Kinetic analyses of crystallization and devitrification: Comparison of activation energies in aqueous solutions of glycerol and dimethyl sulphoxide

I O U R N A L OF

ltllgl mWtM ELSEVIER

Journal of Non-Crystalline Solids 211 (1997) 262-270

Kinetic analyses of crystallization and devitrification: Comparison of activation energies in aqueous solutions of glycerol and dimethyl sulphoxide J.M. Hey, D.R. MacFarlane Department of Chemistr3,, Monash Uni~ersi~', Wellington Rd, Cla,¢ton 3168 Vic., Australia Received 9 November 1995; revised 3 September 1996

Abstract

Crystal growth rates in glass forming 50 w/w% glycerol and 45 w/w% dimethyl sulphoxide (DMSO) aqueous solutions were found to follow Arrhenius behaviour over a broad range of temperatures. Activation energies for ice crystal growth in these solutions were determined from Arrhenius plots of the isothermal crystal growth rate data. The values for the activation energy of crystal growth were found to be the same for each solution, suggesting that the activation energy barrier to transport of water molecules across the ice/solution interface is the same in both solutions. The activation energies were compared to activation energy of crystallization values given by several non-isothermal DSC kinetic analysis techniques. It was found for 45 w/w% DMSO that the activation energy for growth was equal to the activation energy for the whole crystallization process. This was not the case for 50 w/w% glycerol solutions, an observation which highlights the inapplicability of the standard analysis techniques under conditions of continued nucleation.

I. Introduction

The kinetic analyses of crystallization events and devitrification of glass forming liquids has long been of interest in terms of the practical information they can provide and the fundamental perspective that can be developed concerning the underlying processes. Various analytical methods have been developed by a number of groups, as reviewed below, the basis and results of which have not always appeared consistent. This has cast doubt upon the meaningfulness of the kinetic parameters that are produced. We

Corresponding author. Tel.: +61-3 9905 4540; fax: +61-3 9905 4597; e-mail: [email protected].

present an analysis of the various methods designed to resolve a n d / o r characterize their differences. Data for two glass-forming systems of interest to the field of cryobiology are presented. It is believed that the least damaging method of preserving biological specimens at low temperature is through the complete vitrification of the intracellular solution. The solutions studied in this work are also interesting model systems for devitrification analysis, since the samples can be prepared, handled and studied very accurately, and the devitrification event is highly reproducible. The product of the crystallization/devitrification is well known to be pure ice. We thus use data from these solutions to compare and contrast the various analytical methods that have been presented and to highlight their differences.

0022-3093/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0022- 3 0 9 3 ( 9 6 ) 0 0 6 3 7 - 0

J.M. Hey, D.R. MacFarlane / Journal of Non-Co'stalline Solids 211 (1997) 262-270

The equation of Johnson and Mehl [1] and Avrami [2-4] relates the volume fraction transformed, X, to the time of an isothermal treatment, t: X=l-exp

( Jo/J, )m ) -A

tl v

,tudt

dt'

(1)

where m depends on the dimensionality of crystal growth, u is the growth rate, I v is the nucleation rate and A is a collection of time independent constants. If the nucleation and growth rates are independent of time then Eq. (1) can be integrated to give X = 1 - exp( - A ' l , umt ")

(2)

where A' is a new constant, and the values of m and n are determined by the dimensionality of crystal growth and nucleation conditions. The values of m and n for various growth mechanisms and dimensionalities are given in Table 1. Eq. (2) is a specific case of the Johnson-Mehl-Avrami equation (also called the Avrami equation): X= 1 - exp(-(Kt)")

(3)

where K is known as the crystallization constant, and is assumed to follow Arrhenius behaviour: K = K 0 exp( - E a / R T )

(4)

where E~ is the overall activation energy for crystallization. Although the Avrami equation was derived for isothermal transformations only, a large number of analysis techniques for non-isothermal experiments have been developed, based on this equation. These

Table 1 Values of m and n for various crystallization conditions Growth dimensionality

Interface controlled growth

Diffusion controlled growth

??/

m

n

include the methods of Kissinger [5], Augis and Bennett [6], and Ozawa and Chen (as discussed in [7]). Some of these methods have been the subject of discussion and critical review by Henderson [8], Yinnon and Uhlmann [7] and Kelton [9]. Another method of analysis of non-isothermal D S C / D T A experiments has been developed by Matusita and Sakka [10]. This was not based on the Avrami equation, but rather was derived directly from equations for nucleation and growth of crystals. In general, the methods based on the Avrami equation use the fact that most non-isothermal D S C / D T A experiments utilise a constant heating rate, i.e.

