**5. Multiple emulsions**

764 Mass Transfer - Advanced Aspects

Plane Liquid Membrane

Compartment I Compartment II

It has been demonstrated in the presented work and in the literature that the transport of pure water in W/O mixed emulsion or the transfer of pure tetradecane in O/W mixed emulsion is increased when surfactant making micelles is present as the micelles can incorporated the transferred molecules and facilitate their transport through the continuous media. To quantify the transfer, the flux J is introduced and expressed in terms of chemical

according to the following equation:

*j* =− × *L grad*

With L being the factor related to the transferred material diffusion coefficient and the transferred material concentration in the membrane. By using this model, it has been possible to express the changes of the number of material moles in the compartment I

material being transferred between two populations of droplets in direction of the decreasing chemical potential. For example, water being transferred from the pure water droplet to the water+solute droplets or tetradecane being transferred from the pure tetradecane droplets to the hexadecane droplets. A mathematical treatment and thermodynamic considerations developed (Clausse et al., 1995a) allow obtaining the

> 13 13 43 43 2 3 1 3 1 1 1 1 10 <sup>1</sup> ( 0) ( ) ( 0) ( ) 2 ( ) *t I I I <sup>I</sup> II I LD V n t n t n t n t nv V*

In this equation, V and v are the total volume and the molar volume, respectively, and Lp

water transfer within multiple emulsions have been given in the literature (Clausse et al.,

In each studied case, the water transport in W/O mixed emulsion and the tetradecane transfer in O/W mixed emulsion, experimental results are fairly well fitted by the model proposed. Although this model is not perfect, it gives the main physical parameters to be

 are the permeability coefficient and the mean diameter of the droplets, respectively. Furthermore, n1 and n2 are the moles numbers of water and solute or the moles numbers of tetradecane and hexadecane, respectively. Lp is a parameter difficult to know as it is linked to unknown parameters such as the width of the equivalent membrane, the water diffusion coefficient or the tetradecane diffusion coefficient and the water concentration or the tetradecane concentration in the membrane. Nevertheless, it is possible from this model to predict the lapse of time necessary to reach equilibrium, when all the pure water molecules has migrated from the pure water droplets to the water+solute droplets, or when the tetradecane molecules has migrated from the pure tetradecane droplets to the diluted

⎡ ⎤ =− − =− = ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ (8)

μ

<sup>G</sup> (7)

*d*

*<sup>I</sup> n t* = . Similar treatments dealing with

*<sup>I</sup> n t* versus time t. This number decreases versus time ,the

Fig. 19. Picture of mixed emulsion in the model of mass transfer

μ

hexadecane droplets. This point is reached when 1() 0

potential gradient *grad*

following relation:

1995b; Potier et al., 1992).

and *d*

containing pure material, 1( )

Multiple or double emulsions are systems in which a liquid phase is dispersed into globules which in turn are dispersed into another immiscible liquid phase (Garti & Lutz, 2004; Benichou et al., 2007). The liquid phase dispersed as microdroplets in the globules is called the internal phase whereas the liquid phase in which are dispersed the globules is called the external phase. Therefore a multiple emulsion contains three liquids phases and are classified as either Water-in-Oil-in-Water (W/O/W) emulsions or Oil-in-Water-in-Oil emulsions (O/W/O) (Pal, 2011; Muschiolik, 2007). The inner dispersed droplets are separated from the external phase by a layer of the globules phase (Garti & Lutz, 2004).

