**7. Hydrogen and electric field effect to Iron impurities diffusion in the Zr-Fe melt**

Iron and Zirconium diffusion factor dependence on electric field intensity and Hydrogen presence in the molten Zirconium had been analyzed in the terms of molecular dynamics (MD) method. Model system for research of Iron and Hydrogen ions behavior in the Zr-Fe-H melt at Т=2273К temperature and electric field presence contained 516 Zirconium particles, 60 Iron particles and 1Hydrogen particle in cubic cell with a=2.44195 nm cube edge. Integration of the motion equation was carried out by 1.1·10-15s time steps. Interparticle potentials and its parameters had been taken from (Varaksin & Kozyaichev, 1991, Zhou et.al, 2001). Calculation results of impurities migration in the molten Zirconium are compared to experimental data (Lindt et.al, 1999, Ajaja et.al, 2002, Mimura et.al, 1995). Partial radial distribution functions gij(r) for Zirconium-Iron melt are presented on fig. 12. Most probable inter-atomic distance in first coordination sphere is close to sum of atomic radii for Iron and Zirconium (rZr-Fe= 0.29nm, rFe= 0.130nm, rZr= 0.162nm).

This results comparison to computer simulation data for Ta-Fe melt (Pastukhov et.al, 2010, Vostrjakov et.al, 2010) reveals sufficiently close character of the radial distribution function for large dimension atoms, namely Ta-Ta (0.29 nm, rTa = 0.145nm) and Zr-Zr (0.324nm, rZr=0.162nm). Iron and Zirconium diffusion factors in the Zirconium melt in the presence, as well as absence of electric field and Hydrogen at 2273K had been calculated by means of MD method (fig. 13 and 14). Diffusion factor of Iron (DFe) in the Zirconium melts with Hydrogen linearly depends on electric field intensity (E) and Iron concentration (СFe). Hydrogen diffusion factor negligibly decreases from 2.16·10-4 cm2·s-1 to 1.94·10-4 cm2·s-1, if electric field intensity increases from 900 to 1020 v/m. Hydrogen inducing into system at СFe≈0.1% decreases DFe value from 7.86·10-5 to 6.36·10-5cm2s-1, and electric field 1020 v/m intensity applying decreases DFe to 5.23·10-5cm2·s-1 (fig.14).

Molecular Dynamic Simulation of Short Order and

DZr

0

2

4

D\*10-5,cm2

c


6

8

Hydrogen Diffusion in the Disordered Metal Systems 299

DFe

Fig. 13. Dependencies of DFe and DZr on Iron concentration at 2273К (MD - calculation).

0,00 0,02 0,04 0,06 0,08 0,10

c%, Fe

Fig. 14. Iron (СFe =0.1mas.%) diffusion factor dependence on electric field intensity at 1173K

900 950 1000 1050 1100

E,V/m

Calculated values of Еа for different metals (Flynn et.al, 1970) are in quantity agreement with experimental data. Temperature dependence DH at high temperatures is described in the term of theory (Flynn et.al, 1970). Authors (Shmakov et.al, 1998) calculated DH in Zirconium at 2273K without electric field influence as 3.862·10-4cm2·s-1, which low differ

temperature (MD – calculation).

5,10

5,15

5,20

D\*10-5cm2

c


5,25

5,30

from our calculated value DH=5.01·10-4cm2·s-1.

Calculation results of DFe changing in dependence on E value had been compared to evaporation constant rate for the Fe-ions from Zr, calculated by equation (Pogrebnyak et.al, 1987, Vigov et.al, 1987)

Fig. 12. Partial radial distribution functions gij(r) for Zr-Fe melt at 2273К, calculated in terms of MD model.

$$k = \nu \cdot \mathbb{C}\_{Fe} \exp\left[ -\frac{\mathcal{A} + I - \mathcal{W} - (q^3 \cdot E)^{1/2}}{RT} \right],\tag{12}$$

where ν – ion vibration frequency (1013·s-1), СFe – impurity concentration, λ – Fe evaporation heat, I – first ionization potential (V/Å), E – electric field intensity, W – electron exit work, q – ion charge. Values of λ, I, W, R had taken in electron-volt, E – in volt per angstrom. Diffusion factor D directly depends on rate (*k*) and time (t) evaporation of main metal (Kuznetsov et.al, 1968). The log *k* and log DFe on Е dependences (fig. 15) are relatively similar.

Thus assumption is possible, that limiting factor of Fe removal from the Zr melt is diffusion of Fe. The Hydrogen is considered as light intrusion impurity into metals with different cell type. Therefore Hydrogen diffusion is significant problem in researching of high temperature metals refining. Impurities have less action upon Incoherent diffusion. Therefore this kind of diffusion becomes dominating at high temperature (Maximov et.al, 1975).

We had compared Hydrogen diffusion factors in Ta at 3400K (Pastukhov et.al, 2010) and in Zr at 2273K (Ajaja et.al, 2002). This values are 1,7 10-5 and5,01 10-4 cm2·s-1 respectively. The authors (Maximov et.al, 1975) explain such difference due to Hydrogen diffusion activation energy (Ea) dependence on atomic metal mass, its Debye frequency, modulus of elasticity and volume change at Hydrogen addition.

