Individual and collective behavior of emulsion droplets undergoing

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Individual and collective behavior of emulsion droplets undergoing Ostwald ripening Gieberth Rodriguez-Lopez, Yhan O'Neil Williams, and Jhoan Toro-Mendoza Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03959 • Publication Date (Web): 07 Mar 2019 Downloaded from http://pubs.acs.org on March 8, 2019

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Individual and collective behavior of emulsion droplets undergoing Ostwald ripening

Gieberth Rodriguez-Lopez, Yhan O'Neil Williams, and Jhoan Toro-Mendoza

Instituto Venezolano de Investigaciones Cienticas (IVIC), Centro de Estudios Interdisciplinarios de la Fisica. Caracas, 1020 A. Venezuela. E-mail: [email protected]

Abstract Ostwald ripening OR is the dominating phase separation mechanism in nanoemulsions consisting of the mass exchange between separated droplets by solubilization and absorption of molecules. Here, we propose a model based on a stochastic equation for the mass exchange coupled to a Brownian dynamics algorithm. Our model accounts for the simultaneous gain and loss of mass within a medium where the presence of sources and sinks leads to a complex distribution of dissolved oil molecules. Also, a criterion for possible nucleation zones based on the denition of a saturation area around the droplets is found. The predictions of the collective behavior are constructed on the individual contributions of each droplet with its own environment. Individual droplets undergoing molecular exchange exhibited anomalous diusion, while when performing the collective analysis such behavior was disguised. We used reported experiments under diverse conditions to validate and test the scope of our model including the modication to the interfacial tension via Gibbs elasticity, nding close agreement. Our results 1 ACS Paragon Plus Environment

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imply that saturation is not conditional for the occurrence of OR. The ability of this model to extend the limitations imposed by traditional treatments to a broader number of physicochemical conditions make it a useful complementary tool for predicting and understanding experimental results of emulsions experiencing OR.

Introduction and background As soon as formed, emulsions composed of dispersed oil droplets in water start the process of phase separation. The focus of attention has been widely paid to the separation mechanisms governed by collisions and gravity eects. Yet, another mechanism is that of molecular exchange between droplets of dierent sizes mediated by the continuous phase known as Ostwald ripening OR. 15 Traditionally OR is assumed to occur when the smaller droplets render their surface molecules to the continuous medium nally reaching the larger droplets which grow instead. The advent of low energy methods to obtain nanoemulsions has moved the frontier of colloidal applications to template materials, 68 drug carrier, 9 Pickering dispersions, 1012 pesticide delivery, 13,14 foods, 15 among others. Since the smaller the droplet the higher the solubilization, nanoemulsions are prone to phase separate via OR. While important advances in the understanding of how this process aects the separation kinetics, 16 size distribution, 17,18 the presence of micelles as oil transporters, 1921 as well as the role of the interfacial properties, 2228 research on the behavior of the individual droplets while dissolving or growing is scarce. In particular, since the Brownian motion characteristic of nanoscaled particles immersed in liquids explicitly depends on both the size and the specic surrounding medium conditions, it is expected that droplets undergoing OR will deviate from the normal behavior observed in particles with constant size and at homogeneous conditions. The experimental characterization of the emulsions is, by nature, statistical 2 ACS Paragon Plus Environment

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owing to the large number of constitutive droplets typically in the range of 1016 − 1020 droplets/m3 . Therefore, a description of the OR process from the sum of the individual behaviors of each droplet would clarify the comprehension of the underlying complexities of the overall process. The recent perspective studies of active Brownian particles undergoing preferred directional motion due to molecular exchange 29,30 as well as scenarios of possible reverse aging processes 31 increasingly extend the interest in the precise consideration of the dynamic aspects governing OR as a self propelling mechanism or even as motility of living systems. 32,33 For all the above, it is necessary to address the aforementioned challenges with the least number of assumptions as possible. Here, we propose a simple model based on a stochastic equation for the mass exchange of dissolved oil molecules coupled to a Brownian dynamics BD algorithm that allows us to follow the evolution of the individual droplet size, in addition to a detailed description of the droplet's motion. In this sense, the proposed model accounts for the aforementioned complexity of the OR process which arises from the fact that a moving droplet can simultaneously gain and lose mass in a medium where the presence of sources and sinks of oil molecules leads to a non homogeneous distribution of dissolved molecules. This approach evades common assumptions made in the theoretic descriptions of OR such as highly diluted, non interacting, and xed particle systems which permits the prediction of the collective behavior constructed on the individual contributions of each droplet with its own environment. On one hand, the description of the kinetics of the OR process was developed by LisfshitzSlyozov-Wagner LSW. 34,35 This model assumes that the Kelvin eect is the sole responsible for the solubilization and growth of droplets. The Kelvin eect establishes that a droplet of radius a possess a solubility given by ca = c∞ eα/a , c∞ is the solubility of the disperse phase