T = T O+ Ot

3-dimensional 2-dimensional 1-dimensional

constant 3 2 1

nucleation rate 4 3/2 3 2/2 2 1/2

5/2 4/2 3/2

3-dimensional 2-dimensional 1-dimensional

zero nucleation rate 3 3 3/2 2 2 2/2 1 1 1/2

3/2 2/2 1/2

(5)

where T is temperature, TO is the initial temperature, Q is the heating rate and t is time. These methods all lead to the derivation of two parameters, which when plotted against one another, produce a straight line. The slope of this line is taken to be the activation energy of crystallization, E a. To achieve this, each method requires that the variation in the peak crystallization temperature with heating rate must be determined. In each of these methods it is assumed that the volume fraction crystallized at the peak crystallization temperature is constant and independent of the heating rate. Yinnon and Uhlmann [7] have suggested that many of these methods neglected the true temperature dependence of the reaction rate (K(T)), and hence were based on an inappropriate rate equation (Eq. (6). This rate equation is given by differentiation of Eq. (3) with respect to time, followed by substitution for t:

dx/dt=nK(1 ?/

263

- x)( -ln{ 1 -x}) '-l/n

(6)

Yinnon and Uhlmann [7] also suggested that when both nucleation and crystal growth occur in the temperature range of interest, or when these processes cannot be assumed to follow Arrhenius behaviour, the crystallization process should be treated numerically rather than analytically. In a more recent article, Kemeny and Granasy [11] disagreed with the view of Yinnon and Uhlmann regarding the appropriate rate equation (i.e. Eq. (6)), but also stated that the use of kinetic analyses utilizing this equation are

264

J.M, Hey, D.R. MacFarlane / Journal of Non-C~stalline Solids 211 (1997) 262-270

not strictly valid in cases where the nucleation rate is non-zero. Henderson [9] has shown that the transformation rates of systems that undergo nucleation and growth processes are dependent on the thermal history of the system, even though Eq. (6) would predict the same transformation rate for systems with different thermal histories upon being heated to the same temperature. Henderson [9] concluded that non-isothermal analysis methods using Equation Eq. (6) could only be applied to transformations involving nucleation and growth processes in a limited number of cases. Examples of such cases include those in which the nucleation occurs early in the course of the transformation, a condition termed site saturation. Site saturation is realised in cases where the nucleation and growth curves are well separated in terms of temperature [12]. The isothermal growth of ice crystals in 50 w / w % glycerol and 45 w / w % DMSO solutions has been shown, in general, to be time independent [13]. These solutions have been the subject of intense study because they have very similar solute concentration in molar terms, as well as similar devitrification temperatures, Td, but behave quite differently during devitrification [14]. In this work we compare the activation energies of ice crystal growth in these aqueous solutions of glycerol and DMSO, calculated directly from the ice crystal growth measurements, with values of the activation energies of crystal growth and of the entire crystallization process given by applying various kinetic analysis methods to the non-isothermal DSC results of Boutron and Kaufmann [15].

2. Experimental methods

of the video-taped crystal growth experiments. Further details of the instrument design and experimental techniques used in this work are presented elsewhere [13,16]. The growth rate presented here is an average at each temperature over a number of individual measurements. The estimated error range on the data is of the order of the data points shown. Other values for the activation energy of crystal growth and values for the activation energy of the whole crystallization process were determined by applying the kinetic analysis methods of the Ozawa-Chen, Kissinger, Augis-Bennett and Matusita-Sakka to the DSC data of Boutron and Kaufmann [15]. The final equations produced by each of these analysis techniques, and the parameters which are plotted to yield the activation energy in each case, are given below: The final equation given by the Kissinger [5] method is ln(Q/Tp 2 ) = - Ea/RT p + constant

where Tp is the peak temperature of devitrification. Hence, plotting ln(Q/Tp 2) versus 1/Tp should yield a straight line with a slope of - E J R . The AugisBennett [6] method yields

ln(Q/Tp - To) = - E a / R T p + constant

(8)

where TO is the initial temperature of the experiment. Hence, a plot of ln(Q/{Tp - To}) versus 1/Tp should yield a straight line with slope - E a / R . The method of Ozawa and Chen [7] gives I n ( Q / T o) = - EJRTp + constant

