Ostwald Ripening in the Transient Regime: A Cryo-TEM Study

Dec 1, 1999 - The large-time behaviour of coarsening of a particulate assemblage due to Ostwald ripening and coagulation. D. V. Alexandrov. Philosophi...
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Langmuir 2000, 16, 961-967

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Ostwald Ripening in the Transient Regime: A Cryo-TEM Study Y. De Smet,† D. Danino,‡ L. Deriemaeker,† Y. Talmon,‡ and R. Finsy*,† Department of Physical and Colloid Chemistry, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium, and Department of Chemical Engineering, Technion-Israel Institute of Technology 32000 Haifa, Israel Received May 18, 1999. In Final Form: September 13, 1999 The evolution of the droplet size distribution of a tetradecane in water emulsion, stabilized by sodium dodecyl benzene sulfonate (SDBS) was followed by cryo-TEM during a 51 day period. As a reference for the initial distribution of a freshly prepared emulsion, the size distribution of a virtually nongrowing squalane in water emulsion was also monitored. It appears that the initial distribution is right skewed and close to log-normal with a broadness parameter of σ ) 0.5. This distribution evolved rapidly to a more symmetrical one. At the end of the observation the average particle size increased by a factor of about 2.5, but the limiting left skewed distribution, predicted by the LSW (Lifshitz-Slyozov-Wagner) theory, was not obtained. The ripening rate estimated from the slope of an3 (an is the number averaged radius) as a function of time is in agreement with the prediction from the LSW theory.

1. Introduction The physical degradation of emulsions is driven by the spontaneous reduction of the interfacial area between dispersed and continuous phase. Processes such as coagulation and coalescence are well-known and in most situations are thought to be responsible for the coarsening and breakdown of emulsions.1-3 However, with sufficiently high protection against these processes, another one, Ostwald ripening, dominates the coarsening. This process causes a growth of the larger droplets of an emulsion at the expense of smaller ones owing to the higher solubility of the smaller droplets (Gibbs-Kelvin effect) and to molecular diffusion through the continuous phase. A theoretical description of Ostwald ripening was first developed by Lifshitz and Slyozov and by Wagner (LSW theory4-7). The main results are the following. (1) The growth rate (defined as the time derivative of an3) is a constant

v)

d an3 8γVmDC(∞) ) dt 9RT

(1)

γ is the interfacial tension, Vm and D are the molar volume and the molecular diffusion coefficient of the dispersed phase, R and T are the universal gas constant and the absolute temperature, C(∞) is the bulk solubility of the dispersed phase. * Tel: +32 2 629 34 85. Fax: +32 2 629 33 20. E-mail: [email protected]. † Vrije Universiteit Brussel. ‡ Technion-Israel Institute of Technology. (1) Tadros, T.; Vincent, B. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1983; Vol. 1, p 129. (2) Friberg, S.; Yang, J. In Emulsions and Emulsion Stability; Sjo¨blom, J., Ed.; Marcel Dekker: New York, 1996; p 1. (3) Stein, H. The Preparation of Dispersions in Liquids; Marcel Dekker: New York, 1996; p 15. (4) Kahlweit, M. In Physical Chemistry; Eyring, E., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Vol. 10, p 719. (5) Kabalnov, A. S.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 38, 69. (6) Lifshitz, I.; Slyozov, V. J. Phys. Chem. Solids 1961, 19, 35. (7) Wagner, C. Z. Elektrochem. 1961, 65, 581.