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when the surface is at also known as bulk solubility, α(= 2γvm /RT ) the capillary length, γ interfacial tension, vm molar volume of the dispersed phase, R the universal constant of the gases, and T the absolute temperature. 36 One of the most used results of the LSW theory is the expression for the OR rate, which reads

da3 8 c∞ γvm Dm w= = , dt 9 ρRT

(1)

where ρ is the density of the dispersed phase, and Dm is the dispersed phase molecular diusion coecient. Notwithstanding, the LSW theory has validity only in a stationary state of a surface viscous regime being this experimentally corroborated for dierent physicochemical conditions such as a low volume fraction and the absence of surfactant. 3739 More recent developments suggest that OR rates follows an Arrhenius-type model which weights the possible eects of temperature on the activation energy necessary to start the process. 40 It has been also raised that a linear behavior of the droplet size evolution is the ngerprint of the occurrence of OR 1,41 although other separation mechanisms such as occulation and coalescence could exhibit a similar evolution curve. 42 On the other hand, Brownian motion is characterized by the mean square displacement MSD given by hr(t)2 i = 6Dt, where D =

kB T 6πηa

is the diusion coecient of the particle, kB

the Boltzman constant, and η the viscosity of the medium. This linear dependence with time is a characteristic of Brownian particles suspended in a viscous medium. 4347 Remarkably, there are systems in which the particles suspended in complex uids, biological systems, among others, exhibit a deviation from the linear behavior known as anomalous diusion and their MSD follows a power law of the form hr(t)2 i ' Dβ tβ , where Dβ is the generalized diusion coecient and β the anomalous diusion exponent. If β = 1 we have a normal diusion, β > 1 super-diusion and β < 1 sub-diusion. 48 Here, we focus on the individual 4 ACS Paragon Plus Environment

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behavior of droplets which either dissolve or grow, paying particular attention on the MSD and the viscoelastic modulus. Also, a criteria based on the oil-in-water solubility to identify possible nucleation zones is established. Besides the solubilization of the dispersed phase in the medium, 49,50 the presence of surfactant modies the viscoelastic properties of the interface playing a major role on the occurrence of OR. 51 In some cases, the occurrence of OR decreases as observed in experiments and predicted by numerical results. 26,5255 In other cases, counterintuitively, experimental studies show that when reducing interfacial tension, the OR process increases instead of decreasing. 5659 In this sense, we use experiments reported in the literature for diverse systems to validate and test the scope of our proposed model by including the modication to the interfacial tension via Gibbs elasticity. Through this analysis we are able to calculate the rate of OR, the time evolution of the size distribution function, and to dene a solubilization

radius from the use of a xed value for the concentration of oil molecules in the bulk medium. This latter quantity is closely related to the saturation concentration of each oil used.

The model for evolving mass The process of solubilization is studied by Fick's rst and second law. A change in the volume of the droplet occurs due to the variations in the number of molecules n which are dependent on the ow of these through a given area written as cross sectional area through which the molecules move. law as

J

J

dn dt

= JdA, where dA is the

is the ow dened by Fick's rst

= −Dm ∇c(r, t). By combining the last two equations we obtain the number of

molecules dissolved by the Kelvin eect nK ,

dnK ∂c(r, t) = −4πa2 Dm . dt ∂r 5 ACS Paragon Plus Environment

(2)

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Considering the solubilization process to be radial, the form of c(r, t) can be obtained through Fick's second law as

∂c(r, t) 1 ∂ = Dm 2 ∂t r ∂r

r

2 ∂c(r, t)

∂r

.

