NMR Self-Diffusion Studies on PDMS Oil-in-Water Emulsion

short diffusion time of 0.1 s (black circles), the fit to the. Brownian model gives a mean radius R0 ) 0.14 ((0.003). µm and polydispersityσ)0.54 ((...
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Langmuir 2002, 18, 10298-10304

NMR Self-Diffusion Studies on PDMS Oil-in-Water Emulsion Tatjana Garasanin and Terence Cosgrove* School of Chemistry, University of Bristol, Cantock’s Close, U.K. BS8 1TS

Leon Marteaux Dow Corning S.A., 83 Av. Mounier, 1200 Brussels, Belgium

Axel Kretschmer and Andy Goodwin Dow Corning Ltd, Cardiff Road, Barry, South Glamorgan, U.K. CF63 2YL

Klaus Zick Bruker Analytik GMBH D-76287 Rheinststten Silbersteifen, Germany Received June 21, 2002. In Final Form: September 13, 2002 In this paper, standard spin-echo pulsed field gradient (SE-PFG) and stimulated spin-echo pulsed field gradient (STE-PFG) 1H NMR methods have been used to study the translational diffusion of polymericoil-in-water (O/W) emulsions, made with poly(dimethylsiloxane) PDMS fluids and two nonionic surfactants polyoxethylene (4) lauryl ether (Brij-30) and polyoxethylene (23) lauryl ether (Brij-35p). A detailed analysis of the free (Gaussian) diffusion of the PDMS fluid, its restricted diffusion inside the emulsion droplets, and the Brownian diffusion of the droplets themselves are presented. A graphical representation, which relates the square root of the mean-squared displacement of the fluid and emulsion droplets with the experimental diffusion times and the radius of the droplets, helps distinguish the different diffusion regimes. The Brownian diffusion of the emulsion particles becomes the dominant diffusion process with dilution of the system and increase in the molecular weight (viscosity) of the PDMS. These different processes which contribute to diffusion in the system need to be considered if a correct size distribution of the emulsion droplets is to be determined. For a concentrated emulsion system, the condition for restricted diffusion is reached by increasing the diffusion time. Under this condition, the Murday and Cotts model for diffusion inside a spherical droplet can be used to determine the size distribution of the emulsion. In very dilute systems by increasing the diffusion time, the Brownian diffusion becomes the dominant process so that the Stokes-Einstein model can be applied to obtain the size distribution.

Introduction The PFG NMR technique has been used extensively to investigate the size distribution in a variety of emulsion systems by measuring the diffusion of fluid confined inside emulsion droplets.1-14 A detailed review of the SE-PFG NMR technique has been published by Price.15 However, * To whom correspondence should be addressed. E-mail: [email protected]. (1) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (2) Tanner, J. E.; Stejskal, E. O. J. Chem. Phys. 1968, 49, 1768. (3) Neuman, C. H. J. Chem. Phys. 1974, 60, 4508. (4) Murday, J. S.; Cotts, R. M. J. Chem. Phys. 1968, 48, 4939. (5) Packer, K. J.; Rees, C. J. J. Colloid Interface Sci. 1972, 40, 206. (6) Callachan, P. T.; Jolley, K. W. J. Colloid Interface Sci. 1983, 93, 521. (7) Van den Enden, J. C.; Waddington, D.; Van Aalst, H.; Van Kralingen, c. G.; Packer, K. J. J. Colloid Interface Sci. 1990, 140, 105113. (8) Lonnqvist, I.; Khan, A.; So¨derman, O. J. Colloid Interface Sci. 1991, 144, 401. (9) So¨derman, O.; Lonnqvist, I.; Balinov, B. Emulsions - A Fundamental and Practical Approach; Kluwer Academic Publishers: Netherlands, 1992; p 239-258. (10) Li, X.; Cox, J. C.; Flumerfelt, W. AICHE J. 1992, 38, 1671. (11) Balinov, B.; Jonsson, P.; Linse, P.; So¨derman, O. J. Magn. Reson. 1992, 104, 17. (12) Balinov, B.; So¨derman, O.; Warnheim, T. JAOCS 1994, 71, 513. (13) Appel, M.; Fleischer, G.; Karger, J. Macromolecules 1995, 28, 2345. (14) Hakansson, B.; Pons, R.; So¨derman, O. Langmuir 1999, 15, 988.

