Calculation of phase equilibria and evaluation of glass-forming ability of Ni–P alloys

Journal of Alloys and Compounds 282 (1999) 175–181

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Calculation of phase equilibria and evaluation of glass-forming ability of Ni–P alloys Jae-Hyeok Shim, Hun-Jae Chung, Dong Nyung Lee* Division of Materials Science and Engineering, Seoul National University, Seoul 151 -742, South Korea Received 13 March 1998; received in revised form 28 August 1998

Abstract A thermodynamic evaluation of the Ni–P binary system has been made by employing the calculation of phase diagram (CALPHAD) method. A set of the thermodynamic parameters describing the individual phases of this system has been obtained from experimental information available in literature. Phase equilibria and thermochemical properties of this system are calculated using the obtained parameters. Moreover, the glass-forming ability of this system has been evaluated based on a simple kinetic model. The calculated glass-forming ability as well as the calculated phase diagram show good agreement with experimental data in literature.  1999 Elsevier Science S.A. All rights reserved. Keywords: Ni–P alloys; Phase diagram calculations; Glass-forming ability; Kinetic model

1. Introduction The Ni–P system is a typical alloy system which forms an amorphous phase by the reaction between metals and metalloids such as P, B, C and S. Ni–P alloys can form a amorphous phase by electroplating and electroless plating [1,2] as well as melt quenching. Especially, Ni–P alloys films obtained by electroless plating have been widely used for corrosion-resistance, wear-resistance and non-magnetic coatings owing to their superior properties. An amorphous phase is regarded as a state which is of short-range order like liquid. It is thermodynamically metastable, since its Gibbs energy is higher than that of an equilibrium phase. Though a melt-quenching method was first attempted to obtain amorphous phases, solid-state reactions such as mutual diffusion in diffusion couples, mechanical alloying and incidence of high-energy beam can also be utilized. Recently, much research into amorphous thin films obtained using vapor deposition, electroplating and electroless plating has been carried out. Though there are a variety of ways to obtain amorphous phases, the structural and thermodynamic properties of the formed amorphous phases are known to have no relation to their fabrication methods [3]. *Corresponding author. Division of Materials Science and Engineering, Seoul National University, Shinrim-Dong, Kwanak-Ku, Seoul 151-742, South Korea. Tel.: 182-2-8772808; fax: 182-2-8559671.

Great interest of materials scientists has been focused on GFA (glass-forming ability) of alloys for a long time, since a variety of fabrication methods of amorphous alloys were known. Analyses on the GFA of alloys have been attempted by means of thermodynamic factors such as the Gibbs energy of supercooled liquid, chemical short-range order and glass transition temperature, kinetic factors such as nucleation and growth rates and fabrication variables such as a critical cooling rate. Moreover, eutectic composition of an alloy, negative enthalpy of mixing, difference of atomic radii, a T 0 criterion and valence electron concentration have been proposed to explain the GFA [4]. The evaluation of the GFA of alloys is of great importance by judging in advance whether amorphous phases could be formed at given conditions. Though there are a number of evaluation methods of the GFA, most methods, which are empirical based on experiments, have limits in quantitative analyses. T 0 means the composition on which liquid and solid phases in alloys have the same Gibbs energy values at a given temperature. The composition range, within which amorphous phases can be formed, is the interval between two T 0 compositions in a purely thermodynamic aspect [5]. Oh et al. [6] successfully predicted the GFR (glass-forming range) of the Ti–Ni system using the T 0 criterion. Though this approach could be easily applied to alloy systems by using the Gibbs energy expressions evaluated based on thermodynamic modeling, it could not provide important

0925-8388 / 99 / $ – see front matter  1999 Elsevier Science S.A. All rights reserved. PII: S0925-8388( 98 )00826-3