Multiple emulsions are liquid carriers for entrapped and release of active or reactive molecules in pharmaceutics, cosmetics, food and industrial applications. Nevertheless, W/O/W multiple emulsions are more studied because they have higher potential to become commercial products than O/W/O multiple emulsion (Benichou & Aserin 2007). For instance, in cosmetics the release of an encapsulated drug inside an aqueous globule of a W/O/W emulsion can be directly liberated by breaking of the globules via mechanical stress (Muguet et al., 2001; Tejado et al., 2001). In pharmaceutics, drugs can be protected by the membrane until it reaches its target and then released by controlled release (Garti & Lutz, 2004; Hai & Magdassi, 2004; Tejado et al., 2005). Controlled release can by operated via swelling of the globules (or Ostwald swelling) which consists in an increase in size of the inner dispersed droplets due to a difference of chemical potential on both sides of the membrane leading to a water mass transfer from the external phase to the internal phase (Geiger et al., 1998; Grossiord & Stambouli, 2007; Lutz et al., 2009). As a consequence, the volume of the oily globules increases and when the resistance of the membrane becomes insufficient, the globule breaks liberating the active compound.

In water waste treatments which also involve W/O/W emulsions, the toxic compound present in the external phase has to diffuse through the membrane to be entrapped in the inner droplets. In such separation processes, selectivity of the membrane is an important parameter (Kentish & Stevens, 2001; Kumbasar, 2009). Indeed when a compound has to be extracted from a solution containing a variety it is important that only the compound to be extracted diffuses through the membrane. In order to promote selectivity of the membrane, a carrier can be added into the membrane phase (Venkatesan & Meera Sheriffa Begum, 2009; Ng et al., 2010). A carrier is a molecule which can form a complex with the toxic compound at the external interface and transport it to the internal phase where the toxic compound will be entrapped (Hasan et al., 2006; Frasca et al., 2009). High selectivity means that the carrier holds a good affinity with the toxic compound in order to exclusively transport it through the membrane (Kaghazchi et al., 2006).

In separation applications, the instability of the multiple emulsions can reduce significantly the efficiency of the process. Indeed, if Ostwald swelling or coalescence of the globules occurs, such mechanisms can lead to the break of the globules. The extracted molecule is then directly released in the external phase which ruins the extraction process (Yan & Pal, 2001, 2004; Mortaheb et al., 2008).

If O/W/O multiple emulsions have been less extensively studied, they can also finds potential applications in food, cosmetics and controlled delivery drugs (Benichou et al.,

Mass Transfers Within Emulsions Studied by

Differential Scanning Calorimetry (DSC) - Application to Composition Ripening and Solid Ripening 767

three phases composing the multiple emulsion. The signal at around -18°C represents the crystallization of the aqueous phase by breakdown of undercooling. This signal is characteristic of bulk water crystallization as it has been described in section 2. Even if the aqueous phase is dispersed as a multitude of globules their size of 1 mm is not sufficiently tiny enough to involve nucleation delays like those observed in W/O emulsions droplets of a few µm3. Therefore the behavior of crystallization of aqueous globule looks like the crystallization of water bulk phase. The exothermic peak at 17ºC which initial part of the signal is sharp is attributed to the bulk hexadecane crystallization. The hexadecane crystallizes at 17ºC but with a very little degree of subcooling, the melting temperature of hexadecane being of 18°C. The third bell shape peak is attributed to the crystallization of the tetradecane dispersed droplets. The temperature is -12.8°C and is given by the apex of the

Fig. 20. Evolution of the DSC curves of an O1/W/O2 multiple emulsion containing 2 wt% of

Figure 20 also shows a clearly evolution with time of the crystallization peaks of the internal phase and the external phase. Actually, the crystallization peak of the tetradecane exhibits a

Tween 20 in the aqueous membrane

peak which means that nearly 50% of the droplets are crystallized.

2007; Mishra & Pandit, 1989; Laugel et al., 1998, 2000; Yu et al., 2003). They are found also in application in separation processes of hydrocarbons (Krishna et al., 1987; Garti & Kovacs, 1991). In controlled drug delivery applications the challenge consists in the control of the diffusion of the entrapped active molecule (Grossiord et al., 1998). The active matter can diffuse and migrate through the membrane via an osmotic pressure gradient and it can be almost difficult to retain the active molecule inside the inner phase upon prolonged storage, all the more as the emulsifier can form micelles (Benichou et al., 2007). Micelles are capable of solubilizing the active molecule and facilitate the transport of active matter from the internal interface to the external interface.

