3.1 Role of surfactant in geometrical packing

We have already defined, in a limited sense, the word "surfactant" as an amphiphilic molecule which has the capacity to self-organize above a critical concentration. The process of self-assembly is dynamic in nature [13]. For those whose molecules at the air (or oil)-water interface are in exchange equilibrium with bulk solution are called soluble monolayer with a typical residence time on the order of 10�<sup>6</sup> s. On the other hand, for those whose molecules are in less dynamic situation at the monolayer when the interface is expanded or compressed are called insoluble monolayer, and the time taken for such molecular exchange can vary typically from seconds to months (Figure 3).

> most favored structure of these aggregates. A convenient parameter to analyse such diverse structures is the dimensionless number, <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup>, also known as "surfactant parameter" [17]. Close packing of amphiphiles leads to curved interfaces, and the direction of the curvature (either toward the polar or non-polar region) depends upon the value of this parameter. Here, lc is a semiempirical parameter called the critical chain length of the same order (or somewhat less) as the fully extended molecular length of the chain, lmax; a<sup>0</sup> is the optimal head group area; and v is the volume of the hydrocarbon chain [17]. Once these parameters are specified for a given molecule, one may ascertain the most preferred geometrical packing. Gradation of the preferred structure with increasing surfactant parameter has

condition (left), the interface curves toward chain region. For greater than one condition (right), the interface curves toward the polar region. When exactly equal to one (middle), the interface exhibits no preferential

View of the curvature of surfactant aggregates formed at various surfactant parameters, <sup>v</sup>

<sup>3</sup> for spherical to <sup>1</sup>

structures or lameller phases to <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ¼ 1 for vesicles or extended bilayer and finally to a family of "inverted structure" for <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup>>1. Therefore, it can be

concluded that in a non-condensed liquid phase the curvature is a function of the surfactant parameter at the liquid-liquid interface. Here, we find surface tension at work, and consequently it results in a pressure imbalance across a curved surface [6]. The origin of the tendency to minimize the surface energy of oil droplets in water is due to the imbalance in forces acting on a molecule at the interface compared to those acting in the bulk. From the basic fluid dynamics, we find that when a surface is curved there is a difference in pressure on the two sides of the surface which is described by the very important concept of "Laplace pressure" on the two sides of a curved (non-planer) surface, a spherical one being a special case of this

Consider a spherical cap symmetric about z-axis. The pressure exerted on the curved interface by the two bulk phases will be different and will give rise to a force acting along the normal to the interface at each point on the curve. The cap will also feel a force arising from surface tension acting tangentially at all points on the curve. At mechanical equilibrium, these two forces cancel each other along the z-

> <sup>z</sup> <sup>¼</sup> <sup>2</sup>πr<sup>2</sup> c γ r .

<sup>3</sup> ≤<sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≤ <sup>1</sup>

<sup>2</sup> ≤<sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≤1 for various interconnected

<sup>z</sup> <sup>¼</sup> <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>∑</sup>δ<sup>A</sup> <sup>¼</sup> <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>π</sup>r<sup>2</sup>

c.

<sup>2</sup> for ellipsoidal to

a0lc

. For less than one

been made as follows [13]: <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≤ <sup>1</sup>

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

3.2 Derivation of Laplace equation

Force arising from the pressure difference: F<sup>Δ</sup><sup>P</sup>

Force arising from surface tension: F<sup>γ</sup>

<sup>2</sup> for cylindrical or rodlike micelle to <sup>1</sup>

<sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≈ <sup>1</sup>

Figure 4.

curvature.

in general [6].

direction (Figure 5).