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information such as a critical cooling rate as a fabrication variable. Recently, some researchers have made attempts to evaluate the critical cooling rates necessary for amorphous phase formation by combining thermodynamic and kinetic approaches. Uhlmann [7] introduced isothermal transformation kinetics involving a nucleation and growth process and calculated TTT (time–temperature–transformation) curves to estimate the cooling rates necessary to escape crystallization on solidification. Saunders and Miodownik [8,9] evaluated the GFA of some alloy systems by combining isothermal kinetics and thermodynamic modeling, which could be used to determine the driving force on solidification and the Gibbs energy barrier to nucleation. The recent development of thermodynamic modeling and computer calculation of phase diagrams has made it possible to obtain the precise expressions of the Gibbs energies of the phases in alloy systems. This combination method has easiness in prediction of the cooling rate necessary for amorphous phase formation using a few variables and in extension to multicomponent systems as well as binary systems. In the present work, the thermodynamic model parameters of the individual phases of the Ni–P system have been evaluated from the phase diagram and thermodynamic information in literature using the calculation of phase diagram (CALPHAD) method [10] to understand more precisely the thermodynamic behavior of the Ni–P system and to provide the useful information on the Ni–P system. Since the CALPHAD method enables us to evaluate with ease the driving forces of evolution of metastable phases and effects of temperature and composition on phase equilibria as well as to calculate phase diagrams of multicomponent systems [11,12], it has been actively applied to a variety of fields. The present work has been undertaken to evaluate the GFA of the Ni–P system using the thermodynamic property assessed here and the kinetic model proposed by Saunders and Miodownik [8,9]. The results of the present work are expected to be utilized as basic information in alloy design and evaluation of the GFA of multicomponent alloys.

2. Thermodynamic models The Ni–P system with x P ,0.4 consists of liquid, f.c.c. (Ni), Ni 3 P, aNi 5 P2 , bNi 5 P2 , gNi 12 P5 , dNi 12 P5 and Ni 2 P. Though several intermediate phases have been reported in the region with x P .0.4, a concrete phase diagram has not been determined yet [13]. Ni 5 P2 and Ni 12 P5 are divided into the high temperature phases a and g and the low temperature phases b and d, respectively. A regular type solution model has been applied to describe the liquid and f.c.c. (Ni) phases. The molar Gibbs energy of solution can be expressed as follows:

G 5 x oNi GNi 1 x oP GP 1 RT(x Ni ln x Ni 1 x P ln x P ) 1 mo G 1 x Ni x P LNi,P

(1)

where o Gi , called the lattice stability, denotes the Gibbs energy of the component i at hypothetical non-magnetic state. The lattice stability of pure elements has been critically reviewed and reported by SGTE (Scientific Group Thermodata Europe) [14]. The lattice stability values have been accepted without modification in the present work. mo G denotes the Gibbs energy contribution due to magnetic ordering proposed by Inden [15] and Hillert and Jarl [16] and LNi,P , which contributes to excess Gibbs energy of mixing, denotes the interaction between Ni and P atoms. LNi,P can be expressed as a Redlich– Kister polynomial [17], which is a function of composition and temperature. LNi,P is the parameter to be evaluated in the present work. Since the intermediate phases Ni 3 P, aNi 5 P2 , bNi 5 P2 , gNi 12 P5 and dNi 12 P5 show negligible solubility, the stoichiometric compound (line compound) model has been applied to them. Though the Ni 2 P phase has been known to have considerable solubility [18], it is regarded as a stoichiometric compound for lack of systematic and quantitative investigations into the solubility. The Gibbs energy of the stoichiometric compound with the composition ratio of p:q( p1q51) can be expressed as follows: G 2 P o G f.c.c. 2 q o G red Ni P 5 a 1 bT

(2)

The right hand side of Eq. (2) represents the Gibbs energy of formation, referred to f.c.c. Ni and red P, assuming the Neumann–Kopp’s rule is valid. a and b are constants to be evaluated in the present work.

