4. Theory of Ostwald ripening in nano-dispersion

So far, we came to know that the major instability of O/W nanoemulsions is caused due to the molecular exchange of oil molecules between droplets, a process known as Ostwald ripening. Although many specialists on emulsion stability are often unwilling to accept the concept of Ostwald ripening, instead, in most cases, they explain the destabilization under the framework of the more traditional mechanism—coalescence. Despite these disagreements in the literature, the theoretical development of the kinetic regime of the Ostwald ripening is a peculiar form of self-ordering and stimulates curiosity to the researchers till now. The detail mathematics of the theory of Ostwald ripening mechanism is not included in this chapter; rather, we will address a few selected topics of the theory. A more comprehensive discussion on this issue can be found in the reviews given in Refs. [19–21].

The major contribution in the theory of kinetics of Ostwald ripening in its contemporary form was initially formulated by Lifshitz and Slyozov [22] and then independently by Wagner [23] (known as LSW theory). Following publication of their findings, it became the seminal paper on which all subsequent theoretical works have been based. The theory is based on the following assumptions: (1) the particles (here droplets) of dispersed phase are spherical in shape, are fixed in space and are separated from each other by distance which are much larger than their sizes (true diluted system), (2) the mass transport is due only to the molecular diffusion through the dispersion medium and (3) the concentration of the molecularly dissolved species is constant except adjacent to the droplet boundaries.

#### 4.1 Scaling the ripening problem

The ripening mechanism is characterized by two intervals, namely, the transient or short-time regime and the asymptotic limit or long-time regime. The transient limit is composed of a region of random variation of droplet size, whereas the

asymptotic limit (t ! ∞) is ascribed to the region that shows a linear relationship between the cubes of the number average droplet radius (rN 3) with time.

We follow the procedure used in reference [21] by Kabalnov et al. If f(r,t) be the size (radius) distribution function of the polydisperse system, such that f(r,t)dr is the number of particles per unit volume in the size range r to r + dr. The change in the distribution function with time can be expressed as

$$\frac{d\boldsymbol{f}(t,r)}{dt} = \mathbf{0},$$

$$\text{or } \frac{\delta \boldsymbol{f}}{\delta t} + \frac{\delta \boldsymbol{f}}{\delta r} \frac{dr}{dt} = \mathbf{0},$$

$$\text{or } \frac{\delta \boldsymbol{f}}{\delta t} + \frac{\delta}{\delta r} (f\dot{r}) = \mathbf{0},$$

where <sup>r</sup>\_ <sup>¼</sup> dr dt is the velocity of the particles

$$\text{i.e.} \frac{\delta \mathbf{f}}{\delta t} + \frac{\delta \mathbf{j}}{\delta r} = \mathbf{0}, \tag{1}$$

one will become smaller (decay). In this notation the growth rate equation (Eq. (2))

ðÞ¼ <sup>t</sup> <sup>4</sup>πD Ceqð Þ <sup>∞</sup> <sup>α</sup> <sup>r</sup>

In a closed system, the concentration of a substance in the medium and the drop

ð<sup>∞</sup> 0 r 3

The three integro-differential Eqs. (1), (3) and (4) can be solved analytically. It was shown that this system should have an asymptotic solution which is independent of initial conditions. Rather than solving the problem for all times, LSW found an asymptotic solution valid as t ! ∞. Using this approach, the following predictions were made for two-phase mixture undergoing Ostwald ripening in the long-

or <sup>r</sup>\_ <sup>¼</sup> DCeqð Þ <sup>∞</sup> r2

3 ð<sup>∞</sup> 0 r 3 f rð Þ ; <sup>t</sup> dr � � <sup>¼</sup> <sup>0</sup>,

> 4π 3Ceqð Þ ∞

The final element of LSW theory is the mass conservation.

