2012-An Activated-state Model for the Prediction of Solid-phase Crystallization

Journal of Food Engineering 109 (2012) 691–700

Contents lists available at SciVerse ScienceDirect

Journal of Food Engineering

journal homepage: www.elsevier .com/ locate / j foodeng

An activated-state model for the prediction of solid-phase crystallizationgrowth kinetics in dried lactose particles

Debolina Das ⇑, Timothy A.G. Langrish 1

School of Chemical and Biomolecular Engineering (J01), University of Sydney, NSW 2006, Australia

a r t i c l e i n f o

Article history:Received 14 July 2011Received in revised form 9 November 2011Accepted 11 November 2011Available online 28 November 2011

Keywords:Solid-phase crystallizationActivated-state modelFree energy of activationEnthalpyEntropyKineticsLactoseReaction engineeringEyring equationWLF and Avrami equationActivated Rate Theory

0260-8774/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.jfoodeng.2011.11.024

⇑ Corresponding author. Tel.: +61 2 9351 5661; faxE-mail addresses: [email protected]

sydney.edu.au (T.A.G. Langrish).1 Tel.: +61 2 9351 4568.

a b s t r a c t

Molecular-level understanding of solid-phase crystallization growth kinetics has the potential to improvethe fundamental basis for understanding this process. This understanding has been developed here aspart of an activated-state model, which accounts for the effects of the moisture content and the temper-ature on the crystallization rate. Water-induced crystallization (WIC) at different temperatures (15 �C,25 �C, 40 �C) and a constant relative humidity (75%) environment has been used to analyze the changesin enthalpy, entropy and Gibbs free energy of activation for the solid-phase crystallization of lactose. WICshowed that, at higher temperatures, crystallization commences at lower moisture contents. Theenthalpy and Gibbs free energy of activation increased during the crystallization process, which sug-gested that the binding energy needed for the formation of an activated complex increased as the mois-ture content decreased, making the formation of the activated complex more difficult. The entropy ofactivation, on the other hand, decreased with the decrease in the moisture content. From the activatedrate equation, the energy of activation has been estimated to be 39 ± 2 kJ mol�1, which is similar tothe literature value (40 kJ mol�1) for the solid-phase crystallization of lactose. The reaction rate constantat 25 �C is 1.4 � 10�4 s�1, which is similar to the literature value of 1.3 � 10�4 s�1. The WLF equation isinconsistent and hence unreliable in predicting the rates of crystallization at the glass-transition temper-ature from the analysis of experimental data at different temperatures.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Amorphous materials have non-aligned molecular structures,are hygroscopic and are in a higher-energy state relative to crystal-line ones. Spray-dried materials are mainly amorphous (White andCakebread, 1966) and potentially sticky (Bhandari et al., 1997).During spray drying, atomized feed in the form of fine dropletscomes into contact with hot air in the drying chamber and drieswithin a fraction of a second, giving insufficient time for the drop-lets to crystallize. The high hygroscopicity, high solubility, lowmelting point and low glass-transition temperature of the amor-phous spray-dried materials are major factors causing productstickiness, causing them to readily adhere to the walls of spray dry-ers, hindering product flow and stability. An amorphous material isgenerally in a metastable state with respect to the crystalline stateand hence tends to undergo phase transformation during process-ing and storage. A system is said to be in a metastable state at aparticular temperature if small isothermal changes or fluctuationsof its thermodynamic variables cause an increase in the free en-

ll rights reserved.

: +61 2 9351 2854.(D. Das), timothy.langrish@

ergy, while larger deviations cause the system to reach a state ofminimum thermodynamic potential with low free energy. Duringcrystallization, an amorphous material may be considered to passthrough an intermediate state, which acts as a free energy barrierand determines the transformation of the metastable amorphousstate into the stable crystalline state (Kauzmann, 1948).

Solid-phase crystallization kinetics has been widely studied(Takeuchi et al., 2000; Timmerman et al., 2004; Vautaz, 1988;Ibach and Kind, 2007; Roos and Karel, 1992; Jouppila and Roos,1994; Buckton and Darcy, 1996, 1997, 1998a,b; Elamin et al.,1995; Vollenbroek et al., 2010; Paterson et al., 1997) and has beenmodeled using the Williams–Landel–Ferry (Williams et al., 1955)and Avrami equations. The Williams–Landel–Ferry (WLF) equationsuggests that the rate of crystallization is related to the differencebetween the material temperature (T) and its glass-transitiontemperature (Tg). The WLF equation is empirical and is a time–temperature superposition equation. The equation estimates thetemperature shift factor at temperatures other than those forwhich the material was tested and hence is also called a ‘temper-ature-shift’ equation. Although the equation provides a useful cor-relation between the temperature and rate of crystallization, thefunction (T�Tg) is not a physical driving force for the crystallizationprocess. Also, the equation is only valid for temperatures up to100 �C greater than the glass-transition temperature (Williams

http://dx.doi.org/10.1016/j.jfoodeng.2011.11.024

mailto:[email protected]



http://dx.doi.org/10.1016/j.jfoodeng.2011.11.024

http://www.sciencedirect.com/science/journal/02608774

http://www.elsevier.com/locate/jfoodeng

692 D. Das, T.A.G. Langrish / Journal of Food Engineering 109 (2012) 691–700

et al., 1955). The universal constants used in this equation havebeen queried (Peleg, 1992) and are most likely to be product spe-cific (Langrish, 2008). Roos and Karel (1992) showed that theWLF equation makes the reaction appear to be a zero-order reac-tion in terms of the concentration because, in this approach, thecrystallization kinetics are expressed in an integrated form ratherthan as a function of reactant concentration. Therefore, with theabove limitations, it is unclear if the WLF equation is able to predicthigh-temperature crystallization accurately in a quantitativemanner.

Another approach to modeling solid-phase crystallization is theAvrami equation, which assumes that negligibly small ‘droplets’ ofthe stable phase nucleate from the metastable background; thedroplets subsequently grow independently without substantialdeformation, and the stable phase is pictured as randomly placed,freely overlapping, growing spheres (Roos and Karel, 1990).Although it has been shown, by molecular-dynamics simulations,that there is some theoretical basis for applying the Avrami equa-tion to solid-phase crystallization (Novotny et al., 2000; Ramoset al., 1999), often considerable difficulty is faced in applying thistheory in practice (Kedward et al., 2000). The applicability of theAvrami equation has been debated, since this equation does notexplicitly account for the effect of temperature when used as a cor-relation for fitting data from isothermal crystallization experi-ments (Langrish, 2008). Roos and Karel (1992) tested theapplicability of the Avrami equation by measuring the crystalliza-tion rate of amorphous lactose using water-induced crystallizationand found that the equation did not fit the crystallization processvery well.

