2. Thermodynamics and kinetics of nanoemulsion

#### 2.1 Free energy diagram

The thermodynamic stability of a particular system is governed by the change in free energy between it and an appropriate reference state. Nanoemulsion is thermodynamically unstable, which means that the free energy of nanoemulsion is

higher than the free energy of the separate phases (oil and water). For microemulsion, this condition is opposite. To understand the molecular basis of the free energy difference, let us consider that a system (nano-/microemulsion) exists in equilibrium between the initial and final states. The free energy change associated with the formation of the dispersion consists of an interfacial free energy term (ΔGI) and a configuration entropy term (�TΔSconfig) [9]:

$$
\Delta \mathbf{G}\_{\text{formation}} = \Delta \mathbf{G}\_{I} - T \Delta \mathbf{S}\_{\text{config}} \dots
$$

The free energy to increase the contact area (ΔA) at the interface is ΔGI ¼ γΔA, which is always positive; consequently, this term always opposes the formation of the dispersions. However, the interfacial tension (γ) depends on the curvature of the surfactant layer-decreasing as the curvature approaches to its optimum value. A phenomenological description of the dependence of interfacial tension on droplet curvature can be formulated as [10–12]:

$$
\lambda = \lambda\_0 + (\lambda\_\infty + \chi\_0) \frac{(R\_0 - R)^2}{R\_0^2 + R^2},
$$

can assume that thermodynamically stable dispersion can be achieved when droplet radius is close to the optimum (R = R0) and hence the interfacial free energy that

Predicted variation of free energy change with droplet radius for the formation of emulsion state assuming (a) constant interfacial tension and (b) varying interfacial tension with curvature. Taken from Reference [3].

Kinetic stability, as to be contrasted with the thermodynamic stability, is determined by two important factors [3]: (1) Energy barriers: any energy barrier

(or activation energy) that separates the two states (final and initial) will determine the rate of the conversion of one state to another. The height of this energy barrier depends on the forces operating in close proximity of two droplets, such as repulsive hydrodynamic and colloidal (steric and electrostatic) interactions [13]. Nanoemulsion can be made kinetically stable by introducing sufficiently

large energy barrier (typically >20 kT) between the two states, while for

in a nanoemulsion are particularly labile toward the growth over time by a

micelle. In mechanism three, a large number of oil molecules are released from the oil droplets collectively with excess of surfactant molecules to form

microemulsion, there is still an activation energy required (in terms of mechanical agitation or heating the system) to reach the thermodynamically stable state after the components are brought into contact. (2) Mass transport phenomena: droplets

process known as "Ostwald ripening" [8], in which solute molecules (or mass) are exchanged between the droplets via molecular diffusion through the solvent. Three alternative mechanisms have been proposed in the literature suggestive of that micelle which plays an important role in facilitating the mass exchange between the droplets by acting as carriers of oil molecules [8]. In mechanism one, oil molecules are transferred via direct micelle collisions, i.e., the rate is directly proportional to the volume fraction of micelle in solution. Numerous studies indicate a higher rate of mass transport above the CMC of the surfactant used. The lack of such CMC dependence for ionic surfactants however, may stem from the electrostatic repulsion between the droplet and micelle. In mechanism two, oil molecules exit the droplet, are exposed to the continuous phase and are soon captured by micelles in the immediate vicinity of the droplet. For non-ionic surfactant micelles, the higher rate of mass transfer is expected due to their higher solubilization capacity and the absence of electrostatic repulsion between the droplet and uncharged

otherwise opposes the emulsification can be appreciably reduced.

2.2 Kinetic picture

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

Figure 2.

a new micelle.

91

where γ<sup>∞</sup> and γ<sup>0</sup> are, respectively, the interfacial tension values at the planar O/W interface and when the surfactant layer reaches its optimum curvature. R<sup>0</sup> is the droplet radius at the optimum curvature.

On the other hand, configuration entropy (ΔSconfig) which depends on the number of arrangements accessible to the oil phase in an emulsified state is much greater than that in a non-emulsified state, and therefore it always favors the formation of the dispersion. An expression for the ΔSconfig can be derived from the statistical analysis [9]:

$$
\Delta \mathbf{S}\_{\rm config} = -\frac{nk}{\phi} \left( \phi ln \phi + (\mathbf{1} - \phi) \ln \left( \mathbf{1} - \phi \right) \right),
$$

where k is the Boltzmann constant, n is the number of droplets and ϕ is the disperse-phase volume fraction.

The plot for the different free energies with droplet radius assuming that the interfacial tension is the same as that for planer O/W interface is shown in Figure 2(a) [3]. It can be observed from the figure that the interfacial free energy contribution increases (hence unfavorable) with decrease in droplet size (increase in interfacial area), while the configuration of free energy becomes progressively negative (hence favorable) with decrease in droplet size (increase in number of different ways that droplet can be organized). The total free energy, however, becomes increasingly positive with decrease in droplet size, since the interfacial free energy term dominates the configuration entropy term.

