Current Applied Physics 9 (2009) 807–811

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Current Applied Physics

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Correlation between Meyer–Neldel rule and phase separation in Se98�xZn2Inx

chalcogenide glasses

Abhay Kumar Singh, Neeraj Mehta, Kedar Singh *

Department of Physics, Banaras Hindu University, Varanasi-221005, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 May 2008Accepted 23 July 2008Available online 31 July 2008

PACS:61.43.Fs73.61.Jc

Keywords:Chalcogenide glassesActivation energy of crystallizationAmorphous to crystalline phase transitionDSC technique

1567-1739/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.cap.2008.07.013

* Corresponding author. Tel.: +91 542 2307308; faxE-mail address: kedarbhp@rediffmail.com (K. Sing

The present work reports the observation of Meyer–Neldel rule for the thermally activated crystallizationof glassy Se98�xZn2Inx (0 6 x 6 10) alloys. We have observed a strong co-relation between the pre-expo-nential factor K0 of rate constant K(T) of crystallization and activation energy of crystallization Ec in thepresent case. This indicates the presence of compensation effect for the non-isothermal crystallizationprocess in the present glassy system, which is explained in terms of phase separation of the present alloysdue to flaw bonds of these amorphous solids.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

The MN rule describes an exponential relation between the acti-vation energy and the pre-exponential factor. It has been observedin a wide range of materials. These include single crystal, polycrys-talline, amorphous and organic solids and even ionically conduct-ing crystals and glasses [1–8].

This law is obeyed by many processes [1–8] including annealingphenomena and electronic processes in amorphous semiconduc-tors, trapping in crystalline semiconductors, conductivity in ionicconductors, aging of insulating polymers, biological death ratesand chemical reactions. Thus, one can say that this rule is observedin wide range of phenomena in physics, chemistry, biology andelectronics.

In case of chalcogenide glasses also, MN rule is observed by thevariation of activation energy DE of d.c. conduction on changingthe composition of the glassy alloys [9–12] in a specific glassy sys-tem or by either variation of intensity of light [13,14] or variationin applied frequency [15,16].

In this paper, we have reported MN rule in thermally activatednon-isothermal crystallization of chalcogenide glasses. Variousmodels have been proposed for MN rule in activated processes.Fang [17] has proposed a model for MN rule in activated processes.According to this model, the annealing time parameter obeys the

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MN rule. As the rate constant K is also annealing time dependentparameter, our results confirm the validity of the model proposedby Fang [17]. Koga and Sestak [18] have shown that the kineticcompensation effect mathematically results from the exponentialform of the rate constant. A change of activation energy is thuscompensated by the same change in temperature or in the loga-rithm of the pre-exponential factor. However, it should be notedthat the above observations by various workers are highly qualita-tive and it is difficult to get any quantitative information from suchobservations.

Further, Yelone and Movaghar [19] have proposed YM model toexplore the pre-exponential factor for activated process as, if theactivation energy is large as compared to typical excitations ofthe systems, it make necessary to assemble for the reaction to takeplace. As the entropy is preoperational to logarithm of number ofdifferent ways of assembling these excitations, entropy will in-creased as enthalpy increases. The YM model are based on assump-tion explicit or implicit, the excitation process involves manyphonons instead of one phonon model.

Recently, Se–In and Se–Zn based chalcogenide glasses have be-come attractive materials for fundamental research of their struc-ture, properties and preparation [20–22]. This would have manycurrent and potential applications in optics and optoelectronics.In this research paper, we explain the observation of MN rule inthermally activated non-isothermal crystallization of glassySe98�xZn2Inx (0 6 x 6 10) system in terms of phase separation ofthe present glasses.

Se98Zn2

Se96Zn2In2

Se94Zn2In4

Se88Zn2In10

Se92Zn2In6

Endotherm

ic

Exo

ther

mic

DSC heating rate at 15 K/min

Tg1Tg2

Tg1Tg2

Tg1Tg2

Fig. 2. DSC scans of glassy Se98�xZn2Inx (0 6 x 6 10) alloys at heating rate of 15 K/min.