3. Review on phase diagram and thermodynamic information The first investigation into phase equilibria of the Ni–P system was made by Konstantinow [19]. Konstantinow [19] determined liquidus and solidus lines of alloys with 0,x P ,0.34 using metallography and thermal analysis by heating pure Ni and red P in a crucible at 1 atm. Several invariant reactions in the Ni–P system were discovered in his experiments. Yupko et al. [20] made the most systematic investigation into the Ni–P system using DTA (differential thermal analysis), metallography and X-ray diffraction. In the composition range 0,x P ,0.36, they measured first the temperatures, at which the a–b transformation of Ni 5 P2 and the g–d transformation of Ni 12 P5 occur, as well as determined more precisely melting temperatures of alloys and temperatures of eutectic and peritectic reactions. The results of the investigation by Yupko et al. [20] are in good agreement with those by Konstantinow [19]. Recently, Kawabata et al. [21] measured the liquidus line of the f.c.c. (Ni) phase by means of

J. Shim et al. / Journal of Alloys and Compounds 282 (1999) 175 – 181

thermal analysis. The liquidus line reported by Kawabata et al. [21] are slightly displaced toward the high-temperature direction, compared with that by Konstantinow [19]. Koeneman and Metcalfe [22] reported the solubility of P in the f.c.c. (Ni) phase at temperatures ranging from 500 to 9008C using metallography. Recently, Lee and Nash [13] summarized crystal structures and lattice constants of intermediate phases, existence of metastable phases and thermodynamic information as well as assessed the phase diagram of the Ni–P system by compiling a large amount of information which had been published in literature since 1908. The phase diagram assessed by Lee and Nash [13], which is based on the phase diagram reported by Yupko et al. [20], has been generally accepted until now. A few researchers investigated thermodynamic properties of the Ni–P system. Weibke and Schrag [23] measured the enthalpies of formation of Ni–P alloys at 903 K (6308C) by means of high-temperature calorimetry. Myers and Conti [24] reported the enthalpies of formation of Ni–P alloys at 298 K (258C) using the Knudsen-cell effusion method. The results by Myers and Conti [23] are in good agreement with those by Weibke and Schrag [24] within the range of error.

4. Evaluation of thermodynamic model parameters Though experimental data of an alloy system appear as a variety of types such as phase diagram data and thermodynamic data, all of them are determined by only the Gibbs energy values of the individual phases in the alloy system. The evaluation procedure of thermodynamic model parameters represents that the model parameters composing the Gibbs energy of each phase are quantified and optimized so that experimental information reported in literature could be reproduced successfully. Since the evaluation of the lattice stability values of pure elements has been already carried out and the values have been sufficiently examined for a variety of alloy systems, the

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evaluation in the present work has been limited only to the interaction parameters of the solution phases composing the excess Gibbs energies of mixing and the Gibbs energies of formation of the intermediate phases in the Ni–P system. In the present work, the evaluation of the thermodynamic model parameters and the calculation of the phase diagram of the Ni–P system have been carried out using the computer softwares PARROT developed by Jansson [25] and Thermo-Calc developed by Sundman et al. [26], respectively. All the information mentioned in the previous section has been utilized to evaluate the thermodynamic model parameters. The excess Gibbs energies of mixing in the liquid and f.c.c. (Ni) phases were first derived using the liquidus line data [19–21] and the composition of f.c.c. (Ni) in the invariant reaction liquid5f.c.c. (Ni)1Ni 3 P assessed by Lee and Nash [13]. In the next stage, the evaluation of the Gibbs energies of formation of the intermediate phases Ni 3 P, bNi 5 P2 and dNi 12 P5 was made based on the liquidus lines [19,20], the invariant reactions [13] and the enthalpies of formation [23,24]. For aNi 5 P2 and gNi 12 P5 , the evaluation was made based on the temperatures of the a–b and g–d transformations [13] and the enthalpies of formation [23,24]. Then the parameters of the Gibbs energies of all the phase in the Ni–P system have been optimized to the final values at the same time considering all the available information. Table 1 presents the final values of the thermodynamic parameters of the Ni–P system evaluated in the present work (see also Table 2).