This implies that the formation of nanoemulsion becomes increasingly thermodynamically unfavorable as the radius of the droplets fall and where the interfacial tension is similar to that at a planar surface. Interestingly, if the calculation were performed assuming the dependence of the interfacial tension on the curvature, the situation becomes more complex [3]. As shown in Figure 2(b), with decrease in droplet radius from say, 1000 nm, there is an increase in interfacial free energy. But once the droplet becomes smaller below a certain radius, the interfacial free energy decreases and reaches a minimum value, before rising again with further decrease in radius. The interfacial free energy contribution still is the dominant factor, forcing the total free energy to attain a minimum value that is close to the optimum curvature of surfactant layer. Thus, under this consideration, one

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

#### Figure 2.

higher than the free energy of the separate phases (oil and water). For microemulsion, this condition is opposite. To understand the molecular basis of the free energy difference, let us consider that a system (nano-/microemulsion) exists in equilibrium between the initial and final states. The free energy change associated with the formation of the dispersion consists of an interfacial free energy term (ΔGI) and a

ΔGformation ¼ ΔGI � TΔSconfig :

<sup>γ</sup> <sup>¼</sup> <sup>γ</sup><sup>0</sup> <sup>þ</sup> <sup>γ</sup><sup>∞</sup> <sup>þ</sup> <sup>γ</sup><sup>0</sup> ð Þð Þ <sup>R</sup><sup>0</sup> � <sup>R</sup> <sup>2</sup>

where γ<sup>∞</sup> and γ<sup>0</sup> are, respectively, the interfacial tension values at the planar O/W interface and when the surfactant layer reaches its optimum curvature. R<sup>0</sup> is the

On the other hand, configuration entropy (ΔSconfig) which depends on the number of arrangements accessible to the oil phase in an emulsified state is much greater than that in a non-emulsified state, and therefore it always favors the formation of the dispersion. An expression for the ΔSconfig can be derived from the statistical

where k is the Boltzmann constant, n is the number of droplets and ϕ is the

This implies that the formation of nanoemulsion becomes increasingly thermodynamically unfavorable as the radius of the droplets fall and where the interfacial tension is similar to that at a planar surface. Interestingly, if the calculation were performed assuming the dependence of the interfacial tension on the curvature, the situation becomes more complex [3]. As shown in Figure 2(b), with decrease in droplet radius from say, 1000 nm, there is an increase in interfacial free energy. But

once the droplet becomes smaller below a certain radius, the interfacial free energy decreases and reaches a minimum value, before rising again with further decrease in radius. The interfacial free energy contribution still is the dominant factor, forcing the total free energy to attain a minimum value that is close to the optimum curvature of surfactant layer. Thus, under this consideration, one

The plot for the different free energies with droplet radius assuming that the interfacial tension is the same as that for planer O/W interface is shown in Figure 2(a) [3]. It can be observed from the figure that the interfacial free energy contribution increases (hence unfavorable) with decrease in droplet size (increase in interfacial area), while the configuration of free energy becomes progressively negative (hence favorable) with decrease in droplet size (increase in number of different ways that droplet can be organized). The total free energy, however, becomes increasingly positive with decrease in droplet size, since the interfacial free

R2 <sup>0</sup> <sup>þ</sup> <sup>R</sup><sup>2</sup> ,

<sup>ϕ</sup> ð Þ <sup>ϕ</sup>ln<sup>ϕ</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>ϕ</sup> ln 1ð Þ � <sup>ϕ</sup> ,

The free energy to increase the contact area (ΔA) at the interface is ΔGI ¼ γΔA, which is always positive; consequently, this term always opposes the formation of the dispersions. However, the interfacial tension (γ) depends on the curvature of the surfactant layer-decreasing as the curvature approaches to its optimum value. A phenomenological description of the dependence of interfacial tension on droplet

configuration entropy term (�TΔSconfig) [9]:

Nanoemulsions - Properties, Fabrications and Applications

curvature can be formulated as [10–12]:

droplet radius at the optimum curvature.

disperse-phase volume fraction.

<sup>Δ</sup>Sconfig ¼ � nk

energy term dominates the configuration entropy term.

analysis [9]:

90

Predicted variation of free energy change with droplet radius for the formation of emulsion state assuming (a) constant interfacial tension and (b) varying interfacial tension with curvature. Taken from Reference [3].

can assume that thermodynamically stable dispersion can be achieved when droplet radius is close to the optimum (R = R0) and hence the interfacial free energy that otherwise opposes the emulsification can be appreciably reduced.

#### 2.2 Kinetic picture

Kinetic stability, as to be contrasted with the thermodynamic stability, is determined by two important factors [3]: (1) Energy barriers: any energy barrier (or activation energy) that separates the two states (final and initial) will determine the rate of the conversion of one state to another. The height of this energy barrier depends on the forces operating in close proximity of two droplets, such as repulsive hydrodynamic and colloidal (steric and electrostatic) interactions [13]. Nanoemulsion can be made kinetically stable by introducing sufficiently large energy barrier (typically >20 kT) between the two states, while for microemulsion, there is still an activation energy required (in terms of mechanical agitation or heating the system) to reach the thermodynamically stable state after the components are brought into contact. (2) Mass transport phenomena: droplets in a nanoemulsion are particularly labile toward the growth over time by a process known as "Ostwald ripening" [8], in which solute molecules (or mass) are exchanged between the droplets via molecular diffusion through the solvent. Three alternative mechanisms have been proposed in the literature suggestive of that micelle which plays an important role in facilitating the mass exchange between the droplets by acting as carriers of oil molecules [8]. In mechanism one, oil molecules are transferred via direct micelle collisions, i.e., the rate is directly proportional to the volume fraction of micelle in solution. Numerous studies indicate a higher rate of mass transport above the CMC of the surfactant used. The lack of such CMC dependence for ionic surfactants however, may stem from the electrostatic repulsion between the droplet and micelle. In mechanism two, oil molecules exit the droplet, are exposed to the continuous phase and are soon captured by micelles in the immediate vicinity of the droplet. For non-ionic surfactant micelles, the higher rate of mass transfer is expected due to their higher solubilization capacity and the absence of electrostatic repulsion between the droplet and uncharged micelle. In mechanism three, a large number of oil molecules are released from the oil droplets collectively with excess of surfactant molecules to form a new micelle.