One can see that developing potential applications of multiple emulsions requires a good understanding of the mass transfer mechanisms but besides measuring kinetics of mass transfer, it is also important to take into account formulation aspects of the emulsion which have also a direct influence on the mass transfer.

W/O/W emulsions entrapping different compounds, urea or MgSO4 have been studied by using the DSC techniques presented in this chapter (Potier et al., 1992; Raynal et al., 1994 Raynal et al, 1993). As case of study we propose to describe the release of tetradecane within an O/W/O multiple emulsion studied by DSC. It is described how to detect the variations of composition of the external oil phases during time as well as the evolution of the emulsion with time. The influence of formulation parameters such as surfactant concentration and mass ratio of the internal phase on the evolution of the emulsion are also discussed.

#### **5.1 Emulsion preparation**

The tetradecane/Water/hexadecane (O1/W/O2) multiple emulsions were prepared in a two-step emulsification method (Avendaño-Gomez et al., 2005). Firstly, primary simple emulsions O1/W were prepared. Tetradecane was dispersed into an aqueous solution in which was previously dissolved Tween 20. Different concentrations of Tween 20 have been investigated, 2 wt%, 4 wt% and 7 wt%. Tetradecane was then slowly dripped into the aqueous phase under agitation using an Ultra Turrax mixer at a speed of 20,000 rpm. All the O1/W emulsions prepared contained a tetradecane/water ratio of 2/3. The primary emulsions were then sonicated during 10 min. The droplet size distribution was measured using Coulter counter technique and the mean size of tetradecane droplets for all primary emulsion was about 5 µm. In a second step, 19 g of each primary O1/W emulsion was mixed gently in 10.7 g of the hexadecane external phase containing 1 g of Abil EM 90 as surfactant and 0.1 g of decanol as co-surfactant. Each primary emulsion was incorporated slowly in the hexadecane using a Rayneri blender at a speed of 50 rpm and the emulsions were kept under gentle agitation until the duration of the experiments. The globule size of the emulsions was then determined by using a Cannon Microscope and the mean diameter was about 1000 µm for the all emulsions prepared.

#### **5.2 DSC measurements - results and discussions**

Figure 20 presents an example of the evolution of DSC curves obtained for the O1/W/O2 multiple emulsions understudied. The DSC curves are characteristic of a multiple emulsion containing 2 wt% of Tween 20 in the aqueous membrane. On this figure, the initial DSC curve was obtained by submitting the emulsion after 1 minute of its preparation and presents 3 peaks of crystallization. At that time it can be considered that the transfer has not yet started and the three peaks of crystallization corresponds to the crystallization of the

#### Mass Transfers Within Emulsions Studied by Differential Scanning Calorimetry (DSC) - Application to Composition Ripening and Solid Ripening 767

766 Mass Transfer - Advanced Aspects

2007; Mishra & Pandit, 1989; Laugel et al., 1998, 2000; Yu et al., 2003). They are found also in application in separation processes of hydrocarbons (Krishna et al., 1987; Garti & Kovacs, 1991). In controlled drug delivery applications the challenge consists in the control of the diffusion of the entrapped active molecule (Grossiord et al., 1998). The active matter can diffuse and migrate through the membrane via an osmotic pressure gradient and it can be almost difficult to retain the active molecule inside the inner phase upon prolonged storage, all the more as the emulsifier can form micelles (Benichou et al., 2007). Micelles are capable of solubilizing the active molecule and facilitate the transport of active matter from the

One can see that developing potential applications of multiple emulsions requires a good understanding of the mass transfer mechanisms but besides measuring kinetics of mass transfer, it is also important to take into account formulation aspects of the emulsion which

W/O/W emulsions entrapping different compounds, urea or MgSO4 have been studied by using the DSC techniques presented in this chapter (Potier et al., 1992; Raynal et al., 1994 Raynal et al, 1993). As case of study we propose to describe the release of tetradecane within an O/W/O multiple emulsion studied by DSC. It is described how to detect the variations of composition of the external oil phases during time as well as the evolution of the emulsion with time. The influence of formulation parameters such as surfactant concentration and

