Droplet migration in emulsion systems measured using MR methods

Journal of Colloid and Interface Science 296 (2006) 700–709 www.elsevier.com/locate/jcis

Droplet migration in emulsion systems measured using MR methods K.G. Hollingsworth, M.L. Johns ∗ Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK Received 1 August 2005; accepted 24 September 2005 Available online 27 October 2005

Abstract The migration of emulsion droplets under shear flow remains a largely unexplored area of study, despite the existence of an extensive literature on the analogous problem of solid particle migration. A novel methodology is presented to track the shear-induced migration of emulsion droplets based on magnetic resonance imaging (MRI). The work is in three parts: first, single droplets of one Newtonian fluid are suspended in a second Newtonian fluid (water in silicone oil (PDMS)) and are tracked as they migrate within a Couette cell; second, the migration of emulsion droplets in Poiseuille flow is considered; third, water-in-silicone oil emulsions are sheared in a Couette cell. The effect of (a) rotational speed of the Couette, (b) the continuous phase viscosity, and (c) the droplet phase concentration are considered. The equilibrium extent of migration and rate of migration increase with rotational speed for two different emulsion systems and increased continuous phase viscosity, leads to a greater equilibrium extent of migration. The relationship between the droplet phase concentration and migration is however complex. These results for semi-concentrated emulsion systems and wide-gap Couette cells are not well described by existing models of emulsion droplet migration. © 2005 Elsevier Inc. All rights reserved. Keywords: Couette; Migration; Emulsion; Shear

1. Introduction It has been known for some decades that suspensions of solid particles subjected to non-uniform shear will tend to migrate towards regions of low shear [1], effectively leading to demixing of suspensions. At around the same time it was also found that single emulsion droplets undergo migration in shear fields [2]. This study attempted to study the migration phenomena for both non-deformable solid particles and deformable emulsion droplets using similar models. It became apparent that the two effects were rather different: the migration of solid particles being influenced chiefly by collision frequencies, and the migration of deformable emulsion droplets being influenced by asymmetric flow velocities around deformed droplets. The shear-induced migration of solid suspensions has been subsequently well studied [3–9]. There have, however, only been a handful of subsequent studies of emulsion droplet migration to the best of the authors’ knowledge [10,11]. The spatial distribution of emulsion droplets is an important parameter in man* Corresponding author.

E-mail address: [email protected] (M.L. Johns). 0021-9797/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.09.059

ufacturing, because this can strongly influence the rheology of the emulsion, leading to different pumping requirements [12]. Although the shear-induced migration of solid particles has been well studied by MRI, the shear-induced migration of deformable emulsion droplets has not previously been studied using such methods. Previous studies of emulsion droplet migration have relied on the use of dilute systems and optical methods: such methods are limited to those dilute systems which are optically transparent and where the refractive indices of the continuous and discrete phases are closely matched. Emulsions which are of industrial interest are often opaque, have a high droplet phase concentration and thus MRI has a unique contribution to make in this area. Here we explore the possibility of studying the migration under flow of both single droplets and emulsions by MRI methods. Literature survey Single droplets The phenomenon of single droplet migration in shearing flow was reported in the creeping flow regime (Re  1) by Goldsmith and Mason [2] who studied migrations in a

K.G. Hollingsworth, M.L. Johns / Journal of Colloid and Interface Science 296 (2006) 700–709

Poiseuille geometry, where they found that single emulsion droplets suspended in Newtonian fluids migrate away from the wall when they are deformed by shear towards the axis of the tube: an approximate theory was proposed. A more rigorous theoretical treatment was then provided by Chaffey et al. [13, 14]: this treatment was based on an analysis for the interaction of a deformed emulsion droplet with a bounding wall, and that this interaction arises due to the deformed droplet creating an asymmetric disturbance velocity. It was found that the droplet would always migrate away from the bounding wall. Karnis and Mason [15] performed further experimental studies for both Poiseuille and Couette geometries and tested this theory: their results agreed with the theory to within an order of magnitude; the discrepancy between theory and experiment derived mainly from only taking into account the influence of one bounding Couette wall. The Chaffey theory assumes a full development of internal circulation within the droplet and only includes the interaction of the droplet with one wall. Gauthier et al. [16] extended the experiments to single droplets suspended in viscoelastic fluids in Poiseuille flow. Chan and Leal [17] provided a full theoretical treatment of the motion of a deformable drop in both Newtonian and viscoelastic fluids. A thorough theoretical analysis of the velocity contributions due to deformation and viscoelastic rheology was obtained, and the relationship between this work and the earlier theories explored. This theory took into account the effect of both bounding walls of the Couette geometry on the migration of the droplet, though it treated them as plane walls. This was followed by Chan and Leal [18] in which extensive experimental studies were conducted with Couette cells of different radius ratios and comparing theoretical predictions with experimental measurements for Newtonian drops in Newtonian fluids, viscoelastic drops in Newtonian fluids and Newtonian drops in viscoelastic fluids. Good agreement was obtained in all cases for narrow Couette geometries for a wide range of capillary numbers. Deviations from the theory were found for wider gap Couette cells. Multiple droplets (emulsions) The migration properties of a deformable emulsion system containing many droplets were studied theoretically by Loewenberg and Hinch [19], who predicted an expression for the “shear induced” dispersion due to the irreversible interaction of droplet–droplet contact—this built on previous work by Leighton and Acrivos [20,21] and Da Cunha and Hinch [22] who had studied the shear induced dispersion of nondeformable rough solid particles undergoing irreversible collisions. Only two experimental studies of shear-induced dispersion in an emulsion system exist to the best of the author’s knowledge. King and Leighton [10] used a linearised form of the migration velocity of Chan and Leal [17] and combining it with the theoretical shear-induced dispersion model of Loewenberg and Hinch [19]. They used it to predict the radial concentration profiles of the discrete phase in a narrow gap Couette, which could be modelled by a set of self-similar parabolas. The theory was backed up by experimental data on a water-

