PLS method applied to regularities of formability of hydrides of binary transition metal alloys

Journal of Alloys and Compounds 372 (2004) 136–140

PLS method applied to regularities of formability of hydrides of binary transition metal alloys Jin Guo∗ , Wen-lou Wei, Shu-yuan Ma School of Physics Science and Technology, Guangxi University, Nanning 530004, China Received 1 August 2003; received in revised form 29 September 2003; accepted 29 September 2003

Abstract The formability of hydride and hydrogen storage properties of binary transition metal alloys were investigated by partial least square (PLS) method. The results show that formability of hydride is significantly influenced by electron density of element, large electron density of element is favorable for stability of hydride of binary transition metal alloys. Formation enthalpy of transition metal hydride is another important factor affecting the hydrogen storage property of alloy. The criteria for choosing alloys which show reversible hydrogen storage property at normal condition are obtained by PLS method. © 2003 Elsevier B.V. All rights reserved. Keywords: Binary transition metal alloys; Hydrogen storage property; Partial least square

1. Introduction

2. Calculation method

Among functional materials, hydrogen storage alloys have attracted much attention for more than two decades. Most of hydrogen storage alloys are composed of transition metals, and new hydrogen storage alloys are mainly developed among transition metal alloys. Van Mal [1] and Miedema and co-workers [2] proposed some criteria for classification of formability of hydride of binary transition metal alloys by formation enthalpy. Although thermodynamic method is effective, thermodynamic data for some compounds are unknown, particularly for unknown compounds. In previous work, the partial least square (PLS) method, one of chemical pattern recognition methods, was applied to materials design of hydrogen storage alloys [3] and the effect of the simultaneous presence of La, Ce, Pr and Nd on the hydrogen storage properties of AB5 -type alloys [4]. In this work, we discuss the formability of hydride and hydrogen storage properties of binary transition metal alloys by the partial least square method.

2.1. Data preprocessing

∗

Corresponding author. E-mail address: [email protected] (J. Guo).

0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.09.141

Before the calculation is performed, it is convenient to tailor the date in the calibration set in order to make the calculations easier. Variance scaling is used when the variables in a data set are measured in different units (e.g., Å, %, kJ); scaling is accomplished by dividing all the values for a certain variable by the standard deviation for that variable, so that the variance for every variable is unity. For original old }, the standardizing processing is expressed date set {Xki as follow: Xki =

old − X ¯i Xki Si

where ¯i = X

N

1
old Xki N

   Si = 

k=1

N

1
old ¯ i )2 (Xki − X N −1 k=1

k = 1, . . . , N; i = 1, . . . , M.

J. Guo et al. / Journal of Alloys and Compounds 372 (2004) 136–140

137

2.2. Method

Table 1 A–B binary alloys and atomic parametersa

Partial least square method is a pattern recognition methods widely used in physical chemistry [5] and analytical chemistry [6,7]. Detailed information about the PLS method can be found in reference [8]. Here, we briefly present the relevant contents of the PLS method. The PLS model is built on the properties of the nonlinear iterative partial least squares algorithm. It is possible to let the score matrix represent the data matrix. A simplified model would consist of a regression between the scores for X and Y matrix. Let X be an n × m matrix with n samples and m features, Y be a target matrix corresponding to X. In order to calculate the factors of X matrix by the columns of Y matrix, and predict the factors of Y matrix by the columns of X matrix, X and Y are separated as follows:

No.

A

B

RA

XA

ZA /R3A

RB

XB

ZB /R3B

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

Pu Ti Hf Hf Zr La La U Zr Hf Y La Th U La Zr La Y Sc Ti Y Y Sc Th Hf Hf Sc Th Y Pu Y Zr Y Zr Ti Sc U Hf Pu Sc Pu Sc Y Pu Pu Zr La U Ti Pu Ti Pu La Y U Sc Hf Ti Zr Pu Hf Zr Zr