the contribution from the diffusion of the droplets themselves has not been considered explicitly. Callaghan and Jolley6 studied water and fat diffusion in cheese, and van den Enden et al.7 obtained the size distribution of water droplets from the restricted diffusion model in margarine and halvarine products. Balinov et al.11 used computer simulation to compare the Gaussian phase distribution (GPD) and the short gradient pulse (SGP) approximation15 for the cases of diffusion within spheres and between two parallel and infinite planes. The GPD approximation was found to be accurate for most experimental situations, whereas the usefulness of the SGP limit is more restricted. Subsequently Balinov, together with So¨derman, and Warnheim12 determined the size distribution of margarine and low calorie spreads using the GPD approximation. In this paper, we give a general picture of the diffusion processes in emulsion systems where several distinct diffusive processes can be observed. Experimental Section Materials. The nonionic surfactants used were commercial products. Polyoxethylene (4) lauryl ether (Brij-30) was obtained from ICI Surfactants Germany, and polyoxethylene (23) lauryl (15) Price, W. S. Concepts Magn. Reson. Part I, 1997, 9, 299-336.

10.1021/la026109x CCC: $22.00 © 2002 American Chemical Society Published on Web 12/02/2002

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Figure 1. Standard spin-echo pulsed field-gradient experiment (SE-PFG) with rectangular gradient pulses of duration δ and magnitude g inserted into each τ delay. The separation between the leading edges of the gradient pulses is ∆.

Figure 2. Stimulated spin-echo pulsed field-gradient experiment (STE-PFG). The separation between the second and the third π/2 pulse is T. Remaining parameters are defined as in Figure 1.

ether (Brij-35p) was obtained from ICI Surfactants U.K. They were used as received. The poly(dimethylsiloxane) (PDMS) 1000 cS (MW ) 36.k kg/mol, MW/MN ) 2.43) and 350 cS (MW ) 21.9 kg/mol, MW/MN ) 2.05) were obtained from Dow Corning Europe, Brussels, Belgium. The deuterium oxide (D2O) 99.9% (min) was obtained from Aldrich Ltd. Sample Preparation. Concentrated 1000cS emulsions were prepared using the following quantities: 15 g (69.8 wt %) Dow Corning 1000 cS PDMS fluid, 0.6 g (2.8 wt %) Brij-35p, 0.9 g (4.2 wt %) Brij-30, and 5 g (23.2 wt %) D2O. A total of 1.5 g of Brij-35p surfactant was heated at 60 °C in an oven until it melted, and 0.6 g of it was then added to 0.9 g of Brij-30 surfactant in 15 g of the 1000 cS PDMS fluid. The mixture was homogenized with a high speed Silverson mixer (7500 rpm) for 5 min. The solution obtained had a low viscosity and was opaque (white). Water was added to the dispersion while continuously mixing at a fixed speed (max 7500 rpm). Beyond the addition of ∼1 g (5%) of water, the solution became very viscous and gray, and a phase inversion occurred. The emulsion changed from a water-in-oil type (W/O) to an oil-in-water type (O/W).16 Adding more water (drop by drop) and mixing, the solution gradually became less viscous and white again. The preparation of the emulsion was carried out at room temperature ∼25 °C. The weight percent of the emulsion (PDMS) in the sample was 10.0 wt %. A commercial emulsion was prepared by Dow Corning Europe in Brussels, Belgium, and contained 50 wt % Dow Corning 350 cS PDMS fluid, 3.86 wt % polyoxethylene (2-3) lauryl ether (Volpo L3), 1.54 wt % Brij-35p, 43.25 wt % H2O, and 1.35 wt % of other additives. NMR. The STE-PFG 1H NMR measurements were carried out on a Bruker AVANCE 300 spectrometer using a 7.1 T wide bore (89 mm) magnet and a Diff30 diffusion probe with actively shielded gradients. This configuration gives a maximum gradient strength of about 12 T/m with a current of 40 A. Measurements were also performed on a JEOL FX100 high-resolution NMR spectrometer operating at 2.3 T modified to carry out self-diffusion measurements. The current amplifier used to generate the field gradients was an Amkron M-600 (up to 20 A). The unit was calibrated with a sample of known diffusion coefficient (75/25 wt % D2O/H2O mixture at 25 °C with a diffusion coefficient of 2.0 × 10-9 m2 s-1).17 The maximum value of the field gradient of the spectrometer that can be applied is 1 T/m. All experiments were carried out at 25 °C. PCS. The photon correlation spectroscopy (PCS) equipment used to measure the particle size was a “ZetaPlus” (Brookhaven Instruments Corporation). The wavelength of the laser was 678 nm. Measurements were taken at room temperature on very dilute dispersions (∼0.1 wt %).