(2) There exists a critical radius ac. Particles of this size neither grow nor shrink. If a < ac, the particle shrinks, if a > ac, it grows. (3) The scaled size distribution is self-similar:

W(u) ≡ W(a/ac) )

27eu2 exp[1/(2u/3 - 1)]

(u < 3/2) (2) x3 32(u + 3)7/3(3/2 - u)11/3 (u > 3/2) )0

A particular feature of this LSW distribution is that an ) ac. Note that these results are only valid for infinitely diluted systems containing spherical and spatially fixed particles, in which molecular diffusion is the rate determining factor for mass transfer between droplets. Moreover, these results can only be derived analytically in an asymptotically steady state limit for large times. In other words they do not apply to the initial coarsening stage for freshly prepared systems. Nevertheless, in most experimental studies of Ostwald ripening8-16 it is assumed that the investigated systems coarsen according to the predictions of the LSW theory. In a previous paper17 some of us showed on the basis of a computer simulation that the LSW results apply to the initial coarsening stage only if the initial size distribution (of a freshly prepared system) is the limiting (8) Kabalnov, A.; Makarov, K. N.; Pertzov, A. V.; Shchukin, E. D. J. Colloid Interface Sci. 1990, 138, 98. (9) Bremer, L.; De Nijs, B.; Deriemaeker, L.; Finsy, R.; Gelade, E.; Joosten, J. Part. Part. Syst. Charact. 1996, 13, 350. (10) De Smet, Y.; Malfait, J.; De Vos, C.; Deriemaeker, L.; Finsy, R. Prog. Colloid Polym. Sci. 1997, 105, 252. (11) Soma, J.; Papadopoulos, K. D. J. Colloid Interface Sci. 1996, 181, 225. (12) McClements, D. J.; Dungan, S. R. Colloids Surf., A 1995, 104, 127. (13) Kabalnov, A. S. Langmuir 1994, 10, 680. (14) Taylor, P.; Ottewill, R. H. Colloids Surf., A 1994, 88, 303. (15) Taylor, P. Colloids Surf., A 1995, 99, 175. (16) Buscall, R.; Davis, S. S.; Potts, D. C. Colloid Polym. Sci. 1979, 257, 636. (17) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1997, 13, 6884.

10.1021/la9906065 CCC: $19.00 © 2000 American Chemical Society Published on Web 12/01/1999

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aI )

N

∑S /(∑S /a ) i

i)1

Figure 1. Limiting LSW size distribution (dashed) compared to a log-normal size distribution (full) with broadness parameter σ ) 0.5. The sizes are scaled to the number averaged particle size.

stationary LSW distribution given by eq 2. Such a leftskewed size distribution (with a tail toward the smaller particle sizes, see Figure 1) is hardly realistic. Most practical preparation procedures yield rather right-skewed size distributions (with a tail toward the larger particle sizes, see Figure 1). Hence in practice there must be a transient regime in the initial coarsening, whereby the size distribution evolves from right-skewed to the leftskewed LSW distribution. It was shown by simulation17 that for an initial log-normal (right-skewed) distribution, the transient regime corresponds to a coarsening stage whereby the initial average particle size increases by a factor of at least 4. In this contribution some experimental evidence for the evolution of the size distribution in the transient regime is reported. In particular, some results of the monitoring of the coarsening with the cryogenic temperature transmission electron microscopy (cryo-TEM) technique are presented. Possible effects of surfactant micelles on the ripening process are assessed by comparing the aging of emulsions with low and high surfactant concentration. 2. Materials and Methods Materials. The hydrocarbons used were n-tetradecane 99+%, obtained from Sigma-Aldrich, and squalane (2,6,10,15,19,23hexamethyltetracosane) 99% obtained from Riedel-de Hae¨n. These were dispersed in distilled water, and stabilized against coagulation with the anionic surfactant SDBS (sodium dodecyl benzene sulfonate) technical grade, obtained from SigmaAldrich. Emulsification. The emulsions were pre-emulsified with a T25 Ultra-Turrax mixer for 10 min and further homogenized with a 110Y-Microfluidizer for 5 min. All emulsions were stored at room temperature (298 ( 1 K) during the ripening process (except for the period of transport from Brussels to Haifa). Photon Correlation Spectroscopy (PCS). Average particle sizes were monitored with PCS using an ALV goniometer, a Brookhaven BI-9000 correlator and a Lexel Model 95 Argon-ion laser (wavelength in vacuum λ ) 488 nm). Measurements were performed at an angle of 90°. The temperature was 298 K. To obtain a clear sample suitable for PCS measurement, a sample of the emulsion was diluted a thousand times with water, just before particle size measurement. Data analysis was performed with the cumulants method18 yielding intensity averaged particle radii given by19 (18) Koppel, D. J Chem. Phys. 1972, 57, 4814. (19) Finsy, R.; De Jaeger, N. Part. Part. Syst. Charact. 1991, 8, 187.