Mn Cr Mn Mo W Co Tc Mn Mn Cr Fe Ni Mn Cr Os Mo Ru Mn Mn Mn Co Ru Tc Co W Fe Co Fe Os Co Tc Fe Re Co Ni Rh Fe Co Os Re Re Os Ir Rh Ir Ni Pd Co Co Tc Fe Ru Pt Rh Ni Ru Ni Pt Os Pd Pd Ir Tc

1.58 1.46 1.58 1.58 1.6 1.88 1.88 1.56 1.6 1.58 1.8 1.88 1.8 1.56 1.88 1.6 1.88 1.8 1.64 1.46 1.8 1.8 1.64 1.8 1.58 1.58 1.64 1.8 1.8 1.58 1.8 1.6 1.8 1.6 1.46 1.64 1.56 1.58 1.58 1.64 1.58 1.64 1.8 1.58 1.58 1.6 1.88 1.56 1.46 1.58 1.46 1.58 1.88 1.8 1.56 1.64 1.58 1.46 1.6 1.58 1.58 1.6 1.6

1.1 1.5 1.3 1.3 1.4 1.1 1.1 1.7 1.4 1.3 1.2 1.1 1.1 1.7 1.1 1.4 1.1 1.2 1.3 1.5 1.2 1.2 1.3 1.1 1.3 1.3 1.3 1.1 1.2 1.1 1.2 1.4 1.2 1.4 1.5 1.3 1.7 1.3 1.1 1.3 1.1 1.3 1.2 1.1 1.1 1.4 1.1 1.7 1.5 1.1 1.5 1.1 1.1 1.2 1.7 1.3 1.3 1.5 1.4 1.1 1.3 1.4 1.4

0.76 1.28 1.01 1.01 0.97 0.45 0.45 0.79 0.97 1.01 0.51 0.45 0.52 0.79 0.45 0.97 0.45 0.51 0.68 1.28 0.51 0.51 0.68 0.52 1.01 1.01 0.68 0.52 0.51 0.76 0.51 0.97 0.51 0.97 1.28 0.68 0.79 1.01 0.76 0.68 0.76 0.68 0.51 0.76 0.76 0.97 0.45 0.79 1.28 0.76 1.28 0.76 0.45 0.51 0.79 0.68 1.01 1.28 0.97 0.76 1.01 0.97 0.97

1.3 1.36 1.3 1.4 1.41 1.25 1.36 1.3 1.3 1.36 1.27 1.25 1.3 1.36 1.35 1.4 1.34 1.3 1.3 1.3 1.25 1.34 1.36 1.25 1.41 1.27 1.25 1.27 1.35 1.25 1.36 1.27 1.38 1.25 1.25 1.35 1.27 1.25 1.35 1.38 1.38 1.35 1.36 1.35 1.36 1.25 1.38 1.25 1.25 1.36 1.27 1.34 1.39 1.35 1.25 1.34 1.25 1.25 1.25 1.25 1.39 1.35 1.38

1.5 1.6 1.5 1.8 1.7 1.8 1.9 1.5 1.5 1.6 1.8 1.8 1.5 1.6 2.2 1.8 2.2 1.5 1.5 1.5 1.8 2.2 1.9 1.8 1.7 1.8 1.8 1.8 2.2 1.8 1.9 1.8 1.9 1.8 1.8 2.2 1.8 1.8 2.2 1.9 1.9 2.2 2.2 2.2 2.2 1.8 2.2 1.8 1.8 1.9 1.8 2.2 2.2 2.2 1.8 2.2 1.8 2.2 2.2 2.2 2.2 2.2 1.9

3.16 2.39 3.16 2.19 2.15 4.59 2.78 3.16 3.16 2.39 3.87 5.17 3.16 2.39 3.23 2.19 3.33 3.16 3.16 3.16 4.59 3.33 2.78 4.59 2.15 3.87 4.59 3.87 3.23 4.59 2.78 3.87 2.69 4.59 5.17 3.7 3.87 4.59 3.23 2.69 2.69 3.23 3.6 3.7 3.6 5.17 3.84 4.59 4.59 2.78 3.87 3.33 3.75 3.7 5.17 3.33 5.17 3.75 3.23 3.84 3.84 3.6 2.78