intensity at time 2τ in the presence of gradient pulses of strength g in the case of free (Gaussian) diffusion is given by1

Theoretical Background Free Diffusion. In the SE-PFG NMR experiment (Figure 1), the echo attenuation gives information on molecular displacement along the gradient axis (z axis) that has occurred during the time period ∆. This can be related to the self-diffusion coefficient D. The signal (16) Becher, P. (Ed.), Encyclopedia of Emulsion Technology; Dekker: New York, 1981;Vols. 1 and 2. (17) Mills, R. J. Phys. Chem. 1973, 77, 685.

( ) (

I(δ,∆,g,τ) ) I0 exp -

))

2τ δ exp -γ2g2δ2 ∆ - D T2 3

(

(1)

where I0 is the signal intensity observed immediately after the 90°x pulse, τ is the time between the 90°x and 180°y pulses, ∆ is the distance between the leading edges of the gradient pulses, δ is the length of the gradient pulse, D is the self-diffusion coefficient, T2 is the spin-spin relaxation time, and γ is the gyromagnetic ratio (1H, γ ) 26.752 × 107 rad T-1 s-1). The stimulated spin-echo NMR experiment (Figure 2) has the advantage that the time ∆ is made up of two parts, a shorter time (τ) in which T2 spin-spin relaxation takes place and a longer time (T) in which T1, longitudinal relaxation,15 occurs. For systems in which T2 is short but T1 is long, this method makes it possible to extend the diffusion time, ∆, considerably. The signal intensity at time 2τ + T in the case of free diffusion is given by

( ) ( ) ( (

T 2τ 1 exp I(δ,∆,g,τ,T) ) I0 exp 2 T1 T2

exp - γ2g2δ2 ∆ -

))

δ D (2) 3

where I0 is the signal intensity observed immediately after the 90°x pulse. The mean-squared displacement15 is given by

〈(r1 - r0)2〉 )

∫-∞∞ (r1 - r0)2F(r0)P(r0,r1,t) dr0 dr1

(3)

F(r0) is the probability that the spin trajectory begins at r0. P(r0, r1, t) is the diffusion propagator. For the case of (three-dimensional) diffusion in an isotropic and homogeneous medium is

(

P(r0,r1,t) ) (4πDt)-3/2 exp -

)

(r1 - r0)2 4Dt

(4)

Evaluation of eq 3 gives the Einstein equation,18 which in three dimensions is

〈(r1 - r0)2〉 ) 6D∆

(5)

This result can be used as a variable (r ) r1 - r0) to describe the various diffusion regimes in the system. In the PFG NMR experiment, it is the molecular displacement along the gradient axis (z axis) that is (18) Johansson, L. Diffusion and interaction in gels and solutions; Chalmers University of Technology: Go¨teborg, Sweden, 1993.

