ELSEVIER

Journal of Food Engineering 22 ( 1994) 27 l-289 0 1994 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0260-8774/94/$7.00

Water Activity and Food Polymer Science: Implications of State on Arrhenius and WLF Models in Predicting

Shelf Life

K. A. Nelson & T. P. Labuza

Department of Food Science & Nutrition, University of Minnesota, 1354 Eckles Ave., St Paul, Minnesota 55 108, USA

ABSTRACT

The effect of water activity on chemical reactions which are important to food stabi~i~ h as b een studied extensiveIy. Glass transition theory con- siders the effect of the state of a system on reactions and is a relatively new approach for understanding the stability of foods. The role of water in chemical reaction rates is examined based on these two approaches.

The Arrhenius model has traditionally been used to describe the temper~~re dependence of chemical reaction rates. An approach deve- loped by ~iliiams, Landel and Ferry may also be ~pplis~ble for describ- ing the temperature dependence of chemical reactions within some food systems. These two approaches are examined, with emphasis on the applicability and application of each approach.

INTRODUCTION

Understanding the reaction kinetics in food systems is essential for predicting food quality and stability. Moisture and temperature are two impo~ant parameters which affect the rate of reactions in food systems. The influence of moisture on reaction rates has been explained in terms of water activity, and this approach is now well established for control- ling and predicting food stability (Labuza, 1975). The glass transition approach may alternatively be used to understand the relationship between moisture and reaction rates. In this paper, water activity and the gIass transition approach are examined as means for understand~g the role of water with respect to the rate of chemical reactions which occur in food systems.

271

272 K. A. Nelson, T. P. Labuza

This paper will also discuss the applicability of the Arrhenius and the Williams, Landel and Ferry (WLF) models for describing the tempera- ture dependence of reactions within glassy and rubbery food systems.

MOISTURE AND REACTION RATES

Moisture plays an important role in the rate of reactions which are important for food stability. Generally, the rate limiting step in chemical reactions involves achieving the proper molecular orientation or over- coming some type of activation barrier. However, when mobility is re- stricted at low moisture contents or viscosity is very high, translational mobility, the ability of reactants to diffuse through a system, may become rate limiting (Bailley et al., 198 1). Reactant mobility and diffusion within a matrix can be considered from both a water activity and a glass transi- tion theory standpoint.

Water activity and reaction rates

The influence of water activity (a,) on the rate of chemical reactions in food systems has been studied extensively. At low water activities, water is tightly bound to the surface of polar sites making it unavailable to solvate reactants. The upper limit of this water activity region is generally referred to as the monolayer moisture (Labuza, 1975). At higher water activities, water exists in multi-layers or as a condensed phase in capilla- ries and its mobility increases. Increased reaction rates are observed due to increased availability of water to act as a reaction medium in which sufficient reactant mobility occurs to allow reactant interactions. As water activity increases further, the rates of some reactions decrease as the result of the dilution of reacting species in the available aqueous phase. In other words, no additional solute dissolves and increasing moisture content will decrease the liquid phase concentration.

The main exception to this minimum-maximum relationship involves the oxidation of unsaturated lipids. In this case, the reaction rate increases below the monolayer due to the increased catalytic activity of metal ions when sufficient water is removed from the hydration sphere around these ions (Labuza et al., 1966).

Specific water activity regions are often associated with susceptibility to chemical reactions, physical changes and microbial growth. The general relationship between water activity and several food stability phenomena is shown in Fig. 1. This figure serves as a general guide for

Aphelia and WLF yodels to predict she~l~e 273

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Water activity

Fig. I. Relative rate of reactions and moisture content as a function of water activity (Labuza, 197 1).

predicting potential stability problems in food systems within particular water activity regions.

Certainly, water activity provides a general guideline for predicting the stability of food systems and is used extensively throughout the food industry as a quality and safety indicator. The concept of water activity remains useful and should not be discarded from general application. However, there is a need for unproved understanding of the relationship between moisture and reaction rates. The glass transition approach may assist in underst~ding this relations~p.