The physical process of crystallization consists of two mainsteps (Tamman, 1926), nucleation and crystal growth. Both theWLF and Avrami equations do not distinguish between the twoprocesses and instead lump them together. The current level ofempiricism means that considerable experimental work is re-quired for new materials due to the lack of physical understandingthat restricts the prediction of solid-phase crystallization behaviorfrom first principles. A new way of quantitatively modeling thekinetics of solid-phase crystallization may be a reaction engineer-ing approach, by viewing it as an activated-rate process, where itmight be possible to quantitatively express the kinetics of crystalgrowth separately from nucleation in a rate form rather than anintegrated one (Langrish, 2008).

Both the Arrhenius and Eyring equations describe the tempera-ture dependence of reaction rates. The Arrhenius equation is basedon the empirical observation that rates of reactions increase withtemperature, while the Eyring equation is based on a transition-state model. According to the transition-state model, the reactantsreach an unsteady intermediate state on the reaction pathway be-tween reactants and products. An energy barrier exists betweenthe reactants and the products on the reaction pathway that deter-mines the minimum energy required for the reaction to occur. Thisminimum energy is known as the activation enthalpy or activationenergy. The Eyring equation can be expressed as:

lnkT¼ � DH�

R

� �� 1Tþ ln

kB

h

� �þ DS�

R

� �ð1Þ

where k is the reaction rate constant, kB is Boltzmann’s constant, h isPlanck’s constant, DS⁄ and DH⁄ are the entropy and enthalpy of acti-vation, respectively, T is the absolute temperature and R is the uni-versal gas constant. A plot of ln(k/T) as a function of 1/T produces astraight line of the form y = mx + b, where the slope (m) is � DH�

R

� �and intercept (b) is ln kB

h

� �þ DS�

R

� �. The enthalpy and entropy of acti-

vation for the reaction can be calculated from the temperature andthe reaction rate constant.

Moisture plays a key role (Ibach and Kind, 2007; Bianchi et al.,1991; Ando et al., 1992) in the kinetics of solid-phase crystalliza-tion, along with temperature (Thomsen et al., 2005; Langrish,2008; Chiou et al., 2008). The activation enthalpy and entropyare directly related to the change in composition of the material,which is moisture in the case of water-induced crystallization.Chen and Xie (1997) created a drying fingerprint, a unique functionfor each material that expresses the relationship between thechanges in activation energy for drying as a function of the mois-ture content. A similar approach might be taken to create a uniquefunction that relates the activation energy for crystallization to thechange in moisture content and temperature, as a type of ‘‘finger-print’’ for the crystallization kinetics of a material. Sorptionkinetics of various materials, including lactose, have been widelystudied (Paterson and Bronlund, 2004; Jouppila and Roos, 1994a)and effectively modeled before (Jouppila and Roos, 1997; Haqueand Roos, 2004; Omar and Roos, 2007) using a three-parameterGAB equation (Anderson, 1946; de Boer, 1995; Guggenheim,1995) based on a multilayer adsorption model. It represents a ther-modynamic model based on multi-layer condensation, and it hasbeen successfully applied to a wide range of food and other mate-rials (Van den Berg, 1985). In thermodynamic terms, the enthalpyof activation is related to the moisture content of the material bythe energy of interaction between the monolayer and multilayermolecules. The decrease in interaction energy between monolayerand multilayer molecules causes the enthalpy to decrease duringthe process of crystallization and releases moisture. Crystallizationof amorphous sugars releases a quantity of heat, which is depen-dent on the moisture content (Roos and Karel, 1991a). Since thechange in particle moisture content has been successfully modeledbefore with the GAB equation, the corresponding energy (enthalpy)change, from energy-mass equivalence, can be modeled using thesame form of the equation. Also, since the activation enthalpyand entropy follow enthalpy–entropy compensation in lactose,the GAB equation can be used to model the activation entropy aswell, as a function of the particle moisture content. However, theconstants in the equation for modeling the activation enthalpyand entropy for the process are likely to be very different. This pa-per explores the potential for applying a transition-state reactionengineering approach to analyzing water-induced solid-phasecrystallization in comparison with the WLF equation.

2. Materials and methods

Samples of dried lactose powders have been produced using aBuchi-290 laboratory-scale spray dryer (Buchi, Flawil, Switzer-land). The spray-drying conditions were kept at the followingvalues based on previous work by Chiou et al. (2008) and Chiouand Langrish (2008) to produce highly amorphous powders:

Inlet air temperature: 134 �CAspirator rate: 100% (0.0149 kg/s)Atomizer air-flow rate: 52 mm (19.9 L/min)Pump rate: 23% (0.000106 kg/s)Solution concentration: 9.09% w/w

The moisture sorption behavior of the freshly spray-dried lac-tose samples was then determined at different temperatures, rang-ing from 15 �C to 40 �C, using water-induced crystallization (WIC).The method for WIC was to place the sample on a Petri dish andtake mass measurements every minute with an analytical balance(Mettler-Toledo AB204S, four decimal figure analytical ± 0.0001 g)to study the change in moisture content as a function of time. Acontrolled humidity environment with a saturated sodium chlo-ride solution (75% relative humidity, Winston and Bates, 1960)

Fig. 1. Water-induced crystallization at a constant temperature showing thesorption peak, the initial moisture content and the equilibrium moisture content.

Fig. 2. Variation of lactose moisture content with time during solid-phasecrystallization at three different temperatures in water-induced crystallizationwith a relative humidity of 75%.

D. Das, T.A.G. Langrish / Journal of Food Engineering 109 (2012) 691–700 693

and a constant temperature (15 �C, 25 �C or 40 �C) was used. Themoisture contents of the particles were characterized specificallyin terms of the final moisture content, which refers to the moisturecontent of the particles when they are crystalline (the final equilib-rium moisture content), and the moisture sorption percentage,which is the percentage change in weight from the sorption peakto the final equilibrium moisture level. Fig. 1 shows the sorptionpeak, the initial and the final equilibrium moisture content in ageneral sorption curve from a water-induced crystallizationexperiment.

In order to model the rate process for the solid-phase crystalli-zation of lactose, water-induced crystallization experiments wereperformed at three different temperatures of 15 �C, 25 �C and40 �C and a relative humidity of 75% using the same sample offreshly spray-dried lactose.

3. Results and discussion

3.1. Experimental data

The variations in moisture content with time during water-induced crystallization for the same sample at three different tem-peratures from 15 �C to 40 �C have been shown in Fig. 2.

