Microstructural based hydrogen diffusion and trapping models applied to Fe–CX alloys

Journal Pre-proof Microstructural based hydrogen diffusion and trapping models applied to Fe–C-X alloys Andreas Drexler, Tom Depover, Silvia Leitner, Kim Verbeken, Werner Ecker PII:

S0925-8388(20)30420-5

DOI:

https://doi.org/10.1016/j.jallcom.2020.154057

Reference:

JALCOM 154057

To appear in:

Journal of Alloys and Compounds

Received Date: 11 December 2019 Revised Date:

23 January 2020

Accepted Date: 25 January 2020

Please cite this article as: A. Drexler, T. Depover, S. Leitner, K. Verbeken, W. Ecker, Microstructural based hydrogen diffusion and trapping models applied to Fe–C-X alloys, Journal of Alloys and Compounds (2020), doi: https://doi.org/10.1016/j.jallcom.2020.154057. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

A. Drexler: Writing-Original draft preparation, Methodology, Formal analysis, Visualization T. Depover: Writing-Original draft preparation, Methodology, Investigation S. Leitner: Writing-Original draft preparation, Investigation K. Verbeken: Supervision, Reviewing and Editing W. Ecker: Supervision, Reviewing and Editing

Microstructural based hydrogen diffusion and trapping models applied to Fe-C-X alloys Andreas Drexler1, Tom Depover2, Silvia Leitner1, Kim Verbeken2, Werner Ecker1 1

Materials Center Leoben Forschung GmbH, Roseggerstraße 12, 8700 Leoben, Austria

2

Department of Materials, Textiles and Chemical Engineering, Ghent University (UGent), Technologiepark 46, B-9052 Ghent, Belgium

Highlights Parametrized hydrogen diffusion model for Fe-C-V alloys Verification of the deep trapped hydrogen concentrations Microstructural based evolution models for trap densities Interpretation of the main trapping sites related to carbides on a lower scale Simulation of the diffusion depth on the component level

Abstract Hydrogen embrittlement of modern high strength steels consists of different interacting time-dependent mechanisms. One of these mechanisms is hydrogen diffusion and trapping to accumulate hydrogen in critical areas with high mechanical loads. Therefore, understanding hydrogen diffusion and trapping behavior of carbides containing high strength steels is an essential part to effectively increase the hydrogen resistance. For that purpose, a microstructural based model was developed and parametrized to Fe-C-V and Fe-C-Ti alloys. Generalized analytical equations were derived to describe the evolution of different kinds of trap densities with the measured carbide mean radius, annealing temperature or dislocation density. Finally, the models support the idea of hydrogen trapping at carbon vacancies and coherent interface positions. In future, these models are well suited for finite element process simulations of industrial components to predict the local solubility and chemical diffusion as demonstrated in the last section of this work.

Keywords Hydrogen embrittlement (HE); Finite element modeling (FEM); Thermal desorption spectroscopy (TDS); Carbides; Microstructural modeling, Trapping

Introduction Hydrogen embrittlement is a crucial problem for many high strength steels and consists of interacting time-dependent mechanisms. One of these mechanisms is diffusion of hydrogen to critical positions in the microstructures [1]. To avoid hydrogen accumulation along critical fracture paths, alternative alloy designs could be used to enhance hydrogen trapping away from these critical regions, for example, at precipitates. Therefore, understanding trapping of hydrogen at metal carbides is a key and can help to make steels less prone to hydrogen embrittlement (HE).

In order to reduce weight and increase load-bearing capacity of components, steel industry is striving to increase strength levels for many different applications [2,3]. It has been shown, however, that with increasing strength level, steels become more prone to hydrogen induced mechanical degradation and only a few wppm can cause delayed failure of a component [4]. To understand the underlying mechanisms many efforts were put into that topic during the last decade (e.g. HemS [5]). Despite this dedicated research, it is still not fully understood how strength of industrial steel grades can be increased without risking higher susceptibility to hydrogen embrittlement at the same time.

The susceptibility to HE and material strength correlate because the main strengthening

mechanisms

are

grain

boundary

refinement,

precipitation

strengthening, solid solution hardening and cold working. Each of these strengthening mechanisms can be further subdivided. For example, precipitation strengthening summarizes shearing and non-shearing (Orowan) mechanisms [6]. However, all these strengthening mechanisms have in common that they increase the crystal defect concentrations (e.g. interfaces, dislocation densities and point defects) and therefore also the local hydrogen solubility.

The temporary fixture of hydrogen on lattice defects is referred to as hydrogen trapping. Hydrogen trapping plays a key role in the environmental embrittlement of components and can be seen from different points of view [7]: On the one hand, trapping can lead to high hydrogen concentrations along potential fracture paths, whether at grain boundaries or along intense slip bands [7], which lead to HE. On the

other hand, modern alloy design could be used to trap hydrogen at lattice sites far from potential fracture paths; for example at matrix precipitates instead of dislocations or grain boundaries.

For that purpose, the role of irreversible and reversible trapping has been investigated for many decades [8]. According to Pressouyre and Bernstein [9], reversible and irreversible traps could be defined as follows: “A reversible trap is one at which hydrogen has a short residence time at the temperature of interest with an equivalent low interaction energy. For the same conditions, an irreversible trap is one with a negligible probability of releasing its hydrogen”. In other words, irreversible traps should not form networks and have high enough de-trapping or binding energies that the local relaxation time [10] with the surrounding microstructure becomes longer than the chemical diffusion time [11]. The often cited threshold for irreversible traps at room temperature lies above 60 kJ/mol (0.62 eV) [12]. However, in a previous work the “irreversible” nature of traps, especially on a lower scale, was questioned and it was suggested to use this concept with care [13]. The role of deep traps on the chemical diffusion behavior and environmental embrittlement is further discussed in the present work.

To separate the roles of different trap types and to correlate them with microstructural features, a hydrogen diffusion and trapping model in a finite element framework was calibrated for Fe-C-V and Fe-C-Ti model alloys, see previous paper [13]. A generalized Oriani approach considering hydrogen diffusion as rate depending step was implemented and allows modelling thermally activated hydrogen diffusion kinetics and fitting TDS spectra using different trap configurations. This allows a model based decomposition of TDS spectra into individual trapping contributions.