X = TP T + E

(1)

Y = UQT + F

(2)

The simplest model for the relation between X and Y is a linear one: U = BT

(3)

where T and U are score matrices of X and Y , and P T and QT are load matrices of X and Y , and E and F are the residuals of X and Y , respectively. Score and load matrices can be calculated pair-by-pair by an iterative procedure. The elements of B play the role of the regression coefficients. T  = X(P T )−1 is the projection of PLS space, P is m × n orthonormal PLS component matrix. In multi-dimensional PLS space constructed by T  , the samples can be classified according to their target values, so that we can find out regularities of classification by choosing projected plane. In order to describe the factors affecting the formability of binary transition metal hydrogen storage alloys, it is reasonable to select the following set of atomic parameters: 1. metallic radii of element: R; 2. Pauling’s electronegativity: X; 3. atomic electron density; Z/R3 , Z is number of valence electrons. In this work, these atomic parameters listed in Table 1 as features were used to construct the sample matrix X, and then by PLS method, span a six-dimensional space. The pattern recognition method used here allows us to map the pattern in multi-dimensional space [9] onto a plane. In the plane, the samples were classified according to their target values, so that some regularities for formability of hydride of binary transition metal alloys could be found.

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J. Guo et al. / Journal of Alloys and Compounds 372 (2004) 136–140

Table 1 (Continued ) A

B

RA

XA

ZA /R3A

RB

XB

ZB /R3B

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101

Ti Ti Hf Hf Y U U Zr Zr Ti Hf U Hf Th Ti Ti Ti U Zr U Sc Sc Pu Ti U Hf U Sc U Y Hf Th Zr Zr Hf Ho Dy Er

Tc Os Re Os Pd Os Rh Rh Pt Re Pt Ru Ir Pd Ir Pd Rh Ir Pd Pd Pt Ir Pt Ru Re Ru Pt Pd Tc Pt Tc Pt Re Ru Rh Co Co Co

1.46 1.46 1.58 1.58 1.8 1.56 1.56 1.6 1.6 1.46 1.58 1.56 1.58 1.8 1.46 1.46 1.46 1.56 1.6 1.56 1.64 1.64 1.58 1.46 1.56 1.58 1.56 1.64 1.56 1.8 1.58 1.8 1.6 1.6 1.58 1.77 1.78 1.76

1.5 1.5 1.3 1.3 1.2 1.7 1.7 1.4 1.4 1.5 1.3 1.7 1.3 1.1 1.5 1.5 1.5 1.7 1.4 1.7 1.3 1.3 1.1 1.5 1.7 1.3 1.7 1.3 1.7 1.2 1.3 1.1 1.4 1.4 1.3 1.2 1.2 1.2

1.28 1.28 1.01 1.01 0.51 0.79 0.79 0.97 0.97 1.28 1.01 0.79 1.01 0.52 1.28 1.28 1.28 0.79 0.97 0.79 0.68 0.68 0.76 1.28 0.79 1.01 0.79 0.68 0.79 0.51 1.01 0.52 0.97 0.97 1.01 0.54 0.54 0.55

1.38 1.36 1.36 1.36 1.35 1.38 1.35 1.38 1.35 1.35 1.35 1.39 1.38 1.39 1.34 1.36 1.38 1.36 1.38 1.35 1.36 1.38 1.38 1.39 1.36 1.39 1.34 1.38 1.34 1.39 1.38 1.36 1.39 1.36 1.39 1.38 1.34 1.35

1.9 2.2 1.9 2.2 2.2 2.2 2.2 2.2 2.2 1.9 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 1.9 2.2 2.2 2.2 1.9 2.2 1.9 2.2 1.9 2.2 2.2 1.8 1.8 1.8

2.78 3.23 2.69 3.23 3.84 3.23 3.7 3.7 3.75 2.69 3.75 3.33 3.6 3.84 3.6 3.84 3.7 3.6 3.84 3.84 3.75 3.6 3.75 3.33 2.69 3.33 3.75 3.84 2.78 3.75 2.78 3.75 2.69 3.33 3.7 4.59 4.59 4.59

3

(a)

(b)

2

1

0

Y

No.