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Garasanin et al.

measured. The respective mean-squared displacement is

〈(z1 - z0)2〉 ) 2D∆

(6)

Restricted Diffusion. In systems, where the diffusion of molecules is restricted by physical boundaries, eqs 1 and 2 are no longer valid. The attenuation of the signal intensities for the spin bearing molecules confined in a spherical cavity of radius R was first derived by Murday and Cotts,4 using the GPD approximation,19 and is given by

I(R) I /0

[

(

)∑

-2γ2g2 ) exp D

({

∞ m)1

Rm-4

Rm2R2 - 2

×

2 + exp(-R2mD(∆ - δ)) - 2 exp(-R2mDδ)

Rm2D 2δ 2 -2 exp(-RmD∆) + exp(-R2mD(∆ + δ)) Rm2D

})]

(7)

We have omitted the decay due to spin relaxation for simplicity so that I /0 is the initial intensity as in eq 1 (SEPFG) or eq 2 (STE-PFG) but including the relaxation term. Rm is the mth root of the Bessel equation shown below

( )

1 J (R R) ) 2J′3/2(RmR) RmR 3/2 m

(8)

Equation 7 has been evaluated using the software Mathematica v 3.0.20 It was found that the series in eq 7 converges very rapidly so that only the first term needs to be retained. This allowed us to approximate the equation, with the first root of the Bessel equation (R1R ) 2.08). The result is shown below (eq 9) again omitting the decay because of spin relaxation for simplicity:

( )

ln

I(R) ) I0 2

2

2

2

54g2R4γ2(3(e13D(δ-∆)/3R - 2e-13Dδ/3R - 2e-13D∆/3R + e-13D(δ+∆)/3R + 2)R2 - 26Dδ) 2

15379D

(9)

If the product of the diffusion coefficient of the bulk fluid, D, and the diffusion time, ∆, is much smaller than the radius squared of the emulsion droplet, R2, the diffusing fluid cannot diffuse far enough to feel the effect of the boundary, the measured diffusion coefficient is the same as that observed for a freely diffusing species (i.e. D∆ , R2) and is described by eq 1. When D∆ ≈ R2, a certain fraction of the fluid feels the effects of the boundary and we measure an “apparent” diffusion coefficient (Dapp), which is ∆ dependent. When D∆ . R2, virtually all of the emulsified fluid feels the effects of restriction, and in this case, the signal attenuation is sensitive to the diffusion coefficient, shape and dimensions of the droplet and is described by eq 7 and 9. Under this condition, eqs 7 and 9 can be further simplified if Rm2Dδ , 1 or expressed with the first root of the Bessel equation as (13Dδ/3R2) , 1. This reduces eq 7 to exp(-0.2(gRγδ)2) and eq 9 to 0.1978(19) Douglas, D. C.; McCall, D. W. J. Phys. Chem. 1958, 62, 1102. (20) www.wolfram.com.

Figure 3. Condition for the restricted diffusion. The x, y, and z axes are the diffusion coefficient of the fluid, D, the diffusion time, ∆, and the RMS displacement, (〈r2〉)1/2, respectively.

(gRγδ)2. In the case of the emulsion samples studied in this paper, the condition (13Dδ/3R2) , 1 could not be fully reached. A graphical presentation of these regimes in concentrated systems is given in Figure 3. In x and y, we have two independent variables, the diffusion time (∆) and the diffusion coefficient of the fluid (D), respectively. The z axis is the square root of the mean-squared displacement of the fluid ((〈r2〉)1/2), RMS) and it can be compared with the radius of the emulsion droplet. If the radius of the particle is larger than the RMS displacement, there is a low probability that the fluid will travel far enough to feel the boundary of the droplet. The diffusion is unrestricted: this is the region above the shaded surface. Close to the shaded surface, the diffusion is partially restricted as R ∼ (〈r2〉)1/2. Under this surface, the diffusion of the fluid is restricted as the RMS displacement becomes larger than the radius of the particle; that is, there is an increasing probability that fluid will be reflected from the walls of the droplets. By increasing the diffusion time, the displacement of the fluid is increased. For example, if we have a fluid with a diffusion coefficient of D ) 10-12 m2 s-1 confined inside a 1 µm emulsion droplet, effectively, all of the fluid will feel the restriction if the diffusion time (∆) is longer than 1 s. Droplet Diffusion. In a concentrated system, we assume that the Brownian diffusion of the particles is negligible, but in dilute systems, this is not always the case. For a dilute suspension of spherical particles, the diffusion coefficient (D) is given by the Stokes-Einstein equation:18