(3)

For a stationary state, the following boundary conditions apply c(a) = ca and c(as ) = cm . The last condition represents the concentration of oil at the volume dened by the minimum radius of a sphere as in which a droplet would completely dissolve (solubilization radius). √ This can be obtained from c∞ using as = a/ 3 c∞ vm , where vm is the molar volume of the oil molecules. Typical values of as for oil droplets with a radius 100 nm are in the order of p ha2s i, microns. In addition, the time tm as for a molecule to be released by a droplet to reach 2 and the displacement acquired at this time by the droplet tm as = has i /6Dm allows us to nd

ra2s Ddroplet = . 2 has i Dm

(4)

To illustrate this point, the values of ra2s / ha2s i for hexane and octane droplets are 3 × 10−3 and 4 × 10−3 for a 100 nm droplet, respectively. The overlap between two or more regions dened by as can be established as a criteria of nucleation, since the chemical potential of the oil molecules in that region is lower than in the bulk of the continuous phase which is energetically favorable for nucleation. Figure 1 is a schematic representation of the overlapping zones where nucleation is more likely to occur. Finally, by solving Eq. (3), the concentration of oil molecules around the droplet is

c(r) =

1 [aca (as − r) + as cm (r − a)] . r(as − a)


(5)

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Higher density of oil molecules. as

Figure 1: Higher density zones of oil molecules. Areas marked in yellow represent overlapping zones created by the denition of as . Bold yellow regions indicate a high density of oil molecules highlighted by the arrows. Dierentiating Eq. (5) with respect to the coordinate r, evaluating in r = a, inserting in Eq. (2) and taking θ = cm /c∞ , we obtain

α i aas h dnK = 4πDm c∞ θ− 1+ . dt (as − a) a

(6)

Equation (6) represents the contribution of the Kelvin eect to the change of droplet size. The second contribution is the capacity of each droplet to absorb dissolved oil molecules which and can be quantied as follows. For a droplet dispersed in a medium with a local concentration of oil molecules dened as N/V ζ , and a velocity v , where N is the number of oil molecules, V the total volume, and ζ the distribution that governs the form of the local concentration of dissolved oil molecules; we can estimate the number of collisions per unit R∞ area with a wall as N/V ζdt 0 vdv by making use of the kinetic theory of gases, and by assuming that the total number of molecules that collide with the droplet absorb eectively. 7 ACS Paragon Plus Environment

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The applicability of this approach to liquids is an old problem although in our case, it has been demonstrated that, at a surface, the collisions of a second molecular specie with a low number density follows the gas-kinetic expression. 60 Besides, at the interface, the decrease in the molecular velocity of the oil molecules due to the collisions with the surrounding water is compensated by the attraction of the oil molecules due to their proximity to the droplet. Dening nm = N/V , the number of absorbed molecules nA per unit time is

dnA = nm hvi πa2 ζ. dt hvi is calculated as

p

(7)

2kB T /m, where m is the molecular mass. Having the explicit forms

of the contributions of both solubilization, and the capacity to absorb molecules we propose the following equation by summing Eqs. (6) and (7), dn aas h α i = 4πDm c∞ θ− 1+ + nm hvi πa2 ζ. dt (as − a) a

(8)

By using the relation n = 4πa3 /3vm , Eq. (8) describes dierent scenarios for both spatial and time-dependent distribution of oil molecules around the droplet. In this work, a uniform distribution for ζ is assumed. nm is the molecularly dissolved molecules which is related to the saturation value of the oil in water, where two scenarios can be dened. First, for nm = 0 all the droplets would solubilize. In fact, the constant migration of molecules among the droplets throughout the continuous phase means nm 6= 0. Secondly, at high values of nm most droplets would grow. However, this last scenario would require an almost unlimited source of oil molecules. Hence, a droplet may exchange matter as follows: it solubilizes and moves through zones of a higher molecular concentration where these zones act as a source of molecules which in turn counteract the Kelvin eect, leading nally to an evolution of the droplet size.This latter will strongly depend on the solubility which 8 ACS Paragon Plus Environment

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is governed by the interfacial tension and droplet size, and diversely, the concentration of dissolved oil molecules which in turn is aected by their molecular diusion and the diusion of the droplets. Therefore, in our calculations nm is an adjustable parameter with a denite physical signicance, since it allows for the examination of the saturation conditions of the system.