-1

DyCo3 -2

HoCo3 -3

ErCo3

-4 -4

-2

0

2

4

X

Fig. 1. The PLS classification of hydride of binary transition metal alloys: (䊊) forming hydrides, (䊉) unforming hydrides, () predicted hydrides.

Y = −0.56RA + 0.24RB + 0.52XA + 0.16XB ZA ZB + 0.57 3 − 0.11 3 RA RB

(5)

It can be seen from Fig. 1 and Eq. (4) that binary alloys, both elements with high electron density, are easier to form hydrides reacting with hydrogen. Based on Eqs. (4) and (5), three “unknown” alloy hydrides, DyCo3 , HoCo3 and ErCo3 , are predicted, and the predicted result shows that the three alloys can form hydride shown in Fig. 1, which is agreement with experimental.

a

All of A–B binary alloys listed are AB3 and AB suggested by the work of Van Mal [1].

3. Results and discussion 3.1. Formability of hydrides of binary transition metal alloys 101 binary alloys listed in Table 1 are classified for formability of hydride by pattern recognition in the sixdimensional space spanned by six atomic parameters chosen in Table 1. For example, Fig. 1 illustrates a linear mapping of the distribution of alloys forming hydrides and unforming hydrides in the above-mentioned six-dimensional space by PLS method. In Fig. 1, the separation of two classes is rather clear-cut. The coordinate equation can be expressed as: X = 0.14RA + 0.51RB − 0.16XA + 0.82XB ZA ZB − 0.16 3 − 0.07 3 RA RB

(4)

3.2. Formability of hydrogen storage alloys of binary transition metals A characteristic property of hydrogen storage alloys is storing reversibly large quantities of hydrogen. The property is strongly dependent on stability of hydride. If hydrides are extremely stable, high temperature will be needed to show the property of storing reversibly hydrogen at reasonable pressure; if alloy hydrides are unstable, the property is observed must at high pressure. Van Mal [1] classified alloys as four classes according to formation enthalpy of alloy hydrides, H. Class 1 : H(AB3 H4 , ABH3 ) < −41.9 kJ/mol H, stable hydrides are formed. Class 2 : H(ABH3 ) < −41.9 kJ/mol H, H(AB3 H4 ) > −41.9 kJ/mol H, stable hydrides may be formed at high pressure. Class 3 : 0 > H(ABH3 ) > −41.9 kJ/mol H, hydrides may be formed, but the plateau pressures are too low, less than 0.1 MPa.

J. Guo et al. / Journal of Alloys and Compounds 372 (2004) 136–140 2

139

2

(1)

(2)

1

1

0

Y

Y

0

TiFe

TiNi

-1

-1

TiMn -2

-2

-2

(a)

-1

0

1

-3

2

-3

(b)

X

-2

-1

0

1

2

3

X

Fig. 2. The PLS classification of hydrogen storage materials of binary transition metal alloys: (䊉) for Class 3, (䊊) for both Classes 1 and 2, (䊏) for Class 1, (䊐) for Class 2.

Class 4 : H(AB3 H4 , ABH3 ) > 0, hydrides are very unstable and the plateau pressures are much less than 0.1 MPa. These alloys can be regarded as unforming hydrides at reasonable pressure and temperature and classified in Fig. 1(b). Based on the analysis for formability of hydrides of binary transition metals, 57 binary alloys which can form hydrides reacting with hydrogen shown in Fig. 1(a) are classified by PLS again. Fig. 2 shows a pattern recognition classification between alloy hydrides (Class 3) with low plateau pressure (<0.1 MPa) and the other tow class alloy hydrides (Classes 1 and 2) by the PLS method in six-dimensional space. The coordinate equation can be expressed as: X = 0.32RA + 0.19RB + 0.79XA + 0.09XB ZA ZB − 0.36 3 + 0.29 3 RA RB Y = 0.47RA − 0.51RB − 0.37XA − 0.01XB ZA ZB − 0.61 3 + 0.11 3 RA RB