dt C tðÞþ <sup>4</sup><sup>π</sup>

1. Time evolution of average droplet radius: R tðÞ¼ <sup>R</sup><sup>3</sup>

and u is the normalized radius, u ¼ RR, α ¼ 4π=ð Þ 3VmCð Þ ∞ . The salient results of LSW theory are summarized as follows:

ð Þþ 0 4t=9 � ��<sup>1</sup>

> <sup>ω</sup> <sup>¼</sup> <sup>d</sup> dr rN

2. Time evolution of the number of droplets per unit volume:

3. Time invariant droplet size distribution function: <sup>f</sup> <sup>R</sup>; <sup>t</sup> � � <sup>¼</sup> g uð Þ

maximum drop size; Rð Þ 0 is the average radius at the onset of coarsening;

where depending on the approach, R is either the critical radius (rc) or the

1. In the stationary region, the nature of the size distribution function is time

2. The cube of the number-averaged particle size (rN) varies linearly with time:

where rc is the radius of droplet at steady state, D is the diffusion coefficient of the dispersed phase in the continuous phase and α is the characteristic length scale (α = 2γVm/RT). Droplet with radius r < rc will disappear, while droplet with r > rc

<sup>9</sup>RT � � <sup>¼</sup> <sup>4</sup>

<sup>9</sup> <sup>α</sup>DC<sup>∞</sup>,

<sup>3</sup> � � <sup>¼</sup> <sup>8</sup>γDCð Þ <sup>∞</sup> Vm

<sup>0</sup> <sup>u</sup><sup>3</sup>g uð Þdu, <sup>θ</sup> is the dimensionless concentration, <sup>θ</sup> <sup>¼</sup> ð Þ <sup>C</sup> � <sup>C</sup>ð Þ <sup>∞</sup> <sup>=</sup>Cð Þ <sup>∞</sup> ;

rc � 1 � �,

f rð Þ ; t dr ¼ θð Þ 0 : (4)

ð Þþ 0 4t=9 � �<sup>1</sup>=<sup>3</sup>

> R 4

r rc � 1 � �:

can be rewritten as follows:

time limit [20]:

ψ ¼ θ=α

will grow.

97

N tðÞ¼ <sup>ψ</sup> <sup>R</sup><sup>3</sup>

Ð <sup>3</sup>=<sup>2</sup>

invariant.

4π 3 d dtr 3

Importance of Surface Energy in Nanoemulsion DOI: http://dx.doi.org/10.5772/intechopen.84201

size distribution function is interrelated by

d

or θðÞþt

where j ¼ fr\_ is the flux of particles

Now, the growth rate of any droplet is proportional to its size, r, and the concentration, C(t), of the substance of the droplet in the medium (which is a constant according to assumption 3) with respect to its equilibrium value, Ceq :

$$\text{i.e.} \frac{dV}{dt} \text{or} \left(\text{C}(t) - \text{C}\_{eq}(r)\right),$$

$$\text{or} \ \frac{4\pi}{3} \frac{d}{dt} r^3(t) = 4\pi D \left(\text{C}(t) - \text{C}\_{eq}(r)\right),\tag{2}$$

where D is the molecular diffusivity of the disperse phase in the medium.

$$\text{for } r^2 \frac{dr}{dt} = Dr \left( \mathbf{C}(t) - \mathbf{C}\_{eq}(r) \right); \\ \text{i.e.} \\ \dot{r} = \frac{D}{r} \left( \mathbf{C}(t) - \mathbf{C}\_{eq}(r) \right).$$

Now, we introduce a dimensionless quantity θ, which is a measure of the relative concentration by <sup>θ</sup>ðÞ¼ <sup>t</sup> C tð Þ�Ceqð Þ <sup>∞</sup> Ceqð Þ ∞ ; now from the Kelvin equation, we know that Ceqð Þ¼ <sup>r</sup> Ceqð Þ <sup>∞</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup> r .

Therefore, <sup>r</sup>\_ <sup>¼</sup> <sup>D</sup> <sup>r</sup> C tð Þ� Ceqð Þ <sup>∞</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup> r .