Glass transition approach and reaction rates

Glass transition theory applies to amorphous polymers and has also been found to be applicable to low molecular weight sugars (Slade et al., 1989). The glass transition temperature, Tg, is the temperature at which a glass to rubber transition takes place. Water, the most common plasti- cizer in foods, acts to decrease the glass transition temperature. The system properties above and below the glass transition temperature dif- fer quite dramati~~y and such differences have been studied relative to the ~~st~l~ation (Roes & Karel, 1991 bf and the viscosity (Soesanto & Williams, 1981) of sugar solutions. The relationship between the rate of chemical reactions which occur in food systems and the state of the system, either glassy or rubbery, has also been studied. The properties of

274 K. A. Nelson, T. P. Labwa

glassy and rubbery systems may contribute to differences in chemical reaction rates in each of these states.

There is a dramatic change in the local movement of polymer chains at the glass transition temperature, resulting in a number of property differences between glassy and rubbery systems. As a system moves from the glassy to the rubbery state, the viscosity drops dramatically from approximately 1OL2 to lo3 Pa s at the glass transition temperature (Sperling, 1992). The reduced viscosity allows for greater polymer chain and reactant rnob~~. Free volume, defined as the amount of space asso- ciated with a system which is not taken up by polymer chains themselves, also changes between the glassy and rubbery states. The free volume available within a glassy system has been estimated to be between 2% and 11.3% of the total volume (Ferry, 1980), and is believed to increase substantia~y at the glass transition temperature due to a dramatic increase in the thermal expansion coefficient (Ferry, 1980). This increase in free volume should allow for faster diffusion reactions. Rased on the free volume required for diffusion, the size of a diffusing molecule may also be an important factor affecting diffusion rates. Diffusion is a func- tion of the probability of creating a hole within a matrix which is suffi- ciently large for a molecule to occupy. When a molecule is large compared to the available free volume, the probability of creating a hole is low. Thus, a greater degree of free volume redist~bu~on is required in order for diffusion of large molecules to take place compared to smaller molecules (Mauri& et al., 1990) This makes diffusion of large molecules within a region of limited free volume very slow. Reactions which are dependent upon the diffusion of such molecules may also be very slow.

In addition to translational mobility, short range mobility may also be important for chemical reactions. Using electron spin resonance, Roozen and coworkers ( 1990,199 1) found a significant increase in the rotational mobi~ty of spin probes within sucrose-water, glycerol-water and malto- dextrin-water mixtures at a temperature which corresponded to the glass transition temperature, as measured by differential scanning calorimetry. The mobility of protons, as measured by nuclear magnetic resonance, has also been found to be higher in the rubbery state com- pared to the glassy state (Kalichevsky et at., 1992).

Based on the properties associated with the glassy state, it might be expected that chemical reaction rates would be quite slow within the glassy state, or would not occur at all, but would substantially increase in rate within the rubbery state. In fact, this may be the explanation for cessation of reactions at the monolayer determined from moisture sorp- tion isotherms. It is possible that the observed monolayer is not actually a monolayer, but rather a moisture content at which the glass transition is observed at a particular temperature.

Arrhenius and WLF models to predict shedlife 275

Lim and Reid ( 1991) studied the rate of d~sion-contro~ed pro- cesses within frozen systems. Maltodex~s, carbo~e~ylce~ulose (CMC) and sucrose were used as model systems to study the rate of protein aggregation, ascorbic acid degradation and enzymatic hydrolysis at temperatures above and below the TL value for each model system. Figure 2 shows the results of e~ymatic hydrolysis above and below the TL of a maltodextrin DE10 system.

Note that at temperatures below the system Tb ( - lO’C), the reaction was very slow, but the rate increased above the TL, with a large increase in hydrolysis rate occurring 4~5°C above the reported TL. Similar results were found for other DE maltodextrins. Protein aggregation and ascor- bic acid degradation within the maltodextrin systems were also slow within the rubbery state, but increased in rate above the system TL. It was concluded that the maltodextrin system provided good cryostability below Tb.

For the CMC systems, Lim and Reid ( 1991) found that ascorbic acid degradation was observed below the system Tk, possibly due to the porous structure of the CMC matrix which allowed oxygen diffusion even within the glassy state. It may also be possible that the limited free volume within the glassy state was sufficient for oxygen diffusion to occur. CMC also failed to stabilize the system against protein aggrega- tion within the glassy state. This was unexpected considering the high TL value for CMC ( - 1 1°C) and that in order for aggregation to take place, the di~sion of large protein molecules through the matrix would be required.

Lim and Reid ( 1991) found that the Ti for a sucrose model system was very low ( - 33°C) and that all study temperatures were above Tb, within the rubbery state. Respite this, it was observed that sucrose effect- ively stabilized the system against protein aggregation. The glass transi-

Fig. 2.