Computational details Single droplet calculations The discretized form of Eq. (8) is written as follows

ni+1

ai as,i αi ˜ = ni + 4πDm c∞ θ− 1+ ∆t + nm hvi πa2i ζ(t), (as,i − ai ) ai

(9)

where the sub-index i indicates the previous value of the time step ∆t (t0 < t1 < t2 · · · ti ),

˜ = ζ(t)∆t (= ζ(i+1)−ζ(i)), and n0 the initial condition. The last equation ∆t = ti+1 −ti , ζ(t) was solved numerically through the Euler-Maruyama method assuming the random variable follows a uniform distribution. In addition, the Langevin equation, given by

m¨ x = −6πηax˙ + ξ,

(10)

is in turn solved by the BD method. In the high friction limit, we can write that

xi+1 = xi + ξ,


(11)

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where xi is the position of the particle, ξ the Brownian driving force with a mean of zero

(hξi = 0) and the variance σ 2 = 2D∆t(=

kB T ∆t). 3πηak

From the last equation we can obtain

the MSD for a droplet,

MSD = ∆x2 = (xi+1 − xi )2 .

(12)

The β exponent is estimated as follows,

β=

d 2 ∆x . d log t

(13)

A deviation from the Fickian behavior of Brownian motion can be characterized by β and by the elastic and viscous moduli. The latter can be quantied through the generalized Stokes-Einstein relationship and the Laplace transformation, which leads us to

e G(s) =

kB T , πas h∆x2 (s)i

(14)

where ω = 1/t. From the MSD, the complex viscoelastic G∗ module is calculated from

G0 (ω) = |G∗ (ω)| cos

πω 2

(15)

and 00

∗

G (ω) = |G (ω)| sin

πω 2

,

(16)

where |G∗ (ω)| is given by,

|G∗ (ω)| ≈

kB T . πa h∆x2 (1/ω)i Γ (1 + β)

(17)

The simulations make use of the following parameters corresponding to a hexane droplet 10 ACS Paragon Plus Environment

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in water obtained from Sakai et al.: 61 T = 298 K, η = 0.89 × 10−3 Pa.s, c∞ = 1.10 × 10−1 mol/m3 , vm = 1.311× 10−3 m3 /mol, γ = 50.3 mN/m, Dm = 8.5×10−10 m2 /s. Three cases of

nm were evaluated, 0.0 mol/m3 , 4.4 × 10−3 mol/m3 and 3.7 × 10−3 mol/m3 , for an oil droplet with radius a0 = 50 nm, θ = 0.0, using a time step ∆t = 1.0×10−5 s.

N

droplets calculations

A total of 1000 droplets form the N -droplets system using a certain radius distribution. The behavior of each droplet is coupled to the solution of Eqs.(8) and (10).During the numerical evaluation the average radius of the droplets is estimated during every time interval. In addition, the respective histograms are used in order to obtain the time evolution of the droplet size of the system. Also, the MSD of all the droplets is obtained through the Eq. (11) at each time step. From the evolution of the MSD, and the Stokes-Einstein relation, the coecient of collective diusion of the emulsion is obtained. Moreover, the average inuence of hydrodynamic interactions on the diusion of the droplets is quantied in BD simulations through local volume φ as follows, 62

Dφ =

D 1+

2b2 1−b

−

c 1+2c

−

bc(2+2) (1+c)(1−b+c)

,

(18)

where b = (9/8φ)2 and c = 11/16φ. We predict the OR kinetics of octane, hexane, dichoroethane, and benzene. The corresponding values of c∞ of these oils were used: 5.80 × 10−3 mol/m3 , 1.10 × 10−1 mol/m3 , 81.00 mol/m3 and 25.87 mol/m3 , respectively. vm = 1.63×10−4 m3 /mol, 1.31× 10−3 m3 /mol,

79.92 × 10−6 m3 /mol and 88.9 × 10−6 m3 /mol, γ = 51.7 mN/m, 50.3 mN/m, 38.8 mN/m y 35.0 mN/m and Dm = 7.8 × 10−10 m2 /s, 8.5 × 10−10 m2 /s, 1.0 × 10−9 m2 /s and 1.0 × 10−9


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m2 /s.