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Fig. 1. (Left) Schematic cross-section of the arrangement of the Couette cell in the vertical bore NMR superconducting magnet. (Right) Schematic of drop in the Couette cell. (R1 = 2.5 mm, R2 = 8.8 mm for the single drop experiments, R1 = 5.8 mm, R2 = 8.8 mm for the multiple drop experiments.) The Couette shear rate curvatures of the cells (as defined in Eq. (3)), χ , are 0.92 and 0.57, respectively.

in-fluorosilicone oil emulsion system at low discrete phase concentration (1 v%) using optical methods. Agreement of the droplet concentration distributions between experiment and theory was very good. The work of Hudson [11] expands the King and Leighton study by removing the restriction of linearising the migration velocity. The system used in this study consisted of immiscible polymers, and a parallel plate system was used: the study is therefore not relevant to the geometry considered here. 2. Migration theory Chan and Leal [17,18] set out a theoretical description for the migration rate of a single emulsion droplet migrating away from the bounding walls of a Couette cell. The appropriate formulae for Newtonian fluids is outlined below. The dimensionless radial position of the droplet in the Couette, s, is defined as s=

R0 − R1 , R2 − R1

(1)

where R0 is the radial position of the droplet and R1 and R2 are the inner and outer radii of the Couette, respectively (as shown in Fig. 1). The time required, T , for the droplet to migrate from the dimensionless radial position s0 to the dimensionless radial position s is given by dσ T = ηc where Z(s) =

s

ds , Z(s)

(2a)

s0

  1 2 a4 1 + g(κ) (R22 − R12 )2 d 2 R02 R22    1 1 a3 × 2(1 − 2s) + − − h(κ) , (1 − s)2 s 2 R05 Ω 2 R14 R24



(2b)

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 10 + 11κ 2 (8 − κ + 3κ ) , + 140

where the functions g(κ) and h(κ) are given by + 54κ 2 )

g(κ) =

3(16 + 19κ)(54 + 97κ 4480(1 + κ)3

h(κ) =

2(4 + 61κ + 85κ 2 + 25κ 3 ) , 7(2 + 3κ)(1 + κ)2

and (2c)

and where σ is the interfacial tension of the system, d is the gap width, a is the radius of the droplet, κ is the ratio of viscosities of the droplet to the continuous phase, ηc is the viscosity of the continuous phase; Z(s) is thus proportional to the drift velocity of the droplet. A measure of shear rate curvature in a Couette cell, χ , defined as the ratio between the shear rate at the outer wall of the annulus to the shear rate at the inner wall of the annulus for a Newtonian fluid, is given by  2 R1 χ =1− (3) . R22 For the ideal narrow gap Couette, χ → 0. Equation (2) is found to be most accurate under the following conditions: (i) Narrow gap Couette. Equation (2) is based on using a method of reflections of the bounding walls of the Couette. As the Couette becomes wider, the contribution to the droplet’s migration made by the inner cylinder is overestimated. The net result is that the equilibrium position of the droplet lies further towards the inside of the Couette than that which is predicted and the droplet migrates inwards faster than expected—this was demonstrated in Chan and Leal [18]. (ii) Small capillary number. The assumptions made by Chan and Leal [17] in the determination of the formula are strictly only valid for small deformations of the droplet. In their experimental work they found, however, that for the narrow gap Couette the formula gave reasonably good predictions for quite large droplet deformations. 2.1. Droplets in Poiseuille flow The following theory for Poiseuille flow is based on the argument put forward in Chan and Leal [17]. A deformable droplet in Poiseuille flow will migrate away from the wall of the tube in which it is flowing, towards the centre of the tube (the shear rate is minimised at the centre of the tube and is equal to zero): hence the droplet will be undeformed at this position and will not migrate further. The migration can be modelled by the equation  2  D − D02 D0 2Vmax a 3 ηc L (4a) n(κ), − m(κ) ln = 2 D 2B B 4σ where m(κ) = 1 − and n(κ) =