Temperature (“C)

0 50 100 150 200 250

Time (hours) Enzymatic hydrolysis within a maitodextrin DE 10 system (Lim & Reid, 1991).

276 K. A. Nelson, T. I? Labwa

tion approach failed to predict the stability of the sucrose system in this case. As expected, however, the sucrose system showed rapid rates of ascorbic acid degradation within the rubbery state of this system. Note, however, that as the temperature above the TL increased, the water content in the system probably increased due to the melting of ice. Thus, it is difficult to attribute these observed results, as well as the results from other studies involving frozen systems, solely to a temperature effect.

Karmas et al. (1992) found that nonezymatic browning within the glassy state of carbohydrate model systems was very slow below the system Ts. Depend~g on the composition and moisture content of the carbohydrate system, the rate of nonezymatic browning increased sub- stantially at about ZO-75°C above the Tg Figure 3 shows typical results of their work. For systems at a, = 0.12 and a, = 0.33, dramatic changes in the rate of nonenzymatic browning occurred at 50°C and 75°C above the Tg, respectively. They stated that the temperature at which a large increase in brown~g rate occurred could be related to the temperature for which the diffusion coefficient within the system rapidly increased based on changes in free volume. This would be expected to occur at the glass transition temperature, but could occur at temperatures above the Ts depending on the system free volume and the size of the dosing molecules.

A study of the oxidation of orange oil within a maltodextrin Ml00 encapsulating matrix showed that oxidation occurred rather rapidly within the maltodextrin glassy state (Ma et al,, 1992; Nelson & Labuza, 1992). As the water content of the matrix increased, but was still within the glassy state, reaction rates increased, as shown in Fig. 4. In contrast to the expected rate increase within the rubbery state, the system at 75%

T-Tg

Fig. 3. Rate of browning (420 nm) in a lactose/amioca/lysine model system (Karmas et al., 1992).

Arrhenius and WLF models to predict sheIf life 277

relative humidity was within the rubbery state, but showed the greatest stability to oxidation. These findings were attributed to collapse of the matrix within the rubbery state which prevented oxygen diffusion through the matrix. Collapse occurs when a matrix can no longer support itself against gravity and collapses upon itself forming a compact system. Caking and sticking are examples of collapse (Downton et al., 1982).

A rubbery system is characterized as a metastable state. In addition to collapse, crystallization may occur which may subsequently influence chemical reaction rates. Crystallization can occur within the rubber state where viscosity is low such that molecules have sufficient mobility for crystallization to take place. Roos and Karel ( 1991 b) found that the time for crystallization decreased dramatically as temperature above the glass transition temperature ( T- TJ increased for sucrose and lactose solu- tions. Crystallization leads to a cross-linking effect which results in reduced polymer chain flexibility and mobility. Michaels et al. (1963) reported lower gas diffusion rates within glassy and crystalline synthetic polymer matrices compared to rubbery matrices. Crystalline regions were found to impede diffusion since molecules were required to pass through regions of low polymer chain mobility.

Labrousse et al (1992) found that the collapse and crystallization of a lactose-gelatin matrix containing methyl linoleate oil led to an increased rate of oil oxidation. They attributed this to the reduction in free volume upon crystallization which forced the oil to the surface where protection from oxidation was minimal. Numerous researchers have examined the relationship between the collapse and the retention of encapsulated oils and volatiles (Flink & Karel, 1972; Omatete & King, 1978). The extent

0.56

0.44

0.33

0.11 0

0.75:

0 5 10 15 20 25 30

Time (days)

Fig. 4. Limonene oxidation of orange oil encapsulated in maltodextrin Ml00 (Ma et al., 1992).

278 K. A. Nelson, T. P. Labuza

and rate of collapse or crystallization has been found to play a part in the retention and the subsequent stability of these materials.

Based on the properties of glassy and rubbery systems, it would be expected that reaction rates would be very slow within a glassy state and increase within a rubbery system. However, based on the studies reported in the food science literature, it should not be assumed that glassy systems are completely stable, or that rubbery systems are less stable than glassy systems. Molecular size, matrix porosity and changes within the metastable rubbery state such as collapse and crystallization, may contribute to other, more complicated results.

