ON THE DETERMINATION OF THE CRYSTALLIZATION ENERGY IN NON-ISOTHERMAL PROCESSES
ON THE DETERMINATION OF THE CRYSTALLIZATION ENERGY IN NON-ISOTHERMAL PROCESSES BY USING EVOLUTIONARY ALGORITHMS
Ioan Zaharie1, Daniela Zaharie2
1Physics Department, University “Politehnica” Timişoara, Piaţa Regina Maria nr.1, 1900 Timişoara,
e-mail:
2Computer Science Department, West University of Timisoara, bv. V. Parvan, no. 4, 1900 Timisoara,
e-mail:
Abstract. In the present paper the non-isothermal crystallization kinetics of Fe60Gd10Cr10B20 amorphous alloys is investigated by differential thermal analysis (DTA). By X-ray diffraction (XRD) we established the crystalline phases which appeared in the non-isothermal crystallization process. By using an evolutionary algorithm we determined the peak temperatures for each DTA curve and we determined the values of the crystallization energy.
Introduction. The DTA method belongs to the class of experimental methods which allow to characterize the amorphous alloys through the variation with the temperature of a physical or chemical quantity (e.g. heat capacity). The energy changed by the system with the environment during the transformation is proportional with the fraction of the alloy that crystallized. Measuring, at different heating rates, a physical quantity which is proportional to the transformed fraction of the alloy one can find the dependence on temperature and time of the transformed alloy fraction, i.e. the transformation kinetics. The aim of this work is to determine the crystalline phases and the crystallization energies for Fe60Gd10Cr10B20 amorphousalloys. The crystalline phases are determined using XRD while the crystallization energies are computed starting from the peak temperatures. To obtain the peak temperatures, the experimental data have been fitted with a linear combination of some kernel functions [1]. The optimization problem involved in the least squares method have been solved by using an evolutionary optimization algorithm, an adaptive variant [2] of the differential evolution method [3].
Theory. When an amorphous alloy is heated at a constant heating rate, , crystal nuclei are formed and grow until, finally, whole amorphous alloy is crystallized. The rate of change of the volume fraction of crystals, X, precipitated in the amorphous alloy, is expressed by [4]:
(1)
where A is a constant, n, m, k are numerical factors characterizing the crystalli-zation mechanism and is the activation energy for crystalline phases growth. The rate of change of X reaches its maximum at a temperature Ticorresponding to the peak of the DTA curve. From the maximum condition applied in (1) it follows:
(2)
where is the volume fraction of the crystal at the temperature . For bulk nucleation, the exponent k is 1 thus is always equal to 1, while for surface nucleation k is 2/3 and the term can be regarded as constant when comparing with the change of the exponential term. From these remarks it results that the equation (2) can be rewritten as:
(3)
If the mechanism which controls the crystallization is known, one can plot the dependence of ln(/) versus (-). The slope of the plotted line is which divided by produces the crystallization energy. If the mechanism is not known then we use equation (3) to study the recorded DTA curves. The obtained values of the crystallization energy will be analyzed to decide what mechanism was involved in the crystallization of the amorphous alloy. The crystallization process of an amorphous alloy is not based only on one mechanism, so we cannot associate a unique value to the activation energy [5]. The alloy crystallization involves different mechanisms, each one being dominant for a given value of the temperature. Great values of the activation energy suggest the participation of a great number of atoms to the crystallization mechanism.
Experimental setup and data analysis. The amorphous alloy that we analyzed has been obtained by single-roller technique with a cooling rate of the order of 105 to 106[Ks-1]. The samples have been realized of amorphous alloy dilluted in aluminium oxide in such a proportion that the two crucibles have the same value of the heat capacity, Cp. The non-isothermal annealing was realized at constant annealing rates 9 K/min, 12K/min, respectively 15 K/min on a Derivatograph C. To identify the precipitated crystalline phases we recorded the diffractogram X on a DRON using radiation K of Mo with = 0.71 Å. To determine the peak temperatures, the data obtained by DTA have been processed as follows. First, the data have been preprocessed by normalization (such that the area of the region defined by the DTA curve is 1) and then by a linear transformation which brings the data points into [-3, 3]x[0, 3]. Then the prepocessed data have been fitted with a linear combination, , of kernel functions, fi. Due to the particularities of the experimental data we selected as kernel function the bi-Gaussian which depends on three parameters as follows: for x<zi and for x zi. To find the parameters {hi, zi, wi1, wi2}i=1,…,p the least squares method can be used. When the kernel functions have complicated expressions the global minimization problem arised by the least squares method become difficult, thus traditional optimization methods are no more appropriate. To solve the optimization problem we propose the use of an evolutionary method which proved to be a robust and efficient technique in global optimization over continuous domains: the differential evolution algorithms [2],[3].
Results and discussion. Using the ASTM cards in analysing the X diffractogram (Figure 1(a)) we found the following crystalline compounds:1-Fe3B,2-Fe2B, 3-FeB, 4-Fe2Gd, 5-Gd2B5, 6-CrB. This result suggests the presence of 6 crystallization processes. Thus we fitted the experimental data using the model presented in the previous section for p=6. The plots of the data, fitted curve and the kernel functions associated to peaks are presented in Figures 1(b), 1(c) and 1(d). The peak temperatures obtained by the fitting process are presented in Table 1. Starting from the peak temperatures in Table 1 and using relation (3) we found the activation energies presented in Table 2.
Table 1Peak temperatures determined by fitting
[K/min] / Ti1 [K] / Ti2 [K] / Ti3 [K] / Ti4 [K] / Ti5 [K] / Ti6 [K]
9 / 893.913 / 934.530 / 964.404 / 967.370 / 978.061 / 1041.850
12 / 900.755 / 938.484 / 965.860 / 966.554 / 988.317 / 1053.270
15 / 908.357 / 949.193 / 973.490 / 975.151 / 1051.400 / 1104.100
Table 2
The activation energy values
Process / [eV]
n= 1,m = 1,k=2/3 / [eV]
n = 2, m = 1,k = 1 / [eV]
n= 3,m = 2,k=1 / [eV]
n= 4,m = 3,k=1
1 / 2.31 0.24 / 4.79 0.49 / 3.63 0.37 / 3.24 0.33
2 / 2.28 0.84 / 4.73 1.67 / 3.59 1.26 / 3.21 1.12
3 / 3.68 1.86 / 7.52 3.72 / 5.68 2.80 / 5.07 2.48
4 / 3.24 2.77 / 6.64 5.54 / 5.02 4.16 / 4.48 3.69
5 / 0.34 0.25 / 0.85 0.51 / 0.68 0.38 / 0.63 0.34
6 / 0.52 0.31 / 1.22 0.62 / 0.96 0.46 / 0.88 0.41
(a) / (b)
(c) / (d)
Figure 1. (a) X diffractogram for the sample annealing with heating rate 9K/min; (b),(c),(d) Peaks fitting for data recorded at heating rates: 9K/min, 12K/min, 15K/min
Since the smallest values of the crystallization energy are for surface crystallization we can conclude that the alloy crystallization has been realized by surface nucleation. This can be explained by the fact that the alloy has been very well pressed when it has been introduced into crucible so it has been partitioned in very small pieces leading to an increase of its overall surface. Since the DTA treatment was not very appropriate, the registered data did not allow us to associate correct values of the energy to the crystalline phases. Further work will address the study of the applicability of the proposed data analysis method for different amorphous alloys.
Acknowledgements. The authors thank Dr.A.Jianu for supplying the amorphous alloys.
References
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