808 A.K. Singh et al. / Current Applied Physics 9 (2009) 807–811

Phase separation effects are of general interest in glass science. Itis a term used to describe a phenomenon where an initially homo-geneous system, such as a liquid, will un-mix into two or more finelymixed, chemically or structurally different, component or phases.Such structural effects produce usually pronounced changes in glassphysical properties including a lowering of the glass transition tem-perature, a lowering of the optical band gap, an increase of molarvolumes [23]. Phase separation can exist on a variety of lengthscales from nanometers to micrometers or millimeters [23]. Forthe purpose of photonic applications, it is important that phase sep-aration be eliminated or at least minimized to length scales muchsmaller than that the wavelengths of light being used.

The detection of phase separation in a glass is sometimes verydifficult since phase separation may produce only subtle changesin the diffraction pattern, whereas changes in the microstructureare evident only if the separating phases markedly differ in elec-tron density. However, double endothermic and exothermic peakshave been observed in different chalcogenide alloys by variousworkers using DSC technique [24–29], which indicates the pres-ence of phase separation in these glasses.

2. Material preparation

Glassy alloys used in the present study were prepared byquenching technique. The exact proportions of high purity(99.999%) elements, in accordance with their atomic percentages,were weighed using an electronic balance with the least count of10�4 gm. The material was then sealed in evacuated (�10�5 Torr)quartz ampoules (length �8 cm and internal diameter �12 mm).The ampoules containing material were heated to 1173 K and wereheld at that temperature for 12 h. The temperature of the furnacewas raised slowly at a rate of 5–6 K/min. During heating, the am-poules were constantly rocked, by rotating a ceramic rod to whichthe ampoules were tucked away in the furnace. This was done toobtain homogeneous glassy alloys.

After rocking for about 12 h, the obtained melts were cooledrapidly by removing the ampoules from the furnace and droppingto ice-cooled water rapidly. The quenched samples were thentaken out by breaking the quartz ampoules. The glassy natureof the alloys was ascertained by X-ray diffraction (XRD) technique(PHELIPS XRD). The XRD patterns of glassy Se98�xZn2Inx (0 6x 6 10) alloys are shown in Fig. 1. Absence of any sharp peak inXRD patterns of all the glassy alloys confirms the glassy nature ofthese alloys.

10 20 30 40 50 60 70 802 theta

Se98Zn2

Se96Zn2In2

Se94Zn2In4

Se92Zn2In6

Se88Zn2In10

Inte

sity

abr

.uni

t.

Fig. 1. XRD scans of glassy Se98�xZn2Inx (0 6 x 6 10) alloys.

3. Experimental

The glasses, thus prepared, were ground to make fine powderfor DSC studies. This technique is particularly important due tothe fact that: (1) it is easy to carry out; (2) it requires little samplepreparation; (3) it is quite sensitive and (4) it is relatively indepen-dent of the sample geometry.

The thermal behaviour was investigated using differential scan-ning calorimeter (METTLER DSC-25 Model). The temperature preci-sion of this equipment is ±0.1 K with an average standard error ofabout ±1 K in the measured values of crystallization temperatures.

About 5–10 mg of each sample was heated at a constant heatingrate and the changes in heat flow with respect to an empty panwere measured. Four heating rates (5, 10, 15 and 20 K/min) werechosen in the present study. Measurements were made under al-most identical conditions. DSC scans of glassy Se98�xZn2Inx

(0 6 x 6 10) system are shown in Fig. 2 at heating rate of 15K/min. It is clear from these scans that two phases occur simulta-neously during amorphous to crystalline (a–c) transformation inglassy Se98�xZn2Inx (0 6 x 6 10) system. This is confirmed fromdouble endothermic peaks observed at glass transition tempera-ture (Tg) in almost all glassy alloys. Single exothermic peaks are ob-served at crystallization temperatures (Tc) extended in a widerange of temperature. It is also clear from these scans that exother-mic peaks are not sharp as generally observed in case of chalcogen-ide glasses. This also confirms the presence of two or more phasesin the present glasses. Similar DSC scans were obtained at otherheating rates.

4. Theoretical basis

During the isothermal transformation, the extent of crystalliza-tion (a) of a certain material is represented by the Avrami’s equa-tion [30–32]

aðtÞ ¼ 1� exp½ð�KtÞn� ð1Þ

where ‘K’ is rate constant and ‘n’ is the order parameter which de-pends upon the mechanism of crystal growth.

