Jha, N

Mishra, A K

Rafique, S M

IACS Institutional Repository

Indian J. Phys. 75A (5), 519-523 (2001)U P  Aan in te rn a tio n a l jo u rn a lThermodynamic properties and electrical resistivity of liquidMgZn alloysN Jha', A K Mishra•^ S M Rafique' and A N Singh''' R a ja s th an  V idyap ith . P u ru lii-7 2 3  0 01 , W est B engal, India  ^ D ep artm en t o f  P hysics , S ab o u r c | l l e g e ,  S abo u r-8 1 3  210, B ihar, Ind ia  ’ D ep a rtm en t o f  P h y sics , M arw ari C oj^ege, B h ag a lp u r-8 1 2  0 0 7 , B ihar, In d ia  * D ep artm en t o f  P h y sics , P N C o lleg e , P a rsa -8 1 4  219 , B ihar, IndiaReceived 15 November 2000, accepted II July 200JA b s tr a c t  ■ T h e  o b s e rv e d  fre e  e n e rg y  o f  m ix in g  o f  M g Z n  liq u id  a llo y  is d isc u sse d  on (he co m p lex  fo rm a tio n  m o d el T h e  re su lts  fo r lo n g - wavelength co n cen tra tio n -co n cen tra tio n  f lu c tua tions sV^^(O), e lec trica l resis tiv ity  and therm odynam ic activ ity  arc presen ted  as a function o f  concen tration  h as  b een  fu r th e r  u se d  to  o b ta in  th e  c h e m ic a l s h o r t ra n g e  o rd e r  p a ra m e te r  a ,  V alues o f  S^ {^0) an d  su g g e s t  th a t s e g re g a tio n -o rd e r  iranslo rm ation  o ccu rs  in th e  M g Z n  liq u id  a lloy  at a round  30 at % o f  ZnK e y w o rd s  * T h e rm o d y n a m ic  p ro p erty , e le c tr ic a l re s is tiv ity , liq u id  M gZ n a llo y sPACS Nos. : 61.25 Mv, 72.15.C/., 65.50-t-m1. Introductionli is fairly well known that the stability of liquid alloys is dependent on the interactions among the component atoms of ihc alloy [1,2]. Liquid alloys may be considered from two extreme points of view. One is the nearly free electron model and the other is the complex forming model. Complex formation in the liquid phase is a dynamical process characterized by a finite life lime [3]. It is always accompanied by a large heat of mixing, but this quantity docs not always provide an accurate indication of the stoichiometric composition. Of all concentration fluctuations m the long wave length limit, the concentration-concentration fluctuation, has emerged as one of the most importantand powerful microscopic function to understand the nature of interatomic interactions [4]. It is thought that attains ideal values, i,e. random mixing of free atoms in the glass forming ^concentration range [5]. This glass forming range usually lies [ar away from stoichiometric composition [6 ].Some Mg alloys arc particularly interesting due to their glass forming ability and zinc is present in many commercial Mg alloys. Since the availability of Mg alloys enables the optimization of the steps necessary for light alloys production and design [7],Corresponding Authorthe study of the characteristics ol alloys containing Mg is of great importance. At 7 =  923K, MgZn is similar in structure as CaMg and CaMg is a good glass former. MgZn glass was the first transition mctal-frcc metallic glass [8]. It has often been observed that a good glass-forming ability is linked to a negative temperature coefficient of the electrical resistivity [9], There exists a resistivity minima in MgZn alloy [ 10] at temperatures around 200-300K above the liquid temperature. MgZn is also an interesting and significant case for the study of local order and other fundamental properties. Moreover, MgZn alloy gives clear evidence of chemical short-range order (CSRO) [1 L13). The results of extended X-ray absorption fine structure measurements [8] show the occurrence ol CSRO both in liquid and glassy alloys. The phase diagram of MgZn is a complicated one [14,15]. Besides a congruently melting phase MgZn^* there exits a number of incongruently melting compounds Mg2Zn,p Mg2Zn3 and MgZn and the high temperature phase Mg3jZn23 (previously known as Mg^Zn^). We have assumed the complex formation model A^B y where and v  arcintegers) for the study of the thermodynamic properties and electrical resistivity of MgZn. This assumption is being used by various theoreticians [5,6,16-20] to investigate the alloying behaviour of such type of binary alloys.