For efficient trap design, the microstructural differences between alloying concepts with respect to hydrogen must be understood. Vanadium (V) and titanium (Ti) are used in industrial processes to control austenite grain sizes [14] and are known for their strengthening properties in ferrite [15–17]. Even at small concentrations, especially Ti can increase the yield strength by several percent [18].

Both vanadium and titanium form the same NaCl type fcc lattice structure and the orientation relationship of the fcc lattice with the bcc iron is the Baker-Nutting orientation relationship (OR) [19] that is (001)fcc//(001)bcc, [100]fcc//[110]bcc, [010]fcc//[110]bcc with a habit plane of (001)fcc or (001)bcc. V4C3 and TiC as well as VC form thin platelets. Yamasaki and Bhadeshia [20] as well as Takahashi [21] reported TEM images of VC depicting a length to thickness ratio of about 10 for a particle length of 5 to 10 nm. Similar shape and structure of TiC carbides have also been reported by [22].

Despite this similarity, there are microstructural differences: the lattice parameter of TiC is with 4.336 Å [23] larger than that of VC with 4.160 Å. Vanadium also forms a second type of precipitate, V4C3 whose formation kinetic was investigated by [20]. V4C3 and TiC form an fcc lattice structure, which has a carbon vacancy that may act as a potential hydrogen trap, see Takahashi [21,24].

Understanding the interface structures and conditions at which coherency is lost, is necessary to draw conclusions about the differences between V4C3 and TiC with respect to hydrogen trapping and their macroscopic strengthening behavior. The loss of coherency, however, depends on the size, the elastic strain condition and interface configuration of the precipitate. The total energy sum is given by [25] where

is the chemical interface energy and

=

+

, the elastic strain energy

density, times the slab thickness h give the elastic strain energy contribution. Fors [25] used a first principle approach for VN to combine interface properties with Peierls-Nabarro framework continuum mechanical properties and calculate the precipitate size at the coherent/incoherent transition based on the total energy. Later, he compared several MC particles with respect to their structure and energy balance, [23].

The elastic energy density for TiC is much higher than for VC and V4C3 due to the larger lattice mismatch. The lattice mismatch also requires a higher periodicity of interface dislocations for TiC than for VC [23]. This also leads to a smaller critical slab thickness for TiC, at which a semicoherent/incoherent interface becomes more energetically favorable, for V4C3. This implies, that for a comparable particle thickness TiC is more likely to be incoherent than V4C3.

Occurring misfit dislocations and eventual loss of coherency due to the precipitate size has also been reported by means of TEM investigations: It has been reported that TiC already show fringes at a projected size of 5 nm, which indicates a loss of coherency [22]. To the authors knowledge no such study was published for V4C3 yet, but Depover and co-workers documented a decreasing amount of hydrogen trapped at coherent interfaces when the particle sizes exceeded 15-20 nm. This also correlates with theoretical predictions of the higher required interface periodicity predicted by Fors [23]. Comparing the tensile test results in [26] and [18] shows, that for similar particle sizes TiC leads to a more pronounced strengthening than V4C3. That may result from a different dislocation-inclusion interaction mechanism with the TiC particles that lose coherency at about 5 nm compared to VC which is reported to become incoherent at 15-20 nm.

The important role of misfit dislocations at the interface in terms of hydrogen trapping was investigated by [21], showing that larger VC platelets already containing misfit dislocations are favored compared to small coherent particles. However, not only the misfit dislocation cores but also the carbon vacancies at the precipitate/matrix interface have been shown to play a major role for hydrogen trapping [24] for VC and for TiC [27].

In the present work, we combine previous microstructural observations and compare TDS measurements for three different Fe-C-V alloys with measurements from a previous publication on a Fe-C-Ti system to determine differences in trapping behavior between the carbide types. The finite element model based interpretation of the spectra, developed in [13], is applied to split up the spectra in parts of different microstructural contributions and to give a model based interpretation of relevant trapping sites. In the last section of the present work, the parametrized diffusion model is applied to simulate the chemical diffusion coefficients and diffusion depths for Fe-C-V and Fe-C-Ti alloys.

Material and methods 1. Sample material Table 1: Chemical composition of the Fe-C-V alloys (wt%)

Fe-C-V

C

V

Other

Alloy A

0.1

0.570

200-300 ppm Al,

Alloy B

0.190

1.090

5-10 ppm S,

Alloy C

0.286

1.670

10-20 ppm, P, 15-20 ppm

Fe-C-Ti

C

Ti

Other

Alloy A

0.099

0.380

200-300 ppm Al,

Alloy B

0.202

0.740

5-10 ppm S,

Alloy C

0.313

1.340

10-20 ppm, P, 15-20 ppm

Three generic Fe-C-V and Fe-C-Ti materials were produced for the purpose of this study. The materials were cast in a Pfeiffer VSG100 incremental vacuum melting and casting unit under an argon gas atmosphere. Incremental casting was chosen to obtain three alloys for each ternary composition with increasing carbon content. The composition aims at stoichiometric amount of a ternary alloying element V or Ti (cf. Table 2). This carbon variance allows verifying the effect of carbides in different alloys while it also yields a reliable evaluation of their role with varying alloy strength level. The Fe-C-V and Fe-C-Ti alloys were further hot rolled and subsequently austenitized at 1250°C for 10 minutes, followed by a brine quench (7 wt% NaCl). This will be the first condition, referred to as ‘As-Q’. In addition, also a tempering treatment was applied to induce small carbides to assess their interaction with hydrogen. The secondary hardening due to the precipitation of V-based carbides was optimal at 600°C, as demonstrated in [26]. This will be the second condition, referred to as ‘Q&T’. Two different tempering times were applied, i.e. 1 hour and 2 hours, to evaluate the role of carbide growth on the hydrogen trapping ability. The sequence of the applied thermal treatment is schematically presented in Figure 1, where the different conditions are indicated.

Figure 1: Heat treatment process for the as-Q, Q&T-1h and Q&T-2h material states.