(9)

Hence, plotting ln(Q/Tp) versus 1/Tp should yield a straight line with a slope of - E a / R . The MatusitaSakka method [10] yields

ln( Q n / T 2 ) = - m E ~ / R T p +constant Isothermal ice crystal growth rates in 45 w / w % DMSO and 50 w / w % glycerol solutions were determined over a broad temperature range by using a combined DSC-video microscope [13]. The system consisted of a video camera and microscope system mounted over a modified Perkin-Elmer DSC sample head. The DSC used was a Perkin-Elmer DSC-7. In the experiments, the rate of ice crystal growth was determined at each temperature by computer analysis

(7)

(10)

where n and m have the same significance as in Eq. (2), and E G is the activation energy for crystal growth. As pointed out by Matusita and Sakka this method reduces to the Kissinger method (Eq. (7)) when n = m = 1, and is termed by the authors as the 'modified Kissinger method'. Here, a plot of ln(Qn/Tp 2) versus 1/Tp yields a straight line with a slope of - m E c / R .

265

J.M. Hey D.R. MacFarlane / Journal of Non-Crystalline Solids 211 (1997) 262-270

3. Results

-2.0 ~-

. . . . . .

,,,,,

-5.0 p

r=0.996=.

2,

J

-6.5 ~

~ I

4.6

I

4.8

"~0995

t t-8.0

±

;.'~ E/R=(4.06 022).10K ~ a r = 0.996

" ~ " ~

~-9.0

-10

5.2

5.3

5.4

5.5

5.6

1000K/T

5.7

5.8

5.9

p

Fig. 2. Ozawa-Chen and Kissinger plots of the heating rate versus devitrification temperature data of Boutron and Kaufmann [ 15] for 45 w / w % DMSO. The point markers indicate the approximate error range of the data. Lines shown are linear fits, the slopes of which are indicated.

data using the analytical methods described above. For each method of analysis it is necessary to know how the peak devitrification temperature changes with heating rate. For 45 w / w % DMSO solutions these experiments have been performed by Boutron and Kaufmann [15]. Fig. 2 shows the application of the Ozawa-Chen method to the heating rate versus

•

~-

" Augis-Bennett

~

">

-6.0 ~

022).I0 K

I ' '

Matusita-Sakka In(Q3 Tp-2/Kmin "3) I ~ - ~ 1.50

' ........

3Eo/R = (12.36+0.65). 103K

i

E J R = (3.77+0.10).103K ~

I

.

4-

07i .. :0996

,

-4.0 ~k ° ~ o ~

~a/R=(3.88

"''....t

ln(Q. [Tp-Yo]q tmin "1) 0.00 , . . . . ~, ' ' 1 ' '

ln(u/rams -~)

,

''

1

-3.0 ~-

F

-4.5 ~-

-7.0 - 6"-~....

3.1. 45 w / w% DMSO solutions

-3.5~

ln(Q.Tp"2/K'lmin "1)

-1 0

! -4.0 :

Fig. 1 shows that the averages of the isothermal growth rates follow Arrhenius behaviour over almost the whole temperature range investigated, a result that may be expected since the transport properties of aqueous solutions are thought to follow Arrhenius behavior at temperatures close to Tg [17], although Uhlmann [7] has stated that crystal growth rates do not usually follow Arrhenius behavior over broad temperature ranges. It should be noted that this figure does not include the high temperature growth rate data where the growth rate is deceased by the proximity of the liquidus temperature [13]. Fitting an Arrhenius type relation to this data gives an effective activation energy for crystal growth, E~ = 31.5 _+ 0.8 kJ/mol. An activation energy for the whole crystallization event can be calculated from non-isothermal DSC

-+__- Kissinger

In(Q.Tpq/rain q)

Both solutions have Tp in the vicinity of - 9 6 ° C during wanning in the DSC at 10°C/min, however the 50% glycerol solution has Tg = - 117°C, TLiquid.~ = - 2 4 ° C whereas the 45% DMSO solution has a much lower glass transition temperature (Tg = - 1 3 3 ° C ) and TLiquiau~= -34°C. In the following we analyze the behaviour of each solution and then proceed to attempt to characterize the differences.