The tetradecane/Water/hexadecane (O1/W/O2) multiple emulsions were prepared in a two-step emulsification method (Avendaño-Gomez et al., 2005). Firstly, primary simple emulsions O1/W were prepared. Tetradecane was dispersed into an aqueous solution in which was previously dissolved Tween 20. Different concentrations of Tween 20 have been investigated, 2 wt%, 4 wt% and 7 wt%. Tetradecane was then slowly dripped into the aqueous phase under agitation using an Ultra Turrax mixer at a speed of 20,000 rpm. All the O1/W emulsions prepared contained a tetradecane/water ratio of 2/3. The primary emulsions were then sonicated during 10 min. The droplet size distribution was measured using Coulter counter technique and the mean size of tetradecane droplets for all primary emulsion was about 5 µm. In a second step, 19 g of each primary O1/W emulsion was mixed gently in 10.7 g of the hexadecane external phase containing 1 g of Abil EM 90 as surfactant and 0.1 g of decanol as co-surfactant. Each primary emulsion was incorporated slowly in the hexadecane using a Rayneri blender at a speed of 50 rpm and the emulsions were kept under gentle agitation until the duration of the experiments. The globule size of the emulsions was then determined by using a Cannon Microscope and the mean diameter was

Figure 20 presents an example of the evolution of DSC curves obtained for the O1/W/O2 multiple emulsions understudied. The DSC curves are characteristic of a multiple emulsion containing 2 wt% of Tween 20 in the aqueous membrane. On this figure, the initial DSC curve was obtained by submitting the emulsion after 1 minute of its preparation and presents 3 peaks of crystallization. At that time it can be considered that the transfer has not yet started and the three peaks of crystallization corresponds to the crystallization of the

mass ratio of the internal phase on the evolution of the emulsion are also discussed.

internal interface to the external interface.

**5.1 Emulsion preparation** 

have also a direct influence on the mass transfer.

about 1000 µm for the all emulsions prepared.

**5.2 DSC measurements - results and discussions** 

three phases composing the multiple emulsion. The signal at around -18°C represents the crystallization of the aqueous phase by breakdown of undercooling. This signal is characteristic of bulk water crystallization as it has been described in section 2. Even if the aqueous phase is dispersed as a multitude of globules their size of 1 mm is not sufficiently tiny enough to involve nucleation delays like those observed in W/O emulsions droplets of a few µm3. Therefore the behavior of crystallization of aqueous globule looks like the crystallization of water bulk phase. The exothermic peak at 17ºC which initial part of the signal is sharp is attributed to the bulk hexadecane crystallization. The hexadecane crystallizes at 17ºC but with a very little degree of subcooling, the melting temperature of hexadecane being of 18°C. The third bell shape peak is attributed to the crystallization of the tetradecane dispersed droplets. The temperature is -12.8°C and is given by the apex of the peak which means that nearly 50% of the droplets are crystallized.

Fig. 20. Evolution of the DSC curves of an O1/W/O2 multiple emulsion containing 2 wt% of Tween 20 in the aqueous membrane

Figure 20 also shows a clearly evolution with time of the crystallization peaks of the internal phase and the external phase. Actually, the crystallization peak of the tetradecane exhibits a

Mass Transfers Within Emulsions Studied by

the external phase (tetradecane + hexadecane). *C*<sup>I</sup>

the aqueous globule respectively.