It was observed that the sample mass increased due to watervapor sorption and then, after a certain period of time, the samplemass steadily decreased. This characteristic feature of mass losshas been well documented and has been previously assigned tothe crystallization of amorphous lactose (Buckton and Darcy,1995; Burnett et al., 2004; Price and Young, 2004). The lag betweenthe increase in moisture content and the point where moisture lossis observed has been taken as the onset time for crystallization.After the crystallization is complete, the moisture content reachesa constant level. The characteristic moisture sorption peaks were0.088 g/g, 0.070 g/g and 0.036 g/g dry mass for temperatures of15 �C, 25 �C and 40 �C, respectively. It was also observed that thetime for crystallization decreased with an increase in the temper-ature. The crystallization times were 1000 min, 600 min and350 min for temperatures of 15 �C, 25 �C and 40 �C, respectively.Hence, this suggests that less moisture was required to overcomethe activation energy barrier for crystallization at highertemperatures.

Table 1 shows the comparison between the crystallizationtimes and the moisture sorption peaks at 25 �C and different rela-tive humidities for spray-dried lactose obtained from differentauthors. The differences in the moisture content peaks and thecrystallization times here may be mainly due to the differencesin the relative humidities.

Crystallization is assumed to commence after the highest mois-ture content (peak) in Fig. 2 (Buckton and Darcy, 1995; Burnettet al., 2004). Considering that the degree of crystallinity is zero atthe peak of the curve (highest moisture content), correspondingto the start of crystallization, and that the degree of crystallinityis unity at the crystalline state, corresponding to the bottom ofthe curve, the variation of the degree of crystallinity with timehas been shown in Fig. 3.

3.2. Analysis of experimental data

In order to analyze the experimental data, it is necessary to con-sider both the order of the crystallization process and the rate con-stant, which will be presented and discussed further in thefollowing sections.

3.2.1. Reaction orderExperimentally, the order of the reaction for the solid-phase

crystallization of lactose can be determined from data on water-in-duced crystallization at different temperatures. In general, for areaction in which only the mass of a single component changes,the reaction-rate equation can be written as follows:

d½X�dt¼ k½X�n ð2Þ

where n is the order of the reaction, [X] is the moisture content ateach point in time, t is the time from the start of the reaction(min), and k is the reaction-rate constant. An implicit assumptionhere is that the extent of water loss is proportional to the extentof the reaction, and the rate of the reaction is the rate of changein the moisture content after the start of the crystallization process(after the peak in the moisture sorption curve). If the reaction rate(d[X]/dt) is plotted against the moisture content [X] at each pointin time, then the order of the reaction can be determined fromthe shape of the curve. From the water-induced crystallizationexperiments, the solid-phase crystallization results of lactose wereanalyzed to assess if they followed first-order reaction kinetics.Figs. 4–6 shows the results from analyzing the WIC data at 25 �Cusing first-order, second-order and third-order reaction kineticmodels, respectively.

The results show that the plot for the reaction rate against themoisture content [X] using a first-order rate model has given astraight line, suggesting that crystallization follows first-orderkinetics. Plotting the reaction rate with the second and third-orderreaction models has given curved lines, indicating that a second orthird-order kinetic model is not valid for the solid-phase crystalli-zation of lactose. The results of the analysis were same for theother temperatures (15 �C and 40 �C) when first, second and

Table 1Crystallization time and moisture sorption peaks at 25 �C obtained from moisture sorption profiles of spray-dried lactose at differentrelative humidities.

Authors Relative humidity (%) Moisture sorption peak (%) Crystallization time (min)

Burnett et al. (2004) 55 12 350Price and Young (2004) 58 11.5 360Das and Langrish (present work) 75 7.2 560

Fig. 3. Changes in the degree of crystallinity as a function of time during solid-phase crystallization for three different temperatures.

Fig. 4. Analysis of WIC data for the reaction rate as a function of the moisturecontent [X] using a first-order reaction kinetic model.

Fig. 5. Analysis of WIC data for the reaction rate as a function of the square of themoisture content [X] using a second-order reaction kinetic model.

Fig. 6. Analysis of WIC data for the reaction rate as a function of the cube of themoisture content [X] using a third-order reaction kinetic model.


third-order reaction models were used. It has been reported byother authors before that solid-phase crystallization of carbohy-drates follows a first-order reaction kinetic model (Roos, 2003).In 1995, Roos found that carbohydrates are generally miscible withwater and show first-order phase transitions, including crystalliza-tion. However, Schmitt et al. (1999) used a reaction order of threeto estimate the crystallization kinetics of amorphous lactose in thepresence and absence of seed crystals. The results here (first-orderkinetics) are in line with the most recent findings regarding thereaction order.

3.2.2. Reaction rate constantThe statistical thermodynamic version of activated complex

theory has motivated an approach in which the activation processis expressed in terms of thermodynamic functions. According tothe Eyring equation for activated complex theory (Atkins and dePaula, 2002),

lnkT

� �¼ ln

kB

h

� �þ DS�

R

� �� DH�

R

� �1T

ð3Þ

where kB is Boltzmann’s constant (1.381 � 10�23 J/K) and h isPlank’s constant (6.626 � 10�34 J s), T is the temperature in K, R isthe universal gas constant (8.3144 J mol�1 K�1), DS⁄ is the entropyof activation (J mol�1 K�1) and DH⁄ is the enthalpy of activation(J mol�1).

The reaction rate constants (k) of the WIC experiments at 15 �C,25 �C and 40 �C have been calculated from the initial concentrationusing the first-order reaction kinetic equation (Eq. (2)). The enthal-py and entropy of activation have then been obtained from theslope and intercept of the Eyring equation (Eq. (3)) using least-squares linear regression of ln(k/T) as a function of 1/T for the threedifferent temperatures of 15 �C, 25 �C and 40 �C. Each of the exper-iments was repeated three times to measure the uncertainties inthe values of the reaction rate constant. The reaction rate constantk for each of the temperatures (15 �C, 25 �C and 40 �C) has beenfound to be 8 � 10�5 ± 3 � 10�5 s�1, 1.4 � 10�4 ± 7 � 10�5 s�1 and3 � 10�4 ± 1 � 10�4 s�1, respectively. The predicted k at 60 �C hasbeen obtained by extrapolating the line backwards to that temper-ature and has been found to be 7 � 10�4 s�1. Table 2 shows thereaction rate constants obtained from the WIC experiments atthe different temperatures. Fig. 7 shows the linear regression ofEq. (3) (ln(k/T) as a function of 1/T) with the slope and theintercept.