93

The self-assembly of amphiphiles strongly depends on the two key surface forces that act at the interface: the "hydrophobic attraction" between the surfactant tails which induces the association of the molecules and the "hydrophilic repulsion" between the surfactant heads which helps them to remain in contact with water [13]. Though the selection of emulsifier in the preparation of either O/W or W/O nanoemulsions is still made on an empirical basis, however, a semiempirical scale is defined based on the relative percentage of hydrophilic to lipophilic groups in the surfactant molecules, known as hydrophilic-lipophilic balance (HLB number) [14], the HLB number is deduced from the preferential solubility of the surfactant in oil or water and HLB number can vary from 0 (very soluble in oil) to 20 (very soluble in water). Griffin postulated a simple equation to calculate the HLB number for non-ionic surfactant such as fatty acid ester [14], HLB <sup>¼</sup> <sup>20</sup> � <sup>1</sup> � <sup>S</sup> A , where S is the saponification number and A is the acid number (Figure 4). Davies developed a method for calculating the HLB number for surfactants irrespective of their chemical nature, using empirically determined group numbers [15]: HLB ¼ 7þ ∑ðhydrophilic group noÞ � ∑lipophilic group no. Beerbower and Hills used the following expression for the HLB number [16]: HLB <sup>¼</sup> <sup>20</sup> MH MLþMH <sup>¼</sup> <sup>20</sup> VHρ<sup>H</sup> VLρLþVHρ<sup>H</sup> , where MH and ML are the molecular weights of the hydrophilic and lipophilic portions of the surfactants. VH and VL are their corresponding molar volumes, whereas ρ<sup>H</sup> and ρ<sup>L</sup> are the respective densities. After adequately describing the

interactions between the amphiphiles within an aggregate, we need to establish the

#### Figure 3.

Schematic representation of spherical micelle with illustration of different surfactant parameters. Packing factor = <sup>V</sup> lca<sup>0</sup> :The chain volume, v, and chain length lc set the aggregation limit.

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

Figure 4.

3. Formation of oil in water droplets from geometric and force balanced

We have already defined, in a limited sense, the word "surfactant" as an amphiphilic molecule which has the capacity to self-organize above a critical concentration. The process of self-assembly is dynamic in nature [13]. For those whose molecules at the air (or oil)-water interface are in exchange equilibrium with bulk solution are called soluble monolayer with a typical residence time on the order of 10�<sup>6</sup> s. On the other hand, for those whose molecules are in less dynamic situation at the monolayer when the interface is expanded or compressed are called insoluble monolayer, and the time taken for such molecular exchange can vary typically

The self-assembly of amphiphiles strongly depends on the two key surface forces that act at the interface: the "hydrophobic attraction" between the surfactant tails which induces the association of the molecules and the "hydrophilic repulsion" between the surfactant heads which helps them to remain in contact with water [13]. Though the selection of emulsifier in the preparation of either O/W or W/O nanoemulsions is still made on an empirical basis, however, a semiempirical scale is defined based on the relative percentage of hydrophilic to lipophilic groups in the surfactant molecules, known as hydrophilic-lipophilic balance (HLB number) [14], the HLB number is deduced from the preferential solubility of the surfactant in oil or water and HLB number can vary from 0 (very soluble in oil) to 20 (very soluble in water). Griffin postulated a simple equation to calculate the HLB number for

the saponification number and A is the acid number (Figure 4). Davies developed a method for calculating the HLB number for surfactants irrespective of their chemical nature, using empirically determined group numbers [15]: HLB ¼ 7þ ∑ðhydrophilic group noÞ � ∑lipophilic group no. Beerbower and Hills used the fol-

where MH and ML are the molecular weights of the hydrophilic and lipophilic portions of the surfactants. VH and VL are their corresponding molar volumes, whereas ρ<sup>H</sup> and ρ<sup>L</sup> are the respective densities. After adequately describing the interactions between the amphiphiles within an aggregate, we need to establish the

Schematic representation of spherical micelle with illustration of different surfactant parameters. Packing

lca<sup>0</sup> :The chain volume, v, and chain length lc set the aggregation limit.

A , where S is

> <sup>¼</sup> <sup>20</sup> VHρ<sup>H</sup> VLρLþVHρ<sup>H</sup>

,

MLþMH 

non-ionic surfactant such as fatty acid ester [14], HLB <sup>¼</sup> <sup>20</sup> � <sup>1</sup> � <sup>S</sup>

lowing expression for the HLB number [16]: HLB <sup>¼</sup> <sup>20</sup> MH

point of view

Figure 3.

factor = <sup>V</sup>

92

3.1 Role of surfactant in geometrical packing

Nanoemulsions - Properties, Fabrications and Applications

from seconds to months (Figure 3).