Table 1Crystallization temperature Tc of glassy Se98�xZn2Inx (0 6 x 6 10) alloys at differentheating rates

Glassy system Se98�xZn2Inx Tc (K)

5 K/min 10 K/min 15 K/min 20 K/min

x = 0 354 360 364 367x = 2 362 367 371 373x = 4 365 371 374 377x = 6 368 373 375 378x = 10 364 368 372 374

Se98-xZn2Inx

-4.5

-4.3

-4.1

-3.9

-3.7

-3.5

-3.3

-3.1

-2.9

-2.7

-2.5

2.62 2.65 2.68 2.71 2.74 2.77 2.8 2.83

1000 / Tc (K)-1

(βc)

(m

in)-1

x = 0

x = 2

x = 4

x = 6

x = 10

/ΤIn

Fig. 3. Plots of ln (b/Tc) vs 103/Tc for glassy Se98�xZn2Inx (0 6 x 6 10) alloys.

A.K. Singh et al. / Current Applied Physics 9 (2009) 807–811 809

In general, crystallization rate constant K increases exponen-tially with temperature indicating that the crystallization is a ther-mally activated process. Mathematically, it can be expressed as

K ¼ K0 expð�Ec=RTÞ ð2Þ

Here Ec is the activation energy of crystallization, K0 the pre-expo-nential factor and R the universal gas constant. In Eq. (3), Ec andK0 are assumed to be practically independent of the temperature(at least in the temperature interval accessible in the calorimetricmeasurements).

In non-isothermal crystallization, it is assumed that there is aconstant heating rate in the experiment. The relation betweenthe sample temperature T and the heating rate b a can be writtenin the form

T ¼ T i þ bt; ð3Þ

where ‘Ti’ is the initial temperature. The crystallization rate is ob-tained by taking the derivative of expression (1) with respect totime, bearing in mind that the reaction rate constant is a time func-tion through its Arrhenius temperature dependence, resulting in

ðda=dtÞ ¼ nðKtÞn�1½K þ ðdK=dtÞ t�ð1� aÞ ð4Þ

The derivative of ‘K’ with respect to time can be obtained from Eqs.(2) and (3) as follows:

ðda=dtÞ ¼ ðdK=dTÞ ðdT=dtÞ ¼ ðbEc=RT2ÞK ð5Þ

Then Eq. (4) becomes

ðda=dtÞ ¼ nKntn�1½1þ at�:ð1� aÞ ð6Þ

where a ¼ ðbEc=RT2Þ:Augis and Bennett [33] developed a method based on Eq. (6). They

taking proper account of the temperature dependence of the reac-tion rate, and their approach resulted in a linear relation betweenln (Tc � Ti)/b versus 1/Tc. This can be deduced as follows, substituting‘u’ for ‘Kt’ into Eq. (6); the rate of reaction is expressed as

ðda=dtÞ ¼ nðdu=dtÞuðn�1Þ ð1� aÞ; ð7Þ

where

ðdu=dtÞ ¼ u ½ð1=tÞ þ a� ð8Þ

The second derivatives of Eqs. (7) and (8) are given by

ðd2a=dt2Þ ¼ ½ðdu2=dt2Þu� ðdu=dtÞ2

� ðnun � nþ 1Þ�nuðn�2Þð1� aÞ ¼ 0 ð9Þðdu2=dt2Þ ¼ ðdu=dtÞ ½ð1=tÞ þ a� þ u ½ð�1=t2Þ þ ðda=dtÞ�: ð10Þ

Recalling that T = Ti + bt and substituting for (da/dt) = �(2b/T)a,Eq. (10) can be written as

ðdu2=dt2Þ ¼ u ½a2 þ ð2aTi=tT�: ð11Þ

The last term in the above equation was omitted in the originalderivation of Augis and Bennett [33] (Ti� T) and resulted in thesimple form:

ðdu2=dt2Þ ¼ a2 u; ð12Þ

Substitution of (du/dt) and (du2/dt2) from Eqs. (8) and (12), intoEq. (9) gives:

ðnun � nþ 1Þ ¼ ½at=ð1þ atÞ�2: ð13Þ

For E/RT� 1, the right-hand bracket approaches its maximumlimit and consequently u (at the peak) = 1, or

u ¼ ðKtÞc ¼ K0 expð�Ec=RTcÞ ½ðTc � T iÞ=b� � 1 ð14Þ

In logarithm form, for Ti� Tc, we have

lnðb=TcÞ � ð�Ec=RTcÞ þ ln K0 ð15Þ

The values of Ec and K(T) can be evaluated by this equation using theplots of ln b/Tc against 1/Tc.