The predicted value of ln(k/T) at 60 �C has been found from theActivated Rate Theory in Fig. 7 to be �9 ± 1 by extrapolating thebest fit line obtained from the experimental data for the three dif-ferent temperatures. The crystallization rate constant k, has beencalculated from the value of the predicted value of ln(k/T) andhas been found to be (7 ± 2.5) � 10�4 s�1 at a temperature of60 �C and a relative humidity of 75%. From the slope and the inter-cept of the regression line, the enthalpy of activation for solid-

Table 2Crystallization rate constants (k) from the WIC experiments using the Activated Rate Theory.

15 �C 25 �C 40 �C 60 �C

Reaction rate constant k (s�1) from WIC experiments using Activated Rate Theory 8 � 10�5 s�1 1.4 � 10�4 s�1 3 � 10�4 s�1 7 � 10�4 s�1 (predicted)

Fig. 7. Linear regression of the Eyring equation showing the slope and the intercept for the WIC data at three different temperatures.


phase crystallization has been found to be 39 ± 2 kJ mol�1. The en-tropy of activation is �156 ± 6 Jmol�1 K�1.

Table 3 shows the standard uncertainties in the slope and inter-cept of the regression line and the effect of these uncertainties onthe calculated enthalpy and entropy of activation. The standarduncertainty u(y) of a measurement result y is the estimated stan-dard deviation of y.

In 2007, Ibach and Kind studied the crystallization kinetics ofamorphous lactose, whey-permeate and whey powders and foundthat the crystallization rate constant k increased with increasingtemperature. They found that the activation energy for the solid-phase crystallization of pure lactose at relative humidities between70% and 80% was 40 kJ mol�1, which is similar to the experimentalvalue obtained from the WIC experiments here using the activatedrate theory (39 ± 2 kJ mol�1).

3.3. Modeling the activation enthalpy and activation entropy ofcrystallization

The GAB equation has been used to model the thermodynamicvariables, including the enthalpy and entropy of activation, as afunction of the moisture content during the crystallization process.The equation for the enthalpy of activation is shown in Eq. (4).

DH� ¼ m1c1k1X1� k1Xð Þ 1� k1X þ c1k1Xð Þ ð4Þ

Here DH⁄ is the activation enthalpy; X is the moisture content of theparticle; m1, c1 and k1 are the three parameters of the GAB equation.In the equation, m1 is the monolayer activation enthalpy, while c1

and k1 are parameters that depend on the temperature (Van denBerg, 1985). The values of the constants m1, c1 and k1 in the enthal-py equation can be obtained by correlating the moisture content

Table 3Standard uncertainties in enthalpy and entropy of activation.

Coefficients Standard uncertainties

Slope, (m) �4600 ±230y-intercept, (c) 4.99 ±0.77Correlation, (r) �0.8DH⁄ (J mol�1) 39,000 ±2000DS⁄ (J mol�1 K�1) �156 ±6

values at the peak for the WIC experiments at the three tempera-tures with the activation enthalpy value found experimentally(39 ± 2 kJ mol�1) discussed in the previous section. The moisturecontents (X) when crystallization began in the WIC experiment (atthe peak) were 0.036, 0.070 and 0.088 (g/g dry weight) for the tem-peratures of 40 �C, 25 �C and 15 �C, respectively. The activation en-thalpy should fit all the three ‘X’ (moisture content) values in Eq. (4)by using the three parameters m1, c1 and k1. Using the three water-sorption records for the three temperatures, three equations wereobtained as follows:

39000� 2000 ¼ m1c1k1ð0:088Þ1� 0:088k1ð Þ 1� 0:088k1 þ 0:088c1k1ð Þ ð5Þ



Solving Eqs. (5)–(7), the values of the constants m1, c1 and k1 of theenthalpy equation are 21,000 ± 2000, �16.7 ± 2 and 4.7 ± 0.6,respectively.

Similarly, the equation for the activation entropy, as modeledwith the GAB equation, has been shown in Eq. (8), where m2, c2

and k2 are the three constants:

DS� ¼ m2c2k2X1� k2Xð Þ 1� k2X þ c2k2Xð Þ ð8Þ

Here X is the moisture content of the material (g/g dry weight). Theparameter m2 refers to the activation entropy of the monolayer,while c2 and k2 are parameters related to monolayer and multilayerproperties (Kaymak-Ertekin and Gedik, 2004). The values of theconstants m2, c2 and k2 for the entropy equation can be obtainedby comparing the moisture content values at the peaks for theWIC experiments at the three temperatures with the activation en-tropy value found experimentally (�156 ± 6 J mol�1K�1). Equatingthe experimental entropy similarly, in Eq. (8), with the moisturecontent peaks for the three temperatures, three equations havebeen obtained:

�156� 6 ¼ m2c2k2ð0:088Þ1� 0:088k2ð Þ 1� 0:088k2 þ 0:088c2k2ð Þ ð9Þ

Fig. 9. The variation in the entropy of activation for lactose crystallization as afunction of moisture content.

Fig. 10. The variation in the Gibbs free energy of activation for lactose crystalli-zation as a function of moisture content for three different temperatures of 15 �C,25 �C and 40 �C.

Fig. 11. Enthalpy–entropy compensation effect during solid-phase crystallizationof dried lactose particles.




From the least-squares analysis of the experimental moisturecontents and activation entropy for three different temperatures,the constants m2, c2 and k2 of the activation entropy equation are�95 ± 10, �16.7 ± 2 and 4.7 ± 0.6, respectively. Fig. 8 shows thevariation in the enthalpy of activation with the moisture contentduring crystallization using Eq. (4). All the results for different tem-peratures overlap on the same curve, suggesting that this theoryaccounts for these variations in enthalpy with changing tempera-ture and moisture content.

Fitting a function to these data and to this curve, the relation-ship between the enthalpy of activation and the moisture contentcan be written as,

DH� ¼ ð21000Þð�16:7Þð4:7ÞXð1� 4:7XÞð1� 4:7X þ ð�16:7Þ4:7XÞ ð12Þ

The increase in the activation enthalpy (activation energy) withthe decrease in the moisture content during crystallization sug-gested that the binding energy needed for the formation of an acti-vated complex decreased as the moisture content decreased duringthe crystallization process. Fig. 9 shows the variation in the entro-py of activation with the moisture content during crystallizationusing Eq. (8). As with Fig. 8, the consistency of the theory is sug-gested by all the data lying on a single smooth curve for differenttemperatures and moisture contents.

Fitting a function to these data and to this curve, the relation-ship between the entropy of activation and the moisture contentcan be written as,

DS� ¼ ð�95Þð�16:7Þð4:7ÞXð1� 4:7XÞð1� 4:7X þ ð�16:7Þ4:7XÞ ð13Þ

The entropy of activation decreased with the decrease in themoisture content during crystallization. This behavior is expectedbecause entropy measures the degree of randomness or disorderin a system, and, as crystallization proceeds, the particles becomedrier, with more restricted molecular movement. The Gibbs freeenergy of activation has been calculated for each point of moisturecontent during crystallization using the Gibbs–Helmholtz equationEq. (14).