i

i

(3)

i)1

where Si ) Si(λ, ai, m, θ) is the scattering power at angle θ (90° in this study) of a droplet with radius ai and relative refractive index m (1.074 and 1.092 were used for tetradecane and squalane 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.20 Cryo-TEM. Vitrified specimens for direct imaging cryo-TEM were prepared in the controlled environment vitrification system (CEVS) at 25 °C and 100% relative humidity, as previously described.21,22 In brief, a drop was applied onto a perforated carbon film, supported on an electron microscopy copper grid, held by the CEVS tweezers. The sample was blotted, immediately plunged into liquid ethane at its freezing point (-183 °C), and stored under liquid nitrogen (-196 °C) until examination in the microscope. The vitrified specimens were examined in a Philips CM120 microscope operated at 120 kV, using an Oxford CT3500 cooling holder cooled to about -180 °C. Specimens were examined in the low-dose imaging mode to minimize electron beam radiation damage, and recorded digitally at a nominal underfocus of 4-7 µm (to enhance phase contrast) by a Gatan 791 MultiScan CCD camera using the Digital Micrograph 3.1 software package. TEM magnification was calibrated against a grating replica standard (relatively low magnifications were needed for this study). In all cases the TEM specimens were kept in the eucentric position to avoid magnification changes. Image processing was performed by the Adobe Photoshop 5.0 package. The digitized images were analyzed using Micrografx Windows Draw software. A transparent vectorized layer was laid on top of the images, on which the droplet diameters were marked with the aid of the PC mouse. Afterward the length of the marks was extracted by software. From the set of droplet radii {ai; i ) 1, ..., N} the number averaged droplet radius was readily computed as N

aN )

∑a /N i

(4)

i)1

To compare these data with those obtained from PCS measurements, the intensity weighted average droplet radius was calculated as well, with eq 3. Simulations. Simulations were performed using a PC with a Pentium processor according to the algorithm described elsewhere.17 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 size distribution. (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 17). (3) Computation of the number and intensity weighted average radii using eqs 3 and 4 at every time step tj of the simulation.

3. Results and Discussion Four oil-in-water emulsions, each containing 10% oil by volume, were studied. The emulsions were protected against coagulation and coalescence by the surfactant SDBS. The first two were squalane in water emulsions with SDBS concentrations of 0.05 M and 0.15 M, respectively. The other two were tetradecane in water emulsions, also with SDBS concentrations of 0.05 and 0.15 M, respectively. The preparation conditions were the same for all four emulsions. PCS measurements yield hardly any information on the size distribution profiles of the emulsions. With the cumulants method a polydispersity index is obtained, (20) Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles; Wiley: New York, 1983; p 83. (21) Talmon, Y. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 364. (22) Bellare, J.; Davis, H. T.; Scriven, L.; Talmon, Y. J. Electron. Microsc. Technol. 1988, 10, 87.

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Table 1. Number of Images that Were Analyzed, Total Number of Droplets Counted on These Images, Number Averaged and Intensity Weighted Average Radius Determined from the Cryo-TEM Images, and Intensity Weighted Average Radius As Measured with PCS, for the Different Emulsions at All the Times day

emulsion

no. of images analyzed

no. of droplets counted

an (nm) from cryo-TEM

aI (nm) from cryo-TEM

aI (nm) from PCS

4

squalane 0.05 M squalane 0.15 M tetradecane 0.05 M tetradecane 0.15 M squalane 0.05 M squalane 0.15 M tetradecane 0.05 M tetradecane 0.15 M squalane 0.05 M squalane 0.15 M tetradecane 0.05 M tetradecane 0.15 M squalane 0.05 M squalane 0.15 M tetradecane 0.05 M tetradecane 0.15 M squalane 0.05 M squalane 0.15 M tetradecane 0.05 M tetradecane 0.15 M