Rearranging the equation we get

$$\dot{r} = \frac{DC\_{eq}(\infty)}{r} \left( \left( \frac{\mathbf{C}(t) - \mathbf{C}\_{eq}(\infty)}{\mathbf{C}\_{eq}(\infty)} \right) - \frac{\alpha}{r} \right),$$

$$\text{or } \dot{r} = \frac{DC\_{eq}(\infty)}{r} \left( \theta(t) - \frac{\alpha}{r} \right). \tag{3}$$

This is the expression of velocity of particle in size space. The value of particle radius at which the growth rate at any instant of time is zero ð Þ r\_ ¼ 0 is called the instantaneous critical radius (rc); then, it is obvious that rc ¼ α=θð Þt , and for growth rate equation, we can write <sup>θ</sup>ðÞ�<sup>t</sup> <sup>α</sup> <sup>r</sup> >0 or <sup>α</sup> <sup>r</sup> <sup>&</sup>lt; <sup>θ</sup>ð Þ<sup>t</sup> or <sup>r</sup><sup>&</sup>gt; <sup>α</sup> <sup>θ</sup>ð Þ<sup>t</sup> , i.e. <sup>r</sup>>rc, which means that at any instance of time particle with radius greater than the critical radius will exhibit an increase in size where the particle having radius smaller than the critical

asymptotic limit (t ! ∞) is ascribed to the region that shows a linear relationship

df tð Þ ;r dt <sup>¼</sup> <sup>0</sup>,

or δf δt þ δf δr dr dt <sup>¼</sup> <sup>0</sup>,

or δf δt þ δ δr

> i:e: δf δt þ δj δr

Now, the growth rate of any droplet is proportional to its size, r, and the concentration, C(t), of the substance of the droplet in the medium (which is a constant according to assumption 3) with respect to its equilibrium value, Ceq :

where D is the molecular diffusivity of the disperse phase in the medium.

dt <sup>¼</sup> Dr C tðÞ� Ceqð Þ<sup>r</sup> ; <sup>i</sup>:e:r\_ <sup>¼</sup> <sup>D</sup>

or <sup>r</sup>\_ <sup>¼</sup> DCeqð Þ <sup>∞</sup> r

dt <sup>∝</sup>rCtð Þ� Ceqð Þ<sup>r</sup> ,

Now, we introduce a dimensionless quantity θ, which is a measure of the relative

C tðÞ� Ceqð Þ ∞ Ceqð Þ ∞ 

This is the expression of velocity of particle in size space. The value of particle radius at which the growth rate at any instant of time is zero ð Þ r\_ ¼ 0 is called the instantaneous critical radius (rc); then, it is obvious that rc ¼ α=θð Þt , and for growth

<sup>r</sup> >0 or <sup>α</sup>

that at any instance of time particle with radius greater than the critical radius will exhibit an increase in size where the particle having radius smaller than the critical

<sup>θ</sup>ð Þ�<sup>t</sup> <sup>α</sup> r 

<sup>r</sup> <sup>&</sup>lt; <sup>θ</sup>ð Þ<sup>t</sup> or <sup>r</sup><sup>&</sup>gt; <sup>α</sup>

r

dt is the velocity of the particles

i:e: dV

Ceqð Þ ∞ 

<sup>r</sup> C tð Þ� Ceqð Þ <sup>∞</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup>

<sup>r</sup>\_ <sup>¼</sup> DCeqð Þ <sup>∞</sup> r

.

where j ¼ fr\_ is the flux of particles

or 4π 3 d dtr 3

or r <sup>2</sup> dr

concentration by <sup>θ</sup>ðÞ¼ <sup>t</sup> C tð Þ�Ceqð Þ <sup>∞</sup>

r .

