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Useful method to analyze data on overall transformation kinetics
LETTER TO THE EDITOR Journal of Non-Crystalline Solids 356 (2010) 1201–1203
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Letter to the Editor
Useful method to analyze data on overall transformation kinetics I. Avramov a,⁎, K. Avramova a, C. Rüssel b a b
Institute of Physical Chemistry, Bulg. Acad. Sci., 1113 Sofia, Bulgaria Otto-Schott-Institut, Jena University, Germany
a r t i c l e
i n f o
Article history: Received 20 January 2010 Available online 29 April 2010 Keywords: Crystallization; Glass-ceramics; Relaxation
a b s t r a c t The aim of this article is to demonstrate the important source of errors when overall crystallization kinetics data are plotted in coordinates ln[ − ln(1− α)] against lnt (in isothermal case), respectively ln[ − ln (1− α(T))] against ln q (in the case of linear temperature change at a rate q). Due to the specific properties of the logarithmic function, (in particular because ln 0 → − ∞), this plot exaggerates the role of both the initial stage (α → 0) and the stage near the end of the process (α → 1). Unfortunately, these are just the ranges where most grave experimental errors appear. In case the double logarithmic scale is used, data outside the limits − 2 b ln ( − ln(1 − α))b 1, respectively 0.1 b α b 0.9 are not reliable and should not be taken into account. Instead, we propose suitable coordinates for presentation of experimental data, so that the power n in the Kolmogorov–Johnson–Mehl– Avrami equation is determined in a more reliable manner. © 2010 Elsevier B.V. All rights reserved.
When crystallization under isothermal conditions is studied, the time dependence of the transformed fraction α is described in terms of Kolmogorov–Avrami equation [1–4]: n t α = 1 exp τ
ð1Þ
where τ is the characteristic crystallization time of the process, t is the running time and n is the dimensionless power that depends on crystallization mechanism as well as on whether nucleation takes place or crystals are developing around existing active centres. As the value of n plays a key role for the kinetics of the process, it is widely studied. The theoretically expected values of n are summarized in Table 1. The conventional method for the determination of the power n is to plot experimental data in double logarithmic coordinates, i.e. ln [−ln(1 − α)] against ln t. Indeed, Eq.(1) easily transforms to: ln ð− lnð1−αÞÞ = n ln t−n ln τ:
ð2Þ
So, a straight line is expected with a slope equal to the power n. However, in the limiting cases (i.e. at α → 0 and at α →1), the double logarithmic function exaggerates experimental errors because there the value of ln[−ln(1 − α)] tends to infinity (either plus or minus). This is illustrated in the following figures: The solid line in Fig. 1B represents the ideal peak of a DSC study under condition that Eq. (1) is fulfilled with n = 4. The corresponding degree of transformation α
⁎ Corresponding author. E-mail address: [email protected] (I. Avramov). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.03.004
against time (in τ units) is plotted as a solid line in Fig. 1A. Actually, this is the integral of the peak of Fig. 1B with a baseline equal to 0. On purpose, the base line drawn on Fig. 1B is lowered by 1% of the maximum value of the peak, i.e. the shown dotted line has a value of −0.015. It is difficult to notice the “wrong” position of the baseline. Moreover, in real experiments the slight noise will mask this error completely. The integral curve obtained with this “wrong” baseline is shown as solid points in Fig. 1A. In real experiments the differences between the theoretical solid line and the experimentally determined “wrong” solid points will be completely undistinguishable. Meanwhile, the properties of the logarithmic function make the differences between the two curves enormous in coordinates corresponding to Eq. (2), as illustrated in Fig. 2. Here again the result, corresponding to Eq. (1) with n = 4 is shown as a solid line while solid points are the “wrong” integral data. It is seen, that the value of n estimated from the “wrong” points will be completely misleading. Therefore, a new approach is needed. One of the possible ways is to perform sophisticated spectral analyses of the experimental peaks like that in Fig. 1B. This is not only troubling; in many cases the s-shaped curves like that in Fig. 1A are directly obtained and the peaks like that shown in Fig. 1.B can be achieved only indirectly, with a lot of errors. For this reason we proposed [5] a new method, namely to plot the results in coordinates α against ln t. In this case again an S-shaped curve appears, varying between 0 and 1 quite similar to that in Fig. 1. However, the interpretation of results in the new coordinates is much easier. The first and second derivatives of α in respect to ln t are as follows:
