Charactization of Point Defects in B2 NiAl

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Charactization of Point Defects in B2 NiAl

Charactization of Point Defects in B2 NiAl

Saluja S., M.Sc. (Physics), GEC, Panipat

I.INTRODUCTION

B2 alloys often exist in exist over range of compositions on either side of stoichiometric composition [3]. The structure of B2 type is body centered ordered structure where, say A, atom occupies the body centered position and say B atom, occupies the corner site, as is shown in figure 1.It is the most common ordered structure with a simple cubic Bravais lattice. The B2 structure has two atoms per unit cell and thus can be described as two interpenetrating simple cubic sub lattices.

Figure .1.1a: Structure of B2 ordered compound

On the other hand, the disordered structure of the type B2 is shown in figure 2 consisting of two identical lattice points, one at the corner position and the second at the body centered position[1][2].

In contrast with ionic compounds, which normally exist at the stoichiometric composition with limited ranges of homogeneity, most of the B2 intermetallic compounds exist over a wide range of homogeneity[9].

The mechanical properties of B2 compounds are dominated by the presence by the presence constitutional defects, i.e. constitutional vacancies and antisite defects [3].

Defects In B2 Intermetallic Compounds

The homogeneity is retained to high temperatures at even large deviations from the stoichiometric composition, the defect formation in intermetallic compounds is much more complex. Considering a binary inter metallic phase A0.5-xB0.5+x, where A atoms occupy the sublattice-1 and the B atoms occupy the sublattice-2 [1][2]. Six mechanisms of defect formation may occur:

  1. Substitution of B atoms on the sublattice-1 (called B antisite)

  2. Substitution of A atoms on the sublattice-2 (called A antisite)

  3. Creation of vacancies on sublattice-1.

  4. Creation of vacancies on sublattice-2.

  5. Formation of A interstitials.

  6. Formation of B interstitials.

    The overall amount of the defects is the sum of the numbers of defects caused by finite temperature and by the deviation from the stoichiometric composition[1][2]. The interactions among point

    defects in inter metallic compounds are generally small; therefore vacancies and antistructure atoms are

    expected to be more or less independent of each other[7][8].

    Thermal defects in an ordered binary alloy of a fixed composition must appear in a balanced manner in order to preserve the stoichiometry of alloy. Thermal disorder in the B2-type intermetallics is often of a triple defect type[39] and the triple defect formed by two Ni vacancies and one antisite Ni atom is considered to be the dominant thermal excitation in NiAl[10][11]. Here the ab initio calculations of the electronic structure and total energy of NiAl intermetallic compounds containing four types of point defects in different combinations and spatial configurations in order to find the defect formation enthalpies and pair defect interaction energies will be studied on the basis of the values the constitutional defects and the statistics of thermal defects.

    1. GENERAL FORMALISM

      1. Alloy Configuration

        The general B2 NiAl has been considered having the fixed atomic composition Ni1/ 2 Al1/ 2 , where is the deviation from stoichiometry. The alloy components( Al and Ni atoms) and vacancies, i={Al, Ni, V} occupy N lattice sites on the two sublattice (={Al, Ni}) of the B2 structure. Each sublattice has

        N 1 / 2N sites occupied by ni atoms or vacancies.

        In the canonical ensemble[5] the number of Al atoms

        Fig 1.1: three dimensional domain of possible configuratioISnsBfNo:r 978-93-83758-09-8

        the binary off-stoichiometric alloy.

        Points A or V corresponds to the maximally ordered

        n Al n Al

        n Ni n Ni

        • n

        Al

        Al Ni

        • n Ni

          and the number of Ni atoms

          as well as the total number of

          state of the alloy havinganti-site atoms or vacancies.

          Alternatively, the atomic ordere can be specified by the three independent variables: the net concentration

          Al Ni

          lattice sites may vary. The distribution of alloy components between the sublattices may be described

          of vacancies and two long-range order (LRO) parameters

          i

          in terms of site concentration c

          ni / N . However

          xV

          x x ,

          V

          V

          N i A l

          , since the number of lattice sites is not conserved in the

          N i

          presence of vacancies, it is more convenient to use atomic concentrations defined with respect to the total

          N i 2 ( x N i

          x ),

          N i

          A l

          (2.4)

          A l

          2 ( x A l

          • x A l ).

          number of atoms

          n

          Nat n Ni +nAl :

          A l N i

          The atomic concentrations of the alloy components

          x i

          i

          Al Ni

          N a t

          can be uniquely expressed through these variables as

          follows:

          (2.1 If necessary, one can easily transform between the atomic concentration and site concentration using the following relation:

          4xAl

          2x Al Al , 4xAl

          2xAl Al,

          c i

          2 x

          i

          ( 1

          x V )

          4xNi 2x Ni Ni , 4xNi 2xNi Ni ,

          Al Ni

          4xV 2xV Al Ni , 4xAl 2xV Al Ni

          With these definitions one can have six atomic

          concentrations and three constraints:

          i

          i

          x A l x N i ,

          i i

          Al Ni

          (2.5)

          The processes can be expressed in the form of defect reactions:

          x x 1 ,

          (2.2)

          2

          A l A l

          0 V N i

        • V A l ,

          x x

          1 ,

          V V N i ,

          N i

          N i 2

          A l N i A l

          The problem is to find out four concentrations of point defects: two antisite atoms, d={NiAl, AlNi}, and

          V N i

          V A l

          A l N i

          to vacancies d={VNi, VAl}, as a function of temperature T. according to equation 2.2 only three of the the four defect concentrations are independent:

          1

      2. Wagner-Schottky Model

        The equilibrium state of the alloy at temperature T and Pressure p is determinedby the minimum of the gibbs free energy. Here the pure fcc Ni and Al have

        x A l

          • x N i

            • ( xV

          • xV

            )

            been considered, therefore, the energy of alloy

            N i A l

            2

            (2.3)

            N i A l

            formation is defined as

            E ENi Al1 xNi ENi xAl EAl ,

            12 2

            (2.7)

            It is well known that the enthalpy of formation and lattice parameters[10][11] of NiAl are essentially a linear functions of alloy composition. The Wagner- Schottky Model rests on two assumptions:

            1. The enthalpy of alloy formation depends linearly on the defect concentrations,

              H H

              N i A l

              d

              H d x d

              (2.8)

            2. Only the mean-field configurational entropy is taken in to account

      S (1 x ) ln 1 xV x

      ln x

      , (2.9)

      V 2

      i i

      i

      Where i={Al, Ni, V} and ={Al, Ni}. The parameters of the Wagner-Schottky Model are the

      In practice it is common to lineraize the dependence of the defect formation enthalpy at p=0

      defect formation enthalpies

      H H

      d

      Hd Ed pd

      x

      x

      1

      (2.11)

      t

      x

      (2.10)

      xd

      t t

      A l N i N i A l 2

      t

      x

      V N i

      V A l 0

      Where Ed is the defect formation energy and d is the defect formation volume which may be obtained from the results of ab initio calculations performed at T=0 , p=0:

      (2.15)

      compositionconserving combination of single point defects obeying the condition:

      Ed

      (2.12)

      E

      xd

      , d

      xd

      It follows from the above equation that none of the point defects can be alone thermal defects. The simple composition conserving defects are discussed here:

      The minimization of the Gibbs free energy in configurational space ( xV , Ni , Al ) in the Wagner-Schottky Model leads to the following equations for equilibrium defect concentrations:

      4 xV x V

      Al N i exp[ H / T ], (2.13a)

      1. General case:

        There are four composition conserving defects consisting of two types of point defects[13](64) which can be either thermally generated in the

        maximally ordered alloy of any composition including the defect free stoichiometric alloy. The

        (1 xV

        )2

        N i V

        x x

        A l N i

        x x

        D

        exp[ H JN

        / T ],

        defect reactions are as follows:

        Exchange antisite defect (X),

        t t

        (2.13b)

        N iN i

        V A l

        0 NiAl AlNi : xX

        xNi

        x

        Al

        Al Ni

        (2.16)

        x A l x V

        Divacancy or Schottky defect (D):

        N i A l e x p [ H / T ] t t

        A l

        x A l xV N i

        J A

        2.13c)

        0 VAl VNi :

        xD xV

        x

        V

        Al Ni

        (2.17)

        Here HD, HJA and HJN are the enthalpies of the defect reactions (2.6) which are connected with the

        Triple Ni r Simply triple defect (TN):

        0 2V Ni : x 1 xt xt

        (2.18)

        formation enthalpies of four point defects by the following equations:

        D V V

        H H H ,

        Al Ni

        Ni Al TN

        Triple Al Defect (TA):

        2 VNi

        1 t t

        NiAl

        0 2VAl AlNi : xTA xV xAl (2.19)

        H JN H Ni

        H JA H Al

        • HVNi

          Al

        • HV

        • HVAl ,

        • HV .

