Abstract
A comprehensive, yet simple, theoretical model for droplet microemulsions is presented. The model combines thermodynamics of self-assembly with bending elasticity theory and relates microemulsion properties, such as average droplet size, polydispersity, interfacial tension and solubilisation capacity with the three bending elasticity constants, spontaneous curvature (H 0), bending rigidity (kc) and saddle-splay constant (k¯c). In addition, the self-association entropy constant (ks) explicitly determines various microemulsion properties. The average droplet size is shown to increase with increasing effective bending constant, defined as keff=2kc+k¯c+ks, as well as with decreasing magnitudes of H0. The polydispersity decreases with increasing values of keff, but does not at all depend on H0. The model predicts ultra-low interfacial tensions, the values of which decrease considerably with increasing droplet radius, in agreement with experiments. The solubilisation capacity increases as the number of droplets is increased with increasing surfactant concentration. In addition, an enhanced solubilisation effect is obtained as the size of the droplets increases with increasing surfactant concentration, as a result of self-association entropy effects. It is demonstrated that self-association entropy effects favour smaller droplet size as well as larger droplet polydispersity.
Keywords
- surfactant
- self-assembly
- microemulsion
- interfacial tension
- solubilisation
- spontaneous curvature
- bending rigidity
- saddle-splay constant
1. Introduction
Surfactants are amphiphilic molecules consisting of a hydrophilic head group and a hydrophobic tail. Surfactants self-assemble above a certain surfactant concentration in an aqueous solvent to form micelles or bilayers. The driving force for the self-assembly process is the tendency to minimize the interfacial contact area between water and surfactant hydrophobic tails. As a result, the tails make up an interior core of the micelles that is absent of water whereas the hydrophilic head groups are located at the interface adjacent to the aqueous solvent. Hydrophobic components that are usually insoluble in water, such as oil or fat, may be dissolved into a surfactant-water system. The oil molecules become incorporated into the hydrophobic core of the micelles to form swollen micelles. However, there is a limit how much oil that may be dissolved by micelles and above a certain amount of oil two separate liquid phases coexist, i.e. thermodynamically stable oil droplets (discrete phase) dissolved in water (continuous phase) coexisting with excess oil (cf. Figure 1). Such an oil-in-water (o/w) microemulsion phase coexisting with an excess oil phase in a two-phase system is denoted as Winsor I microemulsion. Some surfactants, on the other hand, may form reversed water-in-oil (w/o) droplets that dissolve water (discrete phase) in oil (continuous phase) in the presence of excess water (Winsor II microemulsion). Moreover, a microemulsion phase may also coexist with both excess oil and excess water in a three-phase system (Winsor III microemulsion) [1]. A Winsor III microemulsion does not usually consist of finite-sized droplets. Rather some kind of macroscopic structure is formed, either more or less planar alternating layers of oil and water separated by surfactant monolayers, or some kind of ordered or disordered bicontinuous phase with a system of separated tunnels of water and oil [2].
The microscopic structure of microemulsions (whether Winsor I, II or III), is mainly determined by the chemical structure of the surfactant. Surfactants with large hydrophilic head groups and small hydrophobic (or lipophilic if one prefers) parts tend to curve so as to form ordinary micelles or o/w microemulsion droplets whereas surfactants with a small hydrophilic part and large hydrophobic part (like phospholipids) tend to form w/o microemulsion droplets (cf. Figure 1). The molecular properties determining the curvature of a surfactant monolayer may be summed up in the quantity hydrophilic-lipophilic balance, or shorter the HLB value [3]. As a result, one expects o/w microemulsion droplets to increase in size as the monolayer becomes less curved with decreasing HLB values. For sufficiently low HLB values, a transition from Winsor I to Winsor III is expected, and further decreasing HLB would result in the formation of Winsor II microemulsion. In addition to the volume of the hydrophilic and hydrophobic parts, respectively, the presence of electric charge on the surfactant head group also contributes to increase the HLB value and promotes a positive (oil-in-water) curvature of the droplets.
