The Packing Parameter

As we have seen in the previous section, there are several factors that determine the bulk phase of a surfactant system. One of these is the overall shape of the molecule (e.g., head group area, tail volume, and tail length). The geometric effects that overall molecular shape can have on phase behavior are summarized neatly in a quantity known as the packing parameter. This quantity can be used to predict the likely phase of a particular surfactant system. 

Consider a micelle consisting ofM surfactant molecules. Each molecule takes up a volume v and has a hydrophilic head group area of a . We can write the total volume of the micelle as 

By writing each of these expressions in terms ofM and equating them, we obtain 

for a real molecule the maximum length of the molecule (or the critical length IJ is equal to the radius of the micelle, soR Therefore, we can write 

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The packing parameter ( ) (m) reflects the influence of the Hquid flow rate as shown in Figure 20. reflects the influence of the gas flow rate, staying at unity below 50% of the flooding rate but beginning to decrease above this point. At 75% of the flooding velocity, = 0.6. Sc is the Schmidt number of the Hquid. 

The expanded version of the van Deem ter equation is used to help understand the relationships between the packing parameters and the gas flow. 

It is well known that the aqueous phase behavior of surfactants is influenced by, for example, the presence of short-chain alcohols [66,78]. These co-surfactants increase the effective value of the packing parameter [67,79] due to a decrease in the area per head group and therefore favor the formation of structures with a lower curvature. It was found that organic dyes such as thymol blue, dimidiiunbromide and methyl orange that are not soluble in pure supercritical CO2, could be conveniently solubihzed in AOT water-in-C02 reverse microemulsions with 2,2,3,3,4,4,5,5-octafluoro-l-pentanol as a co-surfactant [80]. In a recent report [81] the solubilization capacity of water in a Tx-lOO/cyclohexane/water system was foimd to be influenced by the compressed gases, which worked as a co-surfactant. 

Different surfactants are usually characterised by the solubility behaviour of their hydrophilic and hydrophobic molecule fraction in polar solvents, expressed by the HLB-value (hydrophilic-lipophilic-balance) of the surfactant. The HLB-value of a specific surfactant is often listed by the producer or can be easily calculated from listed increments [67]. If the water in a microemulsion contains electrolytes, the solubility of the surfactant in the water changes. It can be increased or decreased, depending on the kind of electrolyte [68,69]. The effect of electrolytes is explained by the HSAB principle (hard-soft-acid-base). For example, salts of hard acids and hard bases reduce the solubility of the surfactant in water. The solubility is increased by salts of soft acids and hard bases or by salts of hard acids and soft bases. Correspondingly, the solubility of the surfactant in water is increased by sodium alkyl sulfonates and decreased by sodium chloride or sodium sulfate. In the meantime, the physical interactions of the surfactant molecules and other components in microemulsions is well understood and the HSAB-principle was verified. The salts in water mainly influence the curvature of the surfactant film in a microemulsion. The curvature of the surfactant film can be expressed, analogous to the HLB-value, by the packing parameter Sp. The packing parameter is the ratio between the hydrophilic and lipophilic surfactant molecule part [70] ... 

These considerations imply that a dimensionless group, known as the packing parameter (P given by... 

Example 8.3 Packing Parameter for Spherical Micelles. Show that the packing parameter < of a surfactant has to be less than 1/3 for it to form spherical micelles. 

One can extend the above analysis to come up with the values of the packing parameter for which different shapes of aggregates are favored (Israelachvili 1991). These are summarized in Table 8.2. The results shown also serve as rules of thumb for what one can expect as one changes the chemical conditions of the solvent or the structure of the surfactant or for controlling the shape or aggregation number. For example,... 

What is meant by the optimal head group area of a surfactant What is the packing parameter (P Explain how packing considerations can be used to determine the possible shapes of the micellar aggregates. 

The effect of temperature on g is difficult to predict because effects such as solvatation, entropic thermodynamic have to be taken into account. Thus the phase transition of MCM-41 to MCM-48 can not be explained by using the packing parameter g when crystallization temperature increases. Some complementary studies (synthesis at lower and higher temperatures, XRD or SAXS measurements...) should be made to understand and explain the mechanism of phase transition. 

Figure 2 Micelle curvature in dependence of the packing parameter (left large, right small)...

The packing parameter provides a simple semi quantitative model to interpret the inLuence of the dynamic surfactant structure on the size and shape of the resulting micelles (Israelachvili,... 

The packing parameter of the neighboring surfactant molecules reflects the molecular dimension and is related to the macroscopic curvatures (Gaussian and mean curvature) of the surface imposed by the topology of the coverage relation (127). 

Using the packing parameter, the sample composition can be modified to impose a packing variation toward better solution stability. As an example, if the DMDBPM A-dodecane solution forms a third phase because of the rod-like shape of the aggregates, the addition of a molecule that increases the packing parameter of the extractant will induce a transition toward the sphere and will thus, prevent the third-phase formation. The role of the added modifier can be explained simply using this concept. 

A.2.2 Using the Packing Parameter to Explain the Role of Modifiers... 

A theoretical model predicting the shape-structure relationship between the monomeric units and their aggregates was developed by Israelachvili and was based on statistical mechanics of phospholipids.23 This model predicts the type of the aggregate formed on the basis of the packing parameter (P), which relates the volume of the molecule (V) to its length (1) and to the mean cross-sectional (effective) head group surface area (a) ... 

FIGURE 7.27. Definition of the interfacial mean curvature (H) and Gaussian curvature (K) and the packing parameter (P). Reproduced with permission from WILEY-VCH Verlag GmbH Co.). 

A critical region of the shape parameter (see above) is the range around 5=1, where various phases may coexist and in which small changes of parameters such as hydration and ion concentration may lead to a phase transition. Since frequently the packing parameter cannot be estimated with a sufficient precision, physical techniques such as small-angle scattering with X-rays (SAXS) or neutrons (SANS) must be applied for a reliable determination of the structures. 

Kumar, V.V. Complementary molecular shapes and additivity of the packing parameter of lipids. Proc Natl Acad Sci USA 88 (1991) 444-448. 

Considerations of the packing parameter concept of Israelachvili et al. [1] suggest that double-chain surfactants, which form the basis of measurements described in this article, cannot readily form spherical micelles. With double-chain surfactants, a more likely aggregate structure is the formation of bilayer vesicles, which can be also thought of as a dispersed lamellar phase (La) as such the vesicular dispersed form is likely to be preferentially formed at low concentrations ( 1 mmol dm-3) of surfactant. Furthermore, it is necessary to consider the possibility, unlike in the case of micelles, that such vesicles, formed by self-assembly of surfactant monomers, will not be thermodynamically stable. The instability is then likely to be in the direction of growth to a thermodynamically-stable lamellar phase from the vesicles. This process will be driven, at least initially, by fusion of two vesicles. 

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