9 6 12 7 s 7 8 10 5 s 8 33 s 13 12 s 8 8 7 32

13155 7562 6367 3800 s 12933 1681 2886 7638 s 1051 4966 s 17437 1271 s 10742 5308 1167 1792

18 18 37 29 s 20 56 38 15 s 62 53 s 18 71 s 18 25 92 68

60 62 56 45 s 62 79 68 65 s 90 81 s 55 108 s 55 62 117 101

73 68 70 57 74 66 90 79 75 71 100 91 75 67 114 98 73 69 125 118

14

22

30

51

Figure 2. Typical cryo-TEM image of squalane-in-water emulsion with 0.05 M SDBS (4 days after preparation). “S” denotes the support film. Note segregation of particles according to size in the upper part of the image, resulting from thickness differences between the center and edges of the vitrified film.21

which gives a quantitative measure of the polydispersity, but no actual information about the shape of the distribution. Furthermore, the intensity weighted average and number average radii can easily differ by as much as a factor of 3 (see for example Table 1), depending on the droplet size distribution, scattering angle θ, wavelength λ and relative refractive index m. It should be noted that minute amounts of relatively large particles can determine almost entirely the intensity average radius measured by PCS. For this reason the droplet size distribution was determined by cryo-TEM. Cryo-TEM images of the four emulsions were taken (in Israel) on the 4th, the 14th, the 22nd, the 30th and the 51st day after preparation (in Belgium). A typical image is shown in Figure 2. For each emulsion and for each of the days mentioned above, between 7 and 33 images containing as many droplets as possible and with good contrast between the oil and the water phase, were taken. Table 1 summarizes the number of images analyzed and the number of particles counted for each sample. The images taken on the same day of the same sample were

analyzed and the resulting data were cumulated. In this way average droplet sizes and size distribution profiles are obtained for every emulsion on 4 or 5 different days, distributed over a 51 day time span. Squalane Emulsions. The squalane emulsions were studied for the following reason. Since the Ostwald ripening rate (see eq 1) is predicted to be proportional to the (oil) solubility C(∞) in the continuous (water) phase and since for an alkane the solubility decreases dramatically with increasing number of carbon atoms,23 the Ostwald ripening rate is virtually zero for this emulsion. (From extrapolation of the data in ref 23 we estimate the aqueous solubility of squalane at 3.4 × 10-15 mL/mL). Therefore, possible coarsening is in this case an indication of all possible aging processes, such as coalescence or coagulation or an oil transport mechanism mediated by micelles, different from Ostwald ripening. In contrast, if the average size remains constant in time, the absence of coagulation and coalescence or another transport mechanism is confirmed. Furthermore, in the absence of Ostwald ripening or other aging processes, the initial size distribution is not expected to change during the time span of the investigation, so that the size distributions of these emulsions can be considered as evidence for the distribution of a freshly prepared emulsion. In Figure 3 the evolution in time of the intensity weighted average radii obtained with PCS is reported. The average size fluctuates around 75 nm for the squalane 0.05 M emulsion and 65 nm for the squalane 0.15 M emulsion, confirming the absence of any coarsening (coagulation, coalescence, oil transport by micelles, Ostwald ripening, etc.) during the time span that we studied them. Figures 4 and 5 show the size distributions for the squalane emulsions determined at the different times. Table 1 reports the number and intensity weighted average droplet sizes and, for comparison, the corresponding intensity weighted average droplet radius as determined with PCS on the same day. For the squalane emulsions the droplet size distribution (DSD) remains quite invariant in time and the profile of the right-skewed distribution resembles much more a lognormal than a left-skewed LSW limiting distribution profile. Nonlinear least-squares fits of a log-normal profile, (23) McAuliffe, C. J. Phys. Chem. 1966, 70 (4), 1267.

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Figure 3. Evolution of the intensity weighted average radii obtained with PCS (4, tetradecane 0.05 M; O, tetradecane 0.15 M; ], squalane 0.05 M; 0, squalane 0.15 M).