-5.5

Ozawa-Chen i

•

5,0

r

,

~

[

5.2

5.4

1000K/T

Fig. 1. Arrhenius plot of 45 w / w % DMSO growth rate (u) data. The line drawn is that of a linear fit to the data, the slope of which is indicated.

-1.50 t

-2.25 4.00 - - ' 5.2

-4.00 Ea/R = (3.70e0. -

r=0.995

' t . . . . J ~, I 5.3 5.4 5.5 5.6 1000K/Tp

~

5.7

-6.75

5.8

-9.50 5.9

Fig. 3. Augis-Bennett and Matusita-Sakka (m = n = 3) plots of the heating rate versus devitrification temperature data of Boutron and Kaufmann [15] for 45 w / w % DMSO. Lines shown are linear fits, the slopes of which are indicated.

266

J.M. Hey, D.R. MacFarlane / Journal of Non-Crystalline Solids 211 (1997) 262-270

peak devitrification temperature data of Boutron and Kaufmann [15] for 45 w / w % DMSO solutions. Application of the Kissinger method to the same data is also shown in Fig. 2, the Aug±s-Bennett method is shown in Fig. 3, and finally the application of the 'modified Kissinger' method of Matusita and Sakka is also shown in Fig. 3. The slopes of the fitted curves (EJ R for the first three methods, and mEc/R, where m = 3, for the last) are given in each figure. The value of n is set at 3 (as is the value of m), for the Matusita-Sakka analysis, since the growth of the nuclei is believed to be 3-dimensional, and it has been shown previously that the nucleation rate is negligible in the range of temperatures over which this solution devitrifies [14]. The calculated activation energies are shown in Table 2, along with that calculated by Boutron and Kaufmann [15] from their data. The activation energy given by Boutron and Kaufmann [15] is, within error, the same as that calculated by other methods, however, their calculation method involves only two heating rate-devitrification data points, and so may well be less accurate than the other methods which use all the available data.

3.2. 50 w/w% glycerol solutions Fig. 4 shows the variation in the ice crystal growth rate with the inverse of the isothermal crys-

ln(u/mms -1) -35

~-

Analysis method

F-

.

Ozawa-Chen 33.4 ± Kissinger 32.3 ± Aug±s-Bennett 30.8 ± Matusita-Sakka, E o = 34.3 + n=m=3 Matusita-Sakka, n=4, m=3 Matusita-Sakka, n=2, m=l Boutron and Kaufmann 34.7 Crystal growth rates E o = 31.5 +

56.3±2.4 54,8±2.4 53.2±2.4 =56.7±2.4 =75.9±3.2 ~

0.8

= 112±4.8 48.5 =30.7±1.0

i

E J R = (3.69±0.12). 103K

-6.0 " . . . . i . . . . . . . . . . . . . . . . 4.2 4.3 4,4 4.5 4.6 1000K/T

: ..... 4.7 4.8

4.9

Fig. 4. Arrhenius plot of 50 w / w % glycerol growth rate (u) data. Lines shown are linear fits, the slopes of which are indicated.

tallization temperature for 50 w / w % glycerol solutions. Data collected above - 4 0 ° C was not incorporated in this plot, since the proximity of the liquidus temperature alters the growth kinetics at these temperatures. All growth rate values in this figure are averages over time, except the data collected at the highest temperature ( - 40°C), where the growth rate used was that of the initial portion of the transformation. This is justified since it is believed that the true growth rate for this temperature is that at the initial stages, before the growth rate is affected by the local

&wa-Chen

Kiss! er:

ln(Q.Tpq/min "l)

ln(Q.Tp"2/K'lminq)

-1.0 ~

-7.0 ~

1.8 1.8 1.7 1.8

t

-5.5

Ea/R = (659:t:0.29). 103K

-2.0 ~- ~ " - ~ . . ~

45w/w% glycerol

r

]

E, (kJ m o l - 1 ) 45w/w% DMSO

.... , ....