DG� ¼ DH� � T � DS� ð14Þ

The variation in the Gibbs free energy of activation with mois-ture content during crystallization for different temperatures of15 �C, 25 �C and 40 �C has been shown in Fig. 10.

From the fitted function in Eqs. (12) and (13), the relationshipbetween the Gibbs free energy of activation and the moisture con-tent can be written as,

Fig. 8. The variation in the enthalpy of activation for lactose crystallization as afunction of moisture content.

DG� ¼ 21000ð�16:7Þ4:7Xð1� 4:7XÞð1� 4:7X þ ð�16:7Þ4:7XÞ

� T:ð�95Þð�16:7Þ4:7X

ð1� 4:7XÞð1� 4:7X þ ð�16:7Þ4:7ÞXÞ ð15Þ

Given that m1/m2 = 221 K, from Eqs. (4) and (8), Eq. (15) may besimplified to give,

DG� ¼ DH� 1� T221

� �ð16Þ

It is observed that the free energy increases as the moisturecontent decreases, which suggests that the formation of the acti-vated complex for solid-phase crystallization becomes more diffi-cult as the reaction proceeds. Temperature does not havesignificant effect on the Gibbs free energy of activation, since the

Fig. 12. Inconsistency in the estimated logarithmic values of the rates of crystal-lization at the glass-transition temperature as a function of moisture content fromthe analysis of experimental data at 15 �C, 25 �C and 40 �C.


three temperature curves completely overlap in Fig. 10, as wouldbe expected from previous experience (Crofts and Wang, 1989;Sharma and First, 2009) for activated reactions.

Given that temperature does not have significant effect on theGibbs free energy an isokinetic temperature (compensation effect)was evaluated. In Fig. 8 from Eq. (4), the enthalpy of activation in-creases with a decrease in the moisture content during crystalliza-tion, whereas in Fig. 9 from Eq. (8), the entropy of activationdecreases with a decrease in the moisture content, showing an en-thalpy–entropy compensation effect during the process of lactosesolid-phase crystallization. Hence, solid-phase crystallization oflactose follows enthalpy–entropy compensation and the Gibbs freeenergy, being equal to the difference between the enthalpy of acti-vation and the product of the temperature with the entropy of acti-vation, is not affected by the temperature. Fig. 11 shows theenthalpy–entropy compensation effect for lactose during solid-phase crystallization.

3.4. Inconsistencies in the WLF equation and its outcome

The WLF equation (Eq. (16)) suggests that the ratio of the timefor crystallization at a particular point (hcr) to the time for crystal-lization at the glass-transition temperature (hg) decreases whenthe difference between the material temperature and its glass-transition temperature becomes sufficiently high. The inverse ofthis ratio, r, is a measure of the ‘‘impact’’ of the particle tempera-ture (T) and its glass-transition temperature (Tg) (Chiou et al.,2008):

log r ¼ loghcr

hg

� �¼ log

kg

kcr

� �¼ �17:44ðT � TgÞ

51:6þ ðT � TgÞð17Þ

The effect of the moisture content on this glass-transition tem-perature can be estimated using the Gordon Taylor equation Eq.(17):

Tg ¼w1Tg1 þ kw2Tg2

w1 þ kw2ð18Þ

Here, w1; Tg1 and w2; Tg2 are the weight fractions and glass-transi-tion temperatures of the individual components, water and lactose,respectively. The glass-transition temperature of pure lactose is101 �C (Roos and Karel, 1991b) and that for pure water is �137 �C(Johari et al., 1987). The curvature constant k is equal to 7.42 (Roos,1993), which is determined empirically. The weight fraction ofwater (w1) is related to the moisture content, X, expressed on adry basis, through the equation w1 = X/(1 + X). Since the moisturecontent varies throughout adsorption and desorption (a typicalsorption curve is given in Fig. 1), the corresponding glass-transitiontemperature varies accordingly.

To check for consistency in the WLF equation, the rate of crystal-lization at the glass-transition temperature (kg) estimated from theexperimental data should be same for all the temperatures. Therates of crystallization (k) for three different temperatures (15 �C,25 �C and 40 �C) were assessed using the relation, k ¼ dh=dt whereh is the degree of crystallinity at time t, assuming that the materialwas completely amorphous (degree of crystallinity, h = 0) at thepeak of the sorption curve (Fig. 1) and one hundred percent crystal-line (degree of crystallinity, h = 1) at the bottom of the sorptioncurve when the moisture content becomes constant with respectto time. The WLF equation can be expressed as follows:

log kg� �

¼ logðkÞ � 17:44ðT � TgÞ51:6þ ðT � TgÞ

ð19Þ

The logarithmic value of the rate of crystallization at the glass-transition temperature (logkg) was calculated from the rate of crys-tallization at the each local point (k) of the sorption curve, the

glass-transition temperature (Tg) and the material temperature(T). The kg values for all the crystallization temperatures were com-pared, and it was observed that the values of kg varied widely for15 �C, 25 �C and 40 �C. Fig. 12 shows the inconsistent values forthe rate of crystallization at the glass-transition temperature fromthe analysis of experimental data at the three different tempera-tures using the WLF equation. The rate of crystallization at theglass-transition temperature would be expected to be constantregardless of how it has been obtained, so the inconsistent valuesof kg obtained experimentally here illustrate the unreliability ofthe WLF equation.

3.5. Comparing the WLF equation and Activated Rate Theory

The equation(s) for the Activated Rate Theory relates the rate ofsolid-phase crystallization to the moisture content and the tem-perature of the material and shows how the moisture content inturn changes the enthalpy and entropy of activation, thus affectingthe Gibbs free energy of activation for the crystallization process.The Activated Rate Equation(s) has been summarized below:

lnkT

� �¼ ln

kB

h

� �þ DS�

R

� �� DH�

R

� �1T

ð20Þ

DH� ¼ ð21000Þð�16:7Þð4:7ÞXð1� 4:7XÞð1� 4:7X þ ð�16:7Þ4:7XÞ ð21Þ

DS� ¼ ð�95Þð�16:7Þð4:7ÞXð1� 4:7XÞð1� 4:7X þ ð�16:7Þ4:7XÞ ð22Þ

Eq. (3) shows the correlation between the rate of crystallization(k) with the enthalpy (DH⁄) and entropy of activation (DS⁄) and thetemperature (T) based on the transition state model of Eyringequation. Eqs. (12) and (13) correlate the enthalpy and entropyof activation, respectively, as a function of the moisture contentfor solid-phase crystallization, and the values for the enthalpyand entropy of activation are specific to the solid-phase crystalliza-tion of lactose.

