COMSOL 6.4 - Hydrogen Diffusion in Metals

Hydrogen embrittlement refers to the degradation of metal ductility due to the absorption of hydrogen. The metal becomes more brittle and thus cracks might initiate at lower stress levels. It is important to estimate hydrogen concentration and the speed at which it diffuses into the metal in order to predict and avoid crack formation and propagation.

This model shows how to simulate the uptake and diffusion of hydrogen in a notched metal sample from an aqueous electrolyte. It uses the Transport in Solids interface to model both the concentration-driven and stress-driven diffusion in a solid.

Model Definition

The model geometry consists of a metal part with an initial defect as depicted in Figure 1.

Figure 1: Model geometry.

The part size is L = 20 mm, and the defect is hd = 0.4 mm wide and extends for ld = 10 mm into the specimen.

The electrolyte is located on the left side of the metal part and a hydrogen influx is assigned on the boundary where the solid is in contact with it. The initial defect is also filled with electrolyte.

The solid is constrained on the bottom and right edges with a roller boundary condition. A prescribed displacement of 0.05 mm is applied on the top boundary to produce a state of tensile stress and study its effect on hydrogen diffusion. Although not considered in this example, the symmetry about the xz-plane through the defect could be exploited to reduce the computational size.

The hydrogen diffusion is studied using the mass balance equation

where c is the concentration (SI unit: mol/m3), Γ is the molar flux, and G is a source term. Following Ref. 1, hydrogen can be present both in the interstitial metal lattice and in traps,

To study the diffusion of the hydrogen in the interstitial lattice, the mass balance equation is rewritten as follows,

where only two trap types are considered. A relation between the trap and lattice concentration can be found assuming a low lattice occupancy, see Ref. 1 for details.

Both the concentration gradient (Fick’s law) and the stress gradient drive the diffusion of hydrogen in the metal,

where D is the lattice diffusion coefficient, ΩH is the partial molar volume of hydrogen, and

is the hydrostatic stress. The diffusion of hydrogen in the metal induces swelling strains that are modeled as linearly dependent on the concentration

The electrolyte and the chemical reactions therein are not modeled explicitly; instead an approximation for the flux as a function of the pH and the electrolyte potential, Ve, is used (Ref. 1),

where

and

The electrolyte parameters used in the influx expression are inspired from Ref. 1 and reported in Table 1.

Table 1: MODEL PARAMETERS.
Parameter	Value
Diffusion coefficient (D)	2E-9 m2/s
Metal potential (Vm)	-0.5 V
Equilibrium potential (Veq)	0 V
Electrolyte potential (Ve)	-0.025 V
Hydrogen partial molar volume (ΩH)	2E-6 m3/mol
Temperature (T)	293.15 K
Volmer forward reaction coefficient, acid (αva)	0.48
Volmer forward reaction coefficient, basic (αvb)	0.48
Heyrovsky forward reaction coefficient, acid (αha)	0.33
Heyrovsky forward reaction coefficient, basic (αhb)	0.33
Volmer forward reaction rate, acid (kva)	1E-4 m/s
Volmer forward reaction rate, basic (kvb)	1E-8 mol/m2s
Heyrovsky forward reaction rate, acid (kha)	1E-10 m/s
Heyrovsky forward reaction rate, basic (khb)	9E-10 mol/m2s
Tafel forward reaction rate (kT)	1E-6 mol/m2s
Absorption forward reaction rate (ka)	1E5 m/s
Absorption backward reaction rate (kar)	9E9 m/s
Lattice sites concentration (N)	1e6 mol/m3
Trap 1 concentration (nt1)	2 mol/m3
Trap 2 concentration (nt2)	1 mol/m3
pH	13

Results and Discussion

Figure 2 shows the hydrogen concentration and flux after 2 hours. The concentration is higher at the sharp corner of the crack because of the greater area exposed to the electrolyte.

Figure 2: Hydrogen concentration and hydrogen flux after 2 hours.

Figure 3 shows the flux direction for the concentration-driven (red) and stress-driven (black) contribution to the total flux.

Figure 3: Streamline plot of the concentration-driven (red) and stress-driven (black) hydrogen flux.

The effect of the stress on hydrogen absorption can be seen in Figure 4. The figure compares the time evolution of the hydrogen concentration at the tip of the defect with and without the stress contribution to the flux. The stress-induced flux increases the absorption of hydrogen.

Figure 4: Hydrogen concentration with and without stress contribution to the diffusion flux at the tip of the defect.

The hydrogen concentration after 2 hours along the centerline of the metal sample is shown in Figure 5 for both cases.

Figure 5: Hydrogen diffusion depth along the centerline of the specimen.

Figure 6 below shows the hydrostatic stress and stress-driven flux in the deformed configuration. There is a stress concentration at the crack tip, which enhances the hydrogen adsorption

Figure 6: Hydrostatic stress and stress-driven hydrogen flux in the deformed configuration (100 times amplified).

Notes About the COMSOL Implementation

The two-way coupling is obtained with the multiphysics coupling. It takes into account both the stress-driven diffusion and the swelling deformation due to hydrogen diffusion. If only the stress-driven diffusion effect is of interest, that is, a one-way coupling, a or an subnode to the node in the Transport in Solids interface can be used.

Reference

1. T. Hageman and E. Martinez-Paneda, An electro-chemo-mechanical framework for predicting hydrogen uptake in metals due to aqueous electrolytes, Corrosion Science, vol. 208, 110681, 2022

Structural_Mechanics_Module/Diffusion_in_Solids/hydrogen_diffusion

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