It is possible to tune the HLB value of ionic surfactants by means of adding salt to an aqueous phase and HLB decreases with increasing electrolyte concentration. Similarly, the HLB value of non-ionic ethylene oxide-based surfactants is found to be considerably temperature-sensitive and, as a result, it is possible to observe the sequence of transitions oil-in-water → bicontinuous → water-in-oil microemulsion by means of increasing the temperature [4]. The amount of oil dissolved in the microemulsion increases during this sequence of transitions and, at a certain temperature denoted the phase inversion temperature (PIT), the microemulsion phase contains equal amounts of oil and water. It is also possible to tune the curvature of a surfactant monolayer by means of adding a cosurfactant that is mixed into the layer [5].
In contrast to conventional (macro)emulsions, microemulsions are thermodynamically stable systems. This means that the size of the droplets may fall within a wide range, depending on the surfactant HLB value, from small micelle-like droplets of about 1 nm to about 100 nm, above the size of which the droplets usually transform into a macroscopic bicontinuous structure. In contrast, kinetically stabilized emulsion droplets are usually larger falling in the range 10 nm to 100 μm.
In this chapter, we present a comprehensive, yet simple, theory that rationalizes the structural behaviour of spherical microemulsion droplets. The theory is based on conventional solution thermodynamics combined with bending elasticity theory and it predicts several experimentally available quantities, such as droplet size and polydispersity, interfacial tension and solubilisation capacity. We only consider the case of rather rigid microemulsion droplets that are spherically shaped. More flexible droplets, consisting of interfacial monolayers with bending rigidities approaching zero (see further below), may assume a more spheroidal or ellipsoidal shape [6] or undergo undulatory fluctuations of the droplet interfaces [7].
2. Thermodynamics of self-assembly
Microemulsions are thermodynamically stable equilibrium structures. Hence, the theoretical treatment of microemulsion droplets necessarily needs to take into account thermodynamics of self-assembling surfactant molecules to form an interfacial monolayer that encapsulates the discrete phase (cf. Figure 1). The latter process may be considered in terms of a set of multiple equilibrium reactions
where
where
In order to allow for the spontaneous self-assembly of surfactant molecules, additional contributions to the overall free energy change must be added. These contributions are related to the formation of the surfactant monolayer and can be collected into a single quantity denoted by Δ
Introducing the following free energy parameter [8].
and combining Eqs. (2)–(4) gives the following set of equilibrium conditions
one for each surfactant aggregation number
Summing up the different volume fractions in Eq. (6) gives the total volume fraction
which may be evaluated from the following integral approximation
We have been able to approximately set the lower limit in the integral to zero since the largely curved droplets with low aggregation numbers are too energetically unfavourable to contribute to the integral (see further below). Equation (8) gives the full size distribution of surfactants self-assembled in microemulsion-droplet interfaces; in so far an mathematical expression for the function E(
3. Bending elasticity
The free energy of an arbitrarily shaped surfactant monolayer may be calculated taking into account bending elasticity properties. The curvature at a single point on a surfactant monolayer (most conveniently defined at the hydrocarbon/water interface [cf. Figure 1]) may be defined by considering two perpendicular curves on the interface with radii of curvature,
The Helfrich expression in Eq. (9) is a second-order expansion with respect to
For a spherically curved monolayer, an expression for E is obtained by simply introducing the proper values
where
The three bending elasticity constants
3.1. Spontaneous curvature
The spontaneous curvature
3.2. Bending rigidity
The bending rigidity
From a molecular point of view, it has been demonstrated that
3.3. Saddle-splay constant
The saddle-splay constant
Moreover, according to the Gauss-Bonnet theorem, the last integral in Eq. (10) equals
where
From a molecular point of view,
4. Microemulsion droplets
The size distribution of surfactant monolayers in Eq. (8) may be rewritten so as to give
where
may be evaluated from the geometrical relation
The geometrical relations
Now we may substitute Eq. (11) into Eq. (13) and combine with Eqs. (14) and (15) to arrive at the full size distribution of spherical microemulsion droplets
from which the total volume fraction of droplets (including both surfactant and the discrete phase) may be evaluated so as to give