Results and discussion Surface molecular exchange in a single droplet In this section the simultaneous process of mass gain and loss of a a = 50 nm droplet is evaluated through the analysis of MSD and the viscoelastic properties. The viscoelastic properties referred to here are not based on the quantication of the classical internal dissipative mechanisms, but rather, the deviation from the normal Fickean diusive process. Three cases were studied: (1) a dissolving droplet under the action of the Kelvin eect, (2) absorption of molecules prevails against solubilization, and (3) both Kelvin solubilization and molecular absorption are equilibrated. For the rst case, the absence of oil molecules in the medium is represented by nm = 0 mol/m3 . It can be observed that the droplet dissolves completely at 8 ms as can be seen in Fig. 2a. Also, the MSD shows a deviation from the linear time dependence, and a value of β >1 was found which is indicative of a super-diusive regime (Figs. 3a and 3d). In the Levy ight, super-diusion is caused by the length of the heavy tailed probability distribution. 63 The length of the individual steps are distributed by the following probability density function |x|−1−ν , where 0 < ν < 2. 64 Notwithstanding, the super-diusive behavior observed in this work is predicated on the solubilization of the droplet in the surrounding medium. Solubility governs the time scale of dissolution leading to a simultaneous diminution in droplet size. Thus, a more soluble droplet would present a smaller super-diusive time scale. Besides, the progressive reduction in size due to dissolution of the droplet increases its diusion due to a rise in ∆x. Experimental observations of this phenomenon have not been reported using dynamic light dispersion equipment probably 12 ACS Paragon Plus Environment

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owing to the smaller time intervals used for measurement compared to those observed for the super-diusive process, and the use of the Stoke-Einstein relation. In the second case, the presence of an important amount of oil molecules in the medium is expressed by nm = 4.48×10−6 mol/m3 . The droplet increases its radius up to six times compared to its initial value after 1 s (Fig. 2b). Given the observed trends of the MSD curve (Fig.3b), we found that β < 1 (Fig. 3e) which is indicative of a sub-diusive process. In this case, the droplet loses its ability to diuse in the medium, and in addition, the droplet trajectory is progressively more compact, leading to space connement of the droplet possibly due to the continuous change in the inuence of the surrounding medium on the droplet. For instance, most sub-diusive processes are observed in systems subjected to steric connement, macromolecular crowding, and the presence of a viscoelastic medium and particles or both. The main dierence observed in this study compared to the literature consists in the absence of a visible plateau in the MSD curve. With that in mind, it is also expected that the viscoelastic modulus will not cross their curves, typically observed in solid-like behavior, where viscous and elastic properties intertwine at some point. In the last case, a value of nm = 3.7×10−6 mol/m3 is sucient to equilibrate the loss of mass due to solubilization. It was observed that the droplets exhibit two behaviors which only depend on the dierent seeds used for each run: the droplet may undergo solubilization, or it may increase its size (Fig. 2c). The analysis of the MSD is possible through its classic form hx2 (t)i = 2Dt, taking the value of diusion of 4.7 µm2 /s which can be considered close to the 4.9 µm2 /s of the particle of constant radius and β(t) ≈ 1 (Fig. 3c and 3f). Notwithstanding, the analysis of the MSD may lead to the loss of information in each run because the mechanisms of mass loss and gain compensate one another. As a consequence, an analysis based on only the MSD curve may lead to an erroneous conclusion over the


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evolution of the individual droplets, for droplets with similar sizes could follow dierent pathways. 2.0

8 1.0

a(t)/a0

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(a)

(b) 1.2

4

0.6

2

0.4

0.2 0.0

(c)

6

0.04

0.08 0.0

0.4

0.8

0.0

0.4

0.8

Time (s) Figure 2: The evolution of the normalized droplet radius for a0 = 50 nm with nm : (a) 0, (b) 4.48×10−6 and (c) 3.7×10−6 mol/m3 . Continuous lines represent the evolution of the droplet size, while the yellow spheres indicate the dominating mechanisms contributing to droplet growth, or loss in mass. In our calculation, each run start with a dierent random seed that generates a new Brownian path. For the curves (a) and (b) all the runs ended with a dissolving and growing droplet, respectively. Instead, in (c) it can be observed that the result is strongly aected by the value of the initial random seed, thus generating dierent nal outcomes.