Water activity and the glass transition approach both have strengths and weaknesses in defining the relationship between moisture content and chemical reaction rates. It is important to recognize, however, that it is a relatively simple task to measure the water activity of food systems. A number of inst~mental methods are available for such detestations (Stamp et al., 1984). With the use of controlled relative hu~dity chambers, specific water activities can also be achieved for a pa~icular food system. The measurement of glass transition temperature, on the other hand, requires more effort. Differential scanning calorimetry has been used extensively to measure the Tg of simple food systems, but this method lacks the sensitivity required to measure the Ts of complex food systems. Other Tnethods have, therefore, been examined as a means to measure glass transition temperatures (Bruni & Leopold, 1991; Kali- chevsky et al., 1992). Without the ability to accurately and rapidly deter- mine the Ts for a food system as a function of moisture content, it is diffi- cult to correlate glass transition temperatures and reaction rates within a complex food system. Much work can still be done in this area.

TEMPERATURE DEPENDENCE OF CHEMICAL REACTIONS

In addition to the effect of moisture on reaction rates, it is also useful to understand the temperature dependence of chemical reactions in order to predict product shelf life. Often, shelf life studies are performed at elevated temperatures in order to expedite data collection. In order to predict the rates of degradative chemical reactions at other tempera- tures, a relations~p between the reaction rate and temperature must be established. The Arrhenius relations~p has traditiona~y been used to describe the temperature dependence of chemical reactions (Glasstone, 1946). However, when reactions are diffusion-limited, an alternative approach by Williams et al. ( 1955) may be appropriate.

Arrhenius and WLF models to predict shelf life 279

Arrhenius temperature model

The Arrhenius relationship is the major mathematical model used to describe the temperature dependence of most chemical reactions (Glasstone, 1946). Arrhenius originally developed this empirical rela- tionship for sucrose hydrolysis (3 889). Subsequently, the basis for the relations~p has been derived from ~ermodyna~c and quantum mechanical principles and has been found applicable to many chemical and physical processes such as diffusion. The Arrhenius relationship is

k= k, exp( - E,/RT) (11 where k is the rate constant at temperature T, k,, is a preexponential factor, R is the ideal gas constant and E, is the activation energy. The reference temperature, according to this equation, is absolute zero. Of course, one cannot measure reactions at this temperature.

Kinetic data at several temperatures over a fairly large temperature range are required in order to test the applicability of the Arrhenius model. A plot of ln( k) versus l/ 7: if a straight line, indicates the applic- ability of the Arrhenius model and that the activation energy over the specific temperature range is constant. If the plot deviates from a straight line, either the test methods were poor, or other reactions begin to become critical above some temperature and influence the reaction rate being studied.

It has been stated that the Arrhenius model is applicable for describ- ing the temperature dependence of reactions within the glassy state of a food matrix and also at 100°C above the glass transition temperature, but is not applicable within the rubbery state (Slade et al., 1989). This assumption can be tested by applying the Arrhenius relations~p to rates of chemical reactions within glassy and rubbery systems to determine whether the Arrhenius model provides a good fit of the data.

von Meerwall and Ferguson (1979b) studied the rate of oil diffusion in natural rubber using NMR spectroscopy. They found that an Arrhenius plot of the diffusion coefficient of oil within the rubbery state of the system showed the expected curvature above the glass transition tem- perature. A broad temperature range from - 10°C to 140°C was used in their study. Based on several kinetic models, the Arrhenius plot would be expected to be curved if such a large temperature range were used (Labuza, 1980) due to the extra temperature dependence at high tem- peratures which is not accounted for in the Arrhenius model.

Oilett and Parker ( 1990) reported that the viscosity of anhydrous glucose and fructose measured at temperatures above the system T, did

280 iy. A. Nelson, T. I? Labuza

not show Arrhenius-type temperature dependence. Although the corre- lation coefficients for the Arrhenius plots for glucose and fructose were 0.98 and O-99, respectively, a slight curvature in the Arrhenius plots for both systems was observed, possibly due to the large temperature range for these data (approximately 50°C).

From the data of Lim and Reid (1991), the Arrhenius plot for an enzyme hydrolysis reaction within a partially frozen maltodextrin DE 25 above the system TL was constructed and is shown in Fig. 5. The data span a very limited temperature range of about 10°C above the Ti ( - 13+3”C). The curvature expected within the rubbery state was not observed, and the Arrhenius plot is quite linear ( r2 = O-99) for this reac- tion, with a calculated activation energy of 42.9 kcal/mole (180 kJ/ mole). In order to observe the curvature expected in the Arrhenius plot within the rubbery state, data over a broader temperature range above the system Tg may be required. However, this is impossible for frozen systems since the system melts at slightly higher temperatures, and is essentially a liquid above 0°C.