2. Diffusion and trapping simulations Hydrogen is a very diffusible interstitial atom in iron [28–30] and iron based alloys [31,32] and its mobility in solid materials could be close to the mobility of molecules in liquids. In the body-centered-cubic (bcc) crystal lattice, like in the ferrite phase, hydrogen diffuses by jumping from one tetrahedral position to the next at lower temperatures, and can also occupy octahedral positions with increasing temperatures due to thermal activation [28,33–35]. Describing hydrogen transport in metal alloys, two different diffusion coefficients are important: one is the tracer diffusion coefficient and the other is the chemical diffusion coefficient

[36]. The tracer diffusion

coefficient is always larger than or equal to the chemical diffusion coefficient, ≥

. (1)

While the tracer diffusion coefficient describes the mobility within a perfect defect free single crystal, the chemical diffusion coefficient describes the transport in real materials including crystal defects. The total local hydrogen concentration sample is a sum of the interstitial lattice concentration concentration

,

=

+ ∑!"#

,

.

(2)

in the

and the trapped hydrogen

Hydrogen trapping reduces the chemical transport by local binding of hydrogen to crystal defects, like precipitates, vacancies, dislocations, grain boundaries, etc and make it concentration dependent. An approximation with the effective diffusion coefficient measured with electrochemical permeation should thus be used with care [31,32]. For the trapping, different model approaches are available [37–39] and a long lasting discussion about the appropriateness of the different trapping models can be found in literature [40–42]. From a chemical point of view, trapping of hydrogen could be interpreted as a reversible reaction between an interstitial hydrogen atom $ with a vacant trap site % at a crystal defect, $ + % ⇋ $ + % . (3)

After the reaction, hydrogen is trapped $ at the trap site and an interstitial lattice

position becomes free % . Applying the fundamental equations according to the chemical kinetics of second order would give a rate equation similar to the approach published by McNabb and Foster [38], ' ( '

*+ , -1 − / 0 − )1* , -1 − / 0, =)

(4)

where , is the trap density, , is the density of interstitial lattice sites, / is the site *+ and )1* are occupation of traps, / is the site occupation of the interstitial lattice sites, )

*+ and )1* follow the kinetic type of constants for the forward and backward reactions. )

an Arrhenius relationship. However, even the chemical approach is still a simplification of the trapping and de-trapping mechanisms of hydrogen in the microstructure of real metallic alloys, because it neglects the local distribution of traps and therefore more complex models would be needed instead [43,44]. Note that these models have a number of fitting parameters and this may be the reason why direct comparisons of calculations against experimental data have not been reported yet, but the models have been used to study qualitative trends [40,45–49]. Sometimes the chemical reaction (3) between hydrogen and crystal defects in metallic alloys is much faster in the local microstructure than the long-range chemical diffusion [10]. In that case, hydrogen trapping would always reach local equilibrium and the well-known generalized Oriani equation according Svoboda and Fischer [11] is valid, 23 -#425 0

9

25 -#423

= 0

6 4 7

85

.

(5)

is the binding energy of the specific trap site. For sure, the appropriateness of this

local equilibrium approach has to be proven for each specific case and the interested

reader is referred to the work of Toribio and Kharin [10], of Wang and co-workers [7] or our previous work dealing with hydrogen trapping in Fe-C-Ti alloys. It was shown that the local equilibrium approach is reasonable for hydrogen trapping at carbides in a martensitic matrix [13]. In the sense of the local equilibrium approach used for the simulation of hydrogen trapping at VC, reference is also made to work of Kirchheim [50] and Turk et al. [51]. The local equilibrium approach has only two parameters for each trap and offers a deeper understanding of multiple trapping sites with microstructural evolutions due to

the consideration of trap densities , . Trap densities are the number of available

trapping positions at the microstructural defects and play a crucial role on hydrogen diffusion, trapping and solubility. For the model-based interpretation of complex shaped TDS spectra, a hydrogen diffusion model was implemented into the commercial Finite Element software

Abaqus [52]. The diffusion model considers lattice hydrogen diffusion and local trapping at defects. For that purpose, the default mass-diffusion analysis was extended by the local equilibrium approach in its generalized form for multiple traps [53,54] by using user-defined subroutines (UEL, UVARM). The tracer diffusion coefficient

= 9·

4

<,= 85



@

?

[13] and the density of interstitial lattice positions

, = 8.33 104E FGH/FF³ [55] were taken from literature. The initial interstitial hydrogen concentration

,K ,

which defines the hydrogen concentration after charging

and the set of trapping parameters L = {

9,

, , , } were calibrated independently for

each measured TDS spectra. For the optimization of the TDS spectra an in-house python script was used, based on the optimization algorithm according to “NelderMead” to find the global minimum of the residuum R. The residuum R was defined as O = PQ

∑! RS

∑U

,! − R S,! TUVWX,Y TUX[\V,Y ]P TZ

−

TZ

.

(6)

and corresponds to the difference between simulated jsim and measured hydrogen flux jmeas. In addition, the derivative of the TDS spectrum according to temperature T was considered in the residuum.

3. Thermal-Desorption-Spectroscopy (TDS) TDS was performed to assess the trapping ability of the tempered precipitates. Therefore, a heating rate of 600°C/h was chosen. Electrochemical hydrogen charging

was done in a 1g/L thiourea 0.5 M H2SO4 solution at a current density of 0.8 mA/cm2 for 1 hour. These conditions were chosen since no blisters or any hydrogen induced internal damage was detected, and a complete hydrogen saturation level was reached [18,26,56]. The experimental procedure to perform TDS measurements required one hour between the end of electrochemical hydrogen charging and the start of the TDS test. The delay results from the time that is required to reach a sufficient vacuum level in the analysis chamber, where the mass spectrometer detects the hydrogen signal.

Derivation of evolution equations 1. General aspects Table 2: Ab-initio calculated binding energies / segregation energies for different kinds of hydrogen trapping sites in iron based alloys. The energy values were taken from literature and are given in, both, kJ/mol and eV.