2κa 2 (2 + 3κ)B 2

(4b) 

1 3 16 + 19κ (1 − κ − 2κ 2 ) (1 + κ)2 (2 + 3κ) 14 2 + 3κ

(4c)

where D is the radial distance of the droplet from the tube centre, D0 is the initial radial distance of the droplet from the tube centre, B is the radius of the tube, L is the axial displacement along the tube, Vmax is the maximum velocity in the flow (centreline), ηc is the viscosity of the continuous phase and σ is the interfacial tension between the two phases. 2.2. Multiple droplet migration in Couette flow The first theory for the migration of an emulsion system of many droplets in a Couette was proposed by King and Leighton [10]. Their theory was developed by using a narrow gap Couette and, hence, taking a linearised version of the Chan and Leal [17] drift velocity, Z(s) (Eq. (2)), such that Z(s) = −Zl (s − 0.5)

(5a)

and Zl =

729γ˙ 2 a 4 ηc , 35d 2 σ

(5b)

In a system of many emulsion droplets, where two deformable droplets come into close contact with each other, shear-induced dispersion will occur [18]. The rate of droplet collision is proportional to γ˙ a 2 φ, where φ is the volume concentration of the emulsion droplets and γ˙ is the local shear rate. The diffusion coefficient due to droplet–droplet interactions, Dc , is found to be Dc = γ˙ a 2 φλ,

(6)

where λ is a constant of proportionality. By setting up a differential equation balancing the drift velocity against the dispersion due to droplet–droplet interactions, King and Leighton obtained a concentration profile as a series of self-similar parabolas described by 1 φ(s) = φ0 2P (t ∗ )

 2/3  (s − 0.5)2 3 − , 2 P (t ∗ )2

(7)

where φ0 is the average droplet phase concentration, φ(s) is the discrete phase concentration at position s, t ∗ is the dimensionless time (t ∗ = Zl t/dα), d is the width of the Couette gap and the function P (t ∗ ) is defined by 

   1 1 −3αt ∗ 1/3 1 − . + e P (t ) = α α0 α ∗

(8)

Both the extent and rate of migration are determined by the parameter α, α=

a 729 , · Ca · 35 dλφ0

(9)

with Ca is the capillary number. α0 is a parameter which describes the initial distribution and α determines the extent of migration at infinite time.

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3. Experimental description All images were acquired using a Bruker DMX 300 spectrometer with a magnetic field strength of 7.07 T, corresponding to a 1 H resonance frequency of 300 MHz. A 15 mm innerdiameter bird-cage r.f. coil was used. All chemicals were obtained from Sigma Aldrich. The experiments performed build the measurement technique from the simplest case of a single droplet through to multiple droplets: Experiment A: A single emulsion droplet (large enough to be imaged by MR) migrating within a rotating Couette system. This experiment demonstrates that droplet migration can be observed and quantified. Experiment B: An emulsion system in laminar flow within a pipe. Experiment C: An emulsion within a rotating wide gap Couette, where both deformation and droplet collisions will influence the rate of droplet migration. In experiments B and C, we will not be able to image individual droplets, but will be able to quantify the volume concentration of the droplets, φ, by MRI. 3.1. Experiment A: Couette flow—single drop experiments A Couette cell was formed as shown in Fig. 1. By comparison with Couette cells used in other branches of rheometry, this is a very wide gap (χ = 0.92), but allows the easy introduction of single emulsion droplets. Once the Couette had been assembled, the position of the drop from the bottom of the Couette was measured manually. The Couette was inserted into the r.f. coil of the spectrometer: a long shaft was then inserted to connect the inner tube of the Couette cell to a variable speed motor mounted at a safe distance from the magnet. Further details of this Rheo-NMR device and its use for rheological testing can be found in Hollingsworth and Johns [23]. In the experiments presented here, a 1 µL droplet of 1% Tween 60/24% ethanol in water was used in 1000 cSt silicone oil (PDMS). 24 v% ethanol/0.005 M MnSO4 in water solution was also used to form droplets in order to provide density matching and nulling of the droplet signal, respectively, when required. The radius of the droplet was confirmed by using stationary MR images. Fast 2D RARE imaging [24] was used in the vertical and horizontal planes to locate the droplet (acquisition time 384 ms). Once the droplet was located and its original radial position had been found in a suitably located horizontal RARE image, the Couette was rotated at 0.31 rev/s. Further horizontal RARE images were then acquired at 1-min intervals, up to 10 min, and 5-min intervals, up to 50 min. This was done in order to track the position of the droplet. On each occasion the Couette was stopped, a RARE image was acquired and the Couette was then restarted. 3.2. Experiment B: Multiple droplets in Poiseuille flow A closed flow loop was setup in the NMR spectrometer: a glass tube was used with an outside diameter of 15 mm