© 2 0 0 1  lA C Syi =  (VVj3 / / ? D  /w -l-n j / / jJ ,V3 - { \ V 2 y l  R T ) V~l)(n2/i3 ln^)-vn^ /n-fn2 /nj.N  Jha, A K  Mishra, S  M  Rafique and  A N  Singh520In this paper, the calculated results of free energy of mixing, activity, (0), CSRO and electrical resistivity for MgZn liquidalloy are presented.2. Complex fomuition modelLet a liquid binary alloy containing in all, = C^N atoms A  andN b = atoms B, be assumed to consist of w,/V free atoms of i4, atoms of B and complexes A^B^ which also act as independent scattering centers in the alloy. From the conservation of atoms, therefore,/I] = 1 — C q  ^  f i n ,(T )/I2 — ,n =  Wj - f /?2 . ( 1)Here. and are the concentrations of the first and second species and N  is Avogadro's number. The volume in which the free atoms are randomly distributed can be expressed as^  = ^ all.>y-n^Nn, , (2)where ^aUor is the atomic volume of the alloy and 12,,, the volume of the complex(3),^^2 the atomic volumes of the first and secondspecies ol the alloy, respectively.The number of complexes is obtained through the condition(4)where Gj^ is the Gibb’s energy of mixing and T and P  represent the temperature and pressure, respectively.The conformal solution approximation [21 ] enables us to express G j^ a s=-«3 g + R f^ n ,( \n n ,- \x \n )  + '^ '^ n ,n jW ,j ln ,  (5)/=■! '<7W. are the interaction energies which are independent of the concentration, g  is the formation energy of the complex and the term ( ~ n^g) lowers the free energy of the alloy due to the formation of chemical complexes and R  is the gas constant. Eqs. (4) and (5) yield=  c x p ( - g  /  RT) exp  (y , +  y j  +  >’3 ) - (6)wherey\ R T )^ f i’¥v-\)(n^n2  Z / i - o / i j  / n j ,The interaction energies and g are determined followingthe works by Bhatia and Hargrove [22]. Once the energy parameters are selected, these remain the same for all other thermodynamic investigations. Eq. (6) is solved numerically to obtain the equilibrium value of n y  which in turn is used in (5) to evaluate G^ as a function of concentration.3. Free energy of mixing and thermodynamic activityThe first step is to consider the nature of the chemical complexes ( A f i B v )  AfxBv which might exist in the liquid phase. The phase diagram 115,23] indicates that MgZn2 is the most probable intcrmetallic compound. Once the values of p  and v  arc fixed, then activity data are used to evaluate [24, 25] the values of interaction energies (g and W .^), The energy parameters evaluated for MgZn areg / R T  =  +2.65, W, 2 / /?T = 1 -  0.004 ~ 0.04, 3 / R T  = 0.15,W 2 3 //? r  =  - -0 5 0 .In the light of the complex formation energy, MgZn is a weakly interacting system [2,6,22,24] like AICa, AIMg, MgSn, AgAl and CuSn. Eq. (6 ) has been used to obtain equilibrium values of n  ^ and hence G^ via  eq. (5). The ti j-C  and curves for MgZn are displayed in the upper and lower parts ol Figure 1 respectively. The maximum value of occurs at "0.66. The computed values and observed values of [23] arc in reasonable agreement. It may be observed that the variation of free energy is not symmetrical about C = V2F ig u re  1. U p p er s id e  : N u m b er o f  co m p lex es  (n ,)  versus concen tra tion  of Zn (C^). L o w er side  : F re e  en e ig y  o f  m ix in g  ( G ^ T )  versus concentration o f  Z n (C ,^ ).Eq. (5) for the free energy of mixing can also be used to obtain an expression for the activity a^ for the atom type A as followsThermodynamic properties and electrical resistivity o f liquid MgZn alloys 521R T \n a ^  —( d G / ^  ! d N p^^  (8)(B = Zn).Eqs. (5) and (8 ) yieldIn a^  =  l - «  + ln(«, /r t2 ) + ln w + nsO^BZ/fT")+ n 2(W ,2/ / ? D -  I ’L v .K j IU T . ( „»<7/ In has been computed through (9) as a function of concentration. The computed values of activity of Mg in MgZn alloy are given in Table 1 along with the experimental observations [23], Table I shows that theoretical and experimental values are in reasonable agreement.T able 1. T h e rm o d y n a m ic  a c tiv ity  a o f  M g Z n  liq u id  a llo y s  at 923K .C a lc u la te d O b se rv e d  [23]0 .1 0 .0 0 7 0 .0 1 50 .2 0  031 0  0 5 30 .3 0 .1 1 9 0 1210 4 0 .2 8 1 0  2 1 70  S 0 4 5 2 0 3 3 60 .6 0 .6 0 4 0 4 7 30 7 0  73 1 0 6 1 30 .8 0 .8 3 6 0  7 4 70  9 0  9 2 2 0 8 6 64. Long-wavelength concentration correlation function and chemical short-range orderrhe long-wavelength limit of the three correlation functions, namely the concentration-concentration correlation functions the number-number correlation functions and thenumber-concentration correlation functions have beenwidely used to understand the stability [5,6,25,26) of the binary mixture. The experimental