An improved atomic size factor used in Miedema’s model for binary transition metal systems

Chemical Physics Letters 513 (2011) 149–153

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An improved atomic size factor used in Miedema’s model for binary transition metal systems S.P. Sun a,b, D.Q. Yi a,⇑, Y. Jiang a,⇑, B. Zang a, C.H. Xu a, Y. Li a a b

School of Materials Science and Engineering, Central South University, Changsha 410083, China School of Materials Engineering, Jiangsu Teachers University of Technology, Changzhou 213001, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 11 April 2011 In final form 24 July 2011 Available online 27 July 2011

We improved the atomic size factor term that can be used in extended Miedema’s models for calculating formation enthalpies of binary transition-metal compounds. With this new prefactor, the effects of the atomic size difference between two dissimilar transition metals can be taken into account in a more insightful way. Calculations using the present model were compared with experiments as well as with the original Miedema’s model, showing that it can effectively improve the prediction precision of formation enthalpies. Ó 2011 Elsevier B.V. All rights reserved.

In recent decades, Miedema’s model [1], as an extension to Hume-Rothery’s [2] and Pauling’s [3] schemes of alloy formations, has been widely employed to predict formation enthalpies in binary alloy systems. As defined in the original Miedema’s model, the formation enthalpy of a binary alloy consists of a negative contribution from the electronegativity difference between the two components, a positive contribution from the electron density difference between the two components, and a correction of hybridization if the alloy is composed of a transition element and a non-transition element. Eventually, the formation enthalpy in a binary alloy system can be calculated as

n h io s s s 2 DHfor Miedema ¼ fC A g  C B 1 þ cðC A C B Þ 8 9 2 1=3 2 < = Pð D /Þ þ Q ð D n Þ  ar ws h i.  V2=3 ; A : ðn1=3 Þ1 þ ðn1=3 Þ1 2 ; ws

A

ws

ð1-aÞ

B

where Ci(i = A, B) is the mole concentration of component i. The term in the second bracket describes the atomic surrounding of component A with the component B. The empirical parameter, c, must be 8 for a long-range ordered alloy. C si ði ¼ A; BÞ is the normalized concentration of component i in the contact-surface area, defined as

C si ¼

C i V i2=3 2=3 C i V i þ C j V j2=3

;

ð1-bÞ

where Vi(i = A, B) is mole volume of component i, and V 2=3 reprei sents the contact-surface area of component i. The term in the third bracket evaluates the heat of solution per surface area of the unit ⇑ Corresponding author. Fax: +86 731 88830263. E-mail addresses: yioffi[email protected] (D.Q. Yi), [email protected] (Y. Jiang). 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.07.076

alloy. Dnws is the electronic density difference at the Wigner–Seitz cell boundary, and D/ is the electron chemical potential difference between two components. P, Q, r, a and li are all empirical parameters. The detailed description of Miedema’s model can be referred to Refs. [1,4]. Based on such a few empirical parameters and intrinsic properties of the component elements, Miedema’s theory has been proven effective in predicting some thermodynamic properties of alloys. It was found years later, however, that for a large number of binary intermetallic compounds comprised of two dissimilar transition metals, i.e. one early (of groups IIIB–VIIB) and one late (of groups VIII–IIB in the periodic table), over 90% of the experimentally measured formation enthalpies were higher (or less negative) than the calculated values by Miedema’s model [5]. We compared 381 formation enthalpies measured by Kleppa’s group [5–10] with Miedema’s calculations in Figure 1a. The discrepancy was quantitatively evaluated using mean absolute percentage error (MAPE) defined as

MAPE ¼

 N   1 X yi  ti   100%  N i¼1 ti 

ð2Þ

where N is the total number of data points, ti is an experimental value, and yi the correspondent calculated value. A lower MAPE value suggests a higher systematic precision of prediction. The equality diagonal line in Figure 1a manifested the significant discrepancy between calculations and experiments: most of the calculated values based on the original Miedema’s model are more negative, leading to an unacceptable high value of MAPE = 51.1%. According to Lewis [11], only a prediction model with a MAPE value of less than 50% can be regarded as reasonable. The original Miedema’s model undoubtedly needs to be improved before being applied to transition metal intermetallics.

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S.P. Sun et al. / Chemical Physics Letters 513 (2011) 149–153

Figure 1. Comparison between experiments and calculations of standard formation enthalpies by (a) the original Miedema’s model and (b) Zhang’s model. Experimental formation enthalpies were measured by Kleppa’s group [5–10]. The equality diagonal line represents the 100% agreement between experiments and calculations.