View of the curvature of surfactant aggregates formed at various surfactant parameters, <sup>v</sup> a0lc . For less than one condition (left), the interface curves toward chain region. For greater than one condition (right), the interface curves toward the polar region. When exactly equal to one (middle), the interface exhibits no preferential curvature.

most favored structure of these aggregates. A convenient parameter to analyse such diverse structures is the dimensionless number, <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup>, also known as "surfactant parameter" [17]. Close packing of amphiphiles leads to curved interfaces, and the direction of the curvature (either toward the polar or non-polar region) depends upon the value of this parameter. Here, lc is a semiempirical parameter called the critical chain length of the same order (or somewhat less) as the fully extended molecular length of the chain, lmax; a<sup>0</sup> is the optimal head group area; and v is the volume of the hydrocarbon chain [17]. Once these parameters are specified for a given molecule, one may ascertain the most preferred geometrical packing. Gradation of the preferred structure with increasing surfactant parameter has been made as follows [13]: <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≤ <sup>1</sup> <sup>3</sup> for spherical to <sup>1</sup> <sup>3</sup> ≤<sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≤ <sup>1</sup> <sup>2</sup> for ellipsoidal to <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≈ <sup>1</sup> <sup>2</sup> for cylindrical or rodlike micelle to <sup>1</sup> <sup>2</sup> ≤<sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ≤1 for various interconnected structures or lameller phases to <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup> ¼ 1 for vesicles or extended bilayer and finally to a family of "inverted structure" for <sup>v</sup> <sup>a</sup><sup>0</sup> l = <sup>c</sup>>1. Therefore, it can be concluded that in a non-condensed liquid phase the curvature is a function of the surfactant parameter at the liquid-liquid interface. Here, we find surface tension at work, and consequently it results in a pressure imbalance across a curved surface [6]. The origin of the tendency to minimize the surface energy of oil droplets in water is due to the imbalance in forces acting on a molecule at the interface compared to those acting in the bulk. From the basic fluid dynamics, we find that when a surface is curved there is a difference in pressure on the two sides of the surface which is described by the very important concept of "Laplace pressure" on the two sides of a curved (non-planer) surface, a spherical one being a special case of this in general [6].

#### 3.2 Derivation of Laplace equation

Consider a spherical cap symmetric about z-axis. The pressure exerted on the curved interface by the two bulk phases will be different and will give rise to a force acting along the normal to the interface at each point on the curve. The cap will also feel a force arising from surface tension acting tangentially at all points on the curve. At mechanical equilibrium, these two forces cancel each other along the zdirection (Figure 5).

Force arising from the pressure difference: F<sup>Δ</sup><sup>P</sup> <sup>z</sup> <sup>¼</sup> <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>∑</sup>δ<sup>A</sup> <sup>¼</sup> <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>π</sup>r<sup>2</sup> c.

Force arising from surface tension: F<sup>γ</sup> <sup>z</sup> <sup>¼</sup> <sup>2</sup>πr<sup>2</sup> c γ r .

At equilibrium the forces along z-direction: F<sup>Δ</sup><sup>P</sup> <sup>z</sup> <sup>¼</sup> <sup>F</sup><sup>γ</sup> <sup>z</sup>; or <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>π</sup>r<sup>2</sup> <sup>c</sup> <sup>¼</sup> <sup>2</sup>πr<sup>2</sup> c γ r . So, <sup>Δ</sup><sup>P</sup> <sup>¼</sup> <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>¼</sup> <sup>2</sup><sup>γ</sup> <sup>r</sup> , which is the Laplace equation for spherical surface. For nonspherical interface, two orthogonal radii of curvature (r<sup>0</sup> , r<sup>00</sup>) are needed, the Laplace equation then becomes <sup>Δ</sup><sup>P</sup> <sup>¼</sup> <sup>γ</sup> <sup>1</sup> <sup>r</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup> r <sup>00</sup> <sup>¼</sup> <sup>2</sup><sup>γ</sup> rm, where rm is the mean curvature (inverse of radius) and is equal to <sup>1</sup> rm <sup>¼</sup> <sup>1</sup> 2 1 <sup>r</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup> r <sup>00</sup> . By our convention, if the interface encloses hydrophobic region, the mean curvature would be negative. Therefore, the Laplace pressure always drives the interface in the concave direction if the molecules are expanded or contracted. The high energy required for the formation of nanoemulsion droplets can be understood from the inverse relation of the pressure difference with the radius. The Laplace equation forms the theoretical basis of "Kelvin equation," which describes the effect of surface curvature of a liquid that changes with the equilibrium vapor pressure of the liquid [13]:

size scale of the second phase and accordingly a decrease in total interfacial area. Such a process is known as "Ostwald ripening." The driving force of the ripening process is the dependence of oil solubility on its size, as described by the Kelvin

> 2γVm RTrm

where Vm is the molar volume of the oil and γ is the surface tension. This equation relates the solubility of droplet C rð Þ with an arbitrary radius r to that of an

RTrm

as the capillary length of the drop, typically on the order of �1 nm. The Kelvin equation is derived from the dependence of chemical potential (μ) of the dispersed

> ð<sup>2</sup>γ=<sup>r</sup> 0

Another driving force for Ostwald ripening to occur in nanoemulsions is due to the polymorphic changes during redeposition of solute (such as drug molecules).

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.

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

� � <sup>≈</sup>Cð Þ <sup>∞</sup> <sup>1</sup> <sup>þ</sup>

2γVm RTrm � �,

� � has the dimension of length and is termed

<sup>r</sup> :

Vmð Þ <sup>p</sup> dp<sup>≈</sup> <sup>2</sup>γVm

C rð Þ¼ Cð Þ ∞ exp

Δμð Þ¼ r μð Þr –μð Þ¼ ∞

4. Theory of Ostwald ripening in nano-dispersion

infinite radius <sup>C</sup>ð Þ <sup>∞</sup> . The quantity <sup>2</sup>πVm

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

phase on its size by the relation [19]:

4.1 Scaling the ripening problem

95

equation [18, 19]:

 $\ln\left(\frac{p}{p\*}\right) = \left(\frac{2\sqrt{V\_m}}{r\_n RT}\right)$ , and we immediately obtain the Kelvin radius: 
$$r\_k = \left(\frac{1}{r\_1} + \frac{1}{r\_2}\right)^{-1} = \frac{\frac{\mathcal{V}V\_m}{RT \ln\left(\frac{p}{p\*}\right)}}{RT \ln\left(\frac{p\*}{p\*}\right)}$$
, where  $P$  and  $P^{\infty}$  are, respectively, the vapor pressure over the curved surface with mean curvature  $r\_m$  and a flat surface  $(r = \infty)$ ,  $V\_m$ .

sures over the curved surface with mean curvature rm and a flat surface ð Þ r ¼ ∞ , Vm is the molar volume of the oil, γ is the interfacial surface tension, R is the gas constant and T is the absolute temperature.

### 3.3 The consequences of Kelvin equation are profound

One of them is to govern the process such as the growth of larger droplet in expanse of the smaller one in nanoemulsion. The vapor pressure of a spherical droplet will be greater than that of the same liquid with a flat surface. The smaller the radius, the higher the vapor pressure such that if there is a distribution of droplet size, the smaller one will tend to diminish, while the larger ones will tend to grow. The total energy of the two-phase system thus can be decreased via an increase in the

#### Figure 5.

Resolution of forces on spherical cap symmetrical about the z-axis and part of a spherical interface. The pressures exerted on the interface by the two bulk phases (α and β) will be different if the interface is curved, and this difference (ΔP) will give rise to a force acting along the normal to the interface at each point. The cap will also be subject to a force arising from surface tension acting tangentially at all points around the perimeter of the cap.

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

At equilibrium the forces along z-direction: F<sup>Δ</sup><sup>P</sup>

Nanoemulsions - Properties, Fabrications and Applications

the Laplace equation then becomes <sup>Δ</sup><sup>P</sup> <sup>¼</sup> <sup>γ</sup> <sup>1</sup>

curvature (inverse of radius) and is equal to <sup>1</sup>

<sup>¼</sup> <sup>γ</sup>Vm RTln Pc ð Þ <sup>P</sup><sup>∞</sup>

constant and T is the absolute temperature.