D)

kBT 6πηRH

(10)

where kB is the Boltzmann constant, T is the temperature, η is the viscosity of the surrounding fluid (which in our case is water), and RH is the hydrodynamic radius of the emulsion droplets. For a hard sphere, RH ) R. From the diffusion coefficient, the hydrodynamic radius of the particles can be estimated. By replacing D in eq 1 with the Stokes-Einstein equation (eq 10), the attenuation of the signal intensities becomes

( ) (

)

I(δ,∆,g,τ) 2τ δ kBT ) exp exp -γ2g2δ2 ∆ I0 T2 3 6πηRH

(

)

(11)

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Langmuir, Vol. 18, No. 26, 2002 10301

Figure 4. Diffusion regimes in a dilute oil-in-water emulsion system. The light gray plane is a simulation of the Brownian diffusion of the particles (plane A). The medium gray plane is a simulation of restricted diffusion of the fluid when the condition for the total restriction is satisfied (plane B), plot of R ) RMS. The dark gray plane is a simulation of the diffusion of the bulk fluid with a diffusion coefficient of D ) 10-12 m2 s-1 (plane C). The x, y, and z axes are the radii of the particles, R, the diffusion time, ∆, and the RMS displacement, (〈r2〉)1/2, respectively. The effective RMS displacement (x〈rINT2〉 + 〈rEXT2〉) presented as a mesh surface.

Polydispersity of the droplets can also be taken into account by numerically integrating over all droplet radii. In the expression shown below, the contributions from the various droplets is weighted by the volume of each droplet:9

I ) I0

∫0∞ I(R)R3N(R) dR ∫0∞ R3N(R) dR

(12)

where, I(R) is found from eq 9 for the restricted model and from eq 11 for the Brownian model. N(R) is the relative frequency and is described in our case by the log-normal distribution function as proposed by Packer and Rees5

N(R) )

(

(ln 2R - ln 2R0) 1 exp 2σ2 2Rσx2π

)

2

(13)

where σ is the standard deviation of the ln 2R values. The resultant mean radii, R0, and polydispersities, σ, are obtained from a nonlinear least-squares fit (Mathematica v 3.020) to eq 12. To visually compare the results, relative frequencies, N(R), are plotted as a function of the droplets radius. The area under each curve is normalized as

∫0∞ N(R) dR ) 1

(14)

In this way, plots are relative to the number of particles. However, N(R) can also be normalized to be relative to the volume of the particles if R3 is included in the integral. In Figure 4, the diffusion of oil in a very dilute oil-inwater emulsion system has been simulated. The x axis is the diffusion time (∆), the y axis is the radius of the emulsion droplets (R), and the z axis is the RMS displacement. The plane, which defines the boundary for Brownian diffusion, is marked with a light gray color (plane A). Here,

the RMS displacement of the emulsion droplet is calculated from eqs 10 and 5 (for diffusion in three dimensions) and plotted as a function of its radius (R) and the diffusion time (∆).The height of this boundary surface increases with an increase of the diffusion time and decreases with an increase in the radius (R) of the droplets. If the polymer inside the droplets is very viscous (D ∼ 10-14 m2 s-1), it is effectively immobile inside the droplet. For the whole radius range shown in the graph, Brownian diffusion is the dominant diffusion process in the system. In Figure 4, plane A illustrates the Brownian diffusion in this system. For less viscous polymers, the diffusion of the fluid itself needs to be considered. How much this internal diffusion contributes in the total diffusion measured depends on the radius of the emulsion droplet (R) and the diffusion time (∆). First, we will consider low viscosity polymers (D ∼ 10-9 m2 s-1, order of water) wherein diffusion is restricted for all of the radius range given in the graph when ∆ > 0.1. This implies that the RMS displacement of the polymer is of the order of the radius of the emulsion droplet. The surface plot of the restricted diffusion is presented by the darker gray color (plane B, plot of R ) RMS). The region close to the intercept between plane A and B is critical. Here, the diffusion of the fluid inside the emulsion droplets is of the same order of magnitude as the Brownian diffusion of the droplets. This region corresponds to the range of the droplet radii from ∼100 nm to 5 µm, which is typical for macroemulsions.16 The resultant mean-squared displacement, which corresponds to the diffusion measured by NMR, is a combination of Brownian diffusion (external diffusion) and self-diffusion of fluid confined inside the droplet (internal diffusion). The resultant displacement can be expressed as the sum of their respective meansquared displacements:21