The above equation is derived by Augis and Bennett [33] fromthe classical JMA model [30–32]. Eq. (2) has been used by variousworkers [34–37]. Their results show that Ec values obtained by Eq.(15) are in good agreement with the Ec values obtained by well-known Kissinger’s relation [38] and relation of Matusita and Sakka[39,40]. We have therefore used the method of Augis and BennettEq. (15) to evaluate activation energy of crystallization Ec. Thismethod has an extra advantage that the intercept of ln b/Tc vs.1/Tc gives the value of pre-exponential factor K0 of Arrhenius equa-tion. The values of Tc for glassy Se90�xZn2Inx (0 6 x 6 10) alloys aregiven in Table 1.

5. Results and discussion

According to Eq. (15), the plot of ln (b/Tc) vs. 103/Tc leads to astraight line. This has been verified for glassy Se98�xZn2Inx

(0 6 x 6 10) alloys in Fig. 3. The activation energy of crystallizationEc and pre-exponential factor K0 of the glassy alloys have been cal-culated from the slopes and intercepts of the plots of ln (b/Tc) vs103/Tc. The values of Ec and ln K0 for present glassy alloys obtainedfrom Eq. (15) are given in Table 2.

It is clear from this table that Ec and K0 are composition depen-dent and K0 is not a constant but depends on Ec. Fig. 4 shows theplot of ln K0 vs. Ec for glassy Se98�xZn2Inx (0 6 x 6 10) system.Curve fitting is done by least square method and the square of coef-ficient of correlation (R2) of ln K0 vs. Ec plot is indicated in the fig-ure. It is clear from the figure that ln K0 vs. Ec plot is a straight lineof good correlation coefficient indicating that K0 varies exponen-tially with Ec following the relation:

ln K0 ¼ ln K00 þ Ec=kT0 ð16Þ

Table 2Values of Ec and K0 for glassy Se98�xZn2Inx (0 6 x 6 10) alloys

Glassy system Se90�xZn2Inx Ec (eV) K0 (min)�1 K0 = K00 exp (Ec/k T0) (min)�1

x = 0 1.16 5.5 � 1014 4.5 � 1014

x = 2 1.41 5.2 � 1017 4.9 � 1017

x = 4 1.36 7.3 � 1016 1.2 � 1017

x = 6 1.67 1.1 � 1021 1.1 � 1021

x = 10 1.55 4.0 � 1019 3.1 � 1019

Se98-xZn2Inx

R2 = 0.9974

31

33

35

37

39

41

43

45

47

49

51

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Ec (eV)

lnK

o(m

in)-1

Fig. 4. Plot of ln K0 versus Ec for glassy Se98�xZn2Inx (0 6 x 6 10) alloys.

Se98-xZn2Inx

30

32

34

36

38

40

42

44

46

48

50

x (at %)

lnK

o(m

in-1

)

Experimental

Theoretical

0 2 4 6 8 10 12

Fig. 5. Composition dependence of ln K0 for glassy Se98�xZn2Inx (0 6 x 6 10) alloys.

810 A.K. Singh et al. / Current Applied Physics 9 (2009) 807–811

From the slope and intercept of the line, we have calculated thevalues of (kT0)�1 and K00. Using these values, the expected ln K0

values have been calculated for glassy Se98�xZn2Inx (0 6 x 6 10)system and compared with the reported values (see Table 2). Anoverall good agreement between these two values confirms thevalidity of MN rule in glassy Se98�xZn2Inx (0 6 x 6 10) system(see Fig. 5).

