DEVELOPMENT OF MATHEMATICAL SIMULATION FOR CURIE TEMPERATURE AND ACTIVATION ENERGY OF Co-Sn SUBSTITUTED CALCIUM HEXAFERRITE NANOPARTICLES

SABLE S.N.; DESHPANDE A.D.; MODAK J.P.; NANOTI V.M.; REWATKAR K.G.; GAWALI S.R.

DEVELOPMENT OF MATHEMATICAL SIMULATION FOR CURIE TEMPERATURE AND ACTIVATION ENERGY OF Co-Sn SUBSTITUTED CALCIUM HEXAFERRITE NANOPARTICLES

SABLE S.N.¹*, DESHPANDE A.D.², MODAK J.P.³, NANOTI V.M.⁴, REWATKAR K.G.⁵, GAWALI S.R.⁶
¹Priyadarshini College of Engineering, MIDC Hingna, Nagpur, MS, India.
²Priyadarshini College of Engineering, MIDC Hingna, Nagpur, MS, India.
³Priyadarshini College of Engineering, MIDC Hingna, Nagpur, MS, India.
⁴Priyadarshini College of Engineering, MIDC Hingna, Nagpur, MS, India.
⁵Dr. Ambedkar College, Deeksh Bhoomi, Nagpur, MS, India.
⁶Dr. Ambedkar College, Chandrapur, MS, India.
* Corresponding Author : sharadtz@hotmail.com

Received : 28-02-2012 Accepted : 06-03-2012 Published : 15-03-2012
Volume : 3 Issue : 1 Pages : 28 - 30
Int J Knowl Eng 3.1 (2012):28-30

Conflict of Interest : None declared

Cite - MLA : SABLE S.N., et al "DEVELOPMENT OF MATHEMATICAL SIMULATION FOR CURIE TEMPERATURE AND ACTIVATION ENERGY OF Co-Sn SUBSTITUTED CALCIUM HEXAFERRITE NANOPARTICLES ." International Journal of Knowledge Engineering 3.1 (2012):28-30.

Cite - APA : SABLE S.N., DESHPANDE A.D., MODAK J.P., NANOTI V.M., REWATKAR K.G., GAWALI S.R. (2012). DEVELOPMENT OF MATHEMATICAL SIMULATION FOR CURIE TEMPERATURE AND ACTIVATION ENERGY OF Co-Sn SUBSTITUTED CALCIUM HEXAFERRITE NANOPARTICLES . International Journal of Knowledge Engineering, 3 (1), 28-30.

Cite - Chicago : SABLE S.N., DESHPANDE A.D., MODAK J.P., NANOTI V.M., REWATKAR K.G., and GAWALI S.R. "DEVELOPMENT OF MATHEMATICAL SIMULATION FOR CURIE TEMPERATURE AND ACTIVATION ENERGY OF Co-Sn SUBSTITUTED CALCIUM HEXAFERRITE NANOPARTICLES ." International Journal of Knowledge Engineering 3, no. 1 (2012):28-30.

Copyright : © 2012, SABLE S.N., et al, Published by Bioinfo Publications. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

Abstract

The series of Co-Sn doped Calcium Hexa Ferrite Nanoparticles are synthesized using â€˜Microwave Induced Sol-Gel Combustion Routeâ€™. Their structural, morphological, electrical and magnetic properties are characterized and studied. The mathematical simulation is developed for the select properties like Activation Energy and Curie Temperature to study the influence of doping of Co and Sn on these properties. The substitution of Co2+ (0.72 Ã…) in place of Fe3+ ions is observed to an inverse influence on the Curie Temperature Tc of the samples while it has linear influence on the Activation Energy (âˆ†E) of the samples. The substitution of Sn4+ (0.71 Ã…) in place of Fe3+ ions is found to be has a linear influence on the Curie Temperature Tc of the samples while it has an inverse influence on the Activation Energy (âˆ†E) of the samples. These interpretations found complete matching with those concluded by Mossbauer spectroscopy, Vibrating Sample Magnetometry and others.

Keywords

Mathematical Simulation, Modeling, Nano Magnetic Materials, Activation Energy, Curie Temperature

Introduction

In the advancement of computer interfaced software, ANN Modeling is widely used tool in the studies of material characterization [1] . This mathematical simulation can be used to countercheck experimental findings with those predicted by such models [2] . In the studies of substituted calcium hexaferrite nanoparticles, it is observed that such countercheck using mathematical modeling is not much reported. Keeping in view, the recent bang of success of Artificial Neural Network Modeling, an attempt is made to develop a mathematical model by considering only three properties of the samples, one from each class [3] . This concept can equally be applied to rest of the properties of the samples which do depend upon the composition of Co, Sn and Fe ions in them.

