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FIGURE 4.5 The critical size of a nucleus required for crystal growth depends on a balance between the volume and surface contributions to the free energy of nuclei formation.

nuclei. At a certain critical nucleus radius (r*), the overall free energy has a maximum value given by:

This equation can be rearranged to give an expression for the critical radius of the nucleus which must be achieved for crystallization to occur:

AHfus AT

If a nucleus that has a radius below this critical size is formed, it will tend to dissociate so as to reduce the free energy of the system. On the other hand, if a nucleus that has a radius above this critical value is formed, it will tend to grow into a crystal. Equation 4.3 indicates that the critical size of nuclei required for crystal growth decreases as the degree of supercooling increases, which accounts for the increase in nucleation rate with decreasing temperature.

The rate at which nucleation occurs can be related to the activation energy (AG*), which must be overcome before a stable nucleus is formed (Boistelle 1988):

where J is the nucleation rate, which is equal to the number of stable nuclei formed per second per unit volume of material; A is a preexponential factor; k is Boltzmann's constant; and T is the absolute temperature. The value of AG* is calculated by replacing r in Equation 4.1 with the critical radius given in Equation 4.3. The variation of the nucleation rate predicted by Equation 4.4 with the degree of supercooling (AT) is shown in Figure 4.6. The formation of stable nuclei is negligibly slow at temperatures just below the melting point, but increases dramatically when the liquid is cooled below a certain temperature (T*). In reality, the nucleation rate increases with cooling up to a certain temperature, but then decreases on further cooling. This is because the increase in viscosity of the oil which occurs as the

FIGURE 4.6 Theoretically, the rate of the formation of stable nuclei increases with supercooling (solid line), but in practice, the nucleation rate decreases below a particular temperature because the diffusion of oil molecules is retarded by the increase in oil viscosity (broken line).

temperature is decreased slows down the diffusion of oil molecules toward the liquid-nucleus interface (Boistelle 1988). Consequently, there is a maximum in the nucleation rate at a particular temperature (Figure 4.6).

The type of nucleation described above occurs when there are no impurities present in the oil and is usually referred to as homogeneous nucleation (Boistelle 1988). If impurities such as dust particles are present in the oil, the internal surface of the vessel containing the oil, the interface of an emulsion droplet, the surface of an air bubble, or monoglyceride reverse micelles, then nucleation can be induced at a higher temperature than expected for a pure system (Walstra 1987, McClements et al. 1993e, Dickinson and McClements 1995). Nucleation in the presence of impurities is referred to as heterogeneous nucleation and can be divided into two types: primary and secondary (Boistelle 1988). Primary heterogeneous nucleation occurs when the impurities have a different chemical structure to that of the oil, whereas secondary heterogeneous nucleation occurs when crystals of the same chemical structure are present in the oil. Heterogeneous nucleation occurs when the impurities provide a surface where the formation of stable nuclei is more energetically favorable than in the pure oil (Boistelle 1988). As a result, the degree of supercooling required to initiate droplet crystallization is reduced. On the other hand, certain types of impurities are capable of decreasing the nucleation rate of oils because they are incorporated into the surface of the growing nuclei and therefore prevent any further oil molecules from being incorporated. Whether an impurity acts as a catalyst or an inhibitor of crystal growth depends on its molecular structure and interactions with the nuclei (Boistelle 1988).

4.2.3.3. Crystal Growth

Once stable nuclei have formed, they grow into crystals by incorporating molecules from the liquid oil at the solid-liquid interface (Garside 1987, Boistelle 1988). The crystal growth rate depends on the diffusion of a molecule from the liquid to the solid-liquid interface, as well as its incorporation into the crystal lattice. Either of these processes may be rate limiting, depending on the temperature and the molecular properties of the system. Molecules must encounter the interface at a location where crystal growth is favorable and also have the appropriate shape, size, and orientation to be incorporated (Timms 1991, 1995). At high temperatures, the incorporation of a molecule at the crystal surface is usually rate limiting, whereas at low temperatures, the diffusion step is rate limiting. This is because the viscosity of the liquid oil increases as the temperature is lowered, and so the diffusion of a molecule is retarded. The crystal growth rate therefore increases initially with supercooling, has a maximum rate at a certain temperature, and then decreases on further supercooling (i.e., it shows a similar trend to the nucleation rate) (Figure 4.6). Nevertheless, the maximum rate of nuclei formation may occur at a different temperature than the maximum rate of crystal growth. Experimentally, it has been observed that the rate of crystal growth is proportional to the degree of supercooling and inversely proportional to the viscosity of the melt (Timms 1991).