As described earlier by us, the YM model [19] is supported theincrease entropy due to involvement of impurities. The character-istic interaction length and dimensionality of the volume in whichinteraction take place are varies with enthalpy in localized state. Inglassy Se98�xZn2Inx (0 6 x 6 10) system, it is expected to metallicZn bonds dissolved in Se chains and its makes Zn–Zn, Se–Se homo-nuclear and Se–Zn hetronuclear bonds, where their bond energiesare 204, 104 and 161 KJ/mole, in Se–Zn glassy matrix. Initially bin-ary glass has cross-linked Se2Zn4 heteronuclear metastable statestructure. However, above Tc cross-linked metastable structurehas been relaxed into Se2Zn stable structure [41], due to breakingof dangling bonds of combined stochiometrics. Moreover, theincorporation In in Se–Zn glassy matrix increase the disordered-ness of stochiometrics and it has makes Se–In hetronuclear bondshaving energy 128.4 KJ/mole, with strong fixed metallic Zn–Inbonds, because of fixed amount of Zn. Further incorporation of Inconcentration in ternary matrix structure heavily cross-linkedand steric hindrance increases. Therefore, the expanse of Se chainsand replacement of weak Se–Se bonds by Se–In bonds have beenoccurring. Thus, the enthalpy of the system increases with increas-ing In content. The dangling bonds (or disordered) saturation hasbeen occurring at chemical threshold 6 at. wt.% of In, due to defectsin localized state.

Further addition of Indium in Se–Zn matrix, it forms In–Inbonds and it reducing the Se–In concentration. Therefore, the over-all enthalpy of the system decreases because reducing the danglingbonds or defects of the system, resulting in a decrease pre-expo-nential factor on both side of chemical threshold. This is confirmedfrom the composition dependence of pre-exponential factor K0 ofthe present samples as shown in Fig. 5.

Recently, Widehorm et al. [42] proposed that that the MN rulearises naturally for a quantity where both an intrinsic process aswell as a process involving impurities contributes. The strengthof the latter depends solely on the density of the impurities. Thisleads to a spread in the apparent activation energy of the measuredquantity. In the present case, In is added in binary Se98Zn2 at cost ofSe. Amorphous selenium is believed to consist predominantly of amixture of two structural species, long helical chains and eight-member rings, held to each other by weak forces, perhaps of theVan der Waals type. When In is added in Se, most In atoms areprobably mixed in Se atomic chains, but some of In atoms wouldacts as ionized impurities, since the electron affinity of In(0.30 eV) is lower than that of Se (2.02 eV). The possible phasesin glassy Se98Zn2 alloy are, therefore, Se2Zn4 and Se2Zn. Theaddition of Zn in glassy Se98In2 alloy produces two phases Se2Zn4

and Se2Zn, which affect the crystallization rate. The nucleationand growth of phases of Se–In and Se–Zn may be, therefore,governed by two different transport mechanisms, one is due tointrinsic phases of Se–In and the other is due to impurity phasesof Se–Zn.

The applicability of MN rule for isothermal crystallization ofsome chalcogenide glasses in Se–Ge–Sb system has been observedby Bordas et al. [43,44]. On annealing some Se–Ge–Sb glasses, theyhave observed two crystalline phases: Sb2Se3 and GeSe2. Accordingto Bordas et al. [43,44] such a compensation effect may be attrib-uted to the primary crystallization of Sb2Se3 phase. In our samples,we have also found two phases Se2Zn4 and Se2Zn of the systems.Thus the MN rule in the present glassy system for thermallyactivated crystallization may arise due to nucleation and growthof different phase of Se, Zn and In governed by two transportmechanisms. Our results are, therefore, in good agreement withthe work reported by Bordas et al. and the model proposed byWidehorm et al. for the observation of MN rule in such thermallyactivated phenomena.

6. Conclusions

Glassy Se98�xZn2Inx (0 6 x 6 10) alloys have been prepared byquenching technique. Temperature dependence of crystallizationrate constant K(T) has been studied for various glassy alloys. Wehave find that K(T) is thermally activated. The activation energy,however, depends on the composition.

The activation energy and pre-exponential factor satisfiesthe MN rule for the present glassy system. This shows that theMN rule, which is generally observed for solid state diffusion incrystals and polymers, dielectric relaxation and conduction inpolymers, thermally stimulated processes in polymers andelectronic conduction in amorphous semiconductors; is also

A.K. Singh et al. / Current Applied Physics 9 (2009) 807–811 811

observed for the thermally activated crystallization in chalcogen-ide glasses. The results are explained in terms of nucleation andgrowth of phases of Se–In and Se–Zn may be, therefore, governedby two different transport mechanisms, one is due to intrinsicphases and the other is due to impurity phases as suggested byWidehorm et al.

Acknowledgements

We gratefully acknowledge the financial support from theCouncil of Scientific and Industrial Research [CSIR, Project No.03(1063)/06/EMR-II], New Delhi (India) to carry out this work.We are thankful to Prof. O.N. Srivastava who helped us in variousways in the course of this work.

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