2.6.1. Thermodynamics of Mixing

In food emulsions, we are usually concerned with the interactions of large numbers of molecules in a liquid, rather than between a pair of isolated molecules in a vacuum. We must therefore consider the interaction of a molecule with its neighbors and how these interactions determine the overall organization of the molecules within a liquid (Murrell and Boucher 1982, Murrell and Jenkins 1994, Evans and Wennerstrom 1994). The behavior of large numbers of molecules at equilibrium can be described by statistical thermodynamics (Sears and Salinger 1975, Atkins 1994). A molecular ensemble tends to organize itself so that the molecules are in an arrangement which minimizes the free energy of the system. The free energy of a molecular ensemble is governed by both enthalpy and entropy contributions (Bergethon and Simons 1990). The enthalpy contributions are determined by the molecular interaction energies discussed above, while the entropy contributions are determined by the tendency of a system to adopt its most disordered state.

Consider a hypothetical system that consists of a collection of two different types of equally sized spherical molecules, A and B (Figure 2.6). The free energy change that occurs when these molecules are mixed is given by:

where A£mix and ASmix are the differences in the molecular interaction energy and entropy of the mixed and unmixed states, respectively. Practically, we may be interested in whether the

FIGURE 2.6 System in which two types of molecules may be completely miscible or form a regular solution depending on the strength of the interactions between them and the entropy of mixing.

Immiscible Liquids

FIGURE 2.6 System in which two types of molecules may be completely miscible or form a regular solution depending on the strength of the interactions between them and the entropy of mixing.

resulting system consists of two immiscible liquids or is a mixture where the molecules are more or less intermingled (Figure 2.6). Thermodynamics tells us that if AGmix is positive, mixing is unfavorable and the molecules tend to exist as two separate phases (i.e., they are immiscible); if AGmix is negative, mixing is favorable and the molecules tend to be intermingled with each other (i.e., they are miscible); and if AGmix ~ 0, the molecules are partly miscible and partly immiscible. For simplicity, we assume that if the two types of molecules do intermingle with each other, they form a regular solution (i.e., a completely random arrangement of the molecules) (Figure 2.6 right) rather than an ordered solution, in which the type A molecules are preferentially surrounded by type B molecules or vice versa. In practice, this means that the attractive forces between the two different types of molecules are not much stronger than the thermal energy of the system (Atkins 1994, Evans and Wennerstrom 1994). This argument is therefore only applicable to mixtures that contain nonpolar or slightly polar molecules, where strong ion-ion or ion-dipole interactions do not occur. Despite the simplicity of this model system, we can still gain considerable insight into the behavior of more complex systems that are relevant to food emulsions. In the following sections, we separately consider the contributions of the interaction energy and the entropy to the overall free energy change that occurs on mixing.

An expression for AEmix can be derived by calculating the total interaction energy of the molecules before and after mixing (Israelachvili 1992, Evans and Wennerstrom 1994). For both the mixed and the unmixed system, the total interaction energy is determined by summing the contribution of each of the different types of bond:

where nAA, nBB, and nAB are the total number of bonds, and wAA, wBB, and wAB are the intermolecular pair potentials at equilibrium separation that correspond to interactions between A-A, B-B, and A-B molecules, respectively. The total number of each type of bond formed is calculated from the number of molecules present in the system, the coordination number of the individual molecules (i.e., the number of molecules in direct contact with them), and their spatial arrangement. For example, many of the A-A and B-B interactions that occur in the unmixed system are replaced by A-B interactions in the mixed system. The difference in the total interaction energy between the mixed and unmixed states is then calculated: AEmix = Emix - Eunmixed. This type of analysis leads to the following equation (Evans and Wennerstrom 1994).