Figure 5. Size distributions for the 0.15 M squalane emulsion after 4 (]), 14 (×), 30 (O) and 51 days (0), and the fitted lognormal distribution (dashed line). Table 2. Broadness Parameter σ and Position of the Peak for the Log-Normal Fits to the Squalane Emulsion Size Distributions emulsion

σ

fwhm (nm)

ap (nm)

squalane 0.05 M squalane 0.15 M

0.43 0.57

15 19

14.6 13.2

For both the 0.05 and the 0.15 M emulsion a value for AS,V of 8.2 × 107 m-1 was found. The number of moles ns of SDBS needed to cover the oil-water interface of a volume Ve of emulsion is given by

ns ) φAS,VVe/NaAs

Figure 4. Size distributions for the 0.05 M squalane emulsion after 4 (]), 22 (0), and 51 days (4), and the fitted log-normal distribution (dashed, ×).

given by

W(a) )

1

x2πσ

[

exp -

[ln(a) - ln(a)]2 2σ2

]

(5)

to the average of the curves taken at different times, yields the parameters displayed in Table 2. The position of the peak is denoted by ap, the width of the log-normal DSD is measured by σ. The full width at half-maximum (fwhm) of the log-normal is given by

fwhm ) 2ap sinh(σ‚x2 ln 2)

(6)

The fitted curves are also shown in Figures 4 and 5. It appears that the DSD of the emulsion with the highest surfactant concentration peaks at a slightly lower droplet radius than the one with the lower surfactant concentration. However, the former is also a little broader. To check whether the surfactant concentration was chosen high enough to guarantee complete coverage of the interfacial area, we calculated the ratio of the total interfacial area to the total volume of the oil droplets from the cryo-TEM images for the two squalane emulsions N

AS,V ) 3

N

a2i /∑a3i ∑ i)1 i)1

(7)

(8)

where φ is the oil volume fraction (0.1 in this study), Na is Avogadro’s number, and As is the specific area of a surfactant molecule (0.62 nm2 for SDBS27). Hence it is estimated that 2.2 × 10-5 mol of SDBS are needed per mL of emulsion to cover the oil-water interface completely, whereas for the 0.05 M emulsion there are 5.0 × 10-5 mol of SDBS present per mL of emulsion (1.5 × 10-4 mol for the 0.15 M emulsion); in other words both emulsions contain enough SDBS to cover the oil-water interface completely. The excess of SDBS is aggregated into small spheroidal micelles, as was found by cryo-TEM (not shown). Obviously in the emulsions with the higher surfactant concentration more micelles are present in the continuous phase. However no significant effect of the presence of micelles on the aging of the emulsion is observed in agreement with the results of a more detailed study of the (absence of) effect of SDBS micelles on the ripening process.28 The stability of these squalane emulsions allows us to conclude that the initial DSD of the emulsions prepared as described in Section 2 had a log-normal profile with a width σ of 0.50 ( 0.07. Tetradecane Emulsions. From the solubility of tetradecane in water (∼3.7 × 10-10 mL/mL)8 it was inferred that the Ostwald ripening would be observed over periods of days to weeks. Therefore both the shape of the droplet size distribution and the average droplet size can be (24) Finsy, R. Adv. Colloid Interface Sci. 1994, 52, 79. (25) Hayduk, W.; Laudie, H. AIChE J. 1974, 20, 611. (26) Voorhees, P. W. J. Stat. Phys. 1985, 38, 231. (27) De Smet, Y.; Deriemaeker, L.; Parloo, E.; Finsy, R. Langmuir 1999, 15, 2327. (28) De Smet, Y., Deriemaeker, L., Finsy, R., Langmuir 1999, 15, 6745.

Cryo-TEM Study of Transient Regime Ostwald Ripening

Figure 6. Size distributions for the tetradecane 0.05 M emulsion after 4 (]), 14 (×), 22 (4), 30 (O), and 51 days (0), and the initial DSD (+, right-skewed). The dashed curve is the limiting LSW distribution (left-skewed).