Table 4 compares the crystallization rate constant (k) obtainedfrom WIC experiments using the Activated Rate Theory, and thek value from the WLF equation, with that (k) of the experimentaldata from Haque and Roos (2005). The reaction rate constant (k)values were calculated experimentally from the WIC experimentsat 15 �C, 25 �C and 40 �C using the Activated Rate Theory, as shownin Table 2. At 60 �C, k has been predicted by extrapolating the linein Fig. 7 backwards to that temperature, giving a value for k of7 � 10�4 s�1.

The rates of crystallization (k) for three different temperatures(15 �C, 25 �C and 40 �C) were assessed using the WLF equation

Table 4Comparison of crystallization rate constants from WIC experiments, using the Activated Rate Theory, with the k value from the WLF equation.

15 �C 25 �C 40 �C 60 �C

Reaction rate constant k (s�1) from WIC experiments using Activated Rate Theory 8 � 10�5 s�1 1.4 � 10�4 s�1 3.2 � 10�4 s�1 7 � 10�4 s�1 (predicted)Reaction rate constant k (s�1) using WLF equation 2.5 � 10�5 s�1 6 � 10�5 s�1 1.1 � 10�4 s�1 1.7 � 10�4 s�1 (predicted)Experimental reaction rate constant k (s�1) from Haque and Roos (2005) – 1.3 � 10�4 s�1 – –

Fig. 13. Schematic diagram showing the energy level within the particles duringcrystallization.


Eq. (14) described earlier in Section 3.4. Using the WLF equation,the k values were found to be 2.5 � 10�5 s�1, 6 � 10�5 s�1 and1.1 � 10�4 s�1 at temperatures of 15 �C, 25 �C and 40 �C, respec-tively. The reaction rate constant at 60 �C was found to be1:7� 10�4 s�1 by extrapolating the line to that temperature. Com-paring the reaction rate constants in Table 2 for lactose solid-phasecrystallization from the experimental data of Haque and Roos(2005) shows that the value of k (0.48 h�1 equivalent to1.3 � 10�4 s�1) at a relative humidity of 76.1% at room temperatureis close to the k value (1.4 � 10�4 s�1) at 25 �C and 75% RH from theActivated Rate Theory, but very different from the k value obtainedusing the WLF equation.

The WLF equation is an empirical equation that takes into ac-count the effects of time and temperature on the rate of crystalli-zation, but the function (T�Tg) does not represent a physicaldriving force for the crystallization process. The WLF equation isbased on the principle that the rate of crystallization varies withthis function (T�Tg), which itself varies as crystallization proceeds,because the moisture content varies and changes the glass-transi-tion temperature simultaneously. The equation gives inconsistentvalues of kg experimentally, while it is expected that the crystalli-zation rate at the glass-transition temperature (kg) would be con-stant regardless of the method by which it is obtained. Unlikethe WLF equation, the Activated Rate equation (based on a transi-tion-state model) overcomes these limitations and accounts for theeffects of both the temperature and moisture content (composi-tion). Moisture content acts as a key factor in the physical processof crystallization and affects the enthalpy and entropy of activationfor the process. The experimental crystallization rate constant k(1.3 � 10�4 s�1) in Table 4 obtained by Haque and Roos is closelysimilar to the value of k obtained from the Activated Rate Theory(1.4 � 10�4 s�1) at the same temperature and relative humiditybut significantly different from the k value obtained from theWLF equation 6� 10�5s�1, which supports the better predictiveperformance of the new equation. The Activated Rate equationovercomes the limitations of the WLF equation and provides a uni-fied solution that takes into account the fundamental effects ofboth temperature and moisture content on the kinetics of solid-phase crystallization.

The Activated Rate Theory was not developed to ‘fit’ data to theEyring equation since the Eyring equation itself is a universally-ac-cepted equation that describes the temperature and thermody-namic dependence on reaction rates. We have found thatexperiments conducted on crystallization appear to follow the pat-tern of the Eyring equation. Moreover, the use of lactose for thestudy enabled the contribution of moisture to crystal formationto be understood, which physically corresponds to the activatedstate of the particles from the Eyring equation. In Fig. 2, the so-lid-phase crystallization behavior of lactose was monitored andplotted using Water Induced Crystallization (WIC) at three differ-ent temperatures of 15 �C, 25 �C and 40 �C with a constant relativehumidity of 75%. In all the three cases, it was observed that theplotted curves followed a similar pattern, which is a constant risein moisture content to a peak followed by a steady decline in themoisture content until a constant value is reached, representingthe formation of crystals. The experiments suggested the behaviorof reaching an activated state before subsequently forming crys-tals. Considering these factors, it was deduced that the particles

at the peak moisture content have sufficient activation energy toinitiate the process of crystallization, which starts by expellingthe moisture stored within the high energy state particles as de-picted in Fig. 2 of the paper. This phenomenon enables the mole-cules of the particles to arrange themselves from a high energyamorphous state to a low energy crystalline state by expellingmoisture. So, the mechanism for the process of solid-phase crystal-lization in dried materials can be viewed as an Activated Rate pro-cess wherein the amorphous particles adsorbs moisture until itreaches a peak moisture content, marked by a high energyactivated state, followed by the desorption of water and a steadydecrease in the energy stored within the particles, which corre-sponds to them being arranged into a low-energy structured crys-talline form until a constant moisture content is reached, thatmarks the completion of crystallization. Fig. 13 shows the sche-matic energy levels within the particles during crystallization, cor-responding to the adsorption of moisture, to reach a high energyactivated state at the highest moisture content, and subsequentexpulsion of moisture and a decrease in the internal energy ofthe particles causing them to form lower energy structuredcrystals.

The enthalpy and entropy are related to the activity of thermalenergy lost and gained by the particles in a high energy activatedstate. The enthalpy being the total heat content, and the entropybeing the measure of the randomness or disorder within the parti-cles, both the components are Activated Rate equations directly re-lated to the thermal energy. The moisture content, among othercomponents, is a contributing factor to the phenomenon of thermalenergy lost and gained during the process of crystallization. Theprocess of solid-phase crystallization depends on both the temper-ature and moisture content, and the moisture in itself cannot de-fine the physical process of crystallization. The GAB equation hasbeen used to relate the energy level of the particle to the moisturecontent. Thus the Activated Rate Equations define the phenomenonof solid-phase crystallization, considering the energy level con-tained within the particles irrespective of their chemical composi-tion. Hence, we may also say that the chemical composition relatesthe thermodynamic factors (enthalpy and entropy) to the rate ofcrystallization.