3.3 The consequences of Kelvin equation are profound

For nonspherical interface, two orthogonal radii of curvature (r<sup>0</sup>

So, <sup>Δ</sup><sup>P</sup> <sup>¼</sup> <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>¼</sup> <sup>2</sup><sup>γ</sup>

ln Pc P<sup>∞</sup> <sup>¼</sup> <sup>2</sup>γVm rmRT 

Figure 5.

94

perimeter of the cap.

rk <sup>¼</sup> <sup>1</sup>

<sup>r</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup> r2 �<sup>1</sup> <sup>z</sup> <sup>¼</sup> <sup>F</sup><sup>γ</sup>

<sup>r</sup> , which is the Laplace equation for spherical surface.

<sup>¼</sup> <sup>2</sup><sup>γ</sup>

, where Pc and P<sup>∞</sup> are, respectively, the vapor pres-

<sup>r</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup> r 00 

the interface encloses hydrophobic region, the mean curvature would be negative. Therefore, the Laplace pressure always drives the interface in the concave direction if the molecules are expanded or contracted. The high energy required for the formation of nanoemulsion droplets can be understood from the inverse relation of the pressure difference with the radius. The Laplace equation forms the theoretical basis of "Kelvin equation," which describes the effect of surface curvature of a liquid that changes with the equilibrium vapor pressure of the liquid [13]:

, and we immediately obtain the Kelvin radius:

sures over the curved surface with mean curvature rm and a flat surface ð Þ r ¼ ∞ , Vm is the molar volume of the oil, γ is the interfacial surface tension, R is the gas

One of them is to govern the process such as the growth of larger droplet in expanse of the smaller one in nanoemulsion. The vapor pressure of a spherical droplet will be greater than that of the same liquid with a flat surface. The smaller the radius, the higher the vapor pressure such that if there is a distribution of droplet size, the smaller one will tend to diminish, while the larger ones will tend to grow. The total energy of the two-phase system thus can be decreased via an increase in the

Resolution of forces on spherical cap symmetrical about the z-axis and part of a spherical interface. The pressures exerted on the interface by the two bulk phases (α and β) will be different if the interface is curved, and this difference (ΔP) will give rise to a force acting along the normal to the interface at each point. The cap will also be subject to a force arising from surface tension acting tangentially at all points around the

rm <sup>¼</sup> <sup>1</sup> 2 1 <sup>r</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup> r 00 

<sup>z</sup>; or <sup>P</sup><sup>α</sup> � <sup>P</sup><sup>β</sup> <sup>π</sup>r<sup>2</sup>

, r<sup>00</sup>

. By our convention, if

rm, where rm is the mean

<sup>c</sup> <sup>¼</sup> <sup>2</sup>πr<sup>2</sup> c γ r .

) are needed,

size scale of the second phase and accordingly a decrease in total interfacial area. Such a process is known as "Ostwald ripening." The driving force of the ripening process is the dependence of oil solubility on its size, as described by the Kelvin equation [18, 19]:

$$\mathbf{C}(r) = \mathbf{C}(\infty) \exp\left(\frac{2\gamma V\_m}{RTr\_m}\right) \approx \mathbf{C}(\infty) \left(\mathbf{1} + \frac{2\gamma V\_m}{RTr\_m}\right),$$

where Vm is the molar volume of the oil and γ is the surface tension. This equation relates the solubility of droplet C rð Þ with an arbitrary radius r to that of an infinite radius <sup>C</sup>ð Þ <sup>∞</sup> . The quantity <sup>2</sup>πVm RTrm � � has the dimension of length and is termed as the capillary length of the drop, typically on the order of �1 nm. The Kelvin equation is derived from the dependence of chemical potential (μ) of the dispersed phase on its size by the relation [19]:

$$
\Delta\mu(r) = \mu(\mathbf{r})\text{-}\mu(\infty) = \int\_0^{2\gamma/r} V\_m(p)dp \approx \frac{2\gamma V\_m}{r}.
$$

Another driving force for Ostwald ripening to occur in nanoemulsions is due to the polymorphic changes during redeposition of solute (such as drug molecules).