Evolution of O/W emulsions undergoing Ostwald ripening It was already mentioned that the presence of surfactant modies the viscoelastic response of the droplet surface. Hence, a viscous regime can be dened when the interfacial tension is constant as the surface area of the droplet increase, meanwhile an elastic regime corresponds to the case when the interfacial tension increases as the surface area enlarges. These regimes are dependent on the ability of the surfactant to locate at the newly created surface. It is worth mentioning that these viscoelastic regimes are dierent to those referred to via Eqs. (14-17) which correspond to the deviation of the Fickian behavior of the Brownian motion. 14 ACS Paragon Plus Environment

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1.2

4 (b) 3

0.8

2

0.4

1

0 3 2 1 0

0 0.8 0.4 0.0 0.02 0.04 0.06 0.08 0.0 0.2 0.4 0.6 0.8

b

MSD (mm2)

(a)

0

9

(c)

6 3 0 1.2 1.0 0.8 0.6 0.0 0.2 0.4 0.6 0.8 1.0

Time (s) G'G'' (N/m2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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102 (d) 101 100 10-1 10-2 10-3

102

(e)

G''

(f)

G'

103

G''

G''

G'

G'

104 105

101 102 103 104 105

102

103

104

105

-1

w (s )

Figure 3: The evolution of the mean square displacement (MSD) according to Eq. (12), exponent β according to Eq. (13), and the viscoelastic modulus of hexane in water systems considering: (a) and (d) only the Kelvin eect, (b) and (e) absorption of oil molecules is the dominant process, and (c) and (f) equilibrium between the Kelvin eect and the absorption of oil molecules, as referred to in Fig 2. G0 and G00 represent the elastic and viscous component of the viscoelastic modulus, respectively.


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The results of this section are divided into the viscous and elastic surface regimes owing to the presence of surfactant.

Surface viscous regime We employ our model to estimate both the kinetics and MSD of some emulsions composed of

N droplets following the procedure explained in the computational details section. The rst four emulsions studied were free of surfactant, and were in a viscous regime with respect to the γ . First we study 1,2-dichloroethane-in-water emulsions under the same physicochemical conditions reported by Kabalnov et al. 65 For that, a normal droplet size distribution with a mean radius of 4.49 µm, 17% polydispersity, and nm = 2.46×10−7 mol/m3 is used. Remarkably, the OR rate predicted by our model was 18.1×10−21 m3 /s which is the same value reported by Kabalnov et al. 65 As a comparison, the value predicted by the LSW theory is 6.3×10−21 m3 /s which represents an approximate error of 65% compared to the experimental report. The value of the relation nm /c∞ = 3.06 × 10−9 highlights that the medium is not saturated in this system. Figure 4 shows the evolution of the droplet size distribution of the 1,2 dichloroethane inwater emulsion expressed through the number of droplets as a function of the mean droplet size. The evolution of the mean droplet sizes is represented by white circles. At the initial stages of the OR, an unimodal distribution adequately describes the droplet size disposition. In time, we observe that the distribution shifts not only to the right, but also, there is a progressive change in its form. For instance, intriguingly, after 1620 s a bimodal distribution can be noticed. Nonetheless, an unimodal distribution is restored at the end of the evaluation period. A transition to the right of the mean droplet size indicates that a vast part of the population is concentrated by larger droplets, while on the contrary the smaller droplets 16 ACS Paragon Plus Environment

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reduce in number. We also studied the kinetics of the following oil/water systems octane, hexane, and benzene were the oils used. The tting values estimated for nm were 6.13×10−6 mol/m3 , 6.69×10−6 mol/m3 and 2.04×10−9 mol/m3 , respectively. The corresponding values for

nm /c∞ were: 1.0 ×10−3 , 6.08 × 10−5 , and 7.87 × 10−11 , respectively. The values obtained from the relation nm /c∞ indicate that saturation is not required for the manifestation of OR in contrast to the hypothesis used for LSW theory where supersaturation is responsible for the instantaneous transport of the dissolved molecules. In the table I, we compare the OR rates predicted by our model to the widely accepted LSW theory and the reported experimental ndings. Noticeably, the predictions by this present model are quite precise for these systems. Table 1: Values of calculated OR rate w, compared to experiments, and the LSW theory with units m3 /s (×10−21 ). The experimental values used for comparison were obtained from the references indicated in the oil name. Oil Octane 61 Hexane 61 Benzene 65 1,2-diclororethane 65

