Identifying the Assumptions

The applicability of this molecular mobility approach to food stability is dependent upon several key concepts and assumptions. The first, and most important premise, is that many (if not most) foods contain amorphous components, and exist in either a state of metastable equilibrium or in a nonequilibrium, kinetically labile state. This is true of many complex foods because most of them contain amorphous solid, and supersaturated liquid regions. Biopolymers are typically at least partly

100% Water 0°C

T_E2

T_m1

T_m2

T_g2

T_g1 T_E1

–135°C

100% Solute Composition (%)

Temperature (°C)

L L

FIGURE 2.24 State diagrams for binary systems showing the influence of solute type on the position of T_m^L and on T_g. The extreme left of T_gis fixed at the vitrification temperature of pure water (−135^◦C). The assumptions stated in Figure 2.23 apply here.

amorphous. Many small molecules are difficult to crystallize from solution, and thus exist in an amorphous state when in excess of saturation. This amorphous state may be kinetically constrained, and therefore metastable, or may be slowly changing with time, and therefore in nonequilibrium.

Essentially, the lower (Tg) boundary line in a state diagram defines the limiting conditions for the metastable amorphous phase, and the region of C > CE above Tg and below T_m^s represents a nonequilibrium amorphous state with the upper boundary of this condition being defined by the saturation solubility line (T_m^s) above which exists the simple solution. Note that the part of Tm^L to the left of point E (C < CE) (Figure 2.23) similarly defines the upper limit of the nonequilibrium amorphous state for compositions where ice is not present owing to rapid cooling, or other constraints.

Since it is more difficult to prevent ice crystallization than it is to prevent solute crystallization in foods, the amorphous state for compositions to the left of point E is not readily attained, except for (1) the region of initial undercooling prior to the initial nucleation of ice and (2) the region with C^L less than, but close to C_m^Las defined by the liquidus representing an unfrozen phase that can be termed a nonmaximally freeze-concentrated matrix. It is a major goal of food scientists and technologists to maximize the number of desirable food attributes that depend on metastable equilibrium states, and to find conditions providing acceptable stability for those desirable attributes that depend on the maintenance of nonequilibrium states.

The next key point is to reiterate that the rates of most physical processes, and also of many chemical processes, are governed by molecular mobility, in that they require some form of molecular

temperature and composition that permit metastable or nonequilibrium conditions to exist for useful periods of time. In frozen systems, a particular controversy has addressed the issue of which Tgto employ in the WLF equation. As has been noted, while Levine and Slade recommended the use of T_m^ (T_g^) in their terminology, it is clear that this does not take into account the dilution of the unfrozen phase due to melting at the higher temperature. Nor does it acknowledge that the true glass transition temperature of the maximally freeze-concentrated matrix is T_g^∗. As has been suggested by several workers [74,78,79], it would appear to be more correct to use Tg, the temperature of the glass transition for this more dilute phase. However, given that the parameters of the WLF equation do not have universal values, it emerges that the use of either convention is equally effective in real systems [79]. Given that establishing the true value of Tgis a major challenge, it is indeed fortunate that, as indicated earlier, use of T_m^ in the WLF equation provides sufficient accuracy.

2.10.7 LIMITATIONS OF THECONCEPT

While the Mm approach is useful for predicting many kinds of physical change, its utility is not universal. Examples of where it is unsatisfactory include chemical reactions whose rates are little influenced by diffusion, effects achieved through the action of specific chemicals, and situations where the estimated Mm reflects the properties of a polymeric component, while the process of concern involves smaller molecules whose mobility is little hindered by the loss in mobility of the primary matrix. Also, in the growth of vegetative cells of microorganisms, the mobility of water, and consequently (p/p^o)Tserves as a better predictor.

Returning to the discussion of reaction kinetics, in the last 20 years, there has been an active discussion as to whether the WLF equation or theArrhenius equation provides the better description of the temperature dependence of reaction kinetics in aqueous food systems, particularly at temperatures between Tgand T_m^Lor T_m^s. Consider systems that can form ice. In this region, in taking the molecular mobility approach, there are two factors that might be expected to influence mobility, temperature, and concentration. As temperature is lowered, concentration increases. At first the influence on mobility is primarily that of temperature, but as the temperature continues to drop, the increasing concentration becomes a factor of increasing importance as more ice forms. Figure 2.25 illustrates the effect of temperature and the effect of concentration separately upon viscosity and mobility.

The combined effect is shown in Figures 2.26 and 2.27. Both the Arrhenius equation and the WLF equation properly describe the effects of temperature on kinetics only if concentration is constant.

The effect of concentration on kinetics enters as another term in the analysis. For a first order reaction, concentration does not influence the fractional rate of reaction (i.e., t1/2 is independent of concentration), but for all higher orders of reaction, the relative rate of reaction is concentration dependent. For many reactions in frozen systems, a pseudo first order description is adequate, but this does not guarantee that, especially in the freeze-concentrated zone, the effect of concentration can be ignored when estimating extent of reaction. As previously noted, empirical evidence shows that an equation of the WLF form can provide adequate estimates for rate and extent of reaction as a function of temperature and time, using either T_m^, T_g^∗, or the Tgof the homogeneous glass of composition C_T^L (where T is the storage temperature of interest) as the reference temperature.

Given the limited range of temperatures in the region between Tgand T_m^Lin frozen systems, it is not surprising that the Arrhenius equation also produces a satisfactory fit to the raw data.

Note that another factor that is little discussed is the “equimolal” nature of the unfrozen phase in frozen systems. The presence of ice defines the osmolality of the unfrozen phase, assuming that

0

FIGURE 2.25 Comparison of the effect of concentration on the viscosity of aqueous solutions at two different temperatures: (1) 0^◦C and (2)−40^◦C.

FIGURE 2.26 Predicted viscosities in aqueous systems as a function of temperature: (1) no ice formation on cooling; (2) ice separation such that solution phase concentration tracks line T_m^L; (3) system concentration is C^∗_g. T_m^L defines the concentration. Should the composition (and hence mole ratios) of solutes change as a consequence of reactions, in contrast to an unfrozen system, the amount of ice, and hence the individual concentrations, will adjust to maintain the defined osmolality of the unfrozen phase.

Hence, the evolution of reactant and product concentrations could depend upon the stoichiometry of the reaction in a way different from that of an unfrozen system.

In a system of concentration in excess of T_g^∗, where ice crystallization is not possible, above T_m^s, in the fluid system, Arrhenius kinetics hold. It is not uncommon for an Arrhenius plot incorporating temperatures that span T_m^s to exhibit a change in slope around T_m^s. Between T_m^s and Tgthe system can be described as rubbery. Particularly note that there is a rapid decrease in mobility as the temperature is lowered, and this is reflected by a rapid change in reaction rates. In this region, while it is difficult

4

Log viscosity 2 2

1 3

0