A shift in the slope of an Arrhenius plot indicates a change in the activation energy for a reaction. Due to the property differences between glassy and rubbery systems, a break in the Arrhenius plot at the glass transition temperature of a system would be expected. Figure 6 shows an Arrhenius plot which spans a temperature range which includes a glass to rubber transition. This plot is from work by Karmas et al. (1992) on nonenzymatic browning within a carbohydrate matrix. They referred to the temperature at which a break in the Arrhenius plot was observed as the system-dependent ‘critical temperature’. They attributed this break

y x79.497-2.1621E+4x R2=0.999

O‘b037 0.0038 0.0039

l/T (K-r)

Fig. 5. Arrhenius plot of enzymatic hydrolysis rate (absorbance change/h X 10) within a maltodextrin DE25 system (Lim & Reid, 1991).

Arrhenius and WLF mode& to predict skerf I$e 281

in the Arrhenius plot to a collapse of the carbohydrate matrix above the system glass transition temperature. In the case of the lactose/CMC/ trehalose/xylose/lysine model system in Fig. 6, the break in the Arrhe- nius plot is observed near the glass transition temperature ( Tg = 50°C). As generally expected, the activation energy calculated for nonenzymatic browning within the rubbery system was higher than the activation energy within the glassy state. Note that despite the 40°C temperature range within the rubber state, the Arrhenius plot is quite linear within this region (Y: = 0.99). If all the data, both below and above the Tg, are included in the regression, the activation energy is 42.3 kcalfmole (178 U/mole) and the r2 = 0.985, which is still very good.

In some cases, knowing the glass transition temperature may be useful in understanding and explaining breaks observed in Arrhenius plots. However, a break in an Arrhenius plot may be due to other changes in a system, and is not always the result of a glass to rubber transition. Because the activation energy for reactions in glassy and rubbery systems may be different, caution should be applied when extrapolating rates of reactions within glassy systems to systems within the rubbery state. Although it has been stated that the Arrhenius model is not applic- able for describing the temperature dependence of reaction rates within systems in the rubbery state, it has been seen that over small temperature ranges, the Arrhenius model may adequately predict changes in reaction rate. However, the WLF model has been stated to be a more appropriate approach for modeling the temperature dependence within rubbery systems.

ubbery E, = 45.1 kcalimol

-8

-10 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033

1 /T (K-1)

Fig. 6. Arrhenius plot of nonenzymatic browning (OD at 420 nm) within a lactose/ C~C/tr~halosc/xylose/iysine model system at initial a, =@12 (Karmas et al., 1992)

i-, regression including either glassy or rubbery regions; - -- -. regression inciuding all data).

WLF temperature model

An approach to model the temperature dependence of mechanical and dielectric relaxations within the rubbery state, where it is assumed that the Arrhenius model does not theoreticaIly apply, was suggested by Williams Cal ~kl. ( 1955). This approach is known as the WLF approach and has the following form:

where C, and C2 are system-dependent coefficients {Ferry, 19~U~ and al. is defined as the ratio of the relaxation phenomenon at T to the relax- ation at the reference temperature, Tref (i.e. q/qfrF for viscosity). It has been suggested in the literature that WLF kinetics may also describe the temperature dependence of chemical reaction rates within amorphous matrices above their glass transition temperature (Slade ef al., 1989). For systems where diffusion is free volume dep~nde~t, it was shown by Sapru and Lahuza ( 1993) that the rate of reaction can be expressed using the WLF equation with a shift factor of czX = (~~~~~~~:

Average values for the WLF co~~icients were calculated by Williams et al. (195 5) using the available values for many synthetic polymers, It is quite common in the literature to use the average coefficients, which have the values of C, = 17.44 and C, = 5 1.6, for establishing the applic- ability of the WLF model (Soesanto & Williams, 1981; Roos & Karel, I991 h). Peleg ( 1992) discussed several problems associated with the use of average coefficients in the WLF equations He found disagreement between the use of the average coefficients and litera~re values for actual coefficients for prediction purposes 2f)-30°C above the glass transition temperature,