Rearranging the equation we get

rate equation, we can write <sup>θ</sup>ðÞ�<sup>t</sup> <sup>α</sup>

96

Ceqð Þ¼ <sup>r</sup> Ceqð Þ <sup>∞</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup>

Therefore, <sup>r</sup>\_ <sup>¼</sup> <sup>D</sup>

We follow the procedure used in reference [21] by Kabalnov et al. If f(r,t) be the size (radius) distribution function of the polydisperse system, such that f(r,t)dr is the number of particles per unit volume in the size range r to r + dr. The change in

ð Þ¼ fr\_ 0,

3) with time.

¼ 0, (1)

ðÞ¼ <sup>t</sup> <sup>4</sup>πDr C tðÞ� Ceqð Þ<sup>r</sup> , (2)

C tðÞ� Ceqð Þ<sup>r</sup> :

; now from the Kelvin equation, we know that

� α r

,

: (3)

<sup>θ</sup>ð Þ<sup>t</sup> , i.e. <sup>r</sup>>rc, which means

r

between the cubes of the number average droplet radius (rN

the distribution function with time can be expressed as

Nanoemulsions - Properties, Fabrications and Applications

where <sup>r</sup>\_ <sup>¼</sup> dr

one will become smaller (decay). In this notation the growth rate equation (Eq. (2)) can be rewritten as follows:

$$
\frac{4\pi}{3}\frac{d}{dt}r^3(t) = 4\pi D \text{ C}\_{eq}(\infty)a\left(\frac{r}{r\_c} - 1\right),
$$

$$
\text{or } \dot{r} = \frac{D \text{C}\_{eq}(\infty)}{r^2} \left(\frac{r}{r\_c} - 1\right).
$$

The final element of LSW theory is the mass conservation.

In a closed system, the concentration of a substance in the medium and the drop size distribution function is interrelated by

$$\frac{d}{dt}\left[\mathbf{C}(t) + \frac{4\pi}{3}\int\_{0}^{\infty} r^3 f(r, t) dr\right] = \mathbf{0},$$

$$\text{or } \theta(t) + \frac{4\pi}{3\mathbf{C}\_{eq}(\infty)}\int\_{0}^{\infty} r^3 f(r, t) dr = \theta(\mathbf{0}). \tag{4}$$

The three integro-differential Eqs. (1), (3) and (4) can be solved analytically. It was shown that this system should have an asymptotic solution which is independent of initial conditions. Rather than solving the problem for all times, LSW found an asymptotic solution valid as t ! ∞. Using this approach, the following predictions were made for two-phase mixture undergoing Ostwald ripening in the longtime limit [20]:

1. Time evolution of average droplet radius: R tðÞ¼ <sup>R</sup><sup>3</sup> ð Þþ 0 4t=9 � �<sup>1</sup>=<sup>3</sup>

2. Time evolution of the number of droplets per unit volume:

$$N(t) = \wp\left(\overline{R}^3(\mathbf{0}) + 4t/\mathfrak{Y}\right)^{-1}$$

3. Time invariant droplet size distribution function: <sup>f</sup> <sup>R</sup>; <sup>t</sup> � � <sup>¼</sup> g uð Þ R 4

where depending on the approach, R is either the critical radius (rc) or the maximum drop size; Rð Þ 0 is the average radius at the onset of coarsening; ψ ¼ θ=α Ð <sup>3</sup>=<sup>2</sup> <sup>0</sup> <sup>u</sup><sup>3</sup>g uð Þdu, <sup>θ</sup> is the dimensionless concentration, <sup>θ</sup> <sup>¼</sup> ð Þ <sup>C</sup> � <sup>C</sup>ð Þ <sup>∞</sup> <sup>=</sup>Cð Þ <sup>∞</sup> ; and u is the normalized radius, u ¼ RR, α ¼ 4π=ð Þ 3VmCð Þ ∞ .