703

(i.d. 11 mm). A toluene emulsion was made containing 12 v% toluene and 1 v% Triton X-100 in distilled water. This emulsion was dispersed using a Art Miccra D8 homogeniser for 5 min at 10,800 rpm. Droplet size distribution measurements showed that the largest droplet radius in the emulsion was 10 µm—the method used to make these measurements by NMR is outlined in [25]. The emulsion underwent gravity-driven laminar flow at 21 ml/min (average velocity 0.36 cm/s) through the glass tube. The spatial distribution of the emulsion droplets was measured by using a fast chemical shift imaging (CSI) MRI pulse sequence [26] which produces a NMR spectrum for every pixel in a 2D image. The pipe cross-section in-plane resolution was 234 µm and signal was detected from a 6 mm thick slice. The recycle time of the experiment was 15 s to allow for full T1 relaxation, giving a total experiment time of approximately 30 min. 3.3. Experiment C: Multiple drop migration in Couette flow The system under examination is again water droplets dispersed in viscous silicone oil (PDMS), with viscosities of either 1000 cSt or 12,500 cSt. Two different surfactants were considered: 1 v% Triton X-100 and 1 v% Tween 60 dispersed in distilled water. Since it was necessary for the droplets to be neutrally buoyant over a period of hours the density of the droplet phase was adjusted by adding ethanol. It was found that the best mixture for the 1000 cSt PDMS was to add 25 v% ethanol to the 1% Tween 60 aqueous solution and 24 v% ethanol to the 1 v% Triton X-100 aqueous solution. The aqueous emulsion droplets were formed by initially introducing the discrete phase in 10 µL drops to the required concentration in PDMS, which was then sheared. This process entrains air in the emulsified mixture and the resultant emulsion is left to stand for several hours to allow the air to leave the mixture. After this the emulsion is gently restirred to achieve uniformity of droplet concentration. Several methods of performing this selective imaging were examined, but the method found to work best was to precede a RARE image by an inversion recovery pulse. The T1 relaxation time of the silicone oil (PDMS) was found to be 1.27 s and that of the dispersed water was 1.78 s. Therefore, by having a compound 180◦ pulse [27] at a time 0.671 × T1 (silicone oil) before the RARE sequence, it is possible to blank out the signal due to the silicone oil and collect signal from just the dispersed water. The Couette cell is assembled slowly, filled with the emulsion, and inserted into the r.f. probe. The rheo-NMR motor is then assembled. The Couette is then run at a set rotation rate for given intervals. The Couette is stopped and a further RARE image (acquisition time 384 ms) is taken of the distribution of the discrete droplet concentration. It is important to have a measure of the droplet radii involved in the migration and to compare this measurement between the beginning and end of the experiment so as to determine whether any coalescence has taken place. This droplet size distribution measurement was made using fast Difftrain PFG sequence [28] using the following experimental parameters: δ = 2 ms, 1 = 12.8 ms, inc = 65 ms, g = 10 G/cm.

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Fig. 2. Individual RARE images of 1 µL droplets of 1% Tween 60/25% ethanol solution migrating inwards. (a) Initial position, (b) 280 s, (c) 900 s.

4. Results and discussion 4.1. Experiment A: Couette flow—single drop experiments Several of the individual images for the droplet of 1% Tween 60 solution migrating inwards are shown in Fig. 2. The acquired images are rotated to put the imaged droplet in the same place in all the images, for ease of comparison. The coordinates of the centre of the droplet are then calculated. The overall trajectories for drops migrating inwards and outwards in this configuration are given in Fig. 3: these are typical trajectories, but are found to be highly reproducible. The measured radius of this droplet is 0.54 mm. For the systems in our experiments (water/surfactant in 1000 cSt or 12500 cSt silicone oil (PDMS)), κ = 1 × 10−3 or κ = 8 × 10−5 , and g(κ) = h(κ) = 0.58 (Eq. (2c)) for both values of κ (these functions are insensitive for κ  1). For these conditions, Eq. (2) reduces to    Ω 2 R14 R24 1 2 81a 4 1 Z(s) = 2 + (R2 − R12 )2 140d 2 R02 R22    1 1 8a 3 × 2(1 − 2s) + − (10) − . (1 − s)2 s 2 14R05 The predictions of Eq. (10) are also plotted in Fig. 3. It is clear that in both cases the migration speed is underpredicted by the available theory, while the equilibrium position is well predicted. This is in agreement with the findings of Chan and Leal [18] who found that the droplets migrated much faster in wide gap Couettes than predicted by the theory, developed and verified in narrow gap Couettes (Eq. (10)). It can be seen that there is distinct asymmetry between the speed of the droplet migrating inwards and that migrating outwards. However, a fairly consistent equilibrium position is found and this lies inwards of the centre line of the Couette, as can be found by solving Eq. (10) for the position where there is no migration velocity. Despite the fact that the theory is not strictly applicable to these wide gap systems, it is possible to fit the positions of the droplets reasonably well to an exponential fit of the form s(t) = s∞ + (s0 − s∞ )e−kt ,