O1: dispersed droplets of tetradecane

O2: hexadecane

Differential Scanning Calorimetry (DSC) - Application to Composition Ripening and Solid Ripening 769

Therefore an improved model using the shrinking core model commonly applied to heterogeneous reactions (Stambouli et al., 2007) has been developed. The model has been modified and applied to the multiple emulsions under studied. Owing to the fact that the structure of multiple emulsions is complex and in order to simplify the mathematical treatment of the model, it has been assumed that the internal droplets of tetradecane form only one virtual drop encapsulated in an aqueous spherical shell which represents the membrane. The shrinking model and the typical concentration profile for diffusive transport of tetradecane through the membrane are described in figure 22. For clarity reasons let us define first of all, the notations used in the description of the kinetics model: CI represents the pure tetradecane into the internal virtual drop. CII, is the tetradecane concentration in

interface in the aqueous phase at interface I. *C*II′ is the tetradecane concentration in the aqueous phase at the interface II. C'(r) is the tetradecane concentration in the aqueous phase at radius r. rI and rII are the radius of the internal virtual tetradecane drop and the radius of

Aqueous membrane

Fig. 22. (a) Schematic representation of the tetradecane/water/oil emulsion. (b) A multiple globule containing a virtual drop of tetradecane surrounding by a spherical aqueous

Two partition coefficients can be defined as the ratios between the concentration of tetradecane concentration in the aqueous membrane phase and the oily phases I and II

membrane shell. (c) Profile of tetradecane concentration in the emulsion

considering that equilibrium is assumed at both water/oil interfaces I and II.

′ is the tetradecane concentration at the

**rII**

**0 rI rII**

**r**

(c)

O1: virtual droplet of tetradecane

(a) O2: hexadecane (b)

**rI**

CI

'CI

CII

'CII

progressive decrement in intensity whereas the temperature of the crystallization peaks of hexadecane decreases progressively from 17°C to 10°C. From these results it can be deduced that tetradecane releases gradually over time. On the ultimate DSC curve, the crystallization peak of tetradecane disappears which means that the globules are empty and total tetradecane has been transferred in the external phase. Consequently the external phase composition has changed over time from pure hexadecane to a given tetradecanehexadecane composition when the transfer ends, passing through different intermediate compositions.

Using equation 2, it is then possible to determine the percentage y of tetradecane still entrapped at time t. The values of y versus time and obtained for different amounts of Tween (2, 4 and 7%) are reported on figure 21.

The emulsions initially contained the same amount of encapsulated tetradecane and it can be easily observed that the kinetics depend on the amount of Tween 20 contained in the membrane. The higher the concentration of surfactant, the higher is the release of tetradecane. Nevertheless, it must be emphasized that tetradecane is gradually released through the aqueous membrane and that no globule breaking was observed during the lifetime of the multiple emulsion.

Fig. 21. Evolution of the quantity of tetradecane encapsulated in the globules with time at different concentrations of Tween 20 in the aqueous membranes phase

#### **5.3 Model of kinetics of tetradecane release**

First a model based on the expressions of the flux of tetradecane versus chemical potential gradient has been used (Avendano, 2002). A solubilisation – diffusion model has been developed and the comparison with the experimental data has shown theoretical release times higher: 50% released in 4000 minutes (experiment) and 5000 minutes (model).

progressive decrement in intensity whereas the temperature of the crystallization peaks of hexadecane decreases progressively from 17°C to 10°C. From these results it can be deduced that tetradecane releases gradually over time. On the ultimate DSC curve, the crystallization peak of tetradecane disappears which means that the globules are empty and total tetradecane has been transferred in the external phase. Consequently the external phase composition has changed over time from pure hexadecane to a given tetradecanehexadecane composition when the transfer ends, passing through different intermediate

Using equation 2, it is then possible to determine the percentage y of tetradecane still entrapped at time t. The values of y versus time and obtained for different amounts of

The emulsions initially contained the same amount of encapsulated tetradecane and it can be easily observed that the kinetics depend on the amount of Tween 20 contained in the membrane. The higher the concentration of surfactant, the higher is the release of tetradecane. Nevertheless, it must be emphasized that tetradecane is gradually released through the aqueous membrane and that no globule breaking was observed during the life-

Fig. 21. Evolution of the quantity of tetradecane encapsulated in the globules with time at

First a model based on the expressions of the flux of tetradecane versus chemical potential gradient has been used (Avendano, 2002). A solubilisation – diffusion model has been developed and the comparison with the experimental data has shown theoretical release times higher: 50% released in 4000 minutes (experiment) and 5000 minutes (model).

different concentrations of Tween 20 in the aqueous membranes phase

**5.3 Model of kinetics of tetradecane release** 

compositions.