The salient results of LSW theory are summarized as follows:


$$a = \frac{d}{dr} \left( r\_N^{\;\;\;\beta} \right) = \left( \frac{8 \gamma DC(\infty) V\_m}{\Re T} \right) = \frac{4}{9} a D C^{\infty},$$

where rc is the radius of droplet at steady state, D is the diffusion coefficient of the dispersed phase in the continuous phase and α is the characteristic length scale (α = 2γVm/RT). Droplet with radius r < rc will disappear, while droplet with r > rc will grow.

The comparison between theoretical and experimental Ostwald ripening rates, however, evoked a significant discrepancy in the literature [18], where the latter was found to be several times higher than the former. It has been found that the linear relation of rN <sup>3</sup> with time in the asymptotic limit does not always signify the Ostwald ripening; as a second mechanism, "Brownian-induced coalescence" (particularly if the drop surface coverage is not sufficient to hinder the coalescence) may also be operative, which has the same dependency of the rate over time. The LSW theory assumed that the droplets are fixed in space and the molecular diffusion is the only mechanism of mass transfer. However, for the case of droplets undergoing Brownian motion, one must take into account both the contribution of molecular and convection diffusion as predicted by Peclet number (Pe = rv/D). The velocity <sup>v</sup> of a droplet of mass <sup>M</sup> is approximately given by <sup>v</sup> <sup>¼</sup> <sup>3</sup>kT M <sup>1</sup>=<sup>2</sup> .

<sup>ω</sup>mix <sup>¼</sup> <sup>ϕ</sup><sup>1</sup>

Importance of Surface Energy in Nanoemulsion DOI: http://dx.doi.org/10.5772/intechopen.84201

of insoluble oil (X2 > 0.4), entropy of mixing dominates, Δμ < 0, and the nanoemulsion is thermodynamically stable. For low mole fraction (X2 < 0.17), Laplace pressure between the different size droplets dominates, and the

also has a kinetic energy barrier that prevents further ripening to occur.

5. Stability against coalescence in nano-dispersion

firmed by Kabalnov et al.

Figure 6.

99

ω1 þ ϕ2 ω2 �<sup>1</sup>

nanoemulsion is thermodynamically unstable. At intermediate mole fraction (0.17 < X2 < 0.4), a nanoemulsion is said to be in metastable state. Initially, Laplace pressure will dominate and the system undergoes Ostwald ripening; however, it

The validity of the LSW theory was tested by Kabalnov et al. [18, 19]. The influence of the alkyl chain length of the hydrocarbon on the Ostwald ripening rate of nanoemulsion was systematically investigated from alkyl chain length C9–C16. Increasing the alkyl chain length of the hydrocarbon used for the emulsion results in the decrease in the oil solubility. According to the LSW theory, this reduction in solubility should result in a decrease in the Ostwald ripening rate which was con-

Though the droplets coalescence rate is sufficiently get reduced for the nanoemulsion due to their smaller size, however, as a consequence of Ostwald ripening, it may results in creaming and ultimately leads to phase separation of the nanoemulsion (Figure 6a). The importance of droplet deformation, surfactant transfer and interfacial rheology for the stability of emulsion is well reported in the literature [28–30]. In the presence of thermal noise, the droplets are allowed to explore space by Brownian motion, and the resulting collisions cause the mean droplet radius R to increase. This process is called coalescence. Stabilization against

(a) Schematic elaboration of various destabilization mechanisms for nanoemulsion and (b) cartoon of drop-

drop interaction under two different regions where ligand shell overlaps each other.

where ϕ<sup>1</sup> and ϕ<sup>2</sup> and the ω<sup>1</sup> and ω<sup>2</sup> are the volume fraction and ripening rate of the medium-soluble and medium-insoluble component, respectively. In his treatment [25], Kabalnov came up with three stability regime depending on the mole fraction of the less soluble second component (X2) in the mixture. For high amount