〈r2〉 ) 〈rINT2〉 + 〈rEXT2〉

(15)

However, out of this radius range, the situation is much simpler. For very small droplets (smaller than 100 nm), Brownian diffusion dominates, as 〈rINT2〉 is negligible and for large droplets (∼10 µm) restricted diffusion can dominate (〈rEXT2〉 is negligible). We can conclude that for low viscosity emulsified fluids Brownian diffusion is not always the dominant process and that the degree of domination depends strongly on the particle size. If particles are very large (∼10 µm), and restricted diffusion dominates, the size distribution can be determined using Murday and Cotts4 model. For the critical range (from ∼100 nm to 5 µm), by increasing the diffusion time (∆), the contribution from the inner diffusion of the polymer can be reduced so that Brownian diffusion becomes more dominant. This is especially important for the case of polydisperse emulsion systems. For the intermediate viscosity polymers, the diffusion process in the system is further complicated. In addition to the Brownian and restricted diffusion, depending on the droplets size, we can have partially restricted or unrestricted diffusion. To emphasize this in Figure 4, the surface plot of the bulk diffusion of a fluid with diffusion coefficient of 10-12 m2 s-1, which is in the order of the diffusion of the polymers we used in our work, is shown by the dark gray color (plane C). The RMS displacement of the bulk fluid is calculated from eq 5. To visualize more easily the different diffusion regions, a plot of the resultant RMS displacement (which can be regarded as equivalent (21) Van de Ven, Theo C. M. Colloidal hydrodynamics; Academic Press Limited: London, 1989.

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Garasanin et al. Table 1. Results of the STE-PFG NMR Volume Weighted Size Distribution Determination for 10 wt % 1000 cS PDMS Oil-in-Water Emulsion as a Function of the Diffusion Time ∆ and Volume Weighted Size Distribution Results from the PCS Measurement of the Same Emulsion but Dilute to 0.1 wt %

a

NMR ∆/s

R0/µm ( 2%a

σ ( 5%a

0.1 1.0 2.0

0.14 0.18 0.18

0.54 0.51 0.50

PCS

R0/µm

σ

0.15

0.35

Errors estimated from data fitting.

Figure 5. Signal attenuations (I/I0) versus the magnitude of the field gradient g for 10 wt % 1000 cS PDMS emulsion for a diffusion time of ∆ ) 0.1 (open circles), 1.0 (open squares), and 2.0 s (open triangles). The duration of the field gradient was δ ) 2 ms. The full lines are fits to the model for Brownian particle diffusion. The small filled circles represent the fit of the data with diffusion time of ∆ ) 0.1 s (open circles) using the model for restricted diffusion.

to displacement measured by NMR) is plotted as a mesh surface in Figure 4. It has been calculated using eq 15 as follows. 〈rmesh2〉 ) 〈rbulk,B2〉 + 〈rBrownian,A2〉 when plane B is lower or close to the plane C (bulk or partially restricted diffusion case) and 〈rmesh2〉 ) 〈rrestricted,C2〉 + 〈rBrownian,A2〉 when plane B is higher than plane C (restricted diffusion case). For very large droplets, the mesh surface approaches the plane for bulk diffusion (plane C), and the bulk diffusion of the fluid dominates. In the region close to the intercept between planes for bulk and restricted diffusion of the fluid (plane C and B), partially restricted diffusion dominates. Close to the intercept between the restricted and Brownian diffusion planes (B and A, the critical range from ∼100 nm to 5 µm), the restricted diffusion of the polymer inside the droplet is of the same order of magnitude as the Brownian diffusion. For very small particles, the mesh surface approaches the plane for the Brownian diffusion (plane A). Experimental Results The results for the 10 wt % 1000 cS PDMS emulsion using the STE-PFG sequence22 with the duration of the field gradient, δ ) 2 ms and diffusion times ∆ ) 0.1, 1.0, and 2.0 s are given in Figure 5 (300 MHz Bruker data). The diffusion time has been increased in order to enhance the contribution of the Brownian diffusion of the droplets and diminish the contribution of the partially restricted diffusion of the fluid (as described above). From nonlinear fits to the Brownian model, the resultant mean radius, R0, and polydispersity, σ, of the log-normal distribution function were obtained. The results are shown in Table 1. The fit to the Brownian model for a data obtained with a very short diffusion time of 0.1 s (Figure 5) gives a mean radius for the droplets, R0 ) 0.14 ((0.01) µm and polydispersity σ ) 0.54 ((0.03). These data may also be fitted to the model for restricted diffusion, which indicates that for such a short diffusion time the distance travelled by the droplets is of the same order of magnitude as the size of the particles. A fit to the restricted diffusion model gives a value of R0 ) 0.41 ((0.008) µm and polydispersity σ ) 0.40 ((0.06). These values are larger than from the Brownian model as the polymer has travelled not only the distance allowed (22) Gibbs, S. J.; Johnson, C. S., Jr. J. Magn. Reson. 1991, 93, 395.

