THERMODYNAMICS OF INTERFACES

5.3.1. Gas–Liquid Interface in the Absence of an Emulsifier

In the previous section, the relationship between the molecular structure of emulsifiers and their surface activity and interfacial conformation was highlighted. In this section, math- ematical quantities and thermodynamic relationships which can be used to describe the properties of interfaces are introduced. As a whole, emulsions are thermodynamically un- stable systems because of the unfavorable contact between oil and water molecules (Section 7.2). Nevertheless, their interfacial properties can often be described by thermodynamics because the adsorption–desorption of emulsifier molecules occurs at a rate which is much faster than the time scale of the kinetic destabilization processes (Hunter 1986). Thermody- namically, it is convenient to assume that an interface is a smooth, infinitesimally thin plane which separates two homogeneous liquids (Figure 5.1a). In order to model a real system, it is necessary to decide exactly where this imaginary plane should be located (Hiemenz 1986). For simplicity, consider a system which consists of liquid water in equilibrium with its vapor (Figure 5.5). The volume fraction of water molecules in the liquid water is approximately unity and decreases to approximately zero as one moves up through the interfacial region and into the vapor phase.

The imaginary plane interface could be located anywhere in the interfacial region indicated in Figure 5.5 (Hiemenz 1986, Hunter 1986). In practice, it is convenient to assume that the interface is located at a position where the excess concentration of the substance on one side of the interface is equal to the deficit concentration of the substance on the other side of the interface: cexcess = cdeficit. In our example, the excess concentration corresponds to the amount

of water above the interface, which is greater than that which would have been present if the concentration of water was the same as that in the bulk vapor phase right up to the interface. Similarly, the deficit concentration corresponds to the amount of water which is below the interface, which is lower than that which would have been present if the concentration of the water was the same as that in the bulk liquid phase right up to the interface. This location of the interface is known as the Gibbs dividing surface, after the scientist who first proposed this convention.

5.3.2. Gas–Liquid Interface in the Presence of an Emulsifier

The concept of the Gibbs dividing surface is particularly useful for defining the amount of emulsifier which accumulates at an interface (Hunter 1986). Consider a system which consists of a surfactant solution in contact with its vapor (Figure 5.6). The emulsifier is distributed among the bulk aqueous phase, the vapor, and the interfacial region. The excess

FIGURE 5.6 When an emulsifier is present, the Gibbs dividing surface is conveniently located at the position where c_excess = c_deficitfor the liquid in which the emulsifier is most soluble, and the surface excess concentration is equal to the shaded region.

FIGURE 5.5 From a thermodynamic standpoint, it is convenient to locate the interface (Gibbs dividing surface) separating a liquid and its vapor where c_excess = c_deficit.

emulsifier concentration at the surface (ni) corresponds to the total amount of emulsifier

present in the system minus that which would be present if the emulsifier were not surface active and equals the shaded area shown in Figure 5.6. The accumulation of emulsifier molecules at an interface is characterized by a surface excess concentration (Γ), which is equal to the excess emulsifier concentration divided by the surface area: Γ = ni/A. Food

emulsifiers typically have Γ values of a few milligrams per meter squared (Dickinson 1992, Dalgleish 1996b). It is important to note that the emulsifier molecules are not actually concentrated at the Gibbs dividing surface (which is infinitely thin), because of their finite size and the possibility of multilayer formation. Nevertheless, this approach is extremely convenient for thermodynamic descriptions of the properties of surfaces and interfaces (Hunter 1986). The surface excess concentration is often identified with an experimentally measurable parameter called the surface load, which is the amount of emulsifier adsorbed to the surface of emulsion droplets per unit area of interface.

5.3.3. Liquid–Liquid Interface

For an interface between pure oil and pure water, the Gibbs dividing surface could be positioned at the point where the excess and deficit concentrations of either the oil or the water were equal on either side of the interface, which will in general be different (Tadros and Vincent 1983). For convenience, it can be assumed that the phase in which the emulsifier is most soluble is the one used to decide the position of the Gibbs dividing surface. The surface excess concentration of the emulsifier is then equal to that which is present in the system minus that which would be present if there were no accumulation at the interface (Hunter 1986).

5.3.4. Measurement of the Surface Excess Concentration

The surface excess concentration of an emulsifier can be determined from measurements of the variation in the surface tension of an air–liquid interface as the emulsifier concentration in the bulk liquid is increased (Figure 5.7). There is an equilibrium between emulsifier molecules at the interface and those in the bulk liquid. As the concentration of emulsifier in the bulk liquid is increased, so does their concentration at the interface. The presence of the emulsifier molecules at the interface shields the unfavorable contact between the oil and water molecules and therefore reduces the surface tension. At a certain concentration, the surface tension reaches a constant value because the surface becomes saturated with emul- sifier molecules. For small-molecule surfactants, the saturation of the surface occurs at approximately the same concentration as the surfactant molecules form micelles in the bulk liquid (i.e., the critical micelle concentration) (Section 4.5.2). A relationship between the decrease in surface tension with emulsifier concentration and the amount of emulsifier present at the surface can be derived from a mathematical treatment of the thermodynamics of the system (Hiemenz 1986, Hunter 1986). This relationship is known as the Gibbs isotherm equation, which is given by the following expression for an ideal solution:

Γ = − 1 _ _

RT d

d x

γ

ln( ) for nonionic emulsifiers (5.2)

Γ = − 1 _ _

2 RT d

d x

γ

ln( ) for ionic emulsifiers (5.3)

where x is the concentration of emulsifier in the aqueous phase, R is the gas constant, and T is the absolute temperature. These equations are used to determine the surface excess concen- tration of emulsifiers from experimental measurements of the surface tension versus emulsi- fier concentration, with Γ as the slope of the initial part of the curve (Figure 5.7). The factor 2 appears in the denominator of Equation 5.3 because the counterions associated with the head groups of ionic surfactants also accumulate at the interface (Hunter 1986). Equation 5.3 is only applicable at low ionic strengths where the interaction between the head groups and counterions is strong. As the ionic strength increases, the electrostatic interactions between the head groups and counterions are screened, and so Equation 5.2 becomes more applicable. Knowledge of the surface excess concentration is important for formulating food emulsions because it determines the minimum amount of emulsifier which can be used to create an emulsion with a given size distribution. The smaller the value of Γ, the greater the area of oil–water interface which can be covered per gram of emulsifier, and therefore the smaller

FIGURE 5.7 Dependence of the surface tension on the concentration of emulsifier. This type of plot can be used to measure the critical micelle concentration (CMC), the surface excess concentration, and maximum surface pressure of emulsifiers.

the size of droplets which can be effectively stabilized by the emulsifier. Plots of surface tension versus emulsifier concentration are also useful because they indicate the maximum surface pressure (πmax) which can be achieved when the surface is saturated by an emulsifier,

which has important consequences for the formation and stability of food emulsions (Chap- ters 6 and 7).