An improved calculation method of hydrogen diffusion coefficient in lattice of metals

Scripta METALLURGICA

Vol. 22, pp. 355-357, 1988 Printed in the U.S.A.

Pergamon Journals, Ltd. All rights reserved

AN IMPROVED CALCULATION METHOD OF IlYDROGEN DIFFUSION COEFFICIENT IN LATTICE OF METALS

K. Yang, A.P. Xian, M.Z. Cauand X.J. Wan Institute of Metal Research, Academia SinicB, Shenyang, China (Received (Revised

October January

13, 1987) 4, 1988)

Introduction The electrochemical permeation technique [I] is widely used to obtain the diffusion coefficient of hydrogen in metals in the vicinity of room temperature. Theoretically, the diffusion coefficient can be determined from any point on the hydrogen permeation curve [2]. However, because of the inevitable existence of imperfections in metals that are generally called hydrogen traps, such as dislocations, grain boundaries, impurities, surface, etc., which capture hydrogen atoms and delay the rate of hydrogen diffusion through the lattice of metals, the actual hydrogen permeation time is always larger than the theoretical one. Therefore, the diffusion coefficient of hydrogen in the lattice of metals cannot be determined accurately by the theoretical formula which gives only an effective value. An improved calculation method is developed here to accurately determine the diffusion coefficient of hydrogen in the lattice of metals from the actual permeation curve. Theoretical Analysis In a double cell electrochemical permeation experiment, a constant hydrogen concentration is maintained at the cathode side of the metallic membrane, while zero concentration is kept at the anode side. For the above boundary conditions and an initial condition that hydrogen concentrations are zero at both sides of the membrane before the start of polarization, the hydrogen diffusion flux in the membrane can be deduced by means of a Laplace transformation[2]: JT /J==2 /[(vT )1/2] r~0exp[-(2n+ 1) ' / 4 ]

(1)

where T is a dimensionless parameter and T=Dt/I=; Jr and J= are hydrogen diffusion fluxes at the anode side of the membrane for any time (t) and at the steady state, respectively; 1 is the thickness of the membrane. D is the diffusion coefficient of hydrogen in the membrane, which can be determined from any ratio of J r / J = on the permeation curve corresponding to time (t): D=TI=/t

(2)

However, in an actual hydrogen diffusion process the actual permeation t i m e (t a) is the time (t a) that contains two parts, t o and t d, i.e., ta=to+td

(3)

where t o is the time of hydrogen diffusion in the l a t t i c e of a metal without any traps and t d is the mean delay time of hydrogen diffusion because of traps in the metal. Therefore, from the permeation curve and Eq.(2) only the apparent diffusion coefficient of hydrogen in the membrane can be obtained:

Da=~ I=/t a

(4)

The insertion of Eq.(3) int.oEq.(4) gives the value of D a as Da=Do(l_td/ta)

(5)

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where Do=~12/to, the lattice diffusion coefficient of hydrogen in the membrane. The delay time (t d) of hydrogen diffusion in a metal is a macro-reflection of the interaction between the hydrogen and all the traps in the metal. At a certain temperature, the more the deeper traps in a metal, the larger the t d. In addition, the t d is also related to the effective densities of traps in which hydrogen can be captured effectively, and the larger the densities, the larger the t d. In the early stage of a permeation, the hydrogen concentration in the membrane is not high. Hence, the effective densities of traps in the membrane are kept approximately constant and the t d can be taken as a constant. From Eq.(5) the Da has a linear relationship with the reciprocal of t a. As the permeation time increases, the traps are continuously filled with hydrogen and then t d will begin to decrease gradually if the densities of hydrogen traps in the membranes are not very high. Thus,the relationship between Da and 1/t a will begin to gradually deviate from the linear when the permeation time is large enough. Therefore, in a plot of Da versus I/ta, according to Eq.(5), the lattice diffusion coefficient of hydrogen (DO) in the membrane can be obtained by extrapolating the line, corresponding to the early stage of the permeation, to 1/ta÷0. And the t d can also be determined by the slope of the line. Finally, it should be noticed that a similar result can also be obtained by analyzin~ _the early stages of the decay transient from steady state if the boundary conditions are more certain[11] in the solution of the diffusion equation. Experiments and Results Electrochemical hydrogen permeation experiments for well-annealed a-Fe with a purity of 99.8% have been made at a constant electric potential condition in the temperature range between 290 and 350K. The electrolyte was 0.1N NaOH aqueous solution. The thickness of the membranes was 0.7ram. A thin layer of Pd was electroplated on both sides of the membranes. The Da values for different t a values were obtained according to Eq.(1) and (4) in a range of JT/J~=0.034 to 0.8306 on the permeation curves. The relationships between Da and 1/t a at different temperatures were plotted in Fig.l which agrees well with t h e analysis above mentioned. From Fig.l it can be seen that the deviation from the line occurs early as the temperature increases. The reason is that the densities of hydrogen traps in the membranes used here are not very high and the rate of hydrogen filling of the traps increases with temperature. Both the Do and t d values at four different temperatures were obtained from Fig.l and are given in Table I. A plot of DO versus the reciprocal of absolute temperature is shown in Fig.2. From Fig.2 the lattice diffusion coefficient of hydrogen in a-Fe between 290 and 350K is given by Do=5.12x I 0-8exp[-4.32(kJ/mol)/RT]

