the effect of the transition of an initially log-normal PSD toward the stationary LSW distribution is taken into account. The effect of the finite size of the surfactant layer however is still neglected. (4) The same procedure as in the previous method is used, i.e., accounting for the transition of an initially nonstationary log-normal PSD toward the stationary LSW distribution. The difference from method 3 is that now the effect of the finite layer thickness is taken into account by fitting the data converted with the factor K3 to eq 11 instead of fitting them to eq 4. The results for the Ostwald ripening rates obtained with the four different methods as well as from the raw data fitted to eq 4 are summarized in Table 4. The values of the three conversion factors are plotted in Figure 9 for the Und-NPE1 emulsion as an example. The conversion factors K2 and K3 are clearly not constant during the ripening process, and the factor K3 is larger than both K2 and the constant factor K1. This illustrates once more the significant effect of the transition from a nonstationary regime (initially log-normal PSD) toward the stationary LSW regime. In Table 4 the estimates of the ripening rates calculated with eq 4 using the data from ref 3, i.e., D ) 4.31 × 10-6 and 4.13 × 10-6 cm2/s and Vm ) 211.23 and 227.42 cm3/ mol (respectively for undecane and dodecane), and the estimated value for the interfacial tension (γ ) 10.0 mN/m)8 are also reported. The estimated relative uncertainties on the theoretical rates are of the order of 15%. All experimental rates are larger than the theoretical predictions. The best agreement is achieved for the rates obtained with method 4, i.e., by using a nonconstant factor for the conversion of intensity to number averaged radii and by taking the transition from an initially log-normal PSD toward the stationary LSW distribution as well as the effect of the finite surfactant layer thickness into account. For the undecane emulsions the agreement between the experimental rates, obtained with method 4, and the LSW-predicted rate is within the uncertainties. The experimental rates for the dodecane emulsions are somewhat larger than the LSW prediction. This may be explained by the fact that possible minor effects of coalescence on the observed rates are more probable for dodecane than for undecane emulsions, since the ripening process of a dodecane emulsion (several days) is monitored over time periods lasting three to four times those for the undecane emulsions (typically 20-30 h). Kabalnov et al.8,12 report ripening rates of undecanewater emulsions stabilized with sodium dodecyl sulfate (SDS) ranging from 39 to 56 nm3/s. The oil volume fraction however was 10%, and the raw data analysis was performed with a constant factor for the conversion of intensity averages to number averages, assuming a stationary LSW regime (our method 1). The effect of the oil volume fraction of 10% can be estimated from the work of Voorhees25 as an increase in the ripening rate by a (25) Voorhees, P. W. J. Stat. Phys. 1985, 38, 231.

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Table 4. Summary of the Ripening Rates (in nm3/s) Obtained with the Four Described Methods and Theoretically LSW-Predicted Ostwald Ripening Rates

emulsion

raw data

method 1: constant conversion factor; stationary regime

method 2: nonconstant conversion factor; stationary regime

method 3: nonconstant conversion factor; nonstationary regime

method 4: Method 3 + effect of a surfactant layer

Und-NPE1 Und-NPE2 Und-Tween1 Und-Tween2 Und-Tween3 Dod-NPE Dod-Tween

17.7 16.5 18.3 20.7 25.0 7.5 7.6

10.8 10.0 11.1 12.6 15.2 4.6 4.6

12.5 12.2 14.1 15.6 18.3 5.3 5.4

9.7 9.6 11.0 12.5 14.2 4.3 4.0

8.3 8.5 10.4 11.2 12.7 3.7 3.7

LSW theory 7.7

2.2

3. Third is the preparation procedure. Kabalnov prepared his emulsions by ultrasonification,12 while we prepared ours with a high pressure microfluidizer. In general the initial average droplet size is smaller when using a microfluidizer. Possibly ultrasonification is less efficient in the protection of the oil droplets against coalescence.

Figure 9. Three factors for the conversion of the raw data (aI3) into number averaged radii (aN3) for the Und-NPE1 emulsion (circles, K1; squares, K2; triangles, K3).

factor of 1.75.8 Taking this factor into account, the results of Kabalnov are extrapolated to a 1% volume fraction, yielding now rates ranging from 22 to 32 nm3/s to be compared with our “method 1” results of 10 and 10.8 nm3/ s. At present the exact reasons for this difference by a factor of 2-3 are unclear. The following remaining differences between our work and that of Kabalnov can contribute to the differences in measured growth rates. 1. First is the uncertainty in the estimated oil concentration dependence of the ripening rates. 2. Second is the different nature of the surfactants. Although the surfactant type will affect the interfacial tension, it is however unlikely that the interfacial tension of our NPE emulsions is decreased by a factor of 2-3 compared to a SDS emulsion. Possibly the bulkier NPE molecules offer a better protection against coalescence, which contributes to the observed growth rates.

Experimental ripening rates of dodecane emulsions stabilized by SDS are reported by Kabalnov et al.8 (v ) 17 nm3/s) and by Taylor and Ottewill13 (v ) 5.2-10.9 nm3/ s). Kabalnov’s result is again for an oil volume fraction of 10%, which corresponds to a value of 9.7 nm3/s, assuming an increase by a factor of 1.75 for the ripening rate at the reported oil volume fraction. The results of Taylor and Ottewill are for oil volume fractions in the range 0.10.4%. Both use SDS as a surfactant. The emulsions are prepared by ultrasonification in the work of Kabalnov. Taylor and Ottewill prepare their emulsions by dilution of a SDS/dodecane/pentanol/water microemulsion. Both authors assume a stationary LSW growth regime, and Kabalnov uses a constant factor for the conversion of the intensity to number averages. Therefore these results are to be compared to our “method 1” result, i.e., 4.6 nm3/s. Again the differences are probably due to the differences with our emulsions: different surfactant (NPE-10 and Tween 20 for our emulsions), different method of preparation (ultrasonification in the work of Kabalnov and dilution of a microemulsion in the work of Taylor and Ottewill, microfluidization in our study), and the uncertainty on the oil concentration dependence of the ripening rates. LA9807513