The paper deals with several industrial engineering aspectswithin the area of spray-drying, crystallization and food engineer-ing. The implication of the findings of this paper is found in varioussectors of food engineering, including lactose crystallization, stor-age of dairy powders, production of non-sticky dairy powders,and operation of spray-dryers in the dairy industry. The basic pur-pose of this paper is to understand the contributing factors of thecrystallization process in spray-drying, which is frequently usedin the food industry. Understanding the various design compo-nents for crystallization of spray-dried food materials has been akey motivation. Previous studies done on crystallization duringspray-drying (Chiou et al., 2008) including simulations and exper-iments has given useful insight of the process of solid-phase crys-tallization and understanding the phenomena more fundamentallyfrom both thermodynamic and kinetic point of view can be usefulfor improving spray-drying processes for lactose and better designthe processes of crystallization and drying by including the theoryin the simulations described in Chiou et al. (2008) paper. The re-sults are thoroughly technical and consist of numerical data andtheory sufficient to back the findings of this paper which the indus-try may use to sufficiently broaden the designing of dried foodmaterials.

4. Conclusions

This Activated Rate approach, based on a transition-state model,explains the kinetics of crystallization and relates the rate of solid-phase crystallization to both the temperature and moisturecontent (composition). Water induced crystallization (WIC) exper-iments have shown that, at higher temperatures, crystallizationstarts at lower moisture contents. The enthalpy and the Gibbs freeenergy of activation increased as the moisture content decreasedduring lactose crystallization. On the other hand, the entropy ofactivation decreased with decreasing moisture content. The en-thalpy, entropy and the Gibbs free energy of activation overlapon the same curve (as a function of moisture content) for all thetemperatures, showing that the theory accounts for the variationsin enthalpy and entropy at different temperatures and moisturecontents. The energy of activation for lactose crystallization hasbeen found by the Activated Rate equation to be 39 ± 2 kJ mol�1,which is similar to the activation energy value (40 kJ mol�1) foundby Ibach and Kind in 2007. The entropy of activation was�156 ± 6 J mol�1 K�1. The reaction rate constant (k) from the acti-vated rate theory at 25 �C is 1.4 � 10�4 s�1, which is similar to thatfound by Haque and Roos (2005) of 1.3 � 10�4 s�1, but the valuefrom the WLF equation has been found to be very different. Sincethe WLF equation is a temperature-shift equation, it does not takeinto account the fundamental effect of moisture content on therate of crystallization. The WLF equation is inconsistent in predict-ing the rates of crystallization at the glass-transition temperaturefrom the analysis of experimental data at different temperatures,showing the unreliability of the equation.

References

Anderson, R., 1946. Modifications of the BET equation. Journal of the AmericanChemical Society 68 (4), 686–691.

Ando, H., Ishii, M., Kayano, M., Ozawa, H., 1992. Effect of moisture on crystallizationof theophylline in tablets. Drug Development and Industrial Pharmacy 18 (4),453–467.

Atkins, P.W., de Paula, J., 2002. Molecular reaction dynamics. In: Atkins’, (Ed.),Physical Chemistry, seventh ed. New York, Oxford University Press, 956–961.

Bhandari, B.R., Datta, N., Howes, T., 1997. Problems associated with spray drying ofsugar-rich foods. Drying Technology 15 (2), 671–684.

Bianchi, R., Chiavacci, P., Vosa, R., Guerra, G., 1991. Effect of moisture on thecrystallization behavior of PET from the quenched amorphous phase. Journal ofApplied Polymer Science 43 (6), 1087–1089.

Buckton, G., Darcy, P., 1995. The use of gravimetric studies to assess the degree ofcrystallinity of predominantly crystalline powders. International Journal ofPharmaceutics 123 (2), 265–271.

Buckton, G., Darcy, P., 1996. Water mobility in amorphous lactose below and closeto the glass transition temperature. International Journal of Pharmaceutics 136(1–2), 141–146.

Buckton, G., Darcy, P., 1997. The influence of heating/drying on the crystallisation ofamorphous lactose after structural collapse. International Journal ofPharmaceutics 158 (2), 157–164.

Buckton, G., Darcy, P., 1998a. Crystallization of bulk samples of partially amorphousspray-dried lactose. Pharmaceutical development and Technology 3 (4), 503–507.

Buckton, G., Darcy, P., 1998b. Quantitative assessments of powder crystallinity:estimates of heat and mass transfer to interpret isothermal microcalorimetrydata. Thermochimica Acta 316, 29–36.

Burnett, D.J., Thielmann, F., Booth, J., 2004. Determining the critical relativehumidity for moisture-induced phase transitions. International Journal ofPharmaceutics 287 (1–2), 123–133.

Chen, X.D., Xie, G.Z., 1997. Fingerprints of the drying behavior of particulate or thinlayer food materials established using a reaction engineering model.Transactions of the Institution of Chemical Engineers and the ChemicalEngineer Part C 75 (4), 213–222.

Chiou, D., Langrish, T.A.G., 2008. Stabilization of moisture sorption in spray-driedbioactive compounds by using novel fibre carriers to crystallize the powders.International Journal of Food Engineering 4 (1), article 12.

Chiou, D., Langrish, T.A.G., Braham, R., 2008. Partial crystallization behaviorduring spray drying: simulations and experiments. Drying Technology 26 (1),27–38.

Crofts, A.R., Wang, Z., 1989. How rapid are the internal reactions of theubiquinol:cytochrome c2 oxidoreductase? Photosynthesis Research 22 (1),69–87.

de Boer, J., 1995. The dynamic character of adsorption. In: Rao, M., Rizvi, S. (Eds.),Engineering Properties of Food, second ed. New York, Marcel Dekker Inc., pp.115–251.

Elamin, A., Sebhatu, T., Ahlneck, C., 1995. The use of amorphous model substancesto study mechanically activated materials in the solid state. InternationalJournal of Pharmaceutics 119 (1), 25–36.

Guggenheim, E., 1995. Applications of statistical mechanics. In: Rao, M., Rizvi, S.(Eds.), Engineering Properties of Food, second ed. Marcel Dekker Inc., New York,pp. 115–251.

Haque, M.K., Roos, Y.H., 2004. Water sorption and plasticization behavior of spray-dried lactose/protein mixtures. Journal of Food Science 69 (8), 385–391.

Haque, M.K., Roos, Y.H., 2005. Crystallization and X-ray diffraction of spray-driedand freeze-dried amorphous lactose. Carbohydrate Research 340 (2), 293–301.

Ibach, A., Kind, M., 2007. Crystallization kinetics of amorphous lactose, wheypermeate and whey powders. Carbohydrate Research 342 (10), 1357–1365.