(11)

Fig. 3. Plotted migration trajectories for 1 µL droplet of 1% Tween 60/25% ethanol/0.005 M MnSO4 in 1000 cSt silicone oil migrating both inwards and outwards.

where s(t) is the dimensionless position of the droplet at time t, s0 is the dimensionless initial position of the droplet and s∞ is the dimensionless equilibrium position of the droplet, t is the time in seconds and k is the rate constant of the migration in s−1 . For the drop shown in Fig. 3, for the droplet migrating outward, the dimensionless distances and migration rates are s0 = 0.079, s∞ = 0.286, k = 8.55 × 10−2 s−1 . For the droplet migrating inwards s0 = 0.738, s∞ = 0.229, k = 1.41 × 10−3 s−1 . The droplet diameter corresponds to 0.1 in the dimensionless units. The discrepancy between the measured equilibrium positions of the inward and outward migrating droplets is thus significantly less than a droplet diameter and hence within an acceptable margin of error. Chan and Leal [18] also found that the outward migrating droplet moves to the equilibrium position much faster than the inward migrating droplet for a Couette with shear rate curvature of χ = 0.82 (which is close to χ = 0.96, as used here). It is clear that MRI is a useful method for examining single droplet migration in Couette systems, without the limitations of optical methods (transparent, refractive index matched systems), but that the present theories of droplet migration, developed for similar geometries, are inadequate for wide-gap Couettes.

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Fig. 5. Intensity plots for radially averaged chemical shift images for toluene in water emulsion flowing at 21 ml/min and stationary.

Fig. 4. (a) Slice through the 2D CSI data showing the spectral dimension and one spatial dimension (x axis). (b) Intensity image corresponding to water peak and (c) intensity image corresponding to aromatic peak (x–y slices).

4.2. Experiment B: Multiple emulsion droplets in Poiseuille flow Fig. 4a shows a typical spatial–spectral cross-section through a 3D CSI dataset. The spectral peaks are evident as horizontal bands: the bands from top to bottom are aromatic, water and CH3 respectively. Alternative spatial–spatial orientations are shown in Figs. 4b and 4c which shows the concentration variation across the tube for (b) water and (c) aromatics (toluene). Qualitative observation of Figs. 4b and 4c reveals that both materials are homogeneously distributed across the tube, despite flowing at an average velocity of 0.36 cm/s. The quantitative analysis is done by radially integrating each individual spectral peak to produce material concentration as a function of radial position. Fig. 5 compares the results of the stationary measurement and the measurement at 21 ml/min. It can be seen that there is no significant heterogeneity of concentration across the tube. There is a very slight increase in intensity with radial distance for the flowing emulsion, but since this trend is seen for both the water and toluene peak, this must be due to NMR signal relaxation effects. Thus, there is no significant migration in this setup. We can make an estimate of the anticipated migration if we model a typical droplet in the pipe flow using Eq. (4). For the toluene-in-water emulsion studied here, Eq. (4) reduces to     D 2 − D02 6a 2 Vmax a 3 ηc L D0 . (12) + 1 − = 0.105 ln D 2B 2 19B 2 B 4σ With an interfacial tension of 1.5 mN/m [29], the migration of a 10 µm drop half way between the centreline and the edge

of the tube (D0 = B/2) can be solved by numerical methods over the 30 cm that the flow has to develop and is found to be 1 × 10−9 m (i.e., negligible). Hence it is expected and observed that no migration occurs with a low viscosity continuous phase. The migration effect would be become more significant in a much higher viscosity continuous phase or using a very narrow tube. This is relevant to the recent interest in controlled emulsification by microchannel methods [30], where the diameters of the tubes will be of the order of tens of micrometres, with correspondingly high shear rates. 4.3. Experiment C: Couette flow—multiple drop experiments Examples of the initial water (dispersed droplet phase) droplet distribution and final water distribution images obtained are given in Figs. 6a and 6b: the system used here is 3.2 v% of 1% Tween 60 in 12,500 cSt PDMS rotated at 0.31 rev/s for 30 min. When the same initial sample is rotated at 0.62 rev/s for 5 min, severe coalescence occurs (Fig. 6c) and several large coalesced droplets are clearly visible. The data set, from which Figs. 6a and 6b were extracted, was radially averaged and is presented in Fig. 7. This allows us to examine the concentration effect across the gap in a radial direction. There is clear migration observed in this system and equilibrium (highest droplet concentration) is closer to the inner cylinder as observed for single droplets. The Couette cell used in these experiments has a greater shear rate curvature (χ = 0.57) than that used by King and Leighton for their emulsion experiments (χ = 0.18). The symmetric parabolas of droplet concentration they developed for narrow-gap Couettes (Eqs. (7)–(9)), can clearly not be extrapolated to the wide-gap Couette cell considered in our work; Figs. 6 and 7 are clearly asymmetric with equilibrium positions closer to the inner cylinder. An empirical method of determining the rate of migration and the extent of migration is now proposed. The radially averaged intensity curves are divided by the initial curve, which gives I (r), a measure of how each image differs from the original, as shown in Fig. 8. In order to get a measure of (a) the extent of the migration in any particular dataset and (b) the “centre-line” or equilibrium line of migra-