,

#### 4.2 Adjustment in Ostwald ripening rate

The rate of ripening, according to the LSW model, is directly proportional to the solubility of the oil in the aqueous medium. The presence of amphiphiles (surfactants or co-surfactants) can significantly enhance the oil solubility by allowing them to enter into their hydrophobic core. By replacing the bulk solubility of oil, Cð Þ ∞ by the concentration of oil solubilized by the micelles, and using the micellar diffusion coefficient instead of the molecular diffusion, one would get the Ostwald ripening rate in the presence of micelle. According to the extended LSW theory, the Ostwald ripening rate of nanoemulsion containing a water-insoluble low molecular weight coemulsifier (amphiphile) can be predicted by the following equation [24], <sup>ω</sup> <sup>¼</sup> <sup>64</sup>γDcoCcoð Þ <sup>∞</sup> Vm <sup>9</sup>RT <sup>ϕ</sup>co, where Dco, Ccoð Þ <sup>∞</sup> and <sup>ϕ</sup>co are the molecular diffusivity, bulk solubility (in water) and volume fraction of the coemulsifier in the oil droplet, respectively. Similarly, as the interfacial tension (γ) is incorporated in the ripening rate equation in the capillary length parameter (α = 2γVm/RT), lowering of the interfacial tension by the amphiphiles will lead to smaller capillary length and therefore lower solubility of the oil at the droplet boundary. This will cause lowering of the ripening rate at least for the case of strongly adsorbed amphiphiles. In the absence of micelle, oil molecule transports into and through the aqueous phase separating the droplets. However, in the presence of micelle, it is proposed that mass transfer is still involved through the continuous phase, but the micelles increase the water solubility of the oil, therefore effectively increasing the transport rate [24–26]. Kabalnov [19] has proposed three possible mechanisms to explain the observed effects of micelles on Ostwald ripening; nevertheless, whether the oil molecules are taken up by the micelles directly from the aqueous medium or by fusion/fission of a micelle with a droplet surface is still unclear. Another possible way to slow down the Ostwald ripening rate was proposed by Higuchi and Misra [27] through the incorporation of a second disperse phase (oil) with a much lower continuous phase solubility. Initially, the concentration of the second insoluble phase is equal in all droplets. As the mixed oil nanoemulsion undergoes Ostwald ripening, the more soluble component diffuses from the smaller to larger droplet at a much faster rate than the less soluble one. Thus, time will come when the larger droplets become enriched with the soluble oil and the chemical potential of that component is equal in all the droplets. Thus, there is no driving force for further transfer of the oil to take place, and the state is referred to as pseudo-steady of ripening. As theoretically investigated by Kabalnov et al. in the case of medium-soluble second component, the Ostwald ripening rate of the mixed oil nanoemulsion can be approximated as [25]

Importance of Surface Energy in Nanoemulsion DOI: http://dx.doi.org/10.5772/intechopen.84201

The comparison between theoretical and experimental Ostwald ripening rates, however, evoked a significant discrepancy in the literature [18], where the latter was found to be several times higher than the former. It has been found that the

Ostwald ripening; as a second mechanism, "Brownian-induced coalescence" (particularly if the drop surface coverage is not sufficient to hinder the coalescence) may also be operative, which has the same dependency of the rate over time. The LSW theory assumed that the droplets are fixed in space and the molecular diffusion is the only mechanism of mass transfer. However, for the case of droplets undergoing Brownian motion, one must take into account both the contribution of molecular and convection diffusion as predicted by Peclet number (Pe = rv/D). The

The rate of ripening, according to the LSW model, is directly proportional to the solubility of the oil in the aqueous medium. The presence of amphiphiles (surfactants or co-surfactants) can significantly enhance the oil solubility by allowing them to enter into their hydrophobic core. By replacing the bulk

solubility of oil, Cð Þ ∞ by the concentration of oil solubilized by the micelles, and using the micellar diffusion coefficient instead of the molecular diffusion, one would get the Ostwald ripening rate in the presence of micelle. According to the extended LSW theory, the Ostwald ripening rate of nanoemulsion containing a water-insoluble low molecular weight coemulsifier (amphiphile) can be predicted

