Deuterium atom exposure of self-damaged tungsten#
This validation case is NRA measurements of D in self-damaged W performed by Markelj et al. [25].
The experimental procedure included three main phases. A high-purity polycrystalline W sample of 0.8 mm thickness was pre-damaged with 20 MeV W ions. The pre-damaged sample was then exposed to low-energy (~0.3 eV) D atomic flux of \(5.8\times 10^{18}\,\textrm{m}^{-2}\textrm{s}^{-1}\) at 600 K. The exposure continued until the D fluence of \(1\times 10^{24}\,\mathrm{m}^{-2}\) was reached. Finally, an isothermal desorption of D for 43 h. at 600 K was conducted.
The FESTIM model is based on the approach of Hodille et al. [26]. Only isothermal D exposure and desorption phases are simulated omitting intermediate cooling/re-heating steps. For the surface processes, adsorption of low-energy atoms, desorption of molecules (Langmuir-Hinshelwood recombination), and recombination of an adsorbed atom with an incident atom (Eley-Rideal recombination) are considered.
The D diffusivity in W is defined by scaling the corresponding value for H (Fernandez et al. [24]) by a factor of \(1/\sqrt{2}\). Five types of trapping sites are included to reproduce the experimental data: two intrinsic traps and three extrinsic traps with sigmoidal distribution (\(f\)) within the damaged layer:
\[
f(x)=\dfrac{1}{1+\exp\left(\dfrac{x-x_0}{\Delta x}\right)},
\]
where \(x_0=2.2\,\mu\textrm{m}\), \(\Delta x=0.154\,\mu\textrm{m}\).
The FESTIM results are compared to the experimental data and the results of MHIMS simulation, both taken from [26].
FESTIM model#
Show code cell source
import festim as F
import fenics as f
import numpy as np
import sympy as sp
import h_transport_materials as htm
################### PARAMETERS ###################
# Exposure conditions
Gamma_atom = 5.8e18
T_exposure = 600
t_exposure = 1e24 / Gamma_atom
t_des = 52 * 3600
final_time = t_exposure + t_des
# Sample
L = 0.8e-3 # half thickness, m
# W properties
rho_W = 6.3e28 # W atomic concentration, m^-3
n_IS = 6 * rho_W # concentration of interstitial sites, m^-3
n_surf = 6.9 * rho_W ** (2 / 3) # concentration of adsorption sites, m^-2
nu0 = 1e13 # attempt frequency, s^-1
SP = 0.19
D_H = htm.diffusivities.filter(material=htm.Tungsten, author="fernandez")[0]
D0 = D_H.pre_exp.magnitude / np.sqrt(2) # diffusivity pre-factor, m^2 s^-1
E_diff = D_H.act_energy.magnitude # diffusion activation energy, eV
lambda_IS = 110e-12 # distance between 2 IS sites, m
sigma_exc = 1.7e-21 # Cross-section for the direct abstraction, m^2
lambda_des = 1 / np.sqrt(n_surf)
# Transitions
E_bs = E_diff # energy barrier from bulk to surface, eV
E_sb = 1.545
E_des = 0.87
################### FUNCTIONS ###############
def k_sb(T, surf_conc, t):
return nu0 * f.exp(-E_sb / F.k_B / T)
def k_bs(T, surf_conc, t):
return nu0 * f.exp(-E_bs / F.k_B / T)
def J_vs_left(T, surf_conc, t):