To meet this aim, many recent theoretical efforts [12–16] have been focused on developing extended Miedema’s models by either adjusting some fundamental parameters [12,13], or introducing extra correction factors to the original Miedema’s model [14–16], in an attempt to take into account the neglected but important effects of the atomic size difference between the two dissimilar transition metals. These attempts have led to some interesting results, but many debates remain [14–16]. In particular, Zhang et al. [14] proposed an atomic size influential factor, sZhang ðCÞ;to the original Miedema’s model,

sZhang ðCÞ ¼ 1 

   2=3  C sB V 2=3 A  VB  C sA V A2=3 þ C sB V 2=3 B

;

ð3-aÞ

and the formation enthalpy originally defined in Eq. (1-a) thus became

DHfor Zhang

¼s

Zhang

ðCÞ 

DHfor Miedema :

ð3-bÞ

for Apparently DHfor Zhang can reduce to the original DHMiedema only 2=3 when V 2=3 ¼ V . It was claimed by Zhang et al. that with such a B A 2=3 2=3 correction on the contact-surface area difference, jV A  V B j; the important effect of the atomic size difference on formation enthalpy can be reasonably considered, and consequently the prediction precision on formation enthalpy can be improved by 13–65%. Using Zhang’s model, over 95% of the calculated values were in agreement with the experimental ones, within a statistical error of ±23 kJ/mol-atom [14]. Zhang’s model seems to be successful, for it worked quite well as seen in Figure 1b, resulting in a rather reasonable value of MAPE = 37.8%. However, one serious problem arises: according to Eq. (3-a), the resulted formation enthalpy of a compound AxBy will not be identical to that of ByAx. This is absolutely wrong, and the model is thus not self-consistent. The reason is simply that the atomic size factor sZhang ðCÞ as proposed in Eq. (3-a) does not fulfill the basic commutative law. It was also observed [15] that for some compounds for which formation enthalpies have been already successfully predicted by original Miedema’s model, Zhang’s model failed. In order to restore the commutation, Chen et al. [15] recommended replacing the term of sZhang ðCÞ  V 2=3 that was implicitly i included in Eq. (3-b) with a new factor

3  82 9   B < = C si V 2=3  V 2=3 X B  A 1 41  5V 2=3 : sChen ðC; VÞ ¼ i 2=3 2=3 s s ; 2 i¼A : CAV A þ CBV B

ð4-aÞ

From which, however, one derives

( s

Chen

ðC; VÞ ¼

V 2=3 if V A < V B A ; V 2=3 otherwise B ;

:

ð4-bÞ

It is clear that, to consider the effect of the atomic size difference, Chen et al. simply replaced the V2/3 term in Eq. (1-a) with the smaller one of the two V 2=3 ði ¼ A; BÞ. This is highly questionable i for that such a correction lacks an obvious physical interpretation, and that it cannot possibly yield any reasonable predictions as ‘luckily’ achieved by Zhang’s model and seen in Figure 1b. A possible revision to Chen’s model will be suggested later in this Letter. More recently, Wang et al. [16] proposed another form of the size prefactor as

sWang ðCÞ ¼ 1 

C  C A C B jV A  V B j : CA CA V A þ CB CBV B

ð5Þ

Please note, that Wang et al. [16] chose to directly evaluate the atomic volume difference, |VA  VB|, instead of the contact-surface 2=3 2=3 area difference jV A  V B j, without giving a reason. Here C is a newly-introduced dimensionless parameter to be determined by fitting to experiments. For ordered compounds, the authors claimed that C must be set equal to 2. As shown in Figure 2, the precision of calculation was indeed improved, resulting in a reasonable value of MAPE = 38.1%. It should be notified however that,

Figure 2. Comparison between experiments and calculations of standard formation enthalpies by Wang’s model (revised with C = 1/2).

S.P. Sun et al. / Chemical Physics Letters 513 (2011) 149–153

to achieve this level of precision, we had to set C = 1/2 (instead of 2). A further discussion on this choice of C will be provided. Nevertheless, the fitting parameter lacks well-defined physical meaning, and can be avoided as we will manifest later. In our opinions, all these above models are not satisfactory. We hereby propose a rather simple but more physically insightful solution: while two dissimilar transition metals approach to each other to form a compound, the significant effect of the atomic size difference on the contact surface can be more appropriately estimated using a new form of atomic size factor,

sPresent ðCÞ ¼ 1  d ðCÞ;