values of S^^O) can readily be derived trom the thermodynamic functions.n,p,NScc(0) = RT( d^G, ^ / dc \ ) ^= {i-CA)aAi^ay^ IdC^y'r.p.N  = Q  aaidag /  <9(1 -  . ( 10)Obviously 5cc(0) can be obtained directly from the observed activity data following the last two equalities o f ( 1 0 ) or may be obtained theoretically with the help o f (5) and (10)t 12^CC(O) = i H f - n ' ^  l n  +  ( 2 n / R T ) ^  " i” ) %/=1 i<jwhere a prime denotes differentiation with respect to ,n' = +/IJ +/ij .n ' i = l - p n \ ,/Ij =  -1 _ U ; j '  ,n \ ^ P ! Q ,P  = / ;  /  n, -  u / /i2 -  (/r + 1» -  1)( 1 /  )((/»2 -  «i )W,2 / / ( r+(n, / R T ) (Wi, -  W,2)) -  (Oi + u)iy,2 + Wi3 -  Wj,) (I / n R T \G  =  ( ( / i  +  U - l ) ( y ,  +  V2 +  V ,) +  JU.UW,2 -U.VV23).l2 ! n R T )  + p^  ln ^ + v ^  l n 2 + \ l n ^ - ( p A - v - \ f  In .  (12)The values of computed for MgZn liquid alloyfollowing eq. (1 1 )  arc shown in Figure 2. These are compared with the values obtained directly from the observed activity data [231 using the last two identities in ( 10) and with those computed by Ramchandra Rao e ta !  [26]. It is clear from Figure 2 that the values of (0 ) obtained from activity data and those by Rao e t a l are always lower than 5 /^(0 ) =C^C^). In contradiction of these two values, our computed values of are higher than upto < 0.28. Fora  0.28 , 5^,^0) is smaller than 5^,^/''(0). The segregation (like atoms pairing) is expected upto the region < 0.28 since ^ cc  ^ After Cy^ = 0.28, computed values of <from which existence of chemical ordering leading to complex fonnation (unlike atoms paring) is expected. It may be .seen from Figure 1 that the fraction of chemical complexes is a maximum at = 0.66 but a shoulder in in Figure 2,(11)F ig u re  2 . C on cen tra tio n  flu c tu a tio n s in  th e  lo n g -w av e len g th  lim it (.S'^^(0» an d  ch em ica l sh o r t- ra n g e  o rd e r  ( tti ) versus c o n c e n tra tio n  o f  Zn(CyJ.522 N Jha, A K Mishra, S M Rafique and A N Singhslightly shifts towards larger Zn content at = 0.75. It is because the position of the minimum in (0) depends collectively on and W^j.The non-resemblence of computed values of 5^,^0) with those obtained directly from activity data, may be attributed to the inaccuracies involved in the numerical differentiation of the activity data, particularly at the two ends of . The activity of one of the components becomes negligibly small at the either end of the scale, which creates problems in the numerical differentiation.The Warren-Cowley [27,28] short-range order parameter a ,  for MgZn liquid alloy has been computed to quantify the degree of order, a ,  can be determined experimentally for and Sy,^(g) but it is difficult to determine these parameters for all kinds of binary alloys [25, 29, 30]. a^ can be obtained theoretically [5,25] asa ,  = ( 5 - l ) / ( 5 ( Z - l )  + l),S  =  S c c ( 0 ) / 5 c c ' ^ 0 ) ,(13)(14)Z is the coordination number, which is taken as 12 for our purpose. The computed values of computed through (13) has been shown in Figure 2. The positive value of ( a ,  >0) upto < 0.3 refers to like atoms pairing and negative value of a , ( a  j < 0) for > 0.3 refers to unlike atoms pairing as nearest neighbours. The minimum value of a ,  occurs near = 0.7. This indicates that Zn-rich end is more ordered than Mg-rich end.5. Electrical resistivityWe can express the electrical resistivity of a ternary mixture as [18]1R=^(l2n.Qs) f ( t ie^.V^)jd(q/2kF)(q/2kF)^0X ( X , y , ^ ^ )  Sii(q) + 2(XiXj)' '^V,(q)Vj(q)Sij{q)).  (15)Wfiq)  = -^ T ^ c o s  ( q  r„)- ^ ^ ( s i n  (q r„ ) q  £2 £2q'- q r „  co s  ( q r „ ) ) (16)has been considered. Here r  is the model radius, is them owell depth, e  * (q )  is the modified Hertree dielectric function given by€*(9) = l + ( e ( g ) - l ) ( l - G ( ^ ) ) . (17)The Vashishta and Singwi [31 ] form of exchange G {q) has been used for correlated motion of the conduction electrons.S^fq) are the partial structure factors which have been obtained following the work of Hoshino [32]. The hard sphere diameters <T, , 0*2 occur as the important ingradients. Those a |  ,(72 have been taken which reproduce the first peak of structure factor of pure liquid Mg and Zn at their respective melting points. These diameters arc taken as concentration -  independent. The hard sphere diameter of the complex a   ^ is taken as a parameter [18]. The scattering potential V^(q) for the complex has been estimated as [ 18]fI-¥V ( IS )since V^iq) can not be obtained from first principle as the complex has flexible structure.The resistivity of MgZn liquid alloy calculated through (15) has been depicted in Figure 3 along with the experimcnial observation by Walhout et a l  [33]. There is a fair agreement between the theoretical and experimental values except for puic80.0!