Typical proof of the appIicability of the WLF model, as cited in the food science literature, is made based on fitting the WLF equation to kinetic data shown as a plot of log(k) versus T- Tg. The first step in such a proof invoIves dete~inin~ the rate constant at the glass transition tem- perature, since this is g~neraliy not known. An average value of kg is calculated by solving eqn (3) for aI3 experimental data using the measured T and the average WLF ~oe~i~ie~ts. The ~~l~~lated average k, value is t x en used along with the average WLF ~~e~icients to calcu-

Arrhen~us and WLF models to predict .~h~~~~~ 283

late values of the rate constant over the experimental temperature range using eqn (3). A reasonable fit of the line to the data is stated as proof of the applicability of the WLF model. This was done for establishing the applicability of the WLF approach for the crystallization of sugars (Roos & Karel, 1991 b) and for the viscosity of sugar solutions (Soesanto & Williams, 1981). The disadvantage of this approach is that it assumes that the average WLF coefficients are appropriate for a particular data set, although this may not be the case. Roos and Karel(1992) measured the time for lactose crystallization at several moisture and temperature conditions. They reported that the average WLF coefficients adequately described the temperature dependence of their lactose crystallization data. However, an adequate fit of the crystallization data in the tem- perature range of the measurements can also be made by utilizing other WLF coefficients. Several different values for the WLF coefficients were used to determine an average value for the time for crystallization at the T& using lactose data for systems initially equilibrated to water activities between 0.11 and 0.44, as was done by Roos and Karel(l992) using the average WLF coefficients. The average value of the time for crystalliza- tion at Tf. (8,) was then used to predict the time to cryst~lizatio~ at a specific temperature. This procedure was followed for several different sets of WLF coefficients to construct the lines shown in Fig. 7.

‘*l-----l

Fig. 7.

0 IO 20 30 40 50 60

T-Tg (“C)

Time for crystallization of amorphous lactose ((I, = 0.1 l-044) predicted several different WLF coefficients. Data from Roos and Karel ( 199 1 bi.

bY

284 K. A. Nelson, T. P. Labuza

As observed in Fig. 7, all predicted lines visually fit the crystallization data for the systems equilibrated to water activities between 011 and 0.44 about equally well. However, the time for crystallization at the glass transition temperature, og, was very different, depending on the set of WLF coefficients used. This can be observed as the difference in the intersection of the predicted curves with the y-axis at T- Tg = 0. This is the danger of utilizing the average WLF coefficients, as well as the danger of making predictions at the glass transition temperature using data far above Tg @leg, 1992). Alternative approaches will be suggested for accessing the applicab~i~ of the WLF model.

According to the WLF equation and using the average coefficients, a 10°C increase above the Tg should correspond to a reaction rate increase of approximately 680 times (Slade et al., 1989). Table 1 shows the rate increase expected for other temperature changes above TE. Karmas et al.

(1992) found that the increase in nonenzymatic browning rate within a carbohydrate model system above the Tg was not as large as that pre- dicted using the average WLF coefficients, as shown in Table 1. A simple check of this involves calculating kr values at temperatures above the Ts using eqn (3) using the average coefficients, and comparing these to measured values. It is evident that the rate increases predicted from use of the average WLF coefficients did not describe the temperature dependence of’nonenzymatic browning within this system. This may not mean that the WLF model is not applicable for this reaction, but rather that other WLF coefficients, rather than the average values, must be used for this system.

One approach to determine the applicability of the WLF equation for describing the temperature dependence of a chemical reaction would first be to determine the rate of reaction at the 7’g, if possible. Once kg is

TABLE 1 Rate Increases Predicted Using the WLF Equation with Average Values for C, and Cz, and Observed by Karmas et al. (1992) for Nonenzymatic Browning within a Lactose/

C~C/Trehalose~Xylose/Lysine Model System at Initial a, = 0.12

Temperature rmge Relative rate increase

predicted

Relative rate increase observed

Tg+ Tg+ 10°C 678 21.8 Tg+ lo”C- T,+2O=‘C 110 7-39 Tg + 20°C + Tg + 30°C 3.5 4.06 Tg + 30°C -, Tg + 40°C 1.58 2.82

An-hen&s and WLF models to predict s~e~l~e 285

known, eqn (3), with Ts as the reference temperature, can be rearranged to the following form:

1 1 -I

log2 = -cl 1 -- C,(T- T,) C,

such that a plot of

[ I log%

-I 1

verSuS T-- Tg

gives a straight line with a slope equal to - C,/C, and an intercept of - l/C,, if the WLF model is applicable. For many systems, however, it is not possible to utilize the Tg as the reference temperature because the phenomena at the 76 are so slow that good kinetic rate constants camrot be measured. In such situations, a reference temperature above the Tg should be used to evaluate the appli~ab~ity of the WLF model. In fact, this approach was o~gina~y suggested by Wil~ams et al. (1955) for the same reason.