Calculation results of DFe changing in dependence on E value had been compared to evaporation constant rate for the Fe-ions from Zr, calculated by equation (Pogrebnyak et.al,

Fig. 12. Partial radial distribution functions gij(r) for Zr-Fe melt at 2273К, calculated in terms

r,nm

where ν – ion vibration frequency (1013·s-1), СFe – impurity concentration, λ – Fe evaporation heat, I – first ionization potential (V/Å), E – electric field intensity, W – electron exit work, q – ion charge. Values of λ, I, W, R had taken in electron-volt, E – in volt per angstrom. Diffusion factor D directly depends on rate (*k*) and time (t) evaporation of main metal (Kuznetsov et.al, 1968). The log *k* and log DFe on Е dependences (fig. 15) are relatively similar. Thus assumption is possible, that limiting factor of Fe removal from the Zr melt is diffusion of Fe. The Hydrogen is considered as light intrusion impurity into metals with different cell type. Therefore Hydrogen diffusion is significant problem in researching of high temperature metals refining. Impurities have less action upon Incoherent diffusion. Therefore this kind of diffusion

We had compared Hydrogen diffusion factors in Ta at 3400K (Pastukhov et.al, 2010) and in Zr at 2273K (Ajaja et.al, 2002). This values are 1,7 10-5 and5,01 10-4 cm2·s-1 respectively. The authors (Maximov et.al, 1975) explain such difference due to Hydrogen diffusion activation energy (Ea) dependence on atomic metal mass, its Debye frequency, modulus of elasticity and

*Fe* exp *IW qE k C*

0,0 0,2 0,4 0,6 0,8 1,0

becomes dominating at high temperature (Maximov et.al, 1975).

volume change at Hydrogen addition.

3 1/2 ( )

, (12)

Zr-Zr

Zr-Fe

Fe-Fe

total

*RT*

1987, Vigov et.al, 1987)

of MD model.

0

1

2

3

gij(r)

4

5

6

Fig. 13. Dependencies of DFe and DZr on Iron concentration at 2273К (MD - calculation).

Fig. 14. Iron (СFe =0.1mas.%) diffusion factor dependence on electric field intensity at 1173K temperature (MD – calculation).

Calculated values of Еа for different metals (Flynn et.al, 1970) are in quantity agreement with experimental data. Temperature dependence DH at high temperatures is described in the term of theory (Flynn et.al, 1970). Authors (Shmakov et.al, 1998) calculated DH in Zirconium at 2273K without electric field influence as 3.862·10-4cm2·s-1, which low differ from our calculated value DH=5.01·10-4cm2·s-1.

Fig. 15. Dependence of lgDFe (curve 1) and lgk (curve 2) on electric field intensity.

We have estimated diffusion layer thickness (x) by (13) equation (Flynn et.al, 1970). Calculation were carried out basing on СFe - experimental time – dependence (Mimura et.al, 1995). Value of DFe we calculated by MD – method.

$$\mathbf{C}\_{\{\mathbf{x},t\}} = \mathbf{C}\_0 \text{erfc}\left(\frac{\mathbf{x}}{2 \cdot \sqrt{D\_{\text{Fe}} \cdot t}}\right) \text{\,\,\,\tag{13}$$

Molecular Dynamic Simulation of Short Order and

forces affect, rather than evaporation.

stage of melt.

electric field (fig. 16).

transition to more disordered structure.

corresponds to Ta-Fe without electric field.

Hydrogen Diffusion in the Disordered Metal Systems 301

G/L decreases at temperature increase. Obtained G/L ≈ 1.15 value at СFe= 0.46 and Т = 2450К may indicate, that close to 15% of Iron is being removed due to electro-magnetic

Т,К СFe, mas.% logG logL G/L Remark Issue 2350 0.46 -4.572 -5.066 3.115 L<G [4]

2450 0.46 -4.572 -4.755 1.153 L<G [4]

2900 7.82·10-3 -6.967 -5.406 0.027 L>G [2]

2450 7.82·10-3 -6.967 -6.525 0.361 L >G [2]

2300 7.82·10-3 -6.967 -7.001 1.081 L < G [2]

**8. Hydrogen and electric field effect to Iron impurities diffusion in Ta-Fe melt**  Hydrogen and Iron atoms radial distribution functions and diffusion constants had been found by MD method in the Tantalum melt at 3400K in the presence and absence of outer

The model system was presented by 486 tantalum, 1 iron and 1hydrogen atoms in a cubic cell of 2.13572 nanometers cube edge length. Computer experiment data have found short order of Ta-Fe-H system at 3400K is close to Tantalum structure: first maximum at RTa-Ta ≈ 0.29 nm corresponds to Tantalum atom radius ≈ 0.292 nm. All RDF maxima diffusion is observed at electric field and Hydrogen in the Ta-Fe system. This fact may indicate liquid

Unusual kind of RDF curve for Ta-H atoms pairs obtained at electric field 1020v/m intensity (fig. 17): first maximum of the curve is bifurcated, besides first sub-peak at r1 = 0.22nm corresponds to one of most probable distance Ta-H and second sub-peak at r2 = 0.24nm

Dynamics and local structure of the close to Hydrogen surrounding for the ternary interstitial alloy should depend on Hydrogen concentration, temperature and solvent structure short order. There is no conventional opinion about Hydrogen location in such systems. Since the Hydrogen atom radius is 0.032 nm, it can occupy octahedron (0.0606 nm), as well as tetrahedron (0.0328нм) (Geld et.al, 1985) location. System Ta – H particularity is

Table 3. Dependences of L and G on mean residual Iron content in Zirconium for PAM process. CFe – mean Iron concentration: initial (15min.), middle (90min.), and final (165min.)