Model 1.94 2.64 3.28 18.10

Experiment 1.90 2.80 3.10 18.10

LSW 0.01 0.03 2.20 6.30

Surface elastic regime Based on the precise estimations of the OR rates without surfactant, we test the scope of the model to describe the kinetics of an emulsion in the elastic regime. For that, we study the evolution of the droplet size, and the MSD for the sum of all the droplets of a n-dodecyl-b-maltoside (C12 G2 )/octane/water emulsion as reported by Georgieva et al. For


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this calculation, surface dilatational elasticity EG was estimated by

γ = γ0 + 2EG ln

a(t) , a0

(19)

which is included in Eq. (7) through α(= 2γvm /RT ). Surfactants besides reducing the value of interfacial tension, also confer additional degrees of freedom to the droplets which may enable the dissipation of the thermal uctuations from the surrounding medium. However, this form of dissipation of energy is not accounted for by EG . 1000 droplets form the initial droplet population which obey a log-normal distribution where the value of the average droplet radius used was a ¯ = 1.1 × 10−6 µm. In Fig. 5 the kinetics and droplet size distribution of the n-dodecyl-b-maltoside (C12 G2 )/octane/water emulsion is presented. A linear growth of the cubic radius in time is observed in Fig. 6a. An excellent agreement between the experimental ndings and the simulations based on the model presented in this work is evident throughout the evaluation period. The dotted lines represent a linear t of the simulations. The OR rate estimated using our model was 7.5×10−23 m3 /s while the experimental value reported is 7.8 ×10−23 m3 /s (see Fig. 5a), an error of only 4%. The value of nm = 1.87 × 10−9 mol/m3 , therefore nm /c∞ = 3.23 × 10−7 which means that the number of molecules near the droplet in the local medium is a lot smaller than the saturation point. The dierence between the value of nm /c∞ here compared to the system in absence of surfactant mentioned above, is due to the elasticity of the interface. Elasticity may decrease the dissolution of the droplets in the medium, and thereby the number of molecules near the droplet as dictated by Eq. (10). In this sense, for a less soluble dispersed phase, a smaller quantity of available molecules in the bulk is expected to be required in order to allow the molecular exchange between the smaller and larger sized droplets. Therefore, in that case,

nm is smaller than the saturation concentration. 18 ACS Paragon Plus Environment

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In Fig. 5b, the droplet size distribution is presented. The open white circles represent the mean droplet size. We observe that the distribution evolves from a unimodal log-normal distribution to a bimodal distribution which has been reported as a characteristic behavior of emulsions undergoing OR. 42 Moreover, intriguingly, the bimodal distribution is maintained all through the time lapse. In fact, the fraction of smaller droplet remains narrow but the fraction of larger droplet increases its size distribution. The latter, suggests that the smaller droplets which are continuously replaced by the medium sized drops which shrink for the growth of the larger droplets. An important parameter in the characterization of OR is that of critical radius ac . This is dened as the value at which sizes of droplets below it tend to decrease, meanwhile sizes above it tend to grow. According to Finsy, 66 ac is equivalent to the average size. In his approach, the critical radius is obtained by using dnK /dt = 4πDm c∞ (a/ac − 1)α. Remarkably, within the framework here proposed, the denition of a critical radius will depend both on the distribution and concentration of oil molecules in the medium nm as expressed in Eq.(8), representing a non-trivial problem for the stochastic nature of the model. This problem is even more challenging in the non stationary state that, according to Eq. (6), the parameter

as is an explicit function of time. This problem will be treated elsewhere. Finally, in Fig. 6 the evolution of MSD and apparent average radius of a C12 G2 /octane/water emulsion as a function of the volume fraction of oil-in-water φ is presented. In Fig.6a we observe that an increase in φ leads to a reduction in MSD over time. Linear slopes are observed in each case. From the variation of each slope a collective diusion coecient Dc is estimated, and using the Stokes-Einstein relation an apparent average radius a ¯ is predicted. The evolution of these two parameters are presented in Fig.6b. Clearly, an increase in Dc leads to smaller values of a ¯. Interestingly, from the trends observed in Fig. 6 marked dier-


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Number of droplets (%)

4.5

Time (s) 540 1080 1620 2160 2700

30 4.7 20

4.9

5.0 5.2

10

0 2

0

4

6

8

10

a (mm)

50

Number of droplets (%)

Figure 4: Evolution of the droplet size distribution of 1,2-dichloroethane/water emulsion undergoing OR for nm = 1.87 × 10−9 mol / m3 . The numbers highlighted by the dots and arrows are representative of the average droplet radius at a given time.

