The Crystalline State
4.2 Crystallization and Nucleation
4.2.1.1 Phase Behavior
( ) ( )
φ − ∆
= ∆ Tf T = ∆ T
H H
f Tf f Tf (4.1)
where T is absolute temperature of the system, Tf is the melting point temperature and ∆Hf is the latent heat of fusion.
Often, the driving force for crystallization in melt systems is given simply by the temperature difference, ∆T, with the melting point temperature. In general, the lower the temperature of a system, the greater the rate of crystallization until reduced mass transfer limits nucleation at very low temperatures.
In binary crystallizing systems, sometimes called solution systems, a solvent (usually water) dissolves a solute that eventually crystallizes either during processing or storage. Conditions in the manufacturing process are generally set so that the solute becomes supersaturated in the solvent, after which crystal formation, growth and ripening are controlled to give the desired microstructure. Examples of solution behavior in foods include sugar crystallization in confections, cereal frostings, and milk powders, and salt crystallization during storage of cheese (e.g., cheddar cheese).
Phase behavior of solution systems depends on the nature of the interactions between the two components. Although numerous types of phase diagrams can be found in nature (and foods), many systems of importance in foods exhibit eutectic 4.2.1 Thermodynamic Driving Force
4.2.1.1 Phase Behavior
The Crystalline State 47
phase behavior, where the melting temperature of a mixture falls below the melting points of either of the two individual components. Figure 4.1 shows a generic eutectic phase diagram for a binary system. Above the melting line (solvent) or solubility line (solute), all molecules are in liquid form. At temperatures below or concentrations above the equilibrium phase boundaries, either solvent or solute may form crystals.
The eutectic temperature, represented as the point where the solute and solvent equilibrium curves intersect, defines the point below which a eutectic solid should form. Nominally, a solid with a defined composition (based on the composition at the eutectic point) will be formed at temperatures below the eutectic temperature.
However, in many food systems, because the solvent is water, with a molecular weight often much less than the solute molecules, it is common for no eutectic solid to form at all. In fact, it is common to find the solute, water, continuing to freeze independently even at temperatures well below the eutectic point. The solute does not crystallize simply due to kinetic constraints of limited molecular mobility.
One of the most important uses of phase diagrams in food applications is predict-tion of crystalline yield for a given system. For example, if a sucrose solupredict-tion of certain concentration is allowed to crystallize to equilibrium, the amount of crystalline solid (yield) can be predicted based on the phase diagram. As will be seen later, crystalline phase volume is one of the most important determinants of food material properties.
In solution systems, the driving force for crystallization is the difference in chemical potential between molecules in the liquid state and those in the crystalline state. Therefore, the supersaturation driving force, φ, for crystallization of a solute from solution is given by
φ= = γγ ≈
⎛ ⎞ ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟
⎜ ⎟ ⎝ ⎠ ⎝ ⎠
⎝ ⎠
ln a ln X ln C
RT RT RT
aeq s sX Cs
(4.2)
where a is the activity of the solute in the solution, aeq is the activity of the solute at equilibrium, γ is the activity coefficient of the solution, γs is the activity coefficient of the saturated solution, X is the mole fraction of the solute in the solution, Xs is the mole fraction of the solute in the solution at saturation, C is the concentration of the solute in the solution, and Cs is the concentration of the solute at saturation.
Since activity and the activity coefficient are not easily measured, most food applications simply define the supersaturation driving force in a solution system, s, by the ratio of concentration to the saturation concentration.
s= C
Cs (4.3)
In general, larger supersaturation driving forces lead to more rapid crystalli-zation. However, in systems that exhibit glass transitions, very high driving forces lead to limited molecular mobility, which effectively blocks nucleation.
R.W. Hartel 48
Temperature
Composition
100% A 100% B
Eutectic Point
Figure 4.1. Eutectic phase diagram.
To take into account the effects of other ingredients in foods on crystallization behavior, more complex phase diagrams are needed. As noted previously, sometimes the phase behavior of complex systems can be modeled using ternary diagrams with solvent, solute and impurity on the three points of the triangle phase diagram. In sucrose refining, for example, a ternary diagram expressing the phase relationships among water, sucrose and dissolved impurities may be used to help understand crystallization in practical systems (Mathlouthi and Reiser 1995). Alternatively, sometimes it is sufficient to simply show the effects of one component in a food system on the solubility of the crystallizing component. For example, the effect of corn syrup on depressing sucrose solubility in water is often indicated on a binary-type phase diagram, where a new solubility line is drawn to indicate the new solubility of sucrose in the mixed system. However, in this approach, the meaning of the composition axis of a binary phase diagram is no longer simple since it is a mixed composition of sucrose and corn syrup.
The phase behavior of lipid systems is often quite complex, yet critical to cont-rolling material properties of fat-based foods. In natural fats, mixtures of triacyl-glycerols (TAGs) with different melting points are quite common. Some TAGs have a high melting point, notably those with long-chain, saturated fatty acids. TAGs that contain shorter-chain and/or unsaturated fatty acids typically have lower melting point temperatures, and even may be liquid oils at food product temperatures. Thus, in complex natural fats, some TAGs are crystalline and some are likely to still be liquid. Recently, Wesdorp et al. (2005) outlined the steps necessary to thermodyna-mically model the phase behavior of complex mixed lipid systems. This approach also distinguished between the different polymorphs that can form in lipid crystals.
The Crystalline State 49
Because of the complexities of many food systems, numerous simplifications of phase diagrams have been developed over the years. In lipid systems, diagrams that show lines of constant solid fat content at different temperatures and composition are often used to indicate phase mixing behavior of two fats (Timms 2003). These may be best called pseudo-phase diagrams since, although they represent phase mixing behavior between the two fats, they are not true phase diagrams.
Phase diagrams document the phase behavior expected at equilibrium (see Chapter 7). They can be used to predict crystalline phase volume and which crystalline form (polymorph, hydrate, etc.) is expected in a food that has reached phase equilibrium.
However, in foods that never attain equilibrium, the phase diagram by itself is of limited value. In recent years, certain aspects of nonequilibrium have been charac-terized using the state diagram approach (Slade and Levine 1991).
The state diagram approach generally overlays the phase diagram for a system with the glass transition curve, as shown in Figure 4.2 for the sucrose–water system.
The equilibrium phase boundary for solvent (water) crystallization is seen as the freezing point depression curve, and that for the solute (sucrose) is seen as the solubility curve. Overlaid on top of the standard binary phase diagram is the glass transition curve, which documents the conditions where this sucrose system may be expected to become a glass under certain nonequilibrium conditions.
150
100
50
0
0 20 40 60 80 100
Boiling point elevation
Sucrose solubility
Gl ass transition Freezing point
depression
DILUTE SOLUTION
SUGAR CRYSTAL + SOLUTION
ICE + SOLUTION
GLASS
Composition (wt %)
Temperature (°C)
Figure 4.2. State diagram for sucrose–water (from Hartel 2001, with permission).
4.2.1.2 State Diagrams