, ,~

-40

• Table 2 Comparison of activation energies determined using various methods for crystallization of 45 w / w % DMSO and 50 w / w % glycerol solutions

~T

~....

~

r =0996

3o! 5.4

t-8.0

,90 5.5

5.6 5.7 1000K/Tp

5.8

5.9

Fig. 5. Ozawa-Chen and Kissinger plots of heating rate versus devitrification temperature data of Boutron and Kaufmann [15] for 50 w / w % glycerol. Lines shown are linear fits, the slopes of which are indicated.

J.M. Hey, D.R. MacFarlane/Journal of Non-Co'stalline Solids 211 (1997) 262-270 ln(Q.[Tp-To ]-1/min"I) 0.0~ . . . . ~,,

,

.

....

~ ....

-0.5 -1.0

E / R = (6.404-0.29). 103K

-1.5

"""x

r = 0.996

-2.0

-3.0

•

-3.5 54

,, ~_, 5.5

~,

I , 5.6

, 5.7

5.8

5.9

1000K/Tp Fig. 6. Augis-Bennett plot of heating rate versus devitrification temperature data of Boutron and Kaufmann [15] for 50 w / w % glycerol. Lines shown are linear fits, the slopes of which are indicated.

temperature increases at the interface caused by the enthalpy of crystallization being released. Fitting of an Arrhenius relationship to this data gave E~ = 30.7 _+ 1.0 kJ/mol. Boutron and Kaufmann have also reported [14] heating rate versus devitrification temperature data for 50% glycerol solutions. Hence, activation energies for crystallization can be calculated and compared for this system, in the same manner as described above, Fig. 5 shows the application of the

ln(Qn.T -2/K'2min ")

ll

o.oI-,

5.4

5.5

56

5.7

5.8

5.9

1000K/Tp Fig. 7. Matusita-Sakka plots of heating rate versus devitrification temperature data of Boutron and Kaufmann [15] for 50 w / w % glycerol with n = m = 3 , n = 4 a n d m = 3 , and n = 2 a n d r n = l . Lines shown are linear fits, the slopes of which are indicated.

267

Ozawa-Chen method [7] to the heating rate versus peak devitrification temperature data of Boutron and Kaufmann [15] for 45 w / w % DMSO solutions. Application of the Kissinger [5] method to the same data is also shown in Fig. 5, the Augis-Bennett [6] method is shown in Fig. 6 and the 'modified Kissinger' method of Matusita and Sakka [10] in Fig. 7. The slopes of the fitted curves (Ea/R for the first three methods, and mEt/R, with n = 4, n = 3 and n = 2, for the last) are given in each figure. The calculated activation energies are shown in Table 2, along with that calculated by Boutron and Kaufmann [15] from their data.

4. Discussion Figs. 1 and 4 show that the activation energies for ice crystal growth for the 45 w / w % DMSO solutions and the 50 w / w % glycerol solutions are the same, within error. Since the ice crystallization, under the present conditions, in both of these solutions has been shown to be interface controlled [13,16], this indicates that the activation energy barrier to transport of water molecules across the ice/solution interface is similar in both solutions. This result is not entirely expected, since these solutions have been shown to exhibit markedly different ice crystallization behaviour [ l 3,14]. The differences therefore must arise from the detailed mechanism involved in each case.

4.1. Activation energy calculations in 45 w/w% DMSO solutions Comparison of the activation energies calculated from Boutron and Kaufmann's data [14] shows them to be the same, within error, as the activation energy derived from the crystal growth measurements (Table 2). This shows that for 45 w / w % DMSO solutions the activation energy for crystallization is the same as that for growth. To test the theoretical validity of this, it is necessary to examine Eq. (2). This equation is clearly only valid for I v ---0, and hence its application is limited to cases where nucleation is continuing throughout the transformation.