(2a)3 (μm3)

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(a)

40 30 20 10

Simulation Experiment

0 0

2

4

6

8

10

12

0 1500 1600 3300 6600 8300

(b)

8 4 0

3

Time (10 min)

1

2

3

4

a (μm)

Figure 5: (a) Comparing the evolution of the mean droplet radius of a C12 G2 /octane/water emulsion from simulations done in this work to experimental reports. (b) Evolution of the droplet size distribution using our theoretic model. Again white dots represent the average droplet size at a given time.


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ences between the collective and individual behaviors of the droplets are found. Although individual droplets exhibit anomalous diusive behavior, only Fickian diusion is observed for the collective diusive processes. The latter indicates that the MSD observed for each droplet is hidden as a part of the collective mean. That is, when MSD is taken as an average of all the droplets, the MSD of droplets which solubilize, and those which grow are nullied by one another. In other words, slight deviation from the Fickian behavior is observed. (b)

4

Dc (10-6 m2/s)

φ (%) 0 5 10 15 20

3 2 1

3.6

1.05

3.2 0.90

2.8

0.75

2.4 2.0

0 0

2

4

6

8

0

10

a (μm)

(a)

MSD (10-7 m2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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5

10

15

20

0.60

φ (%)

Time (103 min)

Figure 6: (a) MSD as a function of φ of the C12 G2 /octane/water emulsion. (b) Diusion coecient and apparent mean droplet size as a function of φ for the C12 G2 /octane/water emulsion.

Conclusions The results of applying our model for the size evolution of oil-water emulsion droplets undergoing Ostwald ripening, lead us to the following nal considerations: Non-Fickian behaviors of the mean square displacement of individual droplets were observed, both super-diusive and sub-diusive regimes, depending on if droplets are dissolving or growing. This eect decreases drastically when the collective mean square displacement is analyzed. However, in order to accurately predict OR it is necessary 21 ACS Paragon Plus Environment

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to include these eects, since the collective behavior strongly depends on the tendency of the droplet population to grow or decrease in size. The value obtained for the number of oil molecules dissolved in the medium is below the saturation point in all cases. According to it, saturation is not necessary for OR occurrence. The rates of OR were in close agreement to the experimental results reported in the literature for dierent physicochemical conditions. Importantly, our procedure is able to predict the evolution of the droplet size distribution for both elastic and viscous surface regimes. The measurements of size and any other mean physical observable in experiments based on light scattering methods are able to estimate the collective behavior of OR. Therefore, our observations merit specic measures that allow quantifying the individual behavior of the droplets and its surrounding conditions. Our model allows the inclusion of dierent energy dissipation mechanisms of the droplets, such as surface deformability and internal movement of liquid. These mechanisms can be included in the equation of position directly. Besides, it is necessary to obtain a relation of these mechanisms with the supercial dilational modulus. A more detailed analysis is required to identify the eects owing to volume fraction, mainly the specic eect of both the presence of oil molecules in the continuous phase and the complexities associated with the spatial distribution of interacting droplets. Another challenging task is the application of our procedure to the non-stationary case of Fick's second law, which is a more adequate scheme to describe the molecular exchange among droplets. In addition, the resolution of the equation for the mass 22 ACS Paragon Plus Environment

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evolution presented here for dierent non-at cases of the random distribution function of oil molecules will lead to the direct estimation of the so-called critical radius. To summarize, the scope of this approach covers a wider range of conditions than those imposed by traditional assumptions such as supersaturation, xed particles in space, and mass transport only by means of molecular diusion. Also, it is possible to account for non homogeneous distribution of oil molecules in the medium, direct interactions between droplets, the eect of the volume fraction, and even the occurrence of other phase separation mechanisms such as occulation and/or coalescence. This makes the model a useful and powerful complementary tool for predicting and understanding experimental results of emulsions under OR.

Acknowledgments Thanks are given to J.L Fernandez for his help in the preparation of the gures and to K. Rahn for her encouraging comments. This research was economically supported by project IVIC-1013.

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Individual and collective behavior of emulsion droplets undergoing

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