Figure 7. Size distributions for the 0.15 M tetradecane emulsion after 4 (]), 14 (×), 30 (O), and 51 days (0), and the initial DSD (+, right-skewed). The dashed curve is the limiting LSW distribution (left-skewed).

expected to vary considerably during the time span of about a month. From Table 1 and Figure 3 it is clear that for these emulsions the average radius increases significantly. The number average goes from about 40 to 90 nm for the 0.05 M emulsion, and from 30 to 70 nm for the 0.15 M emulsion. Note that the emulsion with the highest surfactant concentration had the smallest initial average droplet size, something we also observed, although to a lesser extent, for the squalane emulsions treated above. Note also that the relative increase in average size is not significantly different for both emulsions, indicating that there is hardly any effect of the presence of an increased number of micelles. Since we know from the results obtained from the squalane emulsions that our emulsions are very well protected against coalescence or coagulation, the changes in average size of the tetradecane emulsions are attributed to Ostwald ripening. Figures 6 and 7 show the evolution of the size distributions for the tetradecane emulsions. As the preparation procedure was the same as for the squalane emulsion, it is reasonable to assume that these emulsions initially had the same log-normal profiles. The plots in Figures 6 and 7 show the presumed initial lognormal DSD, i.e., the DSD of the squalane emulsion, the DSDs at several times from 4 to 51 days after preparation, and the long-time limit DSD of the classical Ostwald ripening theory (LSW-DSD, see eq 2). The plots confirm the results of the simulation study:17 it takes a long time

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Figure 8. Simulation results for the size distribution of the 0.15 M tetradecane emulsion, started with a log-normal distribution with σ ) 0.50 and ap ) 7 nm. The peaks (of W) of the initial DSD (right-skewed) and the DSDs on days 4, 14, 22, and 51 shift from low to high. The dashed curve is the limiting LSW distribution (left-skewed).

before the stationary regime of the LSW-theory is attained. From the simulations, it was seen that even when the number average size had increased by a factor of 4, the limiting LSW-DSD had not yet been attained. Figures 6-7 are consistent with that. In the cryo-TEM investigation the number averaged radius has increased by about a factor of 2.5 after 51 days, and indeed the DSD at that time evolved from a right-skewed to a symmetric one, but still was far from the LSW-DSD. No significant differences in the ripening behavior are observed for these two emulsions (low and high surfactant concentration) indicating that there is no significant effect of the presence of surfactant micelles. This is again in agreement with the results of the more detailed study.28 Comparison to Computer Simulation. The evolution of the DSD for the tetradecane emulsions was simulated in two ways. (a) In the first the initial DSD was assumed to be lognormal with width parameter σ ) 0.5 (the average of the σ obtained from the log-normal fit to the squalane distributions, determined from the cryo-TEM images). Since the only information concerning the start of the ripening process for the tetradecane emulsions were sizes measured with PCS, these were used to find the only remaining parameter ap of the log-normal profile. This was done as follows. For different values of ap and σ ) 0.5 log-normal distributions were generated. Each distribution was discretized as a set of 5000 particles with different radii {ai; i ) 1, ..., 5000}. For a droplet with a given radius ai the scattering power Si(ai, m, λ, θ) was calculated with the aid of the Mie scattering coefficients for homogeneous spheres,20 using the relative refractive index of tertradecane in water of m ) 1.074, the wavelength λ ) 488 nm and the scattering angle θ ) 90°. The intensity-weighted average particle size, aI, for a given value of ap, was then computed with the aid of eq 3. In this way a table of values of ap, aN, and aI was established. The value of ap corresponding to the initial experimentally measured intensity weighted average aI was then looked up. For the measured value of aI ) 41 and 32 nm that yielded ap ) 9.4 and 7.0 nm (respectively for the tetradecane 0.05 and 0.15 M emulsions). In Figure 8 the simulation results for the DSD are shown for the tetradecane 0.15 M emulsion. As a reference, the LSW-DSD is drawn as well. (b) In the second way the simulations were started with the size distributions obtained from the cryo-TEM images