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J.M. Hey, D.R. MacFarlane / Journal of Non-C~stalline Solids 211 (1997) 262-270

Christian [19] gives a more general equation for the volume fraction of ice transformed:

ation rate is negligible, the same procedure yields from Eq. (16):

X=l-exp(-p~um{Uctm+c't"+q})

ea=Eo

(11)

where n = m + l , N c is the number of nuclei in existence at time t = 0, and the subsequent nucleation rate is given by the power law I v = C't q. Hence, for the case of constant (non-zero) nucleation rate ( q - - 0 ) , the nucleation rate is simply I,, = C', and Eq. (11) becomes X = 1 - e x p ( - A ' u " { Nct m + C ' t ' } )

(12)

If the nucleation rate is sufficiently great, then the contribution of the nuclei present at t = 0 to the rate becomes negligible, and hence Eq. (12) can be approximated by Eq. (2). However, for the cases where the nucleation rate during the transformation is negligible (i.e. I v = C't q = 0 ) Eq. (11) becomes

XAI - e x p ( - a ' u m N ~ t ")

(13)

Eqs. (2) and (13) can both be written as: X = 1 - e x p ( - ( K t ) x)

(14)

where x = m or n, depending on whether the nucleation rate during the transformation is negligible or constant. Hence, from [2,14] for constant (non-zero) nucleation rate:

K"=A'C'u m (n=m+l)

(15)

Similarly, from [13,14] for negligible nucleation:

K m =a'U~u"'

(16)

Assuming that K, C' and u all follow Arrhenius behaviour, [15] gives K" = a' C~ exp( - E N / R T ) u o exp( - m E c / R T ) and if we let K" = K 0 e x p ( - nEa/RT)where Ea, E N and E~ are the activation energies for the total crystallization process, nucleation and growth, respectively, then

Ea = ( E N + m E ~ ) / n

(17)

Eq. (17) thus applies when the nucleation rate is (i) constant over the whole transformation, (ii) high enough to produce significantly more nuclei than were present at t = 0 and (iii) assumed to follow Arrhenius behaviour. Alternatively, when the nucle-

(18)

A similar argument to show that the activation energy for the whole crystallization process should be equal to the activation energy for crystal growth when the nucleation rate is negligible has been put forward by Yinnon and Uhlmann [7]. However, their argument was based on Eq. (2), which clearly is only valid for non-zero nucleation rates. Marotta et al. [20] have given, without derivation, an alternative form of Eq. (18):

E, = (bE N + mEc) / n

(19)

where the value of b is related to the nucleation processes: b = 0 for zero nucleation rate and b = l for a constant nucleation rate with n = m + b. Hence, when n = m (zero nucleation rate), b = 0 and E a = E G. This is the case observed for 45 w / w % DMSO solutions. Direct microscopic observation [16] confirms that, in the temperature range in which this solution devitrifies, the homogeneous nucleation rate is indeed negligible, and so in this case Eq. (13) is valid. Hence, the activation energy for the whole crystallization process would be expected to be the same as the activation energy for crystal growth, as was found experimentally (Table 2). Also, since Matusita and Sakka's 'modified Kissinger' [10] method reduces to the same relationship proposed by Kissinger [5] when n = m = 1, it can be seen from Table 2 that, in the present case, the activation energy is not dependent on the growth dimensionality. It should be noted, however, that the temperature dependence of the nucleation rate is generally nonArrhenian[18,19], although in this case the assumption of Arrhenius behaviour for the crystal growth rate over the region of temperature of interest is validated by Fig. 1.

4.2. Activation energy calculations in 50 w / w % glycerol solutions For the 50 w / w % glycerol solutions, analysis of the non-isothermal DSC data using the Ozawa-Chen, Kissinger, Augis-Bennett, or Matusita-Sakka (n = rn = 3) methods gave the same activation energies,