The crystal growth rate of fats can often be described by the following expression (Ozilgen et al. 1993):

where C0 is the initial fraction of unsolidified oil, C(t) is the fraction at time t, and kG is the crystal growth rate. Thus, a plot of 1/C versus time should give a straight line with a slope equal to kG. One of the major problems in determining kG is that the nucleation rate and the crystal growth rate often have similar magnitudes, and thus it is difficult to unambiguously establish the contribution of each process (Ozilgen et al. 1993).

4.2.3.4. Crystal Morphology

The morphology of the crystals formed depends on a number of internal (e.g., packing of molecules and intermolecular forces) and external (e.g., temperature, cooling rate, mechanical agitation, and impurities) factors. When an oil is cooled rapidly to a temperature well below its melting point, a large number of small crystals are formed, but when it is cooled slowly to a temperature just below its melting point, a smaller number of larger crystals are formed (Moran 1994, Timms 1995). This is because the nucleation rate increases more rapidly with decreasing temperature than the crystallization rate (Timms 1991). Thus, rapid cooling produces many nuclei simultaneously which subsequently grow into small crystals, whereas slow cooling produces a smaller number of nuclei which have time to grow into larger crystals before further nuclei are formed. Crystal size has important implications for the rheology and organoleptic properties of foods. When crystals are too large, they are perceived as being "grainy" or "sandy" in the mouth (Walstra 1987). The efficiency of molecular packing in crystals also depends on the cooling rate. If a fat is cooled slowly, or the degree of supercooling is small, then the molecules have sufficient time to be efficiently incorporated into a crystal (Walstra 1987). At faster cooling rates, or high degrees of supercooling, the molecules do not have sufficient time to pack efficiently before another molecule is incorporated. Thus rapid cooling tends to produce crystals which contain more dislocations and in which the molecules are less densely packed (Timms 1991).

4.2.3.5. Polymorphism

Triacylglycerols exhibit a phenomenon known as polymorphism, which is the ability of a material to exist in a number of crystalline structures with different molecular packing (Hauser 1975, Garti and Sato 1988, Sato 1988, Hernqvist 1990). The three most commonly occurring types of packing in triacylglycerols are hexagonal, orthorhombic, and triclinic, which are usually designated as a, P', and P polymorphic forms (Nawar 1996). The thermodynamic stability of the three forms decreases in the order P > P' > a. Even though the P form is the most thermodynamically stable, triacylglycerols often crystallize in one of the meta-stable states because they have a lower activation energy of nuclei formation (Figure 4.7). With time, the crystals transform to the most stable state at a rate which depends on environmental conditions such as temperature, pressure, and the presence of impurities (Timms 1991).

FIGURE 4.7 The polymorphic state that is initially formed when an oil crystallizes depends on the relative magnitude of the activation energies associated with nuclei formation.
FIGURE 4.8 Comparison of the melting profile of a pure triacylglycerol and a typical edible fat. The edible fat melts over a much wider range of temperatures because it consists of a mixture of many different pure triacylglycerol molecules, each with different melting points.

4.2.3.6. Crystallization of Edible Fats and Oils

The melting point of a triacylglycerol depends on the chain length, branching, and degree of unsaturation of its constituent fatty acids, as well as their relative positions along the glycerol molecule (Table 4.3). Edible fats and oils contain a complex mixture of many different types of triacylglycerol molecules, each with a different melting point, and so they melt over a wide range of temperatures rather than at a distinct temperature, as would be the case for a pure triacylglycerol (Figure 4.8).

The melting profile of a fat is not simply the weighted sum of the melting profiles of its constituent triacylglycerols, because high-melting-point triacylglycerols are soluble in lower melting point ones (Timms 1995). For example, in a 50:50 mixture of tristearin and triolein, it is possible to dissolve 10% of solid tristearin in liquid triolein at 60°C (Walstra 1987, Timms 1995). The solubility of a solid component in a liquid component can be predicted, assuming they have widely differing melting points (>20°C):

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