Defect

Energy ranges

Energy ranges in eV

in kJ/mol Edge: 45.4 [57]

Edge: 0.47 [57]

Screw: 26.1 [57],

Screw: 0.27 [57],

17.8, 18.6 [58]

0.185, 0.193 [58]

Vacancies

38.6 to 57.9 [59]

0.4 to 0.6 [59]

H6V-H1V

29.9 to 59.1 [60]

0.31 to 0.61 [60]

Tilt Σ3 (111): 37.6, Σ5

Tilt Σ3 (111): 0.39, Σ5

(210): 41.5, Σ5 (310): 78.2

(210): 0.43, Σ5 (310):

[61]

0.81 [61]

Twist Σ3: 25.1, Σ9: 65.6,

Twist Σ3: 0.26, Σ9: 0.68,

Σ11: 80.1, Σ17: 91.7 [62]

Σ11: 0.83, Σ17: 0.95 [62]

TiC: 105.2 [63], 125 [27]

TiC: 1.09 [63], 1.30 [27]

V4C3: 116 [27]

V4C3: 1.20 [27]

VC: 56.3 [64]

VC: 0.58 [64]

Precipitates

TiC: 31.0 [63], 48.0 [27]

TiC: 0.32, 0.50 [27]

(Coherent interface)

V4C3: -6 [27]

V4C3: -0.06 [27]

TiC: 44.4, 86.8 [63]

TiC: 0.46, 0.90 [63]

VC: 53.5 kJ/mol [64]

VC: 0.56 eV [64]

TiC: 48.2 [63]

TiC: 0.50 [63]

77 [65]

0.80 [65]

Dislocation

Grain boundaries

Precipitates (Bulk carbon vacancies)

Precipitates (Interface carbon vacancies) Precipitates (Interface misfit dislocations) Voids and Surfaces

The binding energy as well as the trap density affect diffusion and trapping and thus the characteristic peak positions of a hydrogen trap site in measured TDS spectra. Increasing the binding energy and/or the trap density would 1. retard hydrogen effusion from the sample, 2. shift the temperature of the peak maximum to higher values and

3. increase the area or the amount of hydrogen under the TDS peak. The binding energies are often used in literature to investigate the microstructural features of different trapping sites on a lower scale. Note, however, that binding energies are often ambiguous and different defects could offer similar binding energies for hydrogen atoms in steels. For that purpose, a literature surveying of calculated binding energies from ab-initio methods is given in Table 2. According to literature, the deepest trapping sites in steels are carbon vacancies inside of bulk carbides with values well above 100 kJ/mol. The second deepest traps, except free surfaces (e.g. at pores), are vacancies and grain boundaries with binding energies ranging from 60 kJ/mol to 90 kJ/mol. Mixed dislocations, which consist out of edge and screw constituents, can be regarded as shallow traps with binding energies for hydrogen of around 30 kJ/mol. Trap densities and trap occupations are less understood in literature and their role is often underrepresented in the discussion of TDS spectra. Trap densities can increase or decrease depending on the thermal or mechanical processing of the samples and therefore microstructural-based evolution models are needed. For example, aging leads to growth and coarsening of the carbides in the matrix. While during the growth stage trap densities at the interface can increase, they can decrease during the coarsening stage due to the overall reduction of interface area. Another example is the increase of dislocation densities during plastic deformation [55]. Hydrogen trapping at dislocations increases with increasing dislocation densities, although the characteristic binding energies stay in the same range.

4. Trap density equation To simulate hydrogen diffusion and trapping in thermal and mechanical processed materials, microstructural-based evolution equations are necessary. Existing microstructural based models are rare in literature to calculate the trap density based on microstructural information [51,55,66]. For that purpose, evolution equations for trapping at dislocations, carbide interfaces and vacancies are suggested in the present work. Dislocations are shallow hydrogen traps, but providing a huge number of hydrogen trap sites in the microstructure. According to Somerday et al. one can assume one

trap site per atomic plane threaded by a dislocation [55]. This gives the following evolution equation for the trap density of dislocations, , =

√ _ , '

(7)

where ` is the lattice parameter and a is the dislocation density.

Carbides offer different kinds of trap positions at the interface as well as in the bulk [13]. According to Table 2, the most prominent candidates are carbon vacancies at the interface and carbide bulk, coherent trap sites in the interface and misfit dislocations in the interface. Turk and co-workers [51] suggested an evolution for trap densities at carbide interfaces depending on the mean carbide radius. The trap density was expressed as the product of surface trapping capacity, the surface area and the number density of carbides. No additional assumptions were made for the trapping capacity of different defect sites, like trapping on coherent interface positions or carbon vacancies. For that purpose, a more generalized formulation is suggested in the present publication, which assumes that the trapping sites are arranged periodically at an interface. The periodicity depends on the kind of defect and is in the order of the lattice parameter. Following this assumption, one finds that the trap density at interfaces , , b is given by

_

, , b = d? c . (8) c

ae is the effective surface area of the carbides in the volume of the martensitic

matrix, f is the periodicity of trapping sites and ,e is the Avogadro number. The effective interfacial area per volume can be estimated from the measurements and Table 3 gives examples to calculate the effective surface area for different carbide morphologies. Table 3: Examples to calculate the average surface area of different kinds of carbide morphologies. g is the radius and H the mean distance between adjacent carbides.

Morphologies Spheres Cylinders Disks

Average surface area ae 13g Hh 6gℎ ae = h H

ae =

ae =

6g Hh

In general, carbon vacancies are known to be effective hydrogen traps. Their binding energies are above 60 kJ/mol and they can accommodate up to six hydrogen atoms in each vacancy [67]. Furthermore, carbon vacancies can be either intrinsic equilibrium point defects or extrinsic non-equilibrium point defects. To estimate the concentration of the intrinsic vacancies and thus the trap density , ,k

one can

assume following formula,

, ,k

=

k

=l

6 4 m\n

op 5q

l is the entropy factor of 1-10 in most metals,

.

(9) k

is the formation energy for mono

vacancies, rs is Temperature in Kelvin, )t is Boltzmann constant and

k

is the

vacancy concentration in bulk. The equilibrium vacancy concentration decreases exponentially with temperature and vacancy concentration can even be frozen in by quenching the material from higher temperatures. The fact the hydrogen stabilizes vacancies [67] and, hence, shift the equilibrium vacancy concentration to higher values is for the sake of simplicity not considered in this assumption, but it can easily be included by modifying the formation energy.