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Fig. 6. 3.2 v% of a 1% Tween 60 solution in 12,500 cSt silicone oil. (a) Initial distribution of water droplets. (b) Distribution of water droplets after 30 min rotation at 0.31 rev/s. (c) Effect of rotating at 0.62 rev/s for 5 min. The grey density indicates the volume concentration the aqueous droplets.

tion, the zeroth (M0 ) and first (M1 ) moments of the areas where I (r) > 1 are found, r2 M0 =



2πr I (r) − 1 dr,

(13)



2πr 2 I (r) − 1 dr,

(14)

r1

r2 M1 = r1

where r1 and r2 are defined as the innermost and outermost points in the Couette where I (r) = 1. M0 provides a measure of the extent of the droplet migration and the “centre point of the migration,” rc , is the ratio of the two moments: rc =

M1 . M0

Fig. 7. Radially averaged intensity plots for 3.2 v% of 1% Tween 60 dispersed in 12,500 cSt silicone oil rotated at 0.31 rev/s.

(15)

To assess the rate and equilibrium extent to which the droplets migrate, the values for the first moment of the distribution are fitted to the equation M0 = A(1 − e−kt ),

(16)

where t is the time elapsed in seconds, k represents the rate of migration in s−1 and the constant A represents the equilibrium extent of migration. The equation was found to be a reasonably good fit to the experimental data in all cases. 4.4. Comparison of migration results The results of the experiments carried out are given in Table 1. The graphs showing the fit of Eq. (16) are given in Fig. 9a for selected experiments with 3.2 v% of 1% Tween 60 dispersed in 12,500 cSt silicone oil and Fig. 9b for selected experiments with 3.2 v% of 1% Triton X-100 dispersed in 12,500 cSt silicone oil. Table 1 also contains an estimate of the goodness of fit using the degrees-of-freedom adjusted R-square statistic [31]. The droplet radii for the migration experiments are given in Table 2. The central position of the migration, rc , was evaluated using Eq. (15). The experiments performed using 1% Triton X-100 as

Fig. 8. Plot of intensities divided through by the initial intensity distribution, revealing the change in the distributions over time for 3.2 v% of 1% Tween 60 in 12,500 cSt silicone oil rotated at 0.31 rev/s.

the surfactant all have values of rc corresponding to a dimensionless distance, s, of 0.4–0.5. The centre of migration, rc is closest to the centre of the Couette (s = 0.5) at the fastest rotation speed. The 1% Tween 60 experiments show that the central position is similar (0.35–0.6) and that the largest centre of migration corresponds to the fastest rotational speed. According to Eq. (2), the equilibrium position for a single droplet is in-

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Table 1 Extents and rates of migration for multiple droplet experiments (experiments are performed using 12,500 cSt silicone oil (PDMS) as the continuous phase unless otherwise stated) System

Equilibrium extent of migration, A (a.u.)

Rate of migration, k (s−1 )

Centre of migration, rc (s), mm (–)

Quality of fit

3.2 v% of 1% Tween 60, 0.154 rev/s 3.2 v% of 1% Tween 60, 0.31 rev/s 3.2 v% of 1% Tween 60, 0.62 rev/s 3.2 v% of 1% Triton X-100, 0.074 rev/s 3.2 v% of 1% Triton X-100, 0.154 rev/s 3.2 v% of 1% Triton X-100, 0.31 rev/s 3.2 v% of 1% Triton X-100, 0.62 rev/s 3.2 v% of 1% Triton X-100, 0.31 rev/s (1000 cSt) 5 v% of 1% Triton X-100, 0.154 rev/s (12,500 cSt)

13.94

2.43 × 10−3

6.86 (0.35)

0.97

16.56

2.85 × 10−3

6.92 (0.37)

0.85

19.15

1.59 × 10−2

7.59 (0.60)

0.90

13.17

3.51 × 10−3

7.11 (0.44)

0.81

21.79

9.32 × 10−4

7.03 (0.41)