Tween (2, 4 and 7%) are reported on figure 21.

time of the multiple emulsion.

Therefore an improved model using the shrinking core model commonly applied to heterogeneous reactions (Stambouli et al., 2007) has been developed. The model has been modified and applied to the multiple emulsions under studied. Owing to the fact that the structure of multiple emulsions is complex and in order to simplify the mathematical treatment of the model, it has been assumed that the internal droplets of tetradecane form only one virtual drop encapsulated in an aqueous spherical shell which represents the membrane. The shrinking model and the typical concentration profile for diffusive transport of tetradecane through the membrane are described in figure 22. For clarity reasons let us define first of all, the notations used in the description of the kinetics model: CI represents the pure tetradecane into the internal virtual drop. CII, is the tetradecane concentration in the external phase (tetradecane + hexadecane). *C*<sup>I</sup> ′ is the tetradecane concentration at the interface in the aqueous phase at interface I. *C*II′ is the tetradecane concentration in the aqueous phase at the interface II. C'(r) is the tetradecane concentration in the aqueous phase at radius r. rI and rII are the radius of the internal virtual tetradecane drop and the radius of the aqueous globule respectively.

Fig. 22. (a) Schematic representation of the tetradecane/water/oil emulsion. (b) A multiple globule containing a virtual drop of tetradecane surrounding by a spherical aqueous membrane shell. (c) Profile of tetradecane concentration in the emulsion

Two partition coefficients can be defined as the ratios between the concentration of tetradecane concentration in the aqueous membrane phase and the oily phases I and II considering that equilibrium is assumed at both water/oil interfaces I and II.

Mass Transfers Within Emulsions Studied by

Differential Scanning Calorimetry (DSC) - Application to Composition Ripening and Solid Ripening 771

0.282 0.326 0.282

(15)

<sup>∗</sup> <sup>Δ</sup> were fitted using equation 12 using

 : Model : Experimental

t1 = 875 h

<sup>3</sup> <sup>0</sup>

⎛ ⎞ <sup>+</sup> = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

Where 0.282 and 0.326 are the volume fractions of the tetradecane phase and the membrane

Figure 23 presents the experimental and the model results obtained from a multiple emulsion containing 2 wt% of Tween 20 in the membrane phase. The model fits well the experimental data. It is worth noting that the model also fits the experimental data for the multiple emulsions containing 4 and 7 wt% of Tween 20 in the aqueous membrane phase. The time t1 of complete release of tetradecane has been derived from the three formulations and it was obtained, t1= 875h, t1= 451h and t1= 318h for 2, 4 and 7 wt% of Tween 20

> *T D C* ρ

Tetradecane Release (Tween = 2%)

0 100 200 300 400 500 600 700

Time (hr)

Fig. 23. Tetradecane release obtained from the model and compared to the experimental data. Example of an emulsion containing 2 wt% of Tween 20 in the aqueous membrane

<sup>0</sup> 0.387 *Ir mm* <sup>=</sup> calculated from Zo ( <sup>0</sup> <sup>1</sup> *II r mm* <sup>=</sup> ). The values of the effective diffusion coefficient obtained were compared to those of literature (Mandal et al, 1985) and a difference was observed. However, the order of magnitude of the effective diffusion coefficient is in a good agreement with the literature data. Therefore, it can be concluded that the tetradecane release is strongly dependent from the Tween 20 concentration and occurs by a micellar transport mechanism. The difference observed between the diffusion coefficients from the model and literature may be due to the simplifications made for developing the model. A more accurate model can be proposed but new empirical

*II I*

0 0

*<sup>r</sup> <sup>Z</sup> r*

aqueous phase in the multiples emulsions respectively.

respectively. In the model, the values of *<sup>e</sup>*

parameters are needed.