Results and discussion 1. Parametrization The parametrization of the Fe-C-Ti alloys was part of a previous work and the interested reader is referred to the work of Drexler et al. [13]. In the present work, the parametrization section deals with the Fe-C-V alloys. According to Figure 2, the measured TDS spectra of Alloy A, B and C in their respective aging treatment (As-Q, Q&T-1h and Q&T-2h) have similar shape and differ mainly by the amount of hydrogen under the TDS curve. While the as-quenched material shows a single peak in the low temperature range, aging at 600 °C increases significantly the complexity of the TDS spectra. The hydrogen content under the TDS curve is higher for the 1 hour aged material than for the 2 hours. Furthermore, with increasing carbon content from alloy A to C the maximum mass flux increases as well.

a)

b)

c) Figure 2: Fitted TDS spectra of a) alloy A, b) alloy B and c) alloy C. The spectra of the as-quenched (AsQ), quenched & tempered – 1 hour (Q&T – 1h) and – 2 hours (Q&T – 2h) was fitted independent of each other.

The parameters of the thermodynamic based diffusion model were calibrated to the measured TDS spectra. The inverse optimization routine outlined in the previous section was used without any parameter constraints and independently for each set of TDS spectra corresponding to alloy A, alloy B and alloy C, respectively. One of the main issues during calibration was the local re-trapping during heating. The diffusion model considered the one hour of vacuum treatment before temperature ramping, which was necessary in the experiments to reach a sufficient vacuum in the analysis chamber before measurement. The temperature field was assumed homogenous in sample during the calculation. This assumption was verified by an additional heat transfer analysis considering appropriate convection and radiation boundary conditions. With the parametrized diffusion model, a nearly perfect agreement with the measured TDS spectra was reached (as shown in Figure 2).

Figure 3: Line spectra derived for Fe-C-V and Fe-C-Ti alloys as-quenched and heat treated for 1 hour and 2 hours.

The calibrated binding energies

9

and trap densities , are summarized in Figure 3

for alloy A, B and C in form of a line spectrum. The binding energies are defined as the difference in energy between trap site and interstitial lattice position. Therefore,

the density at zero binding energy relates to the available interstitial positions , [55]

and is considered as the reference state for hydrogen trapping. The calibrated trap

densities are all orders of magnitudes smaller than the density of interstitial lattice positions, which is reasonable. The trap densities available for shallow traps are higher than for deep traps. Table 4: Calibrated binding energies for V4C3 and TiC carbides. The values for TiC carbides are taken from a previous work [13]. The averaged binding energies and variations are given in kJ/mol and eV.

Binding energy

Binding energy

Binding energy

Binding energy

[kJ/mol]

[kJ/mol]

[eV]

[eV]

V4C3 carbides

TiC carbides

V4C3 carbides

TiC carbides

Trap one

17 +/- 4

17 +/- 3

0.18 +/- 0.04

0.18 +/- 0.03

Trap two

32 +/- 4

24 +/- 4

0.34 +/- 0.04

0.25+/- 0.04

Trap three

58 +/- 2

58 +/- 3

0.60 +/- 0.02

0.60 +/- 0.03

Number [-]

Trap four

73 +/- 4

76 +/- 5

0.76 +/- 0.04

0.79 +/- 0.05

Trap five

92 +/- 2

103 +/- 9

0.95 +/- 0.02

1.07 +/- 0.09

The calibrated binding energies for each individual trap are in very good agreement and differ less than +/- 5 kJ/mol for all nine Fe-C-V-TDS spectra. Furthermore, the calibrated binding energies for hydrogen trapping at V4C3 carbides are in a reasonable agreement with the binding energies found for hydrogen trapping at TiC carbides (as shown in Table 2) [13]. Assuming that the binding energies are characteristic for the crystal defect sites, one could assume that main trapping positions for hydrogen at V4C3 carbides and TiC carbides are similar. Only trap two and trap five deviate slightly between the different Fe-C-V and Fe-C-Ti alloys.

2. Verification

a)

b)

c) Figure 4: a) Verification of the parametrized diffusion model with an additional room temperature vacuum treatment for 72 hours before recording the TDS spectrum. b) The total hydrogen concentration decreases in the sample during the vacuum treatment. c) Subdivision of the total hydrogen concentration in the sample after 72 hours vacuum treatment into the lattice hydrogen concentration and hydrogen concentrations , trapped at different defect sites u.

For the verification of the calibrated diffusion models, an additional experiment was performed to test, the predictability of the parameter set and model assumptions. To that purpose, a hydrogen charged sample out of Alloy C – Q&T-1h experiences an additional room temperature vacuum treatment for 72 hours before recording the TDS spectrum. During that time, hydrogen can effuse from the surface and creates a cosinusoidal concentration profile in the sample (as shown in Figure 4b). Only the high temperature part of the original TDS spectrum is received in the measurements during the final temperature ramping (as shown in Figure 4a). A comparison with the diffusion simulation of the 72 hours of vacuum treatment gives a very good agreement with the measurements. No additional calibration was performed and the trapping parameters were kept constant. According to the simulation results of Alloy C – Q&T-1h, Figure 4b shows the evolution of the total hydrogen concentration profile in samples after 0, 1 and 72 hours of vacuum treatment. As outlined in the method section, the initial hydrogen distribution after charging is assumed to be homogenous. A concentration level of 5.28 wppm was chosen according to the measurements in [26] and the sample thickness was 1.5 mm. The simulated hydrogen content after 1 and 72 hours

charging is given as a function of depth below the surface. It is seen from the figure that 1) Hydrogen does diffuse out of the sample. 2) Hydrogen content increases rapidly with increasing depth into the material. 3) The longer the vacuum time, the lower the hydrogen content. Vacuum time has an influence on the hydrogen content adjacent to the surface as well as deep inside the material. 4) The local hydrogen concentration at the surface and in the middle section of the sample can differ by a factor 2 to 3. 5) The hydrogen content in the middle of the sample decreases logarithmically with time. In Figure 4c the total hydrogen concentration profile in a sample of Alloy C – Q&T-1h after 72 hours vacuum treatment is split into the hydrogen trap concentrations for each individual trap. Obviously, most of the hydrogen is stored in trap three to five. While the hydrogen content in trap three decreases significantly from the inner part of the sample to the surface, the hydrogen concentrations in trap four and five are hardly affected by the vacuum treatment. According to the calibrated parameters trap four and five have binding energies well above 60 kJ/mol (0.6 eV). In literature, such traps are often called “irreversible” [68]. Note, however, that in the present approach a local equilibrium approach is used for all kind of traps. In other words, the local equilibrium approach is able to describe the high temperature part of the TDS spectrum very well although it does not consider irreversible trapping. Furthermore, the trapped hydrogen concentration of trap one and two becomes almost zero after the vacuum treatment of 72 hours. Before temperature ramping the deep traps with binding energies above 60 kJ/mol stay mainly filled in the model, which is in the agreement the experimental observation and the mobile hydrogen concentration [26] observed with in the experiments can be related to trap one to three.