0.93

42.80

4.27 × 10−3

7.01 (0.40)

0.98

39.73

4.06 × 10−3

7.31 (0.50)

0.93

4.20

7.24 × 10−3

7.13 (0.44)

0.58

10.76

1.74 × 10−3

7.08 (0.43)

0.87

Table 2 Initial and final droplet sizes for droplet migration experiments with 3.2 v% surfactant solution dispersed in 12,500 cSt PDMS

Fig. 9. (a) Plots showing extent of migration for the emulsion system, 3.2 v% of 1% Tween 60 solution dispersed in 12,500 cSt PDMS, rotated at three different rates of rotation. The rotation rates are 0.154, 0.31 and 0.62 rev/s. (b) Plots showing extent of migration for the emulsion system, 3.2 v% of 1% Triton X-100 solution dispersed in 12,500 cSt PDMS, rotated at 0.074 and 0.31 rev/s, respectively.

dependent of the rotation speed of the Couette. The centre of migration, rc , is dependent on the droplet size which can be confirmed by modelling Eq. (2) for the migration of individual droplets of different radii (see Table 3). It is found that the time

System

Initial droplet size (µm)

Final droplet size (µm)

1% Tween 60, 0.154 rev/s 1% Tween 60, 0.31 rev/s 1% Tween 60, 0.62 rev/s 1% Triton X-100, 0.074 rev/s 1% Triton X-100, 0.154 rev/s 1% Triton X-100, 0.31 rev/s 1% Triton X-100, 0.62 rev/s 1% Triton X-100, 0.31 rev/s (1000 cSt) 10 v% of 1% Triton X-100, 0.154 rev/s (12,500 cSt) 5 v% of 1% Triton X-100, 0.154 rev/s

21 21 22 14 16 18 18 16

22 27 84.5 17 18 22 22 23

27

31

16

16

taken for a single droplet to migrate from s0 = 0.7 to 80% of the distance to the equilibrium position is a very strong function of droplet radius, as would be expected from the droplet radius being raised to the fourth power (Eq. (2)). Modelling shows that in narrow gap Couette cells the equilibrium position is less dependent on droplet radius and closer to the centreline (s = 0.5) than in wider gaps. This is due to the diminished influence of the second term of Eq. (2) (which is a strong function of curvature) in narrow gap Couettes [18]. An increase in rc with rotational speed therefore provides evidence for coalescence. For those experiments performed with Triton X-100 as the surfactant in 12,500 cSt, there is a slow coalescence process that leads to an increase of 2–3 µm in each case (Table 2): there is no clear correlation between coalescence and rotation speed. For the migration experiments performed with Tween 60 as the surfactant, it can be seen that the degree of coalescence is related to the rotational speed of the Couette. The migration experiment

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K.G. Hollingsworth, M.L. Johns / Journal of Colloid and Interface Science 296 (2006) 700–709

Table 3 Calculated equilibrium position and time taken to traverse 80% of distance from s0 = 0.7 to equilibrium position for a droplet of 1% Triton X-100/24% ethanol in 12,500 cSt PDMS (the Couette is rotated at 0.62 rev/s) Droplet radius

Equilibrium position, s∞

Time taken

1 mm 200 µm 100 µm 20 µm

0.49 0.44 0.38 0.18

0.05 s 33 s 8.5 min 37 h

at 0.154 rev/s shows a 1 µm increase during the experiment whereas the migration experiment at 0.62 rev/s changes from 22 to 84.5 µm. This corresponds to Fig. 6c where extremely large droplets are visible, far larger than can be detected by PFG-NMR. This significant size increase may well account for its greatly enhanced rate of migration (k = 1.59 × 10−2 s−1 ). The other experiments reported in Table 2 indicate that the emulsions remain reasonably stable over the time of the experiments. With reference to Table 1, the effects of different physical parameters (rotational speed, continuous phase viscosity and droplet phase concentration) are now considered. Effect of rotational speed Examining the experiments with 1% Tween 60 dispersed in 12,500 cSt PDMS it can be seen that both the rate of migration and the equilibrium extent of migration increase as the rotation velocity increases. This is also observed for the equilibrium extent of migration for 1% Triton X-100 dispersed in 12,500 cSt PDMS for rotational speeds between 0.074 and 0.31 rev/s with no further increase at 0.62 rev/s. (The equilibrium extent of migration is more complex as this is determined by the balance between the forces tending to make the droplets migrate inwards and the interactions of the emulsion droplets which tend to disperse the droplets. These results suggest that the equilibrium is dependent on the speed of the rotating Couette.) Effect of continuous phase viscosity Comparing the experiments for 3.2 v% of 1% Triton X100 dispersed in 12,500 and 1000 cSt silicone oil rotated at 0.31 rev/s shows that the equilibrium extent of migration is greater for the 12,500 cSt by a factor of 10. Other things being equal (interfacial tension, droplet size), this is as expected according to Eq. (2) as the force that gives rise to the droplet migration is greater for emulsions with higher viscosity continuous phases. Effect of droplet phase concentration According to the theory set out by King and Leighton [10], increasing the discrete phase concentration should slow the rate of migration and decrease the extent of migration due to an increase in the number of droplet–droplet interactions, which tend to spread the droplets out. Further experiments were thus performed using a dispersion of 1% Triton X-100 in 12,500 cSt PDMS at dispersed phase concentrations of 1, 5 and 10 v%. In the case of the 1 v% dispersion, it was found that the signal