\— RIThwith]0.1005 nm- - - R I Th. with]0.0 0 5 1.0Here, is the Fermi velocity, / n the scattering volumeof the alloy in which free electrons travel, q  the phonon wave vector, Vi (q)  (i = 1,2) = W ^iq)/ e  *{ q)  are the screened form factors of Mg, Zn and V3(^) is the screened form factor of the complex MgZn^. X.  = n^/n e t c  are the concentration functions of different scattering centers, k .^ is the Feri wave vector.In order to compute form factors of Mg and Zn,  the Heine- Abarenkov model potentialFigure 3. Electrical resistivity (R) v e r s u s  concentration of Zn (C ,^JMg and Zn. To see the effect of CT3 , the two values of resistivity were calculated with <73 =0.1005 nm and ^3  = 0.1058 nm and the results are shown in Figure 3. It is found that the rise in thevalue of CT3 notonly increases the electrical resistivity of MgZn liquid alloy, but the maximum shifts towards Mg rich end also The maximum value of resistivity comes at stoichiometric composition. The difference between observed and theoretical resistivity of pure Mg and Zn may be due to the fact that wcThermodynamic properties and electrical resistivity o f liquid MgZn alloys 523have not taken into consideration the non-local contribution in the calculation o f form factors. Also, Zn is the metal which gives most trouble due to d-bonding. There is also a large mismatch at intermediate concentration. Specially, the experimental curve shows no maximum as in the case o f theoretical curve, but only a slight hump. By considering the presence of another complex MgjZnj along with MgZn^ [14] in the intermetallic phase and a phase shift expansion o f form factor V{q) for Zn [34], the mismatch between the computed and experimental values may be minimized.6. Conclcusion(a) In the concentration range 0 ^  Zn< 0.28, the lik^i.e., M g-M g and Zn-Zn atoms arc pairing as nearest! neighbours.Both and a , suggest that segregation-order transformation occurs around 30 at % of Zn. Thus, MgZn liquid alloy is a glass former for a certain range o f composition.exhibits only a weak deepening around the concentration M gZn2. Here small values of the size effect 1.52 and electro negativity 0.0392 arc not responsible for producing asymmetry in 5 ^ ^ 0 )  and(b)(c)a 1 •(d)( 0It is also observed from the low value of electro negativity i.e., 0.0392 as compared with e.g. 0.51 for CsAu. that MgZn posses.ses weaker ionic character.The hard sphere diam eter of the complex plays an important role in the calculation of partial structure factors and thereby inelectrical resistivity.AcknowledgmentThe financial support by the Council o f Scientific & Industrial Research, New Delhi is gratefully acknowledged by A. K. Mishra, The authors are thankful to Dr. R. N. Singh, Prof, of Physics, S. Q. University, Oman for encouragement and Dr. B. B. Sahay, Retd. Prof, o f Physics, B. C. E., Bhagalpur for useful discussions.References[1] R N  Singh and F  Sommer R e p . P r o g . P h y s . 59 1 (1996)[2] L C Prasad and A Mikula J . A l lo y .s  a n d  C o m p o u n d s  299 I7.S(2000)[.t] W Vander Lugt and W Geensma J. N o n - C r \ s i .  S o l id s  61 & 62 187 (1984)(4j V N Singh, L C Pni.sad and M Jha J  In d ia n  C h e m  S o f  76 .18.1 (1V99)f.S| N Jha, S M Rafique, A K Mishra, R N Singh and V N Singh In d ia n  J  P h v s  74 A 461 (2000)16J L C Prasad, S K Chatleqee and V N Singh P h m e u  B217 285(1996)171 R Agrawal, S G Fries. H L Lukas, G Petzow, F Summer, T G Chart and E Effenbcrg Z, M e tu l l 83 216 (1992)18] J Hafner J  P h y s  F18 15.1 (1988)[9] H J Giintherodt and H U Kunji M e ta llic  G l a .w s  ed J J. Gilman, H J Leamy (American Society of Metals, Ohio) p 247 (1978)(10] J Hafner J . N o n -C r y .u .  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Thermodynamic Properties and Electrical Resistivity of Liquid MgZn Alloys

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Thermodynamic Properties and Electrical Resistivity of Liquid MgZn Alloys

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