5. Hydrogen interaction with carbides Interpretation of complex shaped TDS spectra is still a challenge and a deep understanding of the underlying microstructure, hydrogen-defect interaction and diffusion is necessary. For that purpose, a full investigation of the trapping states in steels based on TDS measurements needs additional characterization and evaluation:

1.) experimental characterization of the underlying microstructure, 2.) evaluation of trapping energies (e.g. binding energies ) and

9

or activation energies

3.) evaluation of trap densities , by using modeling approaches.

Point one and two are commonly used in the literature. Especially, the change of TDS spectra with the evolution of microstructure helps to identify major trapping sites, e.g. precipitates or martensitic matrix. Alternatively, trapping energies can be used to distinguish hydrogen trapping even on lower scale and can give deeper insights into the investigated material. Table 5: Summary of the microstructure characterization as the first step of interpretation. The measured values for Fe-C-V were taken from [26] and for Fe-C-Ti were taken from [18].

Fe-C-V

Q&T – 1 hour (Alloy C)

Q&T – 2 hour (Alloy C)

Mean precipitation size

12.5 +/- 0.25 nm

17.5 +/- 0.25 nm

12.3 +/- 4.8 nm

14.2 +/- 5.4 nm

Fe-C-Ti

Q&T – 1 hour (Alloy C)

Q&T – 2 hour (Alloy C)

Mean precipitation size

10 +/- 5 nm

50 +/- 5 nm

5.01 +/- 1.15 nm

8.05 +/- 2.89 nm

Mean distance between adjacent carbides

Mean distance between adjacent carbides

The experimental characterization of the underlying microstructure in the investigated Fe-C-V and Fe-C-Ti alloys was done by using e.g. LOM (Light Optical Microscopy), SEM (Scanning Electron Microscopy) and TEM (Transmission Electron Microscopy). Details of the experimental techniques are given in the work of Depover and Verbeken [18,26]. The carbide size distribution and the mean distance between adjacent carbides were measured and the results are summarized for both alloys in the Table 5. According to these data, the as-quenched material state consists of primary carbides embedded in a martensitic matrix. During aging at 600 °C for 1 hour and 2 hours, the mean carbide sizes increase from 12.5 nm to 17.5 nm and from 10 nm to 50 nm for V4C3 carbides and TiC carbides, respectively. The measured mean carbide size is much smaller for the V4C3 carbides than for the TiC carbides after two hours aging. Due to coarsening of the carbides in the Q&T – 2 hours materials, the

mean distance between adjacent carbides increases as well. While nucleation and growth of secondary carbides are the main mechanisms in the first hour of aging, growth and coarsening become the main mechanisms for 2 hours aging. The aging treatment for two hours leads to a shift of the measured carbide size distribution to higher radii. The decrease of hydrogen content between 1 hour aging and 2 hours aging could be related to the loss of the smallest size class with a radii between 0 – 5 nm and the reduction of the total interfacial area during coarsening. In the context of metal carbides in steels, the most important trapping sites should be situated close to or directly at the interface between the carbides and the matrix. These trapping sites can be easily occupied by hydrogen diffusing in the matrix and have large binding energies according to Table 4. Especially, carbon vacancies (around 44 – 87 kJ/mol), coherent trapping sites at the interface (around 31 to 48 kJ/mol) and misfit dislocations near the interface (around 50 kJ/mol) are most likely favored for hydrogen trapping. In present work, binding energies of Fe-C-V alloys were calibrated to the measured TDS spectra and compared to the binding energies found for Fe-C-Ti alloys previously (as shown in Table 2). The binding energies agree very well with the values found for TiC carbides. Only trap two and trap five are slightly different. Due to the fact that trap one is also visible in the as-quenched state (As-Q) and agrees very well for both alloys, it can be related to the martensitic matrix. Trap two to five arise during tempering and are related to trapping sites at carbides. Tempering for 2 hours (Q&T-2h) reduces the temperature range of the TDS spectra compared to 1 hour aging (Q&T-1h). The high temperature shoulder vanishes. This leads to the conclusion that the deepest trapping site is not occupied any more. The calibrated binding energy of the deepest trap (named trap five) in V4C3 carbides is 92 kJ/mol. Comparing this high value with the calculated values in Table 4, trap five can be related to carbon vacancies in the carbide bulk. The idea is that small carbides, which are visible in the carbide size distribution after one hour of tempering and which can be occupied by hydrogen due to the small diffusion distances vanish after the two hours aging treatment. Anyhow, trap two, three and four relate to the carbide interfaces and stay visible in the TDS spectra even after two hours aging, but the corresponding trap densities are reduced due to reduction of the total interface area.

Figure 5: Mean trap density evolution of trap one in the martensitic matrix of Fe-C-V alloys.

As already outlined, trap densities play a crucial role for diffusion and trapping and thus in the interpretation of TDS spectra. Considering the evolution of the trap densities with the carbide mean radius gives further information about the trapping sites on a lower scale. For example, Figure 5 shows the calibrated trap density of trap one in Alloy C. The error bars were calculated form the variations between Alloy A, B and C and therefore, should give an estimate of the accuracy of the model based calibration. The density of trap one does not evolve significantly with the measured mean radii and has a magnitude of around 7·10-5 mol/mm³. Due to its appearance in the asquenched material state (As-Q), its large magnitude and its low binding energy, trap one relates to the martensitic matrix. There, two different defect sites could contribute the hydrogen trapping: one is the martensitic lath boundaries and the other is the dislocation density. According to measurements published in literature, the dislocation density is around 1014 to 1015 m-² in martensitic phases [69,70]. Applying formula 7 to this calibrated trap density, the estimated dislocation densities for trap 1 would give 1019 m-², which is much higher than experimentally observed values for dislocation densities. Therefore, trap one with a binding energy of around 17 kJ/mol is dominated by trap sites in the martensitic lath boundaries.

a)

b) Figure 6: Evolution of the mean trap densities and deviations for alloy C with the measured mean carbide radius. The results are given for a) V4C3 and b) TiC.