Figure 6. Relative frequency N(R) versus radius of the particles R for the 1000 cS PDMS emulsion. Black circles correspond to a ∆ ) 0.1 s, white circles correspond to a ∆ ) 1.0 s, and black triangles correspond to a ∆ ) 2.0 s, NMR data. The empty squares are the PCS results. Data normalized using eq 14.

by the size of the droplet but also from the diffusion of the droplet itself and corresponds to the mesh region around the interception between all three planes A, B, and C in Figure 4. However, for such a small diffusion time, the diffusion of the polymer inside the largest droplets is only partially restricted which increases the effective diffusion coefficient. For the longer diffusion times of 1.0 and 2.0 s, the restricted model gives unfeasible results which indicates that the distance travelled by the droplets is now much larger than their size. This corresponds to the mesh region close to the plane A under the dark gray triangle in Figure 4 (D). The contribution from the Brownian diffusion of the droplets is significantly larger than the diffusion of the polymer itself, which allows us to obtain a correct value for the particle size using the Brownian model. In Figure 6 and Table 1, the resultant particle size distributions from NMR and PCS are compared. For a short diffusion time of 0.1 s (black circles), the fit to the Brownian model gives a mean radius R0 ) 0.14 ((0.003) µm and polydispersity σ ) 0.54 ((0.03). When the diffusion time is substantially increased to 1.0 and 2.0 s (white circles and black triangles, respectively), the resultant radius distributions are very close giving the same value (Table 1). As the diffusion time is increased, the Brownian diffusion of the emulsion particles becomes the dominant diffusion process, and diffusion of the internal fluid becomes negligible, giving larger values for the particle size. The results from PCS measurements for the 0.1 wt % 1000 cS emulsion (white squares) give a mean radius for the emulsions particles of 0.15 µm and polydispersity of 0.35. The main difference between the NMR results (with enough large diffusion times 1.0 and 2.0 s) and PCS is that a higher proportion of larger particles are found.

Studies on PDMS Oil-in-Water Emulsion

Figure 7. Signal attenuation (I/I0) versus the duration of the field gradient δ for the concentrated 50 wt % commercial 350 cS PDMS emulsion (circles) and for a bulk 350 cS PDMS fluid (triangles). The magnitude of the field gradient g ) 0.81 T/m. Black circles and triangles are the data with a diffusion time of ∆ ) 2.04 s, and the white circles and triangles are the data with a shorter diffusion time ∆ ) 0.74 s. The solid lines are the fit to the model for restricted diffusion of the emulsion data, and the dotted lines are a Gaussian diffusion model for the bulk PDMS data.

Figure 8. Relative frequency N(R) versus radius of the particles R for the commercial 50 wt % 350 cS PDMS emulsion. Black circles correspond to a diffusion time of ∆ ) 2.04 s for which the diffusion is restricted, and white circles correspond to ∆ ) 0.74 s for which the diffusion is partially restricted. Data normalized using eq 14.