ð6-aÞ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u B uX V 2=3 s t ; d ðCÞ ¼ Ci 1  i V 2=3 i¼A 

where V 2=3 ¼ can derive

PB

sPresent ðCÞ ¼ 1 

s i¼A ðC i

2=3

ð6-bÞ 2=3

2=3

 V i Þ ¼ C sA V A þ C sB V B . From which, one

qffiffiffiffiffiffiffiffiffiffiffi C sA C sB 2=3

2=3

C sA V A þ C sB V B

2=3 jV 2=3 A  V B j:

ð7Þ

The present prefactor not only naturally fulfills the basic commutative law, but also directly evaluates the contact-surface area difference, i.e. jDV 2=3 j (not the atomic volume difference jDV i j as i in Wang’s model). Furthermore, our proposed method is superior to Wang’s approach for it does not require any new fitting parameters: both C si and V 2=3 are fundamental parameters of a compound. i The formula for the parameter d was enlightened by the derivation of d that has been often used to measure the atomic radius difqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   Pn P ri 2 ferences in alloys as d ¼ , where r ¼ ni¼1 ðC i  r i Þ i¼1 C i 1  r [17]. To validate the present model, we recalculated formation entropies using this new prefactor for the same group of 381 binary transition metal compounds and compared the results with Kleppa’s experiments. As shown in Figure 3a, a much improved agreement with experiments was achieved, with MAPE = 37.3%, as compared to the performance of the original Mediema’s model in Figure 1a. The prediction precision is even slightly better than that of Wang’s model which relies on a parameter fitting to match a number of experimental data. Encouraged by this, we revisited Chen’s model defined in Eq. (4-a) and proposed a revision to the factor scher(C,V) as

( " #) 2=3 B C s jV 2=3 1X A  VB j  V 2=3 1  s i 2=3 i 2 i¼A C A V A þ C sB V B2=3  Chen-R  2=3 ¼ s ðCÞ  V i :

sChen-R ðC; VÞ ¼

ð8-aÞ

151

One can thus obtain

Chen-R

s

   2=3 2=3  1 V A  V B  ðCÞ ¼ 1   s 2=3 : 2 C A V A þ C sB V B2=3

ð8-bÞ

Comparing Eqs. (8-b) and (7), one sees that the revised Chen’s model qffiffiffiffiffiffiffiffiffiffiffi represents a specific case of our present model, with C sA C sB ¼ 1=2. Calculations using the revised Chen’s factor were performed for those compounds and the results were then compared with experiments in Figure 3b. Unsurprisingly, the prediction precision that is almost the same as our present qffiffiffiffiffiffiffiffiffiffiffi model was achieved, with MAPE = 37.6%, suggesting that C sA C sB ¼ 1=2 happened to be an overall good choice for the large group of 381 compounds. A further comparison between Eqs. (5) and (8-b) reveals a fact that only by setting the parameter C equal to 1/2, can the two formulas take on the similar form. This shall help to explain why, in order to achieve a reasonable prediction accuracy, the parameter C in Wang’s model must be fitted to be 1/2, not 2 as originally claimed by Wang et al. [16]. It is natural to expect that the three models (i.e. Wang’s, Chen’s, and the present model) would give about the same level of prediction accuracy, as we have already shown above. Our present model is preferable to the other two models, for it gives insight into the physical meaning of the atomic size factor, requires no new empirical parameters, and is also easy to understand and use. As we aware, many other methods [18–21] can be used to predict thermodynamic properties of alloy system, including ab initio calculations. To further evaluate the present model, we calculated formation enthalpies for another series of transition-metal intermetallics and compared the results in Table 1 with the most recent ab initio predictions [21–32], all available experimental measurements [5–7,9,33–50], and as well the original Miedema model’s predictions. The temperatures of experiments were also provided. As seen in Table 1, despite being performed at temperature T = 0 K, for most of the intermetallics (except for the Mo-Ni system), ab initio calculations can yield results comparable to experimental measurements at higher temperatures. The temperature effect can be thus regarded as second order. Neither ab initio calculations nor the Miedema-based methods (including the present one) considered the temperature effect. Nevertheless, it is evident from Table 1 that compared to both ab initio and experimental results, the present model without involving any new empirical parameters can predict the formation enthalpies better than the original Miedema’s model. In contrast to an ab initio approach, the present model does not require beforehand knowledge of crystal structures and an enormous amount of computation time.

Figure 3. Comparison between experiments and calculations of standard formation enthalpies by (a) our present model and (b) the revised Chen’s model.