In Figure 6 a and b the evolution of the calibrated trap densities for trap two to five are given as a function of the measured mean radii for V4C3 carbides and TiC carbides, respectively. Due to the evolution of these trap sites during tempering, they can be related to trap sites at the carbide interface. At a first glance, the trends of the

calibrated trap densities appear different: While the trap density of e.g. trap two increases from 12.5 nm to 17.5 nm, the trap density at the TiC carbides decreases from 10 nm to 50 nm. However, the measured mean radii are different as well and the trend between two support points are just generated by interpolation just in order to illustrate the trends. To estimate the trap density at the interface according equation 8, a disk shaped morphology is assumed for both kinds of investigated carbides. Assuming an array of misfit dislocations at the interface of V4C3 carbides and TiC carbides, a periodicity of 7.53 nm and 3.70 nm is given in literature, respectively [23]. According to the difference in periodicity, equation 8 gives a much lower trap density for V4C3 carbides of around 10-10 mol/mm³ compared to TiC carbides with around 10-9 mol/mm³. However, the calculated trap densities for misfit dislocations are much smaller than the calibrated trap densities found for trap two. Other possible trap sites at the interface with similar binding energies are coherent interface positions directly at the interface. Directly at the interface, these trap sites offer binding energies of around 30 kJ/mol to 50 kJ/mol. The periodicity of coherent trap sites can be approximated by the lattice parameter of the matrix at the interface. According to Fors and Wahnström [23] the lattice parameter is 0.2913 nm for V4C3 carbides and 0.2993 nm for TiC. By inserting these values into equation 8, the magnitude of the trap density is 10-7 mol/mm³, which agrees very well with the calibrated trap densities of trap two (as shown in Figure 6). Furthermore, the difference between the trap density in V4C3 carbides and TiC carbides is predicted very well. Trap three and four relate to carbon vacancies at the interface of the carbides, due their large calibrated binding energies of 58 kJ/mol to 76 kJ/mol and their correlation with the mean radius. Less is known in literature about the density of carbon vacancies at the interface and no periodicity could be found to estimate the trap density. However, the suggested models for trap densities could be used to give a rough estimation of the periodicity of carbon vacancies at the carbide interface. Therefore, the periodicities of carbon vacancies at V4C3 carbide and TiC carbide interfaces are 1.4 nm and 3.3 nm, respectively. The value for V4C3 carbides gives a number density of 0.51 nm-2 at the interface of V4C3 and is in very good agreement of the experimentally observed density of 0.6 +/- 1 nm-2 by Takahashi et al. [24]

Furthermore, the carbon vacancy concentration at the interface of V4C3 carbides seems to be higher than of TiC carbides. The calibrated trap density of trap five has the lowest value of both alloys. Its magnitude is in the order of 10-10 mol/mm³ and was related to carbon vacancies in the carbide bulk. Due to the quenching after the aging treatment at 600 °C, it is assumed that the vacancy concentration in small TiC carbides is close to equilibrium at 600 °C. For TiC carbide bulk, the vacancy formation energy of 0.9 eV is given in literature [71]. This gives a calculated vacancy concentration for TiC carbides of around 10-10 mol/mm³, which is very close to the calibrated value from the measured TDS spectra (as shown in Figure 6b). For the V4C3 carbide bulk, the authors are not aware of vacancy formation energies. To that purpose, based on the theoretical considerations and the calibrated trap densities, a rough estimation of around 0.95 eV is suggested for the vacancy formation energy of V4C3 carbide bulk. Table 6: Microstructural defects corresponding to the hydrogen traps in Fe-C-V alloys and Fe-C-Ti alloys.

Number [-]

Fe-C-V alloy

Fe-C-Ti alloy

Trap one

Martensitic lath boundaries

Martensitic lath boundaries

Trap two

Coherent interface

Coherent interface

Carbon vacancies in

Carbon vacancies in

carbide/matrix interface

carbide/matrix interface

Carbon vacancies in

Carbon vacancies in

carbide/matrix interface

carbide/matrix interface

Carbon vacancies in carbide

Carbon vacancies in

(<20 nm)

carbide (<20 nm)

Trap three

Trap four

Trap five

Although, all the suggested models for trap densities at carbide interfaces and in carbide bulk are based on simplified assumptions, the discrepancies with the calibrate trap densities are small. The results of trap densities, binding energies and experimental characterizations give a full picture of hydrogen trap sites on a lower scale at metal carbides embedded in a martensitic matrix (as shown in Table 6). Carbon vacancies and coherent trap sites are the most dominant hydrogen traps for the investigated carbides and bind the largest amount of hydrogen. The hydrogen trapping capacity strongly depends on the effective interface area of carbides rather than their volume fraction. Thus, smaller carbides are better hydrogen traps for a

given carbide volume fraction and most likely increase the precipitation strength. However, both kinds of carbides offer traps with critical binding energies below 60 kJ/mol, which release trapped hydrogen more easily in case of fracture and could lead to increased hydrogen concentration along the fracture paths. The presented models in combination with thermo-kinetic calculations could also be used in future to predict local solubility and chemical diffusion for new alloy designs with optimized hydrogen resistivity.

6. Simulation of local solubility and chemical diffusivity

a)

b) Figure 7: Calculated chemical diffusion coefficient for a) Fe-C-V – Alloy C and b) Fe-C-Ti-Alloy C. The physical meaningful chemical diffusion coefficients are compared with the measured apparent diffusion coefficients by using electrochemical permeation.

The calibrated diffusion and trapping models such as for Fe-C-V and Fe-C-Ti offer the possibility to study the role of local properties in industrial components, like the local hydrogen concentration

v

and

the chemical diffusivity

.