From Figure 6, it can be seen that the most probable radius is the same for both techniques. A full comparison of the NMR method with other sizing methods will be published separately. In Figure 7, we present data (100 MHz JEOL NMR) from a concentrated 50 wt % 350 cS PDMS commercial emulsion (circles) and bulk 350 cS PDMS (MW/MN ) 2.05) fluid (triangles) using the STE-PFG sequence22 for two different values of the diffusion time. The value of the field gradient was g ) 0.81 ((0.01) T/m and diffusion times 0.74 and 2.04 s (white and black symbols, respectively). Considering that the PDMS fluid is very polydisperse (MW/ MN ) 2.05), its average diffusion coefficient has been estimated using the Gaussian1 model and is 2.64 ((0.2) × 10-12 m2 s-1. The signal attenuation from the emulsion (circles) becomes smaller as the diffusion time increases as opposed to the attenuation of the bulk fluid (triangles), which gets larger. This shows that, by increasing the diffusion time, the diffusion of the PDMS inside the droplet becomes more restricted. The resultant size distributions obtained using the model for restricted diffusion are given in Figure 8. It can be seen that as the diffusion time is increased the distribution has moved toward smaller effective radii and the polydispersity is decreased. This is

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Figure 9. Signal attenuation (I/I0) versus the duration of the field gradient δ for the 1000 cS PDMS emulsion. The diffusion time was ∆ ) 2.04 s, and the magnitude of the field gradient was g ) 0.81 T/m. Filled circles are the data from the concentrated 60 wt % 1000 cS emulsion. Empty circles are the data from the dilute 10 wt % 1000 cS emulsion.

Figure 10. Relative frequency N(R) versus radius of the particles R for the 1000 cS PDMS emulsion. White circles correspond to a 60 wt %, and black circles correspond to a 10 wt %. Data normalized using eq 14.

because the droplet restricts PDMS fluid diffusion. For a diffusion time of ∆ ) 0.74 s (white circles), a mean radius for the emulsion droplets R0 ) 0.25 ((0.01) µm and polydispersity σ ) 0.64 ((0.09) are obtained. For the longer diffusion time of ∆ ) 2.04 s (black circles), a mean radius for the emulsion droplets R0 ) 0.14 ((0.01) µm and polydispersity σ ) 0.57 ((0.08) are obtained and are the best estimates as full restriction is measured. In Figure 9, the results (100 MHz data) from a 10 (white circles) and 60 wt % (black circles) 1000 cS emulsion are presented. The diffusion time used was 2.04 s, sufficiently long for Brownian diffusion to be dominant in the dilute system and for the diffusion of the PDMS molecules in the concentrated emulsion system to become restricted. The value of the field gradient was g ) 0.81 ((0.01) T/m. The resultant size distributions are shown in Figure 10. The fit to the Brownian diffusion model of the 10 wt % emulsion ) 0.18 ((0.01) µm and data gives a mean radius Rout 0 polydispersity σ ) 0.42 ((0.02) (black circles). The fit to the model for restricted diffusion of the 60 wt % emulsion ) 0.16 ((0.01) µm and polydispersity of σ ) gives Rinn 0 0.62 ((0.09) (white circles). The result from the dilute system gives a larger value as the outer radius of the droplets (hydrodynamic radius) is measured. The difference between resulting outer and inner mean radius of - Rinn the droplets (Rout 0 0 ) is related to the interfacial thickness though this number exceeds the extended length of the surfactants considerably, and other factors such as

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a stationary PDMS layer near the interface and hydrodynamic effects need to be investigated. Conclusions The contributions from all of the diffusion processes in an emulsion system need to be considered if a correct size distribution is to be determined. For a concentrated emulsion system, an increase of the diffusion time enables the condition for restricted diffusion to be satisfied. This allows us to apply the Murday and Cotts4 model for diffusion inside the spherical particle to determine the

Garasanin et al.

size distribution. By increasing the diffusion time in very dilute systems, the Brownian diffusion of the particles dominates so that the Stokes-Einstein18 model can be applied for size distribution analysis. Acknowledgment. T.G. would like to acknowledge BRUKER for measurements with their 300 MHz NMR, Dow Corning Ltd., Barry and the University of Bristol for funding. LA026109X