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Release fraction

$$K\_1 = \frac{C\_1'}{C\_1} \tag{9}$$

$$K\_2 = \frac{C\_{\text{II}}'}{C\_{\text{II}}} \tag{10}$$

For simplification, it is considered that the internal dispersed droplets of tetradecane are immobile inside the aqueous globules and are able to release tetradecane in the external phase through the membrane. Therefore, it is assumed that the release of tetradecane occurs via a pure molecular diffusion or pure micellar diffusion.

The expression of the tetradecane flow through the aqueous spherical membrane shell of radius r is expressed as equation (11) where De is the effective diffusion coefficient of tetradecane in the aqueous membrane phase.

$$J(r) = -4\Pi r^2 D\_\epsilon \left(\frac{dC'}{dr}\right) \tag{11}$$

From mass balance equations and integration between r=r1 and r=r2 assuming that the volume of the aqueous membrane does not vary during the tetradecane release, the time t of release can be expressed by the following equation

$$t = \frac{\rho\_T (r\_1^0)^2}{2D\_c \Lambda C} \left[ (Z\_0 - \mathbf{x})^{\frac{2}{3}} - (1 - \mathbf{x})^{\frac{2}{3}} + 1 - Z\_0^{\frac{2}{3}} \right] \tag{12}$$

where *C CC I II* <sup>∗</sup> Δ=−′ ′ ; ρT and VT are respectively the density of tetradecane and the volume of the virtual tetradecane phase I ; <sup>3</sup> <sup>0</sup> 0 0 *II <sup>r</sup> <sup>Z</sup> r* ⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ , 0 *I r y <sup>r</sup>* <sup>=</sup> where <sup>0</sup> *Ir* is the initial radius of the

*I*

*I*

virtual tetradecane droplet.

If we considered the time t1 at which the internal tetradecane phase is completely release, (r1=0 and x= 1), then t1 is expressed as

$$t\_1 = \frac{\rho\_T \left(r\_l^0\right)}{2D\_\epsilon \Delta C} \left[ (Z\_0 - 1)^{\frac{2}{3}} + 1 - Z\_0^{-\frac{2}{3}} \right] \tag{13}$$

Finally it is possible to write the time t of release in the equivalent following form as

$$t = \frac{t\_1}{\left[ (Z\_0 - 1)\frac{2}{3} + 1 - Z\_0^{\frac{2}{3}} \right]} \left[ (Z\_0 - \alpha)^{\frac{2}{3}} - (1 - \alpha)^{\frac{2}{3}} + 1 - Z\_0^{\frac{2}{3}} \right] \tag{14}$$

Taking into account the value of Z0 from equation 15, it was possible by using equation (14) where t1 is the fitting parameter and the least-squares method to obtain the best fitting of the experimental data relating to the tetradecane release x versus time.

1

2

via a pure molecular diffusion or pure micellar diffusion.

of release can be expressed by the following equation

2 *T e*

*D C* ρ

> 2 *T I e*

<sup>3</sup> <sup>3</sup> 0 0

1 1

⎣ ⎦

experimental data relating to the tetradecane release x versus time.

*Z Z*

( )

∗

tetradecane in the aqueous membrane phase.

where *C CC I II*

of the virtual tetradecane phase I ;

(r1=0 and x= 1), then t1 is expressed as

virtual tetradecane droplet.

*<sup>C</sup> <sup>K</sup> C*

For simplification, it is considered that the internal dispersed droplets of tetradecane are immobile inside the aqueous globules and are able to release tetradecane in the external phase through the membrane. Therefore, it is assumed that the release of tetradecane occurs

The expression of the tetradecane flow through the aqueous spherical membrane shell of radius r is expressed as equation (11) where De is the effective diffusion coefficient of

From mass balance equations and integration between r=r1 and r=r2 assuming that the volume of the aqueous membrane does not vary during the tetradecane release, the time t

( ) ( )