          (2.14)

          2 Al Ni

          The formation enthalpies of a composition conserving defect HCD for the exchange, triple Ni and

          Ni Al Ni

          Expressed in terms of site concentrations, the set of equations (2.13) take the usual form of mass action

          triple Al defects is given by:

          law for the quasichemical reactions (2.6).

          H X H Ni

          • H

          Al

          Al Ni

          H JN H JA,

          Thermal Defects

          HTN H Ni 2HV H JN H D,

          (2.20)

          Al Ni

          The various thermal defects have been discussed here. In addition to the the constitutional defects the

          HTA H Al HV

          H JA H D,

          Ni Al

          thermal defects at finite temperature also added to the point defect concentration. Since the alloy composition is fixed they can appear only in the

          The formation energy of the composition-conserving

          defect is then the sum of the formation energies of the constituent.

      2. Off- stoichiometric Alloys

      The presenceof the constitutional defects in the maximally ordered atate of an off-stoichiometric

      alloy gives an additional possibility to satisfy the equation (2.15) The process of the formation of such defect may be considered as a transition from stable

      state to a unstable branch, and

      hence called

      ISBN: 978-93-83758-09-8

      concentrations of thermal point defects are as follows

      interbranch defect. The defect reactions and the Interbranch Ni Defect (IN):

      t 1 t

      5 1 2

      NiAl 2VAl :

      xIN xNi

      xV (2.21)

      Al

      2 Al

      xTN 2

      2 exp[HTN / T ]

      Interbranch Al Defect (IA):

      1 2

      2V Al : x xt

      xt

      (2.22)

      And

      x 23 exp[H

      / T ]

      (2.28)

      Ni Ni IA VNi

      2 AlNi

      IA 1 2 IA

      The formation enthalpies are given by: Depending on the assumption whether interbranch Al

      H IN 2HV

      H IA H Al

        • H

          Ni

          Al Al

      • 2HV

      H D H JN ,

      H JA H D .

      (2.23)

      defect dominates or interbranch Ni defect dominates..

      if the formation enthalpy of an interbranch Al defect is sufficiently small, one can expect a competition

      Ni Ni

      They characterize the relative stability of the antisite and vacancy branches in the Ni-rich and Al-rich NiAl, respectively. Since the antisite and vacancy branches in Al-rich B2 transition metal aluminides are competitive in energy, the interbranch Al defect may become the lowest-energy thermal defect in this compositional region

      Thermal Defect Concentration

      between triple and interbranch Al defects in the Al- rich NiAl.

      The effective formation enthalpy of the composition conserving defect can be derived from the equations 2.25- 2.28. and is given by and is based on Arrhenius formula:

      H

      d ln xt

      In this section the thermal defect equations will be given which were

      concentration derived using

      eff

      CD

      CD

      dT 1

      pconst

      (2.29)

      mass action law. Here the triple defect is considered as dominant thermal defectin stichiometric NiAl. The

      The formation enthalpies of

      single point defect

      defect concentration in the maximally ordered are:

      found for the nearly stoichiometric and off-

      stoichiometric alloys expressed t rough the formation enthalpies of composition-conserving defects.

      c ,

      0 c

      0,

      0

      xNi

      , xV

      Al

      (2.24)

      0,

      0

      Ni 2,

      0

      For nearly stoichiometric alloy(NiAl), the equilibrium concentration of triple defcts in the stoichiometric is given by:

      x 25/3 exp[H / 3T ].

      TN TN

      (2.25)

      dxTN

      d

      0

      1 / 3,

      -2/3,

      <0,

      0.

      (2.26)

      For concentrated Ni-rich NiAl the estimate of the concentration of thermal triple defects is given by:

      x 25/ 2 ( )1/ 2 exp[H / 2T ] (2.27)

      TN TN

      Whereas for Al-rich NiAl,

      The method of calculation used here is the a 54-site

      The ratio RTN

      ISBN: 978-93-83758-09-8

      should be close to one if the dominant

      cubic supercells to simulate isolated defects, and 108 site supercells to evaluate defect interactions.

    2. RESULTS

The thermal defects can be interpreted as composition conserving defects. The calculated equilibrium atomic concentrations of defects, xd , in NiAl at 1300K and zero external pressure is shown here in the figure 1.4(a) as a function of deviation from stoichiometry.

Here it has been seen that thermal defects are mainly vacancies on the Ni sublattice and antisite Ni atoms, which are the constituents of triple defect. On the Al- rich side, the concentration of thermal antisite Ni atoms quickly becomes very small with increasing

thermal excitations are of triple defect type, while RIA should be close to one if the dominant thermal excitations arethe interbranch Al defects. Thus we find that the thermal excitations in NiAl at 1300K are mainly of triple defect type for deviation >0.023.