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Figure 9. Simulation results for the size distribution of the 0.05 M tetradecane emulsion, started with the experimental (cryo-TEM) DSD of the 0.05 M squalane emulsion (0, left-hand side curve). ], DSD on day 4; the other four DSDs (day 14, 22, 30, and 51) are almost indistinguishable. The dashed curve is the limiting LSW distribution (left-skewed).

for the 0.05 and 0.15 M squalane emulsions. The actual profile was retained, but the DSD was translated such that the intensity-weighted average size was that of the corresponding tetradecane emulsions, as measured immediately after preparation with PCS (in the same way as described above). In Figure 9 the simulation results for the DSD are shown for the 0.05 M tetradecane emulsion. For both tetradecane emulsions the two ways of simulation (different initial DSDs) give more or less the same results for the evolution of the DSD. The only difference seen is that for the simulations started with the actual DSD coming from the cryo-TEM study of the squalane emulsions, the right-hand side of the DSD fluctuates a little. This can be attributed however to the small fluctuations in the experimental initial DSD at the larger size ranges. The experimental cryo-TEM distributions (Figures 6 and 7) have many features in common with the simulated ones (Figures 8 and 9). For both emulsions, the first measured DSD, on the 4th day after the emulsion preparation, clearly had already evolved away from the initial log-normal DSD. For both emulsions, the different experimentally determined DSDs, covering a period of 51 days after emulsion preparation, are very close to one another. This means that the first move away from the initial log-normal happens rather fast, but the subsequent evolution toward the stationary LSW-DSD takes a long time. The fast changes seen initially can be ascribed to the fast breakdown of the small droplets (of which there are plenty in the initial log-normal DSD), so that the position of the peak shifts rather fast to the right-hand side, and the shape of the distribution becomes more symmetrical. The log-normal DSD however has a long tail (longer than the LSW tail) toward the larger sizes, and it takes a long time to shift this toward the left-hand side.17 This implies that initially a relatively large amount of oil is already present in a few large droplets with sizes larger than 1.5 an, whereas in the limiting (stationary) LSW-DSD no such droplets are present. The amount of oil present in the initially relatively large number of smaller droplets, which break down rapidly, is transported rapidly to the larger ones. However, this amount is relatively small, thus it takes a long time before the majority of the droplets are larger than an, and simultaneously smaller than 1.5 an, as in the limiting LSWDSD. Therefore the last DSD, after 51 days, still is far

De Smet et al.

Figure 10. Ratio of the intensity weighted average size obtained with PCS to the intensity weighted average size extracted from the images with the aid of eq 3, plotted for all four emulsions at all times (tetradecane, 0.05 M, ], 0.15 M, 0, squalane, 0.05 M, 4, 0.15 M, ×).

from the limiting LSW-DSD, due to the presence of those large droplets (for which a/an > 1.5). It would probably take several additional months before stationary conditions (and thus the LSW-DSD) are attained. Comparison Between PCS and Cryo-TEM Data. In the review given in ref 21 the reader is warned that, due to the sample preparation procedure typical for cryoTEM, part of the larger particles may not be detected, leading to qualitative and/or quantitative distortion of the DSD. To detect such possible distortions the number average sizes, extracted from the images, are transformed into intensity weighted average sizes as explained in Section 2, in order to make a comparison with PCS measurements (see also Table 1). In Figure 10 the ratio of the size obtained with PCS to the intensity weighted average size extracted from the images with the aid of eq 3 is plotted at all the times at which cryo-TEM images were recorded, for all four emulsions. The conclusion is readily drawn: the PCS results indeed are on average about 10% greater than the sizes obtained with cryo-TEM. Note that although the average difference is 10%, it is as high as 22-33% in some cases. The inaccuracy of the cryo-TEM also contributes to the scatter of the data. It is known that the intensity weighted average particle size obtained by PCS is strongly biased by the presence of a few large particles in the presence of large amounts of smaller ones.24 Therefore, the difference between PCS and cryo-TEM average sizes can be attributed to only a very minute amount of relatively large particles, discarded by the sample preparation procedure for cryo-TEM, but detected during the PCS measurements. Keeping this in mind, it can be concluded that the PCS and cryo-TEM measurements are in fair agreement. Ostwald Ripening Rates. Although not the main objective of this study, the Ostwald ripening rates of the tetradecane emulsions are also estimated from the number average sizes determined by cryo-TEM. The LSW theory (eq 1) predicts a linear increase of a3n with time. From the slope of the plots of a3n (obtained from the cryo-TEM images) with time (see Figure 11), the Ostwald ripening rates are estimated as v ) 0.2 and 0.7 nm3/s respectively for the 0.05 and the 0.15 M emulsion. These results are to be compared to the value of 0.2 nm3/s estimated from the LSW theory (eq 1) with values for γ ) 10.2 mN/m,8 C(∞) ) 3.7 × 10-10 mL/mL,8 Vm ) 260 cm3/ mol, D ) 3.82 × 10-6 cm2/s, and T ) 298 K. The diffusion coefficient D was calculated using the Hayduk-Laudie