The size distribution according to Eq. (16) is plotted in Figure 2 for some different values of volume fraction of droplets. In accordance with Eq. (16), the formation of finite-sized microemulsion droplets may be considered as the result of bending elasticity properties as well as self-association entropy effects. The latter contribution always favours small droplets. Since the driving force towards smaller droplets due to the entropy of self-assembly increases in magnitude with decreasing droplet volume fraction, the average size of the droplets depends on
Equation (17) may be solved so as to give
where we have introduced the dimensionless parameter [6, 23]
as well as the quantity
Except for values of
The entropy parameter
5. Average droplet size
The following expression for the (volume weighted) average droplet radius may be derived from the size distribution in Eq. (16)
where the quantity
only depends on
Combining Eqs. (23) and (25) gives the following rather simple expression for the average droplet radius
According to Eqs. (23) and (26), the average droplet size depends on
In addition to the three bending elasticity constants, the droplet size also depends on the self-assembly entropy parameter
6. Polydispersity
Like average droplet size, microemulsion droplet polydispersity may be predicted from the present model. From the size distribution function in Eq. (16), we may derive the following expression for the average of the squared radius
where
Hence, the polydispersity in terms of the relative standard deviation equals
in accordance with Eqs. (23) and (27) and where the definition of
The polydispersity according to Eqs. (29) and (30), respectively, is plotted in Figure 5. Notably, the droplet polydispersity does not at all depend on spontaneous curvature, but is seen to be a sole function of
7. Interfacial tension
The (planar) interfacial tension between the water and oil phases in a microemulsion in equilibrium with excess solvent phase is an experimentally available quantity. Hence, it has been observed that unusually small interfacial tensions are, in general, generated in microemulsion systems. The values are found to decrease in magnitude as the droplet size increases and may approach as low values as 10−3–10−2 mNm−1 in non-ionic surfactant microemulsion systems at the phase inversion temperature (PIT) [10, 25].
From our present model we obtain the planar interfacial tension
Similar to droplet size, the interfacial tension depends on the three bending elasticity constants as well as the entropy parameter
By means of combining Eqs. (26) and (31), we may eliminate
According to Eq. (32), the interfacial tension must decrease in magnitude as the size of the droplets increases (cf. Figure 7). This behaviour agrees very well with experimental observations that surfactants forming larger microemulsion droplets are, in general, found to have smaller interfacial tensions.
Since
The relation between interfacial tension, droplet radius and polydispersity may also be expressed in terms of the dimensionless parameter [27]
Combining Eqs. (26), (30) and (31), the following approximate expression for
In the limit
8. Solubilisation
A notable property of microemulsion systems is the ability to dissolve hydrophobic components which otherwise are insoluble in an aqueous solvent. A quantitative measure of the solubilisation capacity may be considered as the ratio of the volume of oil (
Since the surfactant is exclusively located at the droplet interface whereas the oil molecules are confined to the interior of the droplets,
where
as a function of the single quantity
The second equality in Eq. (40) is obtained by means of employing the expression in Eq. (26) for the average droplet radius.
In Figure 8, we have plotted the average droplet radius 〈
The solubilisation capacity
9. Summary
A comprehensive theoretical model for the formation of microemulsion droplets has been presented. The theory is based on thermodynamics of self-assembling surfactant molecules aggregated in the droplet interfaces, combined with bending elasticity properties of the surfactant monolayer as taken into account by the three parameters spontaneous curvature, bending rigidity and saddle-splay constant. It relates properties depending on surfactant chemical structure with experimentally available properties of the droplets, such as average size, polydispersity, interfacial tension and solubilisation capacity. It has recently been demonstrated that all three constants
Likewise, it is possible to determine the various bending elasticity constants by means of measuring the average droplet radius, polydispersity and/or interfacial tension of a microemulsion system. We have included the important contribution of entropy of self-assembly and, as a result, we obtain quantitative expressions that differ from previously derived ones [7, 24, 27] where self-association entropy effects were omitted. Nevertheless, the previously derived expressions for average droplet radius, polydispersity and interfacial tension are all recovered as special cases from our model in the limit 2
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