At low temperatures, triple defcts dominate for the stoichiometric and Ni-rich compositions, whereas interbranch defects decome dominant for the Al-rich compositions. Fig 2.7 shows that the region in which triple dfect dominateexpands towards Al-rich compositions with increasing temperature.

The cross-over concentration B can be estimated and given as:

1

off-stoichiometry. But antisite Al atoms appear in a relatively large amount.B

exp[(H IA HTN ) / 4T ] (2.31)

4

In summary, the calculations here show that the onset of defect interactions at low temperatures does not affect the staistics of the constitutional defects in NiAl but leads to a defect ordering which is consistent with experimental observations. At 1300 K the equilibriumconcentration of Ni vacancies in Al- rich NiAl alloys with more than 52.3 at.% Al decreases with temperature. It has been shown in our previous work84 and independently by Meyer and Fa¨hnle22 that this kind of thermal behavior can be associated only with the interbranch Al defect, in which two constitutional Ni vacancies are replaced by one antisite Al atom.

Fig: 1.5 Ratios of the thermal defect concentrations at 1300 K

In fig.1.5 the calculated ratios between thermal defect concentrations have been shown

xt

Thus for Al-rich alloy the equilibrium vacancy concentration decreases only up to a certain temperature, below which the interbranch Al defects dominate over the triple defect, while above that temperature the vacancy concentration starts to increase because the triple defect become dominant for this alloy composition.

Fig 1.6: Boundary between two regions of NiAl in which triple defect(TN) or interbranch Al defects (IA) dominate respectively( solid line).

It has been found that the ground state of off- stoichiometric NiAl alloy is formed by antisite Ni atoms on the Al sublattice in Ni-rich NiAl and by

vacancies on the Ni sublatticein Al-rich NiAl. According to the the calculated defect interactions,

RTN

RIA

VNi ,

t

2x

NiAl

xt

2x

t

VNi .

AlNi

(2.30)

the constitutional defects of the same kind must form ordered structures in the ground state, in which they tend to avoid each other at the shortest possible distance on their sublattice.

The dominant thermal defects are shown to be triple defects in Ni-rich and stoichiometric NiAl and

interbranch Al defects in Al-rich NiAl. Since an interbranch Al defect annihilates two constitutional vacancies and creates only one antisite Al atom, the equilibrium amount of vacancies as well as the total number of point defects must decrease with temperature in Al-rich NiAl at low temperatures. The calculated defect interactions indicate that all three constituents of a triple defect may exist individually in NiAl without forming a bound complex. In contrast, a strong attractive interaction was found for the following defect pairs in NiAl: Ni on the Al sublattice and Al on the Ni sublattice, and vacancy on the Al sublattice and Al on the Ni sublattice. Although the defect interactions are shown to have only minor effects on the equilibrium defect concentrations, they may be of importance for understanding the kinetic processes and metastable defect arrangements in NiAl.

REFRENCES

[1]. James K Wessel and Sikka Vinod, The Handbook of advanced marterials, John Wiley & sons, 2004.

[2]. Wolfgang pfeiler, AlloyPhysics, 200

[3]. I.Baker, Material Science & Engineering A192/193, pp1-13, 1995

[4]. Chaturvedi D.K, Allnatt A.R, Philosophical Magazine A, 69, 821, 1994.

[5]. Belova I.V, Murch. G.E, Philosophical Magazine, 151, 46, 2002.

[6]. Gosain K. L, Chaturvedi D.K., Belova I.V. & Murch G.E., Defect & Diffusion forum, Trans Tec publication, pp 247-248, 2005, 2006.

[7]. Flynn C.P., Point Defects and Diffusion (Oxford: Clarendon), 1972.

[8]. De Groot SR & Mazur P, Non Equilibrium Thermodynamics (Amsterdam: North Holland), 1962.

[9]. C.R.Kao, Sungtae Kim, Y.A.Chang, Material Science & Engineering A 192/193, pp 965-979, 1995.

[10]. Noebe, R.D., Bowman, R.R., and Nathal,

M.V., Int. Mater. Rev. 38, 193, 1993

[11]. Bardley.A.J., Taylor.A., Proc.R.Soc.London,

Ser A 173, 232, 1939.

[12]. G.v. Hevesy and W. Seith-Freiburg, Diffusion in metallen, Z. Electrochem 37 , 528- 31, 1931

[13]. A. Fick, Uber diffusion, Progg. Ann. 94 (1855) 59.

ISBN: 978-93-83758-09-8

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