J.M. Hey, D.R. MacFarlane / Journal of Non-Co'stalline Solids 211 (1997)262-270

within error (Table 2). All of these were significantly greater than the activation energy for rystal growth calculated directly from crystal growth rate data. The activation energy given by Boutron and Kaufmann [ 15] is calculated using similar reasoning to Kissinger [5], but for the same reasons as stated above for the DMSO solution, their activation energy calculation may well be less accurate than that given by the other methods. The choice of the value of n for the Matusita-Sakka analysis is more complicated for this solution than for the 45 w / w % DMSO solution, since this solution devitrifies in a temperature region in which the nucleation rate is believed to be relatively high [14]. It is obvious from Eqs. (11)-(19) that the nucleation rate during devitrification is nonzero, since a negligible nucleation rate would give E~--E~ from each of the analysis methods, as was found for the 45 w / w % DMSO solution. However, none of the calculated activation energies for crystallization are even approximately equal to the activation energy for crystal growth. This then suggests, from Table 1, that n = m + 1. To further complicate deduction of the value of n, the dimensionality of the ice crystal growth mechanism for aqueous solutions of glycerol in the temperatures of interest is uncertain. Although the ice crystals that grow in 50 w / w % glycerol solutions at temperatures slightly above those at which the solution devitrifies can be observed as spherulites [13,21,22], they are in the form of evanescent spherulites. These spherulites have been observed to be partly transparent. Luyet and Rapatz [21,22] have suggested that the evanescent nature of these crystals may stem from incomplete crystallization of hexagonal ice, and X-ray diffraction studies [23] have indicated that the evanescent spherulites may consist of long thin needles of ice radiating from a centre. This would then suggest that the crystal growth is one dimensional, and hence m = 1 and n = 2. At the devitrification temperature, direct observation of individual spherulites in this solution was not possible, even by optical microscopy, due to their small size [16]. However, it has been observed that the 50 w / w % glycerol solution does not become completely opaque during isothermal crystallization experiments in the region of Td, even after all the enthalpy of crystallization had been released [16]. This suggests that either incomplete formation of hexagonal ice crystals

269

also occurs at these lower temperatures, or that the ice crystals formed at these temperatures are too small to cause the sample to become completely opaque. It would seem then, from the results presented above, that the Ozawa-Chen, Kissinger, or AugisBennett methods give invalid activation energies for crystallization for the 50 w / w % glycerol solution, since the value they give is equivalent to E~ from the Matusita-Sakka method with n = m = 3. This is in agreement with Henderson [8] and Yinnon and Uhlmann [7] who state that these analysis methods are generally not valid when nucleation is continuing throughout the transformation. It is also apparent that the Matusita-Sakka method, with any reasonable values of m and n, does not give the same value for E C as that determined directly from crystal growth measurements. It is generally proposed that the value for n can be calculated from isothermal DSC traces [24], however, this method supposes that the extent of transformation can be directly related to the area under the isothermal crystallization curve. In practice, however, some crystallization often occurs before the instrument has thermally equilibrated, and some may also occur during the cooling step, hence, in many cases [25], an accurate determination of the onset of crystallization is difficult. For these reasons no attempts were made in this work to calculate n from isothermal crystallization measurements.

5. S u m m a r y The fact that the activation energies for crystal growth, E C, for each solution are equal, within error, leads to the conclusion that the activation energy barrier to water molecules crossing the ice/solution interface is similar in both solutions. However, comparison of E c for these two solutions with activation energies for the whole crystallization process, E a, determined from kinetic analyses of the heating rate versus devitrification temperature data of Boutron and Kaufmann [15] showed differences in the crystallization behaviour of the two solutions. All of the analysis techniques employed showed Ea to be the equal to E G obtained directly from crystal growth measurements in the case of the 45 w / w % DMSO.

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J.M. He3', D.R. MacFarlane / Journal of Non-Crystalline Solids 211 (1997)262-270

For the 50 w / w % glycerol solution, however, this was not found to be the case. For this solution, all of the analyzed methods gave the same value for E a (or E 6 in the case of the Matusita-Sakka method), which was not equal to the value determined for E 6 from crystal growth measurements. Hence, it is concluded that nucleation is continuing throughout the transformation in this case. This is supported by the correlation of the devitrification temperature and the temperature of homogeneous nucleation in the 50 w / w % glycerol, as shown by the TTT curve [14]. Hence, in the analysis method of Matusita and Sakka, m cannot be equal to n. However, each of the other analysis techniques gave E a to be the same as E 6 for the Matusita-Sakka method with m = n = 3; this suggests that none of these methods gives a meaningful value for the activation energy in this case. This is in agreement with the views of Henderson [8] and others [7,11] who have stated that these methods do not apply in cases where the nucleation rate is non-zero during the transformation. The reason for this may be the strongly non-Arrhenian nature of the temperature dependence of the nucleation rate, in particular near and above its maximum in temperature.

6. Conclusion Non-isothermal DSC kinetic analysis methods are meaningful under s o m e circumstances, as expected from theory. In these cases the E a from the DSC methods and direct growth measurements are the same. However, under circumstances of continued

nucleation during devitrification these DSC methods do not produce meaningful activation energies.

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