The chemical diffusion coefficient in the framework of localized trapping was derived in the work of Svoboda and Fischer [11] for a single trap and can be easily extended for multiple traps. In general, the chemical diffusion coefficient depends on temperature, hydrogen concentration, the number of trap sites and trapping parameters of the investigated material. Therefore, the chemical diffusion coefficient is a local quantity in industrial components. On the one hand, the temperature dependency is related to the Arrhenius behavior of the tracer diffusion coefficient as well as to the thermal activation of trapping. With increasing temperature, the equilibrium concentration of the trapping reaction given in equation 3 is shifted to the so-called reactants side, which leads to a stepwise increase of the chemical diffusion coefficient with increasing temperature. On the other hand, the concentration dependency is related to the occupation of the available trap sites. Once all the trap sites are occupied, the effective trap does not contribute anymore to the chemical diffusion. The chemical diffusion coefficients as a function of the local hydrogen concentration are given for the Fe-C-V and Fe-C-Ti alloys in Figure 7. No generalized conclusion can be drawn due to the strong hydrogen concentration dependency. Due to the different numbers of effective traps and their parameters considered in the diffusion models, the concentration dependency differs significantly for the three different aging states. While only one trap in the As-Q material leads to a single step in the concentration profile, five or four traps in the Q&T-1h and Q&T-2 materials lead to more steps, as shown in Figure 7. Therefore, a direct comparison between the physical meaningful chemical diffusion coefficient and the constant effective diffusion coefficient measured by permeation experiments [18,26] is very difficult and only possible in simple cases. For example, the chemical diffusion coefficient for the As-Q material is concentration independent for a large concentration range. In this range, the chemical diffusion coefficient derived from the parametrized diffusion model is in good agreement with the measured effective diffusion coefficient. The difference is less than one order magnitude. A comparison for the Q&T-1h and Q&T-2h materials is not reasonable, because of the strong dependency of the chemical diffusion coefficient on the local hydrogen concentration in the sample during the permeation experiment. Nevertheless, the intersection point of the effective diffusion coefficient

with the chemical diffusion coefficient is in the range between 0.5 wppm to 2 wppm. This seems to be a reasonable for electrochemical permeation experiments. In industrial components (e.g. for car bodies, fasteners or line pipes) the hydrogen concentration is often heterogeneously distributed [72]. For demonstration reasons, the parametrized diffusion models for both alloys are used to simulate hydrogen diffusion and outgassing. In the present paper, the “industrial component” is represented by a one dimensional diffusion model with a length of 100 mm. According to the work of Vecchi et al. [73,74], hydrogen uptake at the surface is modeled by a surface flux of 10-15 mol/mm². Natural boundary conditions are used on the other surface side. For the simulation of the outgassing the hydrogen concentration is set to 0 wppm. The initial hydrogen concentration is assumed to be zero.

a)

b) Figure 8: Depth profile of the hydrogen concentration for a) Fe-C-V and b) Fe-C-Ti alloy C after 48 hours of charging and after 48 hours of outgassing.

Figure 8 shows the hydrogen concentration profile after 48 hours of charging and 48 hours of degassing. It is seen from the simulation that during charging hydrogen diffuses into the component to a depth of around 0.1 mm, hydrogen concentration is highest at the surface and hydrogen concentration decrease rapidly with increasing depth. After the charging, an outgassing of the hydrogen is simulated. Hydrogen effuses from the surface of the model. For a better comparison the concentration profiles are included in Figure 8. It can be seen that the hydrogen concentration decreases significantly, the position of the highest hydrogen concentration does not occur anymore at the surface and is shifted to inner parts and hydrogen diffuses to depth of around 0.3 mm. The reason for the shift of the hydrogen maximum to inner parts and the deeper hydrogen penetration during outgassing are the two different hydrogen concentration gradients. While one is pointing to the surface and the other to the inner parts of the model. In general the hydrogen concentration profile in the sample strongly depends on the charging conditions at the surface and the microstructure of the investigated material.

Conclusions The following conclusions can be made from the present paper dealing with hydrogen diffusion in ternary alloys containing carbides with different size distribution: A model-based evaluation of complex shaped TDS spectra allows a better interpretation of hydrogen trapping on a lower scale and delivers parametrized diffusion models to simulate hydrogen distribution in industrial components. An inverse calibration method is performed to determine the effective trap densities and binding energies of the hydrogen diffusion model. The parametrized diffusion model is verified by an additional TDS spectra recorded after a 72 hours vacuum treatment. It was shown that the diffusion model and localized trapping approach is suitable to describe the deep (“irreversible”) trapped hydrogen very well without additional calibration. Therefore, the definition of irreversible traps should be used with care [13].

A comprehensive literature overview of hydrogen segregation energies in iron and steels is given in the present paper. Based on the literature survey the deepest traps in steels are carbon vacancies in bulk carbides, which are difficult for hydrogen to occupy by diffusion. Most likely, hydrogen occupies the deep traps available at the interface between carbides and matrix. Hydrogen can easily diffuse into these traps from the matrix. In addition to the binding energies, trap densities have a detrimental effect on the shape of TDS spectra and, therefore, on hydrogen diffusion in steel components. For that purpose, generalized evolution equations are derived and parametrized with the help of the present diffusion model. The evolution equations correlate the trap densities of dislocations, carbon vacancies and coherent interface positions with the microstructural evolution during thermomechanical treatments. The microstructural diffusion model applied to Fe-C-V alloys revealed coherent trap sites and carbon vacancies as the most effective traps at the interface of carbides. The effective trap in the martensitic matrix is most likely dominated by hydrogen trapping at the lath boundaries. Carbon vacancies in carbide bulk are difficult to be occupied by hydrogen diffusion and therefore, are only visible in for very small size class. Finally, the parametrized diffusion models for Fe-C-V and Fe-C-Ti alloys are used to simulate the hydrogen depth profile during charging and outgassing on a component level. The diffusion depth strongly depends on the chemical diffusion coefficient, which is in the case of hydrogen diffusion strongly concentration dependent and a local quantity in an industrial component. Anyway, the use of effective diffusion coefficients for simplified hydrogen diffusion simulations is not recommended for industrial steel grades.

Acknowledgment The authors gratefully acknowledge the financial support under the scope of the COMET program within the K2 Center “Integrated Computational Material, Process and Product Engineering (IC-MPPE)” (Project No 859480). This program is supported by the Austrian Federal Ministries for Transport, Innovation and Technology (BMVIT) and for Digital and Economic Affairs (BMDW), represented by the Austrian research funding association (FFG), and the federal states of Styria, Upper Austria and Tyrol. TD holds a postdoctoral fellowship of the Research Foundation - Flanders (FWO) via grant 12ZO420N. The authors also wish to thank the Special Research Fund (BOF), UGent (BOF15/BAS/062 and BOF01P03516).

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