5. Calculation of glass-forming ability In order to evaluate the GFA of the Ni–P system, the present authors have chosen the model of the isothermal transformation kinetics involving nucleation and growth, which was proposed by Uhlmann [7] and later modified by Saunders and Miodownik [8,9]. This model incorporated

Table 1 Summary of thermodynamic parameters of the Ni–P system (J / g-atom) Liquid LNi,P 5 2381589.0179.1144T 1(104201.3290.7228T )(x Ni 2x P )178364.1(x Ni 2x P )2 f.c.c. (Ni) LNi,P 5 2176522.328.6002T Ni 3 P G20.750 o G f.c.c. 20.250 o G Pred 5 250383.123.4658T Ni aNi 5 P2 G20.714 o G f.c.c. 20.286 o G Pred 5 255392.725.0530T Ni bNi 5 P2 G20.714 o G f.c.c. 20.286 o G Pred 5 255292.725.1300T Ni gNi 12 P5 G20.706 o G f.c.c. 20.294 o G Pred 5 256690.624.9716T Ni dNi 12 P5 G20.706 o G f.c.c. 20.294 o G Pred 5 256590.625.0501T Ni Ni 2 P5 G20.667 o G f.c.c. 20.333 o G Pred 5 262790.424.4249T Ni

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Table 2 Summary of invariant reactions in the Ni–P system Reaction

Temperature (K)

Composition (x P )

Liquid5f.c.c.1Ni 3 P

1141 1143 1244 1243 1446 1443 1298 1298 1399 1398 1273 1273 1368 1373 1387 1383

0.190 0.19 0.230 0.235

Liquid1aNi 5 P2 5Ni 3 P Liquid5bNi 5 P2 bNi 5 P2 5aNi 5 P2 bNi 5 P2 1liquid5dNi 12 P5 dNi 12 P5 5gNi 12 P5 Liquid5dNi 12 P5 1Ni 2 P Liquid5Ni 2 P

H

J

0.286

0.311

0.294

0.294 0.320 0.31

0.294 0.333 0.333

1 ] 4

(3)

(4)

where fT 5 C exp(2EH /RT )

0.250 0.25 0.250

0.286

where t is the time taken to transform volume fraction X of crystalline solid, h is the viscosity of liquid, a 0 is an atomic diameter, f is a structural constant, Nv is the number of atoms per unit volume, G * is the Gibbs energy barrier to nucleation and Gm is the Gibbs energy driving force for the liquid–crystal transformation. The constants have been taken as X510 26 , a 0 50.28310 29 m, f 50.1 and Nv 55310 28 atom / m 3 . In order to apply the Eq. (3) to a real alloy system, h, G* and Gm for the alloy system should be derived. Since it is very difficult to measure experimentally the viscosity of supercooled liquid, there have been few measurements of it. Though Nishi and Yoshihiro [28] made measurements for the Ni–P alloys, the viscosity in the vicinity of the glass transition temperature is difficult to derive due to a high measured temperature range close to the liquidus temperature. The viscosity of supercooled liquid in the temperature range between the liquidus temperature T m and the glass transition temperature T g can be generally described using a Doolittle-type expression involving the relative free volume fT [29] such that

h 5 A exp(B /fT )

0.0032 0.0032 0.286 0.286

kinetic factors with thermodynamic factors. Thermodynamic inputs necessary for the kinetic equation can be derived from the thermodynamic parameters evaluated in the present work. The TTT curves on crystallization of liquid can be calculated using the following equation derived from the combination of the nucleation and growth equations with the Johnson–Mehl–Avrami kinetics [27] (see Ref. [7] for the detailed deviation procedure): 9.3h a 0 X exp(G * /KT ) ]]]]]] t 5 ]] ]] kT f 3 Nv [1 2 exp(Gm /RT )] 3