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S.P. Sun et al. / Chemical Physics Letters 513 (2011) 149–153

Table 1 Comparison of these calculated formation enthalpies for some binary transition metal alloys using our present model and the original Miedema’s model with literature data. Intermetallics

DHexp (kJ/mol-atom)

DHab-initio (kJ/mol-atom)

DHPresent (kJ/mol-atom)

DHMiedema (kJ/mol-atom)

Ag2Dy AgDy Ag2Er AgEr Ti3Pd Ti2Pd TiPd

22.3[7],1373 K 37.3[7],1373 K 24.0[7],1373 K 44.9[7],1373 K  – 53.3[33],1477 K 53.1[34],298 K 60.3[35],1373 K – 58.7[34],298 K 58.1[34],298 K 0.6[36],1073 K 2.6[36],1073 K 29.3[37],1202 K 26.8[38],298 K 34.0[37],1460 K 33.9[38],298 K 31.60[39],1300 K 33.10[40],1477 K 32.7[35],1373 K 42.9[37],1513 K 34.7[38],298 K 43.6[39],1300 K 42.2[40],1477 K 47.9[41],1573 K 45.2[42],1233–1644 K 48.3[5],1477 K 36.7[41],1423 K 33.8[42],1233–1644 K – – 14.38[43],323–998 K 14.1[44],1373 K 24.3[6],1372 K – 12.3[44],1373 K 26.1[9],1373 K 15.7[45],1123 K 22.6[9],1373 K 8.6[9],1373 K 28.9[28],298 K 16.7[9],1373 K

32.58[21] 32.81[21] 34.12[21] 34.49[21] 36.20[22] 45.00[22] 53.30[22]

31.45 34.70 31.62 35.05 60.55 75.17 91.05

39.64 42.88 39.36 42.89 63.56 79.34 96.64

57.90[22] 59.10[22] 60.80[22] 8.84[23] 10.06[23] 28.96[23]

85.51 82.39 75.88 5.62 6.89 34.21

90.78 87.44 80.44 6.28 7.76 39.26

36.10[23]

43.49

51.18

48.73[23]

31.25

36.92

52.99[24]

47.99

61.08

31.80[24]

37.37

45.36

43.36[24] 11.37[25] 20.24[25]

43.85 14.58 18.20

57.46 22.32 27.96

18.21[25] 16.13[25] 12.93[26] 18.47[27] – 29.6[28]

21.72 25.75 9.14

33.05 37.24 11.55

31.53 9.67

42.38 10.84

1.73[29] 1.51[30] – 11.42[31]

2.52

2.95

50.69 22.01

51.49 28.34

– –

20.68 21.93

23.54 26.59

–

21.93

25.92

50.17[32]

31.11

45.08

Ti2Pd3 Ti3Pd5 TiPd2 MoNi4 MoNi3 NiTi2 NiTi

Ni3Ti

HfNi

Hf2Ni HfNi2 Cu5Zr Cu51Zr14

Cu8Zr3 Cu10Zr7 ScMn2 Sc4Os11 TiMn2 V2Hf VIr Fe2Hf Fe7Nb6 Co3Nb Co3Ta

NiY

28.1[9],1373 K 12.3[9],1373 K 39.4[46],1723 K 6.2[9],1373 K 7.4[9],1373 K 12.4[47],1273 K 8.5[9],1373 K 21.7[48],1200 K 32.9[9],1373 K 35.38[49],887–1224 K 36.6[50],298 K

To summarize, we have proposed a new and improved atomic size factor used in Miedema’s model for binary transition metal systems. The new Miedema-based model was validated by calculating formation enthalpies of a large number of binary intermetallic compounds, and comparing the results to correspondent experimental measurements. Compared to the original Miedema’s model, the present model predicted formation enthalpies with much improved precision for most compounds. Compared to other available Miedema-based models, the present model is fairly simple, does not rely on any new empirical parameters, can provide more insightful physical meaning and achieve the same (if not better) prediction precision. Additionally, based on the present model, Chen’s and Wang’s model have been revised and reinterpreted.

Acknowledgment This support of the National Key Basic Research Program of China under Grant No. 2005CB623705 is gratefully acknowledged. References [1] A.R. Miedema, P.F. de Châtel, F.R. de Boer, Physica B 100 (1980) 1. [2] W. Hume-Rothery, G.W. Mabbot, K.M. Channel-Evans, Trans. R. Soc. London, Ser. A 233 (1934) 1. [3] L. Pauling, The Nature of the Chemical Bond (3rd ed.), Cornell University Press, Ithaca, New York, 1960. [4] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals, North-Holland, Amsterdam 1988. [5] Q. Guo, O.J. Kleppa, J. Alloys Compd. 321 (2001) 169. [6] S.V. Meschel, O.J. Kleppa, J. Alloys Compd. 350 (2003) 205.

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