to noise in the measurement was too poor for quantitative results. The 5 v% dispersion was rotated at a rate of 0.154 rev/s. Comparison with the 3.2 v% dispersion at the same rotational speed (Table 1) shows that the equilibrium extent of migration is less (10.76 for the 5 v% case compared with 21.79 for the 3.2 v% case) although the rate of migration is nearly the same (k = 1.74 × 10−3 for the 5 v% case and k = 9.32 × 10−4 for the 3.2 v% case). This rate for the 3.2 v% emulsion at 0.154 rev/s is less than for any of the other 3.2 v% emulsions, however. For the 10 v% dispersion, no significant migration was noted at 0.154 rev/s and higher rotation rates resulted in gross coalescence of the emulsion. 4.5. Comparison with the theory of Chan and Leal If the measured droplet sizes and interfacial tensions are compared with the migration rates that would be predicted from Eq. (2) for a single drop, it is found that the migration rates measured here are far in excess of those predicted for a single droplet. Table 3 gives the equilibrium positions and the times taken to traverse 80% of the distance from s0 = 0.7 to the equilibrium position for droplets of 1% Triton X-100/24% ethanol of different radii being rotated at 0.62 rev/s. This measure is used as strictly it takes an infinite amount of time to actually reach the equilibrium position. The results demonstrate the sensitivity of the migration rate to droplet size. A 200 µm droplet only requires 33 s to migrate significantly whereas a 100 µm droplet requires 8.5 min. The discrepancy between the experimentally measured droplet sizes and those implied by the migration behaviour shows that the migration behaviour of semiconcentrated emulsions is not adequately described by Eq. (2): indeed the authors are not aware of a theoretical or modelling study for such an effect. It is clear that the theoretical framework for analysing this system is not presently available, but that measurements of emulsion droplet migration by magnetic resonance are able to provide detailed information about the migration behaviour of industrially important semi-concentrated emulsion systems. 5. Conclusion This paper has demonstrated how MRI methods can be used to track the migration of emulsion droplets in Couette and Poiseuille flows, both for single large droplets and for emulsion systems. Whilst we have demonstrated trends in the observed rate and degree of migration, it is clear that the existing theoretical framework for droplet migration cannot adequately describe the behaviour of semi-concentrated emulsions and non-ideal geometries (e.g., wide gap Couette cells), which are likely to be of practical interest in emulsion processing: by contrast, the theories for single droplets migrating in narrow gaps have been confirmed by optical methods. This work thus provides a basis for further studies of migration within semi-concentrated emulsions.

K.G. Hollingsworth, M.L. Johns / Journal of Colloid and Interface Science 296 (2006) 700–709

Acknowledgment The EPSRC are thanked for the provision of a Quota Studentship award for Kieren Hollingsworth. Appendix A. Nomenclature a A B Ca d D D0 Dc I (r) k L M0 M1 r rc R0 R1 R2 s s0 s∞ t t∗ T Vmax Z(s) Zl

radius of emulsion droplet equilibrium extent of migration breadth of a deformed emulsion droplet, radius of tube capillary number width of Couette gap deformation of emulsion droplet, radial distance of droplet from tube centre in Poiseuille flow initial radial distance of droplet from tube centre in Poiseuille flow diffusion coefficient due to droplet–droplet interactions radial intensity function exponential rate constant length of a deformed droplet, axial displacement along tube in Poiseuille flow zeroth moment first moment radial distance centre of migration radial distance from centre of Couette to centre of droplet radius of inner rotor radius of outer stator dimensionless droplet position initial dimensionless droplet position equilibrium droplet position time reduced time time maximum velocity in Poiseuille flow migration velocity function migration velocity function for the narrow gap Couette

Greek symbols α γ˙ δ 1 inc

migration parameter shear rate pulsed field gradient duration diffusion time of first difftrain point diffusion time between difftrain points

ηc κ λ σ φ φ0 χ Ω

709

viscosity of the continuous phase ratio of droplet viscosity to the continuous phase viscosity dimensionless diffusivity due to droplet–droplet interactions interfacial tension discrete phase volume fraction average discrete phase fraction measure of Couette shear curvature angular velocity of Couette

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