(m=/s)

(6)

Eq.(6) shows t h a t the a c t i v a t i o n enerEv for hydrogen diffusion in a - F e is 4.32kJ/mol. This value is much lower t h a n t h e previous results[3-6]-and well c o n s i s t e n t with some new results obtained by t h e new method [7-8]. Table 2 lists t h e comparison of the results of t h e r e f e r e n c e s and the present work. Since the t r a p e f f e c t is e l i m i n a t e d completely in t h e c a l c u l a t i o n of Do, it is much more reliable to obtain t h e diffusion coefficient of hydrogen in metals by the method developed here. Since the interaction between the hydrogen and the traps is a thermal active process, and the t d is a macro-reflection of this interaction, the average binding energy between the hydrogen and the traps in metals can also be determined from the temperature dependence of t d. From Fig.2 the binding energy in the membranes is calculated to be 21.4kJ/mol that is much closer to the energy between the hydrogen and the dislocation, 24kJ/moI[9] and that between the hydrogen and the grain boundary, 26kJ/moI [10]. Therefore, dislocations and grain boundaries may be the dominant hydrogen traps in the a-Fe used here. References I. M.A.V. Devanathan and Z. Stachurski: Proc. R. Soc., A270, 90(1962). 2. B.S. Chaudhari and T. P. Radhakrishana: Surf. Tech., 2__22,353(1984). 3. K. Yamakawa, T. Tsuruta and S. Yoshizawa: Boshoku Gijutsu, 3__00,443(1981). 4. W. Beck, J. O'M. Bockris, J. Mcbreen and L. Nanis: Proc. R. Soc., A290, 220(1966). 5. H. Hagi, Y. Hayashi and N. Ohtani: Trans. Jpn. Inst. Metals, 20, 349-~-979). 6. S. Asano, K. Hara, Y. Nakai and N. Ohtani: J. Jpn. Inst. Metals, 3_~8,626(1974). 7. M. Nagano, Y. Hayashi, N. Ohtani, M. Isshiki and K. Igaki: Trans. Jpn. Inst. Metals, 2_22,423(1981).

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M. Nagano, Y. Hayashi, N. Ohtani, M. Isshiki and I<. Igaki: Scripta Met., 16, 973(1982). R. Gibala: Trans. TMS-AIME, 239, 1574(1967). I.M. Bernstein: Scripta Met., 8, 348(1974). L. Nanis and T. K. Govindan Namboodhiri: J. Eletrochem. Soc., 119, 691(1972).

Table l

T

The lattice diffusion coefficient of hydrogen (Do) in a-Fe and the corresponding delay time (td). OK:

Doxl09(m=/s): t d (s):

290

313

333

353

8.52

9.78

10.71

11.79

16.2

6.4

4.6

3.2

Tab]e 2 Comparisons of previous results with present work on the activation energy for hydrogen diffusion. Q (kJ/mol): Ref:

4.86

5.57

6.70

7.49

(3)

(4)

(5)

(6)

4.32 present

4.20

3.85

(7)

(8)

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Fig.l

The plotting of Da versus ]/t a at different temperatures.

Fig.2

The plotting of D O and t d versus t h e reciprocal of absolute t e m p e r a t u r e .