= − −− + ⎢ − ⎥ Δ ⎢ ⎥ ⎣ ⎦

*<sup>r</sup> t Zx x Z*

<sup>3</sup> <sup>0</sup>

If we considered the time t1 at which the internal tetradecane phase is completely release,

( ) ( )

*t ZZ*

*<sup>t</sup> t Z <sup>x</sup> <sup>x</sup> <sup>Z</sup>*

<sup>⎡</sup> <sup>⎤</sup> <sup>=</sup> <sup>⎢</sup> <sup>−</sup> <sup>−</sup> <sup>−</sup> <sup>+</sup> <sup>−</sup> <sup>⎥</sup> ⎡ ⎤ <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup> ⎢ ⎥ − +−

Taking into account the value of Z0 from equation 15, it was possible by using equation (14) where t1 is the fitting parameter and the least-squares method to obtain the best fitting of the

<sup>3</sup> <sup>3</sup> 1 00 1 1

= −+ ⎢ − ⎥ <sup>Δ</sup> <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

0 0 *II I*

*<sup>r</sup> <sup>Z</sup> r* ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

*r*

∗

Finally it is possible to write the time t of release in the equivalent following form as

*D C* ρ

<sup>3</sup> 0 2 <sup>2</sup> <sup>2</sup> <sup>I</sup> <sup>3</sup> <sup>3</sup> 0 0 ( ) 1 1

<sup>∗</sup> Δ=−′ ′ ; ρT and VT are respectively the density of tetradecane and the volume

, 0 *I I r y*

<sup>0</sup> <sup>2</sup> <sup>2</sup>

⎡ ⎤

( ) ( )

<sup>1</sup> <sup>3</sup> <sup>3</sup> <sup>3</sup> 0 0 <sup>2</sup> <sup>2</sup>

2 2 2

1 1

*<sup>r</sup>* <sup>=</sup> where <sup>0</sup>

⎡ ⎤

<sup>2</sup> () 4 *<sup>e</sup> dC Jr rD dr* ⎛ ⎞′ =− Π ⎜ ⎟

*<sup>C</sup> <sup>K</sup> C*

I

′ <sup>=</sup> (9)

′ <sup>=</sup> (10)

⎝ ⎠ (11)

(12)

*Ir* is the initial radius of the

(13)

(14)

2

I

II

II

$$Z\_0 = \left(\frac{r\_{ll}^0}{r\_l^0}\right)^3 = \frac{0.282 + 0.326}{0.282} \tag{15}$$

Where 0.282 and 0.326 are the volume fractions of the tetradecane phase and the membrane aqueous phase in the multiples emulsions respectively.

Figure 23 presents the experimental and the model results obtained from a multiple emulsion containing 2 wt% of Tween 20 in the membrane phase. The model fits well the experimental data. It is worth noting that the model also fits the experimental data for the multiple emulsions containing 4 and 7 wt% of Tween 20 in the aqueous membrane phase. The time t1 of complete release of tetradecane has been derived from the three formulations and it was obtained, t1= 875h, t1= 451h and t1= 318h for 2, 4 and 7 wt% of Tween 20

respectively. In the model, the values of *<sup>e</sup> T D C* ρ <sup>∗</sup> <sup>Δ</sup> were fitted using equation 12 using

<sup>0</sup> 0.387 *Ir mm* <sup>=</sup> calculated from Zo ( <sup>0</sup> <sup>1</sup> *II r mm* <sup>=</sup> ). The values of the effective diffusion coefficient obtained were compared to those of literature (Mandal et al, 1985) and a difference was observed. However, the order of magnitude of the effective diffusion coefficient is in a good agreement with the literature data. Therefore, it can be concluded that the tetradecane release is strongly dependent from the Tween 20 concentration and occurs by a micellar transport mechanism. The difference observed between the diffusion coefficients from the model and literature may be due to the simplifications made for developing the model. A more accurate model can be proposed but new empirical parameters are needed.

Fig. 23. Tetradecane release obtained from the model and compared to the experimental data. Example of an emulsion containing 2 wt% of Tween 20 in the aqueous membrane

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