If the glass transition temperature is known, the WLF constants at the reference temperature can be transformed to correspond to the glass transition temperature, as shown by Peleg ( 1992):

In these relations~ps, 6 is the temperature difference between TrT;,, and Tg, and Ci, and Ck are the transformed WLF coe~cients at the glass transition temperature. These values can then be compared to the widely used average WLF coefficients.

Figure 8 was constructed using the approach of eqn (5) and nonenzy- matic browning rates obtained by Karmas et al. within a carbohydrate model system over a 50°C temperature range above the glass transition temperature. Note that the line is quite linear ( r2 = 0*99), indicating the applicability of the WLF model. From the linear regression for the plot, the values of C, and CZ were determined to be 18-O and 180.2, respect- ively, using 55°C as the reference temperature. When transformed to correspond to the glass transition temperature, 5O”C, using eqn (6), the values for C ‘I and C ‘2 were 18.5 and 175.2, respectively. These are quite different from the average WLF coefficients often cited in the food science literature.

286 K. A. Nelson, T. P. Lu~u~u

0.0 0.1 0.2 0.3

1 U-T,,)

Fig. 8. WLF plot for nonenzymatic browning rates within a la~t~s~/CMC/trehalos~/ xylose/lysine model system at initial u, = 0.12 (Karmas et al., 1992).

An alternative method to determine the applicability of the WLF model for describing the temperature dependence of a chemical reaction is possible when kinetic data at many temperatures are available, the TE is known, but k, cannot be measured. In this approach, it is first assumed that the WLF relationship describes the temperature dependence of the reaction. Then, initial estimates of $ C, and C, are made and iterative nonlinear regression is used to optimize these values, according to the relationship of eqn (3). If kinetic data are available at many temperatures over a wide range (at least 30”C), it is probable that there is only one fit of the data set and that the values determined for k,, C, and C, are probably quite close to actual values. von Meerwall and Ferguson (1979a) used this three parameter optimization method to fit the diffu- sion of a diluent within a rubbery matrix with the WLF equation. They found that experimental precision was very important for the accuracy of such a fit. An additional disadvantage of this method is that it assumes, rather than proves, that the WLF model applies to the system.

CONCLUSIONS

Understanding the relationship among moisture, temperature and chemical reaction rates can be very useful for food stability prediction purposes. Both water activity and glass transition theory can be used to understand the influence of water on rates of chemical reactions. Water activity essentially considers the state of water in a food. Its relationship to chemical reaction rates is fairly complex and dependent upon the particular chemical reaction of interest. Glass transition theory generally

Arrhenius and WLF models to predict shelf life 287

considers the state of a food matrix, which is, of course, a function of water in the food. Based on the properties of the glassy state, glassy systems should be stable to chemical reactions, whereas rubbery systems should be fairly reactive. However, it was shown that the opposite result may be observed in some systems due to the porosity associated with glassy systems or the collapse and crystallization associated with rubbery systems. Therefore, stability to chemical reactions is a function of the state of a matrix, as well as several other system-dependent properties.

Several approaches for determining the theoretical and practical applicability of the Arrhenius and WLF temperature dependence models were discussed. The WLF approach should theoretically apply to reactions within rubbery systems. However, the Arrhenius rela- tionship has also been shown to adequately describe the temperature dependence of chemical reactions over narrow temperature ranges within the rubbery state.

Data over a broad temperature range are required in order to differ- entiate between the WLF and the Arrhenius models. Both may be used successfully in describing the temperature dependence of chemical reactions over limited temperature ranges, but the WLF approach provides a physical explanation for changes in rate observed for temper- ature changes within rubbery systems.

ACKNOWLEDGEMENTS

Published as paper number 20,107 of the contribution series of the Minnesota Agricultural Experimental Station based on research conducted under Projects 377-1572 and 377-1578. This research was partially funded by a USDA National Needs Fellowship.

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