> 9RT

are the molecular diffusivity, bulk solubility (in water) and volume fraction of the coemulsifier in the oil droplet, respectively. Similarly, as the interfacial tension (γ) is incorporated in the ripening rate equation in the capillary length parameter (α = 2γVm/RT), lowering of the interfacial tension by the amphiphiles will lead to smaller capillary length and therefore lower solubility of the oil at the droplet boundary. This will cause lowering of the ripening rate at least for the case of strongly adsorbed amphiphiles. In the absence of micelle, oil molecule transports into and through the aqueous phase separating the droplets. However, in the presence of micelle, it is proposed that mass transfer is still involved through the continuous phase, but the micelles increase the water solubility of the oil,

therefore effectively increasing the transport rate [24–26]. Kabalnov [19] has proposed three possible mechanisms to explain the observed effects of micelles on Ostwald ripening; nevertheless, whether the oil molecules are taken up by the micelles directly from the aqueous medium or by fusion/fission of a micelle with a droplet surface is still unclear. Another possible way to slow down the Ostwald ripening rate was proposed by Higuchi and Misra [27] through the incorporation of a second disperse phase (oil) with a much lower continuous phase solubility. Initially, the concentration of the second insoluble phase is equal in all droplets. As the mixed oil nanoemulsion undergoes Ostwald ripening, the more soluble component diffuses from the smaller to larger droplet at a much faster rate than the less soluble one. Thus, time will come when the larger droplets become enriched with the soluble oil and the chemical potential of that component is equal in all the droplets. Thus, there is no driving force for further transfer of the oil to take place,

and the state is referred to as pseudo-steady of ripening. As theoretically investigated by Kabalnov et al. in the case of medium-soluble second

approximated as [25]

98

component, the Ostwald ripening rate of the mixed oil nanoemulsion can be

velocity <sup>v</sup> of a droplet of mass <sup>M</sup> is approximately given by <sup>v</sup> <sup>¼</sup> <sup>3</sup>kT

4.2 Adjustment in Ostwald ripening rate

Nanoemulsions - Properties, Fabrications and Applications

by the following equation [24], <sup>ω</sup> <sup>¼</sup> <sup>64</sup>γDcoCcoð Þ <sup>∞</sup> Vm

<sup>3</sup> with time in the asymptotic limit does not always signify the

M <sup>1</sup>=<sup>2</sup> .

ϕco, where Dco, Ccoð Þ ∞ and ϕco

linear relation of rN

$$a\_{\rm mix} = \left(\frac{\phi\_1}{\alpha\_1} + \frac{\phi\_2}{\alpha\_2}\right)^{-1},$$

where ϕ<sup>1</sup> and ϕ<sup>2</sup> and the ω<sup>1</sup> and ω<sup>2</sup> are the volume fraction and ripening rate of the medium-soluble and medium-insoluble component, respectively. In his treatment [25], Kabalnov came up with three stability regime depending on the mole fraction of the less soluble second component (X2) in the mixture. For high amount of insoluble oil (X2 > 0.4), entropy of mixing dominates, Δμ < 0, and the nanoemulsion is thermodynamically stable. For low mole fraction (X2 < 0.17), Laplace pressure between the different size droplets dominates, and the nanoemulsion is thermodynamically unstable. At intermediate mole fraction (0.17 < X2 < 0.4), a nanoemulsion is said to be in metastable state. Initially, Laplace pressure will dominate and the system undergoes Ostwald ripening; however, it also has a kinetic energy barrier that prevents further ripening to occur.

The validity of the LSW theory was tested by Kabalnov et al. [18, 19]. The influence of the alkyl chain length of the hydrocarbon on the Ostwald ripening rate of nanoemulsion was systematically investigated from alkyl chain length C9–C16. Increasing the alkyl chain length of the hydrocarbon used for the emulsion results in the decrease in the oil solubility. According to the LSW theory, this reduction in solubility should result in a decrease in the Ostwald ripening rate which was confirmed by Kabalnov et al.