∂α =n ∂ln t
n t n τ t e τ
ð3Þ
LETTER TO THE EDITOR 1202
I. Avramov et al. / Journal of Non-Crystalline Solids 356 (2010) 1201–1203
Table 1 Values of n (see Eq. (1)) in dependence of growth dimension, on growth kinetics and on nucleation conditions. Growth dimension
Active places Size ∼ t
pffiffi Size ∼ t
Size ∼ t
Nucleation pffiffi Size ∼ t
One Two Three
1 2 3
1/2 1 3/2
2 3 4
3/2 2 5/2
respectively ∂2 α = ∂ðln t Þ2
n t n τ : t n 1 nt e τ
ð4Þ
It is seen that the α vs ln t curve has a maximum at t=τ, respectively at α=1 −1/e= 0.63. The slope at this point is α’ðt = τÞ = ne = 0:368n. To derive n from a single derivative point determined from experimental curve seems unreliable. Therefore it is advisable to draw the maximum slope to the integral curve, as shown by the dotted line in Fig. 3. It starts from the onset point ln t1 and has an end point ln t2, at α= 1. The parameter n is easily determined by the difference e n= : ln t2 ln t1
ð5Þ
Additionally, the characteristic time of the process (crystallization time) τ can be estimated as the abscissa of the inflection point of this curve, it should appear as the point at which α = 0.63. Again the solid
Fig. 2. Data from Fig. 1A in coordinates ln[− ln(1 − α)] against ln t. It is seen that determination of n as a slope of the expected straight line could be quite misleading for the “wrong” data. Only data in the interval − 2 b ln[− ln(1 − α)] b 1 is more or less reliable.
points are according to the “wrong” data. It is seen that in these coordinates there is no problem to determine n. It is natural to ask why the same result cannot be obtained from α against t plot. The answer is that Eq. (1) has two independent variables: τ and n. The point is that when abscissa is in log coordinates, the difference between two points is becoming dimensionless, i.e. ln t2 − ln t1 = ln tt12 . Therefore, the result given by Eq. (5) is no more depending on τ. In the case of non-isothermal crystallization, Ozawa [6] proposes to analyze data in coordinates ln (− ln (1 − α(T))) against logarithm of the constant heating or cooling rate q. According to [6], at a given temperature T, the degree of transformation depends on q through certain temperature function χ(T) in the form: χðT Þ αðT Þ = 1 exp n : q
ð6Þ
Again, it can be shown that in coordinates α(T) against ln q the slope is maximal at χqðTn Þ = 1, so that the power n can be determined in a similar way from Eq. (5). Note that α(T) is the transformation degree at a given temperature and different rates of linear temperature change. For that reason collection of sufficiently large number of
Fig. 1. Dependence of degree of transformation α on time t (in τ units) (Fig. 1A). The line is according to Eq. (1) with n = 4. Solid points simulate “wrong” experimental results with some minor systematic error. These points were obtained as integral of the experimental peak shown in Fig. 1B with a “wrong” baseline shown with dotted line in the figure.
Fig. 3. Dependence of the degree of transformation α on logarithm of time. The maximum slope is shown as a dotted line. It is seen that n can be easily determined from the intercepts at α = 0 and α = 1. The same value will be obtained from the absolutely correct curve (solid line) as well as from the “wrong” data (solid squares).
LETTER TO THE EDITOR I. Avramov et al. / Journal of Non-Crystalline Solids 356 (2010) 1201–1203
experimental points for analyses neither in double logarithmic coordinates nor in coordinates α(Τ) against ln q is not an easy task. Conclusions The main conclusion is that analyses in coordinates α against ln t (in isothermal case) or α(Τ) against ln q (in the non-isothermal case) permit more reliable interpretation. In case the double logarithmic scale is used, data outside the limits − 2 b ln (− ln (1 − α)) b 1, respectively 0.1 b α b 0.9 are not reliable and should not be used.