G_atom = Gamma_atom * f.conditional(t <= t_exposure, 1, 0)
phi_atom = SP * G_atom * (1 - surf_conc / n_surf)
phi_exc = G_atom * sigma_exc * surf_conc
phi_des = 2 * nu0 * (lambda_des * surf_conc) ** 2 * f.exp(-2 * E_des / F.k_B / T)
return phi_atom - phi_exc - phi_des
def J_vs_right(T, surf_conc, t):
phi_des = 2 * nu0 * (lambda_des * surf_conc) ** 2 * f.exp(-2 * E_des / F.k_B / T)
return -phi_des
################### MODEL ###################
W_model = F.Simulation(log_level=40)
# Mesh
vertices = np.concatenate(
[
np.linspace(0, 5e-8, num=100),
np.linspace(5e-8, 5e-6, num=400),
np.linspace(5e-6, L, num=500),
]
)
W_model.mesh = F.MeshFromVertices(vertices)
# Materials
tungsten = F.Material(id=1, D_0=D0, E_D=E_diff)
W_model.materials = tungsten
distr = 1 / (1 + sp.exp((F.x - 2.2e-6) / 1.54e-7))
traps = F.Traps(
[
F.Trap(
k_0=D0 / (n_IS * lambda_IS**2),
E_k=E_diff,
p_0=nu0,
E_p=0.85,
density=1e-4 * rho_W,
materials=tungsten,
),
F.Trap(
k_0=D0 / (n_IS * lambda_IS**2),
E_k=E_diff,
p_0=nu0,
E_p=1.00,
density=1e-4 * rho_W,
materials=tungsten,
),
F.Trap(
k_0=D0 / (n_IS * lambda_IS**2),
E_k=E_diff,
p_0=nu0,
E_p=1.65,
density=0.19e-2 * rho_W * distr,
materials=tungsten,
),
F.Trap(
k_0=D0 / (n_IS * lambda_IS**2),
E_k=E_diff,
p_0=nu0,
E_p=1.85,
density=0.16e-2 * rho_W * distr,
materials=tungsten,
),
F.Trap(
k_0=D0 / (n_IS * lambda_IS**2),
E_k=E_diff,
p_0=nu0,
E_p=2.06,
density=0.02e-2 * rho_W * distr,
materials=tungsten,
),
]
)
W_model.traps = traps
W_model.T = T_exposure
BC_left = F.SurfaceKinetics(
k_sb=k_sb,
k_bs=k_bs,
lambda_IS=lambda_IS,
n_surf=n_surf,
n_IS=n_IS,
J_vs=J_vs_left,
surfaces=1,
initial_condition=0,
t=F.t,
)
BC_right = F.SurfaceKinetics(
k_sb=k_sb,
k_bs=k_bs,
lambda_IS=lambda_IS,
n_surf=n_surf,
n_IS=n_IS,
J_vs=J_vs_right,
surfaces=2,
initial_condition=0,
t=F.t,
)
W_model.boundary_conditions = [BC_left, BC_right]
# Exports
export_fluences = [
5.22e22,
1.25e23,
4.8e23,
6.3e23,
1e24,
]
export_times = [fluence / Gamma_atom for fluence in export_fluences]
export_times += [export_times[-1] + 20 * 3600, export_times[-1] + 52 * 3600]
derived_quantities = F.DerivedQuantities(
[
F.TotalVolume(field="retention", volume=1),
F.AdsorbedHydrogen(surface=1),
F.AdsorbedHydrogen(surface=2),
],
show_units=True,
)
TXT = F.TXTExport(field="retention", filename="./FESTIM_sim.txt", times=export_times)
W_model.exports = [derived_quantities] + [TXT]
W_model.dt = F.Stepsize(
initial_value=1e-5,
stepsize_change_ratio=1.25,
max_stepsize=500,
dt_min=1e-6,
milestones=export_times,
)
W_model.settings = F.Settings(
absolute_tolerance=1e7,
relative_tolerance=1e-13,
final_time=final_time,
traps_element_type="DG",
)
W_model.initialise()
W_model.run()
Show code cell output
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Defining initial values
Defining variational problem
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19.8 % 7.1e+04 s Elapsed time so far: 62.7 s