Cryo-TEM Study of Transient Regime Ostwald Ripening

Figure 11. Evolution of an3 (retrieved from the cryo-TEM data) with time for the 0.05 M (O) and 0.15 M (4) tetradecane emulsions.

equation.25 A multiplicative correction factor of 1.75 was used to account for the effect of the finite volume fraction.26 Kabalnov8 found an experimental ripening rate of 1.0 nm3/s for a 10% tetradecane in water emulsion, stabilized with SDS (sodium dodecyl sulfate). His results were obtained from PCS measurements whereby the intensity weighted average sizes (obtained with the cumulants method) were converted to number averages by multiplying by a constant factor of 1.18. Note that a linear dependence for an3 only holds when the initial size distribution is already a stationary LSWDSD, which was not the case here. A procedure for the determination of the limiting rates in the case of transient ripening is presented elsewhere.27 However, with the limited number of data (4 or 5 estimates of an) such a procedure cannot be applied in this study. Even the estimation of the Ostwald ripening rate from the slope of a linear fit may lead to relatively large uncertainties. The difference in rate between the 0.05 and 0.15 M emulsions (0.2 and 0.7 nm3/s respectively) is not confirmed by the estimates of the ripening rates (0.1 and 0.2 nm3/s) from the PCS data (about 40 data points [t, an]), following the method described in ref 27 (see also Figure 12). The latter rates compare well to the theoretical estimate of 0.2 nm3/ s. 4. Summary The size distribution determined by cryo-TEM appeared to be quite different from the long-time limiting distribution predicted by the LSW theory. An evolution of the

Langmuir, Vol. 16, No. 3, 2000 967

Figure 12. Evolution of an3 (retrieved from the PCS data using the method described in ref 27) with time for the 0.05 (0) and 0.15 M (]) tetradecane emulsions.

initially right-skewed distribution close to log-normal with a broadness parameter σ ) 0.5 toward a more symmetrical one was observed in agreement with the prediction of the computer simulation model. At the end of the observation the average size increased by a factor of 2.5, but the limiting left-skewed distribution predicted by the LSW theory was not obtained yet. No significant difference in ripening behavior was observed for the emulsions with low (0.05 M) and high (0.15 M) surfactant concentration indicating that the presence of SDBS surfactant micelles did not affect the oil-transport mechanism significantly. The average particle sizes determined by PCS are on the average 10% greater than those obtained by cryoTEM. The difference can be attributed to very minute amounts of relatively large particles discarded by the sample preparation procedures for cryo-TEM, but detected in the PCS measurements. The Ostwald ripening rates estimated from the slope of an3 as a function of time are for the tetradecane emulsions in agreement with the LSW predictions. Acknowledgment. The marking of almost 100 000 particles on the cryo-TEM pictures was made possible thanks to the generous participation of Mr. G. Vereyken and Prof. P. Coppens. This research was partially financed by the Fonds voor Wetenschappelijk Onderzoek (F.W.O.) Belgium. The work at the Technion was supported in part by a grant from the Fund for the Promotion of Research at the Technion. LA9906065