(6)

(7)

It can be seen from Fig. 2(a) that alloy hydrides with low plateau pressure are separated from all of 57 binary alloy hydrides and located at X (Eq. (6)) > 0 region. In other words, hydride plateau pressure will be decreased with X (Eq. (6)) increasing. In fact, there are little differences of atomic radii and Pauling’s electronegativity between transition metal elements. Evidently, atomic electron density, Z/R3 , is an important factor influencing on property of hydrogen storage alloy. Large electron density of B atom is

favorable for stability of hydride of binary transition metal alloys. From above argument, for example, we can “predict” the change describing stability of TiFe hydride when Mn and Ni are substituted for Fe. Because electron density of Fe element is smaller than that of Ni element and larger than that of Mn element, stability of hydride will be increased while Ni element substituted for Fe element and decreased while Mn element substituted for Fe element. There are two kinds of alloy hydrides shown in Fig. 2(a) part 1. One (Class 1) is stable, another (Class 2) show their reversible hydrogen storing property at high pressure though their hydrides are also fairly stable. Because of little difference of stability between the two kinds of alloy hydrides, above-mentioned three atomic parameters, atomic radii, Pauling’s electronegativity and atomic electron density, cannot classify the two kinds of alloys completely. So, formation enthalpy of transition metal hydride H(MH) is added. The classification for 33 alloys shown in Fig. 2(a) part 1 is obtained by PLS and traced on Fig. 2(b). In Fig. 2(b), the separation of two classes is rather clear-cut. The boundary equations between two classes are RA ZA + 2.21X − 0.013HA − 0.19HB − 0.76 3 RB RA ZB − 0.57 3 + 0.06 = 0 RB

3.62

RA ZA − 0.67X − 0.05HA − 0.03HB − 2.1 3 RB RA ZB + 0.66 3 − 9.14 = 0 RB

(8)

5.27

(9)

140

J. Guo et al. / Journal of Alloys and Compounds 372 (2004) 136–140

The boundary line equations reveal that stability of hydrides of binary transition metals is significantly influenced by electron density and formation enthalpy of element, especially by electron density. Alloys located near to boundary line can show reversible hydrogen storage property more easily than those far away from boundary line at normal condition.

Acknowledgements This work was supported by the National Nature Science Foundation of China (50171023), the key Project of China Ministry of Education (03104) and the Nature Science Foundation of Guangxi (0144033, 0249004, 2000220).

References 4. Conclusion For binary transition metal alloys, formability of hydride is significant influenced by electron density of element, large electron density of element is favorable for stability of hydride of binary transition metal alloys. Besides, formation enthalpy of transition metal hydride is another important factor effecting the hydrogen storage property of alloy. The criteria for choosing alloys which show reversible hydrogen storage property at normal condition are given by Eqs. (8) and (9).

[1] H.H. Van Mal, Philips Res. Repts. Suppl. 1 (1976) 61. [2] K.H.J. Buschow, P.C. Bouten, A.R. Miedema, Rep. Prog. Phys. 45 (1982) 937. [3] J. Guo, C.H. Li, H.L. Liu, N.Y. Chen, Mater. Sci. Eng. B 96 (1997) 47. [4] J. Guo, C.H. Li, H. Liu, N.Y. Chen, J. Electrochem. Soc. 144 (1997) 2276. [5] C. Nianyi, C. Li, S. Yao, X. Wang, J. Alloys Compd. 234 (1996) 130. [6] W. Linbery, J.A. Persson, Anal. Chem. 643 (1983) 55. [7] H.L. Liu, N.Y. Chen, W.C. Lu, J.W. Zhu, Anal. Lett. 27 (1994) 2195. [8] P. Geladi, B.R. Kowalaski, Anal. Chim. Acta 1 (1986) 185. [9] O. Strouf, Chemical Pattern Recognition, Research Studies Press, Letchwoth, Hertfordshire, UK, 1986, pp. 13–104.