Reference

(5)

0.333

Present work [13] Present work [13] Present work [13] Present work [13] Present work [13] Present work [13] Present work [13] Present work [13]

where EH is the hole formation energy and A, B and C are constants. EH can be estimated by means of a direct relationship for T g [29]. In the present work, EH for Ni–P alloys has been approximated at 29 900 J / mol, assuming T g for the Ni–P system is 618 K (3458C) reported by Chen [30]. A and C have been approximated at 3.33310 23 and 10.1, respectively, assuming B is unity and fT and h are 0.03 and 10 12 Ns / m 2 , respectively at T g . The Gibbs energy driving force for the liquid–crystal transformation Gm is defined as the following equation: ] ] Gm 5 x NiG liquid 1 x PG liquid 2 Gcrystal (6) Ni P where x Ni and x P are the mole fractions of Ni and P in the ] ] crystalline phase, respectively, G liquid and G liquid are the Ni P partial Gibbs energies of Ni and P in the liquid phase, respectively, and Gcrystal is the Gibbs energy of the crystalline phase. The Gibbs energy functions in Eq. (6) can be obtained from the thermodynamic model parameters evaluated in the present work. The Gibbs energy barrier to nucleation G * can be described as follows: 16p G * 5 ]] (s 3m /G 2m ) 3N

(7)

where N is Avogadro’s number and sm the molar interfacial energy. sm is directly related to the molar enthalpy of fusion H fm such that

sm 5 a H mf

(8)

where a is a proportional constant. H fm can be obtained in a similar way to evaluate Gm . a has been empirically estimated to be 0.41 [9]. The critical cooling rate R c necessary for amorphous phase formation with a melt-quenching method can be evaluated from the TTT curve calculated using Eq. (3) as shown in Fig. 1. R c can be approximated as follows:

J. Shim et al. / Journal of Alloys and Compounds 282 (1999) 175 – 181

179

Fig. 1. Schematic diagram of a TTT curve on crystallization.

Tm 2 Tn R c 5 ]]] 5t n

(9)

where T n and t n are the temperature and the time at the nose of the TTT curve, respectively. Since the cooling rate calculated directly from the isothermal transformation curve is somewhat overestimated compared with that from the CCT (continuous cooling transformation) curve, the right side of Eq. (9) has been divided by a factor of 5 to emulate continuous cooling. In the composition range with R c ,1310 7 K / s, which hae been generally known to be a maximum available cooling rate for melt quenching, the amorphous phase formation may be possible.

6. Calculation results and discussion The phase diagram of the Ni–P system calculated using the thermodynamic model parameters (Table 1) evaluated in the present work is given in Fig. 2. The calculated phase diagram is in good agreement with the measured values reported [19–21]. The calculated liquidus line of f.c.c (Ni) shows better agreement with those by Yupko et al. [20] and Kawabata et al. [21] than that by Konstantinow [19]. The calculated phase equilibria in the composition range 0.2, x P ,0.4 show good agreement with the results of thermal analyses by Konstantinow [19] and Yupko et al. [20]. The dashed line in this figure denotes the T 0 line on which the Gibbs energies of the liquid and f.c.c. (Ni) phases are equal. The T 0 criterion previously mentioned implies that the amorphous phase can be formed in the composition range 0.113,x P ,0.25, which is the interval between T 0 and Ni 3 P at T g (618 K). However, since this prediction comes from a purely thermodynamic approach, the GFR will shrink to a narrow range if kinetic factors are involved. Fig. 3 shows the calculated solubility of P in f.c.c. (Ni). The result is in good agreement with that of Koeneman and Metcalfe [22]. The maximum solubility of

Fig. 2. Calculated phase diagram of the Ni–P system in comparison with experimental data [19–21].