Development of Mathematical Model

The properties chosen for such modeling are Crystallite Size (D), Curie Temperature (Tc) and Activation Energy (âˆ†E). These properties eventually depend upon composition and combination of Co, Sn and Fe ions in the sample as tabulated in the following [Table-1] .
As it is discussed earlier that the structural, magnetic and electrical properties are jointly decided by the substitution as well as combination of Co and Sn ions with Fe ions; this modeling will remain beneficial to counter check dependence of these properties of the sample on its constituent elements.
Using the table, inputs (causes) and outputs (responses/effects) can be decided. As a pilot study of simulation and mathematical modeling, in this case only two responses and four causes are selected [4] . The inputs (causes) selected to develop this sort of model are â€˜No. of atoms of Co, Sn and Fe in each molecule of an individual sampleâ€™. The number of atoms of Ca and O are ignored as they remain constant in every molecule of the sample. The outputs (responses) are selected as â€˜Curie Temperature (Tc) as of magnetic properties and Activation Energy (âˆ†E) as of electrical propertiesâ€™.
For this research module, using these causes and effects (inputs and outputs), an equation for the mathematical model can be written as-

$T_{c}=k\left [ (Co^{x}m)(Sn^{y}m)(Fe^{z}m) \right ]\; \; \; \; \; (1)$

for the dependence of composition of Co, Sn and Fe on magnetic property i.e. Curie Temperature T_c of the sample

$\Delta E=k\left [ (Co^{x}e)(Sn^{y}e)(Fe^{z}e) \right ]\; \; \; \; \; (2)(eV)$

for the dependence of composition of Co, Sn and Fe on electrical property i.e. Activation Energy âˆ†E of the sample
(The constant k represents the compositional effect of Ca and O ions on Curie Temperature Tc and Activation Energy âˆ†E. This of course along with other causes which are known to influence the phenomenon but somehow could not be quantified due to instrumentation inadequacies or other reasons. It is called â€˜Curve Fitting Constantâ€™)
The equation (1) can be solved by taking the log of both sides as-

$log(Tc_{1})=log\, k+x_{m}log(Co_{1})+y_{m}log(Sn_{1})+z_{m}log(Fe_{1})$

$log(Tc_{2})=log\, k+x_{m}log(Co_{2})+y_{m}log(Sn_{2})+z_{m}log(Fe_{2})$

$log(Tc_{3})=log\, k+x_{m}log(Co_{3})+y_{m}log(Sn_{3})+z_{m}log(Fe_{3})$

$log(Tc_{4})=log\, k+x_{m}log(Co_{4})+y_{m}log(Sn_{4})+z_{m}log(Fe_{4})$

Using these equations, a matrix equation can be simplified as-

$\begin{bmatrix} log\, 659.30\\ log\, 606.11\\ log\, 544.99\\ log\, 465.97\end{bmatrix}=\begin{bmatrix} 1 & log\, 1 & log\, 1 & log\, 10\\ 1 & log\, 2 & log\, 2 & log\, 8\\ 1 & log\, 3 & log\, 3 & log\, 6\\ 1 & log\, 4 & log\, 4 & log\, 4 \end{bmatrix}\begin{bmatrix} log\, k\\ x_{m}\\ y_{m}\\ z_{m}\end{bmatrix}$

This can be simplified as-

$\begin{bmatrix} 6.4912\\ 6.4071\\ 6.3008\\ 6.1441\end{bmatrix}\: \begin{bmatrix} 1 & 0 & 0 & 2.3026\\ 1 & 0.6931 & 0.6931 & 2.0794\\ 1 & 1.0986 & 1.0986 & 1.7918\\ 1 & 1.3868 & 1.3868 & 1.3863 \end{bmatrix}\begin{bmatrix} log\, k\\ x_{m}\\ y_{m}\\ z_{m}\end{bmatrix}$

$Let\, M=\begin{bmatrix} 1 & 0 & 0 & 2.3026\\ 1 & 0.6931 & 0.6931 & 2.0794\\ 1 & 1.0986 & 1.0986 & 1.7918\\ 1 & 1.3868 & 1.3868 & 1.3863 \end{bmatrix}$

$\small M^{-1}=\begin{bmatrix} 0 & 0 & 0 & 0\\ -2.0855 & 7.6214 & -8.3143 & 2.7784\\ 2.0855 & -7.6214 & 8.3143 & -2.7784\\ 0 & 0 & 0 & 0 \end{bmatrix}\times 10^{15}$