1203
Acknowledgments The authors gratefully acknowledge the financial support from Project TK-X-1713/07 (Bulgarian Ministry of Science and Education). References [1] [2] [3] [4] [5] [6]
A. Kolmogorov, Izv. Acad. Sci. USSR, Ser. Math. 1 (1937) 355. W. Johnson, R. Mehl, Trans. AIME 135 (1939) 416. M. Avrami, J. Chem. Phys. 7 (1939) 1103. M. Avrami, J. Chem. Phys. 8 (1940) 212. I. Avramov, K. Avramova, C. Rüssel, J. Cryst. Growth 285 (2005) 394. T. Ozawa, Kinetics of Non-isothermal Crystallization, Polymer 12 (1971) 150.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Letter to the Editor
Useful method to analyze data on overall transformation kinetics I. Avramov a,⁎, K. Avramova a, C. Rüssel b a b
Institute of Physical Chemistry, Bulg. Acad. Sci., 1113 Sofia, Bulgaria Otto-Schott-Institut, Jena University, Germany
a r t i c l e
i n f o
Article history: Received 20 January 2010 Available online 29 April 2010 Keywords: Crystallization; Glass-ceramics; Relaxation
a b s t r a c t The aim of this article is to demonstrate the important source of errors when overall crystallization kinetics data are plotted in coordinates ln[ − ln(1− α)] against lnt (in isothermal case), respectively ln[ − ln (1− α(T))] against ln q (in the case of linear temperature change at a rate q). Due to the specific properties of the logarithmic function, (in particular because ln 0 → − ∞), this plot exaggerates the role of both the initial stage (α → 0) and the stage near the end of the process (α → 1). Unfortunately, these are just the ranges where most grave experimental errors appear. In case the double logarithmic scale is used, data outside the limits − 2 b ln ( − ln(1 − α))b 1, respectively 0.1 b α b 0.9 are not reliable and should not be taken into account. Instead, we propose suitable coordinates for presentation of experimental data, so that the power n in the Kolmogorov–Johnson–Mehl– Avrami equation is determined in a more reliable manner. © 2010 Elsevier B.V. All rights reserved.
When crystallization under isothermal conditions is studied, the time dependence of the transformed fraction α is described in terms of Kolmogorov–Avrami equation [1–4]: n t α = 1 exp τ
ð1Þ
where τ is the characteristic crystallization time of the process, t is the running time and n is the dimensionless power that depends on crystallization mechanism as well as on whether nucleation takes place or crystals are developing around existing active centres. As the value of n plays a key role for the kinetics of the process, it is widely studied. The theoretically expected values of n are summarized in Table 1. The conventional method for the determination of the power n is to plot experimental data in double logarithmic coordinates, i.e. ln [−ln(1 − α)] against ln t. Indeed, Eq.(1) easily transforms to: ln ð− lnð1−αÞÞ = n ln t−n ln τ:
ð2Þ
So, a straight line is expected with a slope equal to the power n. However, in the limiting cases (i.e. at α → 0 and at α →1), the double logarithmic function exaggerates experimental errors because there the value of ln[−ln(1 − α)] tends to infinity (either plus or minus). This is illustrated in the following figures: The solid line in Fig. 1B represents the ideal peak of a DSC study under condition that Eq. (1) is fulfilled with n = 4. The corresponding degree of transformation α
⁎ Corresponding author. E-mail address: [email protected] (I. Avramov). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.03.004
against time (in τ units) is plotted as a solid line in Fig. 1A. Actually, this is the integral of the peak of Fig. 1B with a baseline equal to 0. On purpose, the base line drawn on Fig. 1B is lowered by 1% of the maximum value of the peak, i.e. the shown dotted line has a value of −0.015. It is difficult to notice the “wrong” position of the baseline. Moreover, in real experiments the slight noise will mask this error completely. The integral curve obtained with this “wrong” baseline is shown as solid points in Fig. 1A. In real experiments the differences between the theoretical solid line and the experimentally determined “wrong” solid points will be completely undistinguishable. Meanwhile, the properties of the logarithmic function make the differences between the two curves enormous in coordinates corresponding to Eq. (2), as illustrated in Fig. 2. Here again the result, corresponding to Eq. (1) with n = 4 is shown as a solid line while solid points are the “wrong” integral data. It is seen, that the value of n estimated from the “wrong” points will be completely misleading. Therefore, a new approach is needed. One of the possible ways is to perform sophisticated spectral analyses of the experimental peaks like that in Fig. 1B. This is not only troubling; in many cases the s-shaped curves like that in Fig. 1A are directly obtained and the peaks like that shown in Fig. 1.B can be achieved only indirectly, with a lot of errors. For this reason we proposed [5] a new method, namely to plot the results in coordinates α against ln t. In this case again an S-shaped curve appears, varying between 0 and 1 quite similar to that in Fig. 1. However, the interpretation of results in the new coordinates is much easier. The first and second derivatives of α in respect to ln t are as follows:
∂α =n ∂ln t
n t n τ t e τ
ð3Þ
LETTER TO THE EDITOR 1202
I. Avramov et al. / Journal of Non-Crystalline Solids 356 (2010) 1201–1203
Table 1 Values of n (see Eq. (1)) in dependence of growth dimension, on growth kinetics and on nucleation conditions. Growth dimension
Active places Size ∼ t
pffiffi Size ∼ t
Size ∼ t
Nucleation pffiffi Size ∼ t
One Two Three
1 2 3
1/2 1 3/2
2 3 4
3/2 2 5/2
respectively ∂2 α = ∂ðln t Þ2
n t n τ : t n 1 nt e τ
ð4Þ
It is seen that the α vs ln t curve has a maximum at t=τ, respectively at α=1 −1/e= 0.63. The slope at this point is α’ðt = τÞ = ne = 0:368n. To derive n from a single derivative point determined from experimental curve seems unreliable. Therefore it is advisable to draw the maximum slope to the integral curve, as shown by the dotted line in Fig. 3. It starts from the onset point ln t1 and has an end point ln t2, at α= 1. The parameter n is easily determined by the difference e n= : ln t2 ln t1
ð5Þ
Additionally, the characteristic time of the process (crystallization time) τ can be estimated as the abscissa of the inflection point of this curve, it should appear as the point at which α = 0.63. Again the solid
Fig. 2. Data from Fig. 1A in coordinates ln[− ln(1 − α)] against ln t. It is seen that determination of n as a slope of the expected straight line could be quite misleading for the “wrong” data. Only data in the interval − 2 b ln[− ln(1 − α)] b 1 is more or less reliable.
points are according to the “wrong” data. It is seen that in these coordinates there is no problem to determine n. It is natural to ask why the same result cannot be obtained from α against t plot. The answer is that Eq. (1) has two independent variables: τ and n. The point is that when abscissa is in log coordinates, the difference between two points is becoming dimensionless, i.e. ln t2 − ln t1 = ln tt12 . Therefore, the result given by Eq. (5) is no more depending on τ. In the case of non-isothermal crystallization, Ozawa [6] proposes to analyze data in coordinates ln (− ln (1 − α(T))) against logarithm of the constant heating or cooling rate q. According to [6], at a given temperature T, the degree of transformation depends on q through certain temperature function χ(T) in the form: χðT Þ αðT Þ = 1 exp n : q
ð6Þ
Again, it can be shown that in coordinates α(T) against ln q the slope is maximal at χqðTn Þ = 1, so that the power n can be determined in a similar way from Eq. (5). Note that α(T) is the transformation degree at a given temperature and different rates of linear temperature change. For that reason collection of sufficiently large number of
Fig. 1. Dependence of degree of transformation α on time t (in τ units) (Fig. 1A). The line is according to Eq. (1) with n = 4. Solid points simulate “wrong” experimental results with some minor systematic error. These points were obtained as integral of the experimental peak shown in Fig. 1B with a “wrong” baseline shown with dotted line in the figure.
Fig. 3. Dependence of the degree of transformation α on logarithm of time. The maximum slope is shown as a dotted line. It is seen that n can be easily determined from the intercepts at α = 0 and α = 1. The same value will be obtained from the absolutely correct curve (solid line) as well as from the “wrong” data (solid squares).
LETTER TO THE EDITOR I. Avramov et al. / Journal of Non-Crystalline Solids 356 (2010) 1201–1203
experimental points for analyses neither in double logarithmic coordinates nor in coordinates α(Τ) against ln q is not an easy task. Conclusions The main conclusion is that analyses in coordinates α against ln t (in isothermal case) or α(Τ) against ln q (in the non-isothermal case) permit more reliable interpretation. In case the double logarithmic scale is used, data outside the limits − 2 b ln (− ln (1 − α)) b 1, respectively 0.1 b α b 0.9 are not reliable and should not be used.
1203
Acknowledgments The authors gratefully acknowledge the financial support from Project TK-X-1713/07 (Bulgarian Ministry of Science and Education). References [1] [2] [3] [4] [5] [6]
A. Kolmogorov, Izv. Acad. Sci. USSR, Ser. Math. 1 (1937) 355. W. Johnson, R. Mehl, Trans. AIME 135 (1939) 416. M. Avrami, J. Chem. Phys. 7 (1939) 1103. M. Avrami, J. Chem. Phys. 8 (1940) 212. I. Avramov, K. Avramova, C. Rüssel, J. Cryst. Growth 285 (2005) 394. T. Ozawa, Kinetics of Non-isothermal Crystallization, Polymer 12 (1971) 150.