 0.01 -6.09 -6.728 4.348 2.2·10-4 -7.902 -8.386 3.047

 0.01 -6.09 -6.418 2.128 2.2·10-4 -7.902 -8.076 1.491

 2.68·10-3 -7.44 -5.871 0.027 0.92·10-3 -7.903 -6.335 0.028

 2.68·10-3 -7.44 -6.990 0.355 0.92·10-3 -7.903 -7.454 0.356

 2.68·10-3 -7.44 -7.466 1.062 0.92·10-3 -7.903 -7.930 1.065

In the equation С0 and С(х,t) are impurity concentrations in initial and refined Zirconium (Flynn et.al, 1970) and t is time of refining. Calculated (*x)* value is equal 7·10-2cm. By the order of value it's close to data on Silicon borating (Filipovski et.al, 1994), which is 1.6-1.8·10- 2cm. It should be noted, that zone thickness of Zirconium shell interaction with molten Uranium is 0.2·10-2cm (Belash et.al, 2006).

We carried out calculation of Iron removal rate (G) from Zirconium by Iron concentration decrease during matched time intervals of plasma-arc-melting (PAM) with Hydrogen basing on the experimental data of (Mimura et.al, 1995) for 9.5 Pa and 50% of Hydrogen concentration in residual Argon. The data are shown in table 3. Mean residual Iron concentration at 15, 90 and 165 minutes of melting compiled 0.46, 0.01 and 2.2·10-4 mas.% respectively. These calculations had been compared with Iron evaporation rate from Zirconium melt obtained from Langmuir equation:

$$L = 0.0583 \gamma C\_{Fe} cp \sqrt{\frac{M}{T}}\tag{14}$$

This equation parameters are following: Iron activity coefficient in Zirconium (γ) is equal to 0.052 from [43], (СFe) - concentration of Iron in Zr, ( *р* ) – Iron vapor pressure at 2273K, (M) – Iron atomic mass, (T) – Kelvin temperature. Temperature of melting [46] is indicated in the 2350 – 2450К limits. Calculation of L by these temperatures founds L < G in both cases. Ratio

Fig. 15. Dependence of lgDFe (curve 1) and lgk (curve 2) on electric field intensity.

1995). Value of DFe we calculated by MD – method.

Uranium is 0.2·10-2cm (Belash et.al, 2006).

Zirconium melt obtained from Langmuir equation:

We have estimated diffusion layer thickness (x) by (13) equation (Flynn et.al, 1970). Calculation were carried out basing on СFe - experimental time – dependence (Mimura et.al,

( ,) 0 , <sup>2</sup> *x t*

In the equation С0 and С(х,t) are impurity concentrations in initial and refined Zirconium (Flynn et.al, 1970) and t is time of refining. Calculated (*x)* value is equal 7·10-2cm. By the order of value it's close to data on Silicon borating (Filipovski et.al, 1994), which is 1.6-1.8·10- 2cm. It should be noted, that zone thickness of Zirconium shell interaction with molten

We carried out calculation of Iron removal rate (G) from Zirconium by Iron concentration decrease during matched time intervals of plasma-arc-melting (PAM) with Hydrogen basing on the experimental data of (Mimura et.al, 1995) for 9.5 Pa and 50% of Hydrogen concentration in residual Argon. The data are shown in table 3. Mean residual Iron concentration at 15, 90 and 165 minutes of melting compiled 0.46, 0.01 and 2.2·10-4 mas.% respectively. These calculations had been compared with Iron evaporation rate from

> 0.0583 *Fe <sup>M</sup> L C cp <sup>T</sup>*

This equation parameters are following: Iron activity coefficient in Zirconium (γ) is equal to 0.052 from [43], (СFe) - concentration of Iron in Zr, ( *р* ) – Iron vapor pressure at 2273K, (M) – Iron atomic mass, (T) – Kelvin temperature. Temperature of melting [46] is indicated in the 2350 – 2450К limits. Calculation of L by these temperatures founds L < G in both cases. Ratio

*<sup>x</sup> C C erfc D t* 

*Fe*

(13)

(14)


G/L decreases at temperature increase. Obtained G/L ≈ 1.15 value at СFe= 0.46 and Т = 2450К may indicate, that close to 15% of Iron is being removed due to electro-magnetic forces affect, rather than evaporation.

Table 3. Dependences of L and G on mean residual Iron content in Zirconium for PAM process. CFe – mean Iron concentration: initial (15min.), middle (90min.), and final (165min.) stage of melt.