P in f.c.c. (Ni) is x P 50.0032 at 1141 K (8688C). The calculated enthalpies of formation of Ni–P alloys at 298 K (258C) are given in Fig. 4, in comparison with the measured ones [23,24]. The calculated TTT curves of the liquid–f.c.c. (Ni) and liquid–Ni 3 P transformations at the eutectic composition x P 50.19 are given in Fig. 5. It can be seen that Ni 3 P first crystallizes from liquid. The nose of the TTT curve of the liquid–Ni 3 P transformation lies at 850 K (5778C) and 0.729310 25 s. In this case, the critical cooling rate necessary for amorphous phase formation is evaluated to be R c 58.03310 6 K / s from Eq. (9). Since this value is smaller than the maximum available cooling rate 10 7 K / s, the amorphous phase formation may be possible by means

Fig. 3. Calculated solubility of P in the f.c.c. (Ni) phase in comparison with experimental data [22].

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Fig. 4. Calculated enthalpies of formation of Ni–P alloys at 298 K referred to Ni (f.c.c.) and P (red) in comparison with experimental data [23,24].

of a melt-quenching method at this composition. The critical cooling rates necessary for amorphous phase formation in the vicinity of the eutectic composition calculated from Eq. (9) are given in Fig. 6. In the calculation of the cooling rate, the TTT curve of the first crystallized phase has been selected. R c has a minimum value near the composition x P 50.175, which is slightly off the eutectic composition. It should be noted that the simple kinetic model can explain the empirical phenomenon that amorphous phases can be easily formed at eutectic compositions of alloys. In this figure, the dashed line represents the maximum available cooling rate necessary for amorphous phase formation. Therefore, the range of the R c

Fig. 5. Calculated TTT curve of an alloy with eutectic composition on crystallization.

Fig. 6. Calculated critical cooling rate of Ni–P alloys necessary for the amorphous phase formation.

curve below the dashed line becomes the GFR of Ni–P alloys. In this case, the GFR lies in 0.161,x P ,0.192. Recently, Wachtel et al. [31] reported that the GFR of Ni–P alloys lies in 0.165,x P ,0.21, using a melt-quenching method. Fig. 7 compares the GFRs of Ni–P alloys calculated based on the kinetic model and the T 0 criterion and reported by Wachtel et al. [31]. While the GFR from the T 0 criterion as a purely thermodynamic approach appears very wide, the GFR from the kinetic model in the present work gives reasonable agreement with the experimental range [31]. However, the GFR from the kinetic model appears somewhat narrow in the composition range, in which Ni 3 P crystallizes first, compared with the ex-

Fig. 7. GFRs calculated based on in the kinetic model and the T 0 criterion in the present work and reported by Wachtel et al. [31] in the calculated Ni–P phase diagram.

J. Shim et al. / Journal of Alloys and Compounds 282 (1999) 175 – 181

perimental range. Since tetragonal Ni 3 P has a more complicated structure than f.c.c. (Ni), it is probable that the crystallization of Ni 3 P tends to be suppressed. Therefore, the experimental GFR of Ni–P alloys appears wider than the theoretical GFR in the crystallization range of Ni 3 P.

7. Summary The phase diagram and the thermodynamic property of the Ni–P system have been compiled and analyzed in terms of the thermodynamic models of the individual phases. The calculated phase diagram and thermodynamic property from the evaluated thermodynamic model parameters in the present work show good agreement with the experimental data. The GFA of the Ni–P system has been evaluated based on the simple kinetic model describing a nucleation and growth process. The calculated GFR gives reasonable agreement with the experimental range. The extension of these approaches to multicomponent alloys will provide useful tools in prediction of phase equilibria and GFRs of multicomponent alloys bearing the Ni–P system.

Acknowledgements This study has been supported by the Academic Research Fund of Ministry of Education, Republic of Korea, in 1997.

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