Therefore, we can write-

$\small M^{*}M^{-1}\begin{bmatrix} log\, k\\ x_{m}\\ y_{m}\\ z_{m}\end{bmatrix}=\begin{bmatrix} 0 & 0 & 0 & 0\\ -2.0855 & 7.6214 & -8.3143 & 2.7784\\ 2.0855 & -7.6214 & 8.3143 & -2.7784\\ 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} 6.4912\\ 6.4071\\ 6.3008\\ 6.1441\end{bmatrix}\times 10^{15}$

$\small \begin{bmatrix} log\, k\\ x_{m}\\ y_{m}\\ z_{m}\end{bmatrix}=\begin{bmatrix} 0 & 0 & 0 & 0\\ -2.0855 & 7.6214 & -8.3143 & 2.7784\\ 2.0855 & -7.6214 & 8.3143 & -2.7784\\ 0 & 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} 6.4912\\ 6.4071\\ 6.3008\\ 6.1441\end{bmatrix}\times 10^{15}$

$\small \begin{bmatrix} log\, k\\ x_{m}\\ y_{m}\\ z_{m}\end{bmatrix}=\begin{bmatrix} 0\\ -2.23\\ 2.23\\ 0\end{bmatrix}\times 10^{13}$

Therefore,

$\small log\, k=0\, x_{m}=-2.23\times 10^{13};y_{m}=2.23\times 10^{13}\, z_{m}=0$

Hence,

$\small k=1\, x_{m}=-2.23\times 10^{13};y_{m}=2.23\times 10^{13}\, z_{m}=0$

Using these values, the equation (1) can be written as-

$\small T_{c}=\left [ (Co^{-2.23\times 10^{13}}) (Sn^{2.23\times 10^{13}})(Fe^{0})\right ]\; \; \; \; \; (3)$

(K)
In the same way by simplifying, the equation (2) can be written as-

$\small \bigtriangleup E=\left [ (Co^{-2.23\times 10^{14}}) (Sn^{2.23\times 10^{14}})(Fe^{0})\right ]\; \; \; \; \; (4)$

(eV)

Interpretation of Model

1. The equations (3) and (4) represent the quantitative influence of various inputs like ionic radius Co²⁺ and Sn⁴⁺ ions on Curie Temperature $\small (T_{c})$ and Activation Energy $\small (\Delta E)$ of the samples.
2. The equation (3) is very useful from the point of view of deciding relative influence of Sn and Co. The index of Co is â€“ $\small 2.23\times 10^{13}$ . The negative sign indicates that as Co would increase, TC would decrease. Further, index of Sn is $\small 2.23\times 10^{13}$ . The positive sign indicates that as Sn would increase, Tc would also increase. Hence it can be interpreted that the influence of Co is to decrease Tc while the influence of Sn is to increase Tc. Similar interpretation can be drawn from equation (4) $\small \Delta E$ for as -The influence of Co is to increase $\small \Delta E$ while that of Sn is to decrease $\small \Delta E$ .
3. In the absence of this mathematical modeling, such quantitative interpretation of influence of Co, Sn and Fe on $\small T_{c}$ and $\small \Delta E$ would not have been possible. Thus, this model supports the study of â€˜Synthesis and Characterisation of Magnetic Nano Materialsâ€™ and it has significant importance in it.
4. We can further have an advance mathematical model of $\small \frac{\partial T_{c}}{\partial Co},\frac{\partial T_{c}}{\partial Sn}$ i.e. first order derivatives of Tc with respect to Sn and Co at some specific value of Co, Sn and Fe. That would certainly represent the rate of increase in Tc with respect to Sn and Co. Similar derivatives $\small \frac{\partial \Delta E}{\partial Co},\frac{\partial \Delta E}{\partial Sn}$ , can further be simplified to get better information about influence of Co and Sn on $\small \Delta E$ .
5. The substitution of Co²⁺ (0.72 Ã…) in place of Fe³⁺ ions has an inverse influence on the Curie Temperature $\small T_{c}$ of the samples while it has linear influence on the Activation Energy $\small \Delta E$ of the samples.
6. The substitution of Sn⁴⁺ (0.71 Ã…) in place of Fe³⁺ ions has a linear influence on the Curie Temperature $\small T_{c}$ of the samples while it has an inverse influence on the Activation Energy $\small (\Delta E)$ of the samples.

References

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[2] Tho K.K. et al. (2004) Modelling and Simulation in Materials Science and Engineering, 12, 1055.
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[3] Mike Mesterton-Gibbons (1995) A Concrete Approach to Mathematical Modelling, ISBNâ€“10: 0471109606, Wiley Interâ€“Science Paperback Series.
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[4] Mark M. Meerschaert, Mathematical Modeling, 3rd Edition, ISBN: 978-0-12-370857-1.
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[5] Jagat Narain Kapur, Mathematical Modelling, New Age International, 1988-259.
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