Johari, G.P., Hallbrucker, A., Mayer, E., 1987. The glass-liquid transition ofhyperquenched water. Nature (London) 330, 552.

Jouppila, K., Roos, Y.H., 1994. Water sorption properties and time-dependentphenomena of milk powders. Journal of Dairy Science 77 (7), 1798–1808.

Jouppila, K., Roos, Y., 1997. Water sorption isotherms of freeze-dried milk products:applicability of linear and non-linear regression analysis in modelling.International Journal of Food Science and Technology 77 (6), 2907–2915.

Kaymak-Ertekin, F., Gedik, A., 2004. Sorption isotherms and isosteric heat ofsorption for grapes, apricots, apples and potatoes. Food Science and Technology37 (4), 429–438.

Kauzmann, W., 1948. The nature of the glassy state and the behavior of liquids atlow temperatures. Chemical Reviews 43 (2), 219–256.

Kedward, C.J., MacNaughtan, W., Mitchell, J.R., 2000. Crystallization kinetics ofamorphous lactose as a function of moisture content using isothermaldifferential scanning calorimetry. Journal of Food Science 65 (2), 324–328.

Langrish, T.A.G., 2008. Assessing the rate of solid-phase crystallization for lactose:the effect of the difference between material and glass-transition temperatures.Food Research International 41 (6), 630–636.

Novotny, M.A., Rikvold, P.A., Kolesik, M., Townsley, D.M., Ramos, R.A., 2000.Simulations of metastable decay in two- and three-dimensional models withmicroscopic physics. Journal of Non-Crystalline Solids 274 (1–3), 356–363.

Omar, E.A.M., Roos, Y.H., 2007. Water sorption and time-dependent crystallizationbehavior of freeze-dried lactose–salt mixtures. International Journal of FoodScience and Technology 40 (3), 520–528.

Paterson, A.H.J., Bronlund, J.E., O’Donnell, A.M., 1997. Amorphous LactoseDetermination in Lactose Powders by Adsorption. Paper presented at theCHEMECA Conference, Rotorua, New Zealand.

Paterson, T., Bronlund, J., 2004. Moisture sorption isotherms for crystalline,amorphous and predominantly crystalline lactose powders. InternationalDairy Journal 14 (3), 247–254.

Peleg, M., 1992. On the use of the WLF model in polymers and foods. CriticalReviews in Food Science and Nutrition 32 (1), 59–66.

Price, R., Young, P.M., 2004. Visualization of the crystallization of lactose from theamorphous state. Journal of Pharmaceutical Sciences 93 (1), 155–164.

Ramos, R.A., Rikvold, P.A., Novotny, M.A., 1999. Test of the Kolmogorov–Johnson–Mehl–Avrami picture of metastable decay in a model with microscopicdynamics. Physical Review B 59 (14), 9053–9069.

Roos, Y.H., Karel, M., 1990. Differential scanning calorimetry study of phasetransitions affecting the quality of dehydrated materials. BiotechnologyProgress 6 (2), 159–163.


Roos, Y.H., Karel, M., 1991a. Water and molecular weight effects on glass transitionin amorphous carbohydrates and carbohydrate solutions. Journal of FoodScience 56 (6), 1676–1681.

Roos, Y.H., Karel, M., 1991b. Plasticizing effect of water on thermal behavior andcrystallization of amorphous food models. Journal of Food Science 56 (1), 38–43.

Roos, Y.H., Karel, M., 1992. Crystallization of lactose. Journal of Food Science 57 (3),715–777.

Roos, Y.H., 1993. Water activity and physical state effects on amorphous foodstability. Journal of Food Processing and Preservation 16 (6), 433–447.

Roos, Y.H., 2003. Thermal analysis, state transitions and food quality. Journal ofThermal Analysis and Calorimetry 71 (1), 197–203.

Schmitt, E.A., Law, D., Zhang, G.G.Z., 1999. Nucleation and crystallization kinetics ofhydrated amorphous lactose above the glass transition temperature. Journal ofPharmaceutical Sciences 88 (3), 291–296.

Sharma, G., First, E.A., 2009. Thermodynamic analysis reveals a temperature-dependent change in the catalytic mechanism of bacillus stearothermophilustyrosyl-trna synthetase. The Journal of Biological Chemistry 284 (7), 4179–4190.

Takeuchi, H., Yasuji, T., Yamamoto, H., Kawashima, Y., 2000. Temperature andmoisture-induced crystallization of amorphous lactose in composite particleswith sodium alginate prepared by spray-drying. Pharmaceutical Developmentand Technology 5 (3), 355–363.

Tamman, G., 1926. The States of Aggregation. Transactions. In: Mehl, R.F. (Ed.), VanNostrand, New York, pp. 220.

Thomsen, M.K., Lauridsen, L., Skibsted, L.H., Risbo, J., 2005. Temperature effecton lactose crystallization, maillard reactions, and lipid oxidation in wholemilk powder. Journal of Agricultural and Food Chemistry 53 (18),7082–7090.

Timmerman, I.L., Bolzen, N., Trunk, M., Steckel, H., 2004. A calorimetric study onamorphous lactose during and after re-crystallisation at different relativehumidity. Paper presented at the Respiratory Drug Delivery IX.

Van den Berg, C., 1985. Development of BET-like models for sorption of water onfoods: theory and relevance. In: Simatos, D., Multon, J.L. (Eds.), Properties ofWater in Foods in Relation to Quality and Stability. Martinus Nijhoff Publishers,Dordrecht, pp. 119–131.

Vautaz, G., 1988. Preservation of skim-milk powders: role of water activity andtemperature in lactose crystallisation and lysine loss. In: Seow, C.C. (Ed.), FoodPreservation by Moisture Content. Elsevier Applied Science, New York.

Vollenbroek, J., Hebbink, G.A., Ziffels, S., Steckel, H., 2010. Determination of lowlevels of amorphous content in inhalation grade lactose by moisture sorptionisotherms. International Journal of Pharmaceutics 395 (1–2), 62–70.

White, G.W., Cakebread, S.H., 1966. The glassy state in certain sugarcontaining foodproducts. Journal of Food Technology 1 (1), 73–83.

Williams, M.L., Landel, R.F., Ferry, J.D., 1955. The temperature dependence ofrelaxation mechanisms in amorphous polymers and other glass-formingliquids. Journal of the American Chemical Society 77 (14), 3701–3707.

Winston, P.W., Bates, D.H., 1960. Saturated solutions for the control of humidity inbiological research. Ecology 41 (1), 232–237.

2012-An Activated-state Model for the Prediction of Solid-phase Crystallization

Documents

Transcript of 2012-An Activated-state Model for the Prediction of Solid-phase Crystallization