AEmix = nXA XBw

A XB

where n is the total number of moles, w is the effective interaction parameter, and XA and XB are the mole fractions of molecules of type A and B, respectively. The effective interaction parameter is a measure of the compatibility of the molecules in a mixture and is related to the intermolecular pair potential between isolated molecules by the expression

where z is the coordination number of a molecule and NA is Avogadro's number. The effective interaction parameter determines whether the transfer of a molecule from a liquid where it is surrounded by similar molecules to one in which it is partly surrounded by dissimilar molecules is favorable (w is negative), unfavorable (w is positive), or indifferent (w = 0). It should be stressed that even though there may be attractive forces between all the molecules involved (i.e., wAA, wBB, and wAB may all be negative), the overall interaction potential can be either negative (favorable to mixing) or positive (unfavorable to mixing) depending on the relative magnitude of the interactions. If the strength of the interaction between two different types of molecules (wAB) is greater (more negative) than the average strength between similar molecules (wab < [wAA + wBB]/2), then w is negative, which favors the intermingling of the different types of molecules. On the other hand, if the strength of the interaction between two different types of molecules is weaker (less negative) than the average strength between similar molecules (wAB > [waa + wBB]/2), then w is positive, which favors phase separation. If the strength of the interaction between different types of molecules is the same as the average strength between similar molecules (wAB = [wAA + wBB]/2), then the system has no preference for any particular arrangement of the molecules within the system. In summary, the change in the overall interaction energy may either favor or oppose mixing, depending on the relative magnitudes of the intermolecular pair potentials.

An expression for ASmix is obtained from simple statistical considerations (Israelachvili 1992, Evans and Wennerstrom 1994). The entropy of a system depends on the number of different ways the molecules can be arranged. For an immiscible system, there is only one possible arrangement of the two different types of molecules (i.e., zero entropy), but for a regular solution, there are a huge number of different possible arrangements (i.e., high entropy). A statistical analysis of this situation leads to the derivation of the following equation for the entropy of mixing:

ASmix is always positive because XA and XB are both between zero and one (so that the natural logarithm terms are negative), which reflects the fact that there is always an increase in entropy after mixing. For regular solutions, the entropy contribution (-ZASmix) always decreases the free energy of mixing (i.e., favors the intermingling of the molecules). It should be stressed that for more complex systems, there may be additional contributions to the entropy due to the presence of some order within the mixed state (e.g., organization of water molecules around a solute molecule) (Chapter 4).

2.6.4. Free Energy Change on Mixing

For a regular solution, the free energy change on mixing depends on the combined contributions of the interaction energies and the entropy:

We are now in a position to investigate the relationship between the strength of the interactions between molecules and their structural organization. The dependence of the free energy of mixing on the effective interaction parameter and the composition of a system consisting of two different types of molecules is illustrated in Figure 2.7. The molecules are completely miscible when the free energy of mixing is negative and large compared to the thermal energy, are partly miscible when AGmix ~ 0, and are completely immiscible when the free energy of mixing is positive and large compared to the thermal energy. Figure 2.7 indicates that mixing occurs even when the effective interaction parameter is zero, because of the contribution of the entropy of mixing term. This accounts for the miscibility of liquids in which the interactions between the two types of molecules are fairly similar (e.g., two nonpolar oils). Two liquids are completely immiscible when the effective interaction parameter is large and positive. The above approach enables us to use thermodynamic considerations to relate bulk physicochemical properties of liquids (such as immiscibility) to molecular properties (such as the effective interaction parameter and the coordination number).

The derivation of Equation 2.13 depends on making a number of simplifying assumptions about the properties of the system that are not normally valid in practice (e.g., that the molecules are spherical, that they all have the same size and coordination number, and that there is no ordering of the molecules within the mixture) (Israelachvili 1992). It is possible

FIGURE 2.7 Dependence of the free energy of mixing (calculated using Equation 2.13) on the composition and effective interaction parameter of a binary liquid. When AGmix is much less than -RT, the system tends to be mixed; otherwise, it will be partly or wholly immiscible.

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