19.9 % 7.2e+04 s Elapsed time so far: 62.9 s
20.0 % 7.2e+04 s Elapsed time so far: 63.2 s
20.2 % 7.3e+04 s Elapsed time so far: 63.4 s
20.3 % 7.3e+04 s Elapsed time so far: 63.7 s
20.4 % 7.4e+04 s Elapsed time so far: 63.9 s
20.6 % 7.4e+04 s Elapsed time so far: 64.2 s
20.7 % 7.5e+04 s Elapsed time so far: 64.5 s
20.9 % 7.5e+04 s Elapsed time so far: 64.7 s
21.0 % 7.6e+04 s Elapsed time so far: 65.0 s
21.1 % 7.6e+04 s Elapsed time so far: 65.2 s
21.3 % 7.7e+04 s Elapsed time so far: 65.5 s
21.4 % 7.7e+04 s Elapsed time so far: 65.7 s
21.6 % 7.8e+04 s Elapsed time so far: 66.0 s
21.7 % 7.8e+04 s Elapsed time so far: 66.3 s
21.8 % 7.9e+04 s Elapsed time so far: 66.5 s
22.0 % 7.9e+04 s Elapsed time so far: 66.8 s
22.1 % 8.0e+04 s Elapsed time so far: 67.0 s
22.3 % 8.0e+04 s Elapsed time so far: 67.2 s
22.4 % 8.1e+04 s Elapsed time so far: 67.5 s
22.5 % 8.1e+04 s Elapsed time so far: 67.7 s
22.7 % 8.2e+04 s Elapsed time so far: 68.0 s
22.8 % 8.2e+04 s Elapsed time so far: 68.2 s
23.0 % 8.3e+04 s Elapsed time so far: 68.5 s
23.0 % 8.3e+04 s Elapsed time so far: 68.8 s
23.1 % 8.3e+04 s Elapsed time so far: 69.1 s
23.2 % 8.3e+04 s Elapsed time so far: 69.3 s
23.3 % 8.4e+04 s Elapsed time so far: 69.5 s
23.4 % 8.4e+04 s Elapsed time so far: 69.8 s
23.6 % 8.5e+04 s Elapsed time so far: 70.0 s
23.7 % 8.5e+04 s Elapsed time so far: 70.3 s
23.8 % 8.6e+04 s Elapsed time so far: 70.6 s
24.0 % 8.6e+04 s Elapsed time so far: 70.8 s
24.1 % 8.7e+04 s Elapsed time so far: 71.0 s
24.3 % 8.7e+04 s Elapsed time so far: 71.3 s
24.4 % 8.8e+04 s Elapsed time so far: 71.5 s
24.5 % 8.8e+04 s Elapsed time so far: 71.8 s
24.7 % 8.9e+04 s Elapsed time so far: 72.0 s
24.8 % 8.9e+04 s Elapsed time so far: 72.3 s
25.0 % 9.0e+04 s Elapsed time so far: 72.6 s
25.1 % 9.0e+04 s Elapsed time so far: 72.8 s
25.2 % 9.1e+04 s Elapsed time so far: 73.1 s
25.4 % 9.1e+04 s Elapsed time so far: 73.3 s
25.5 % 9.2e+04 s Elapsed time so far: 73.6 s
25.6 % 9.2e+04 s Elapsed time so far: 73.9 s
25.8 % 9.3e+04 s Elapsed time so far: 74.1 s
25.9 % 9.3e+04 s Elapsed time so far: 74.4 s
26.1 % 9.4e+04 s Elapsed time so far: 74.6 s
26.2 % 9.4e+04 s Elapsed time so far: 74.9 s
26.4 % 9.5e+04 s Elapsed time so far: 75.1 s
26.5 % 9.5e+04 s Elapsed time so far: 75.4 s
26.6 % 9.6e+04 s Elapsed time so far: 75.6 s
26.8 % 9.6e+04 s Elapsed time so far: 75.9 s
26.9 % 9.7e+04 s Elapsed time so far: 76.2 s
27.0 % 9.7e+04 s Elapsed time so far: 76.4 s
27.2 % 9.8e+04 s Elapsed time so far: 76.7 s
27.3 % 9.8e+04 s Elapsed time so far: 76.9 s
27.5 % 9.9e+04 s Elapsed time so far: 77.2 s
27.6 % 9.9e+04 s Elapsed time so far: 77.4 s
27.7 % 1.0e+05 s Elapsed time so far: 77.7 s
27.9 % 1.0e+05 s Elapsed time so far: 77.9 s
28.0 % 1.0e+05 s Elapsed time so far: 78.2 s
28.1 % 1.0e+05 s Elapsed time so far: 78.4 s
28.3 % 1.0e+05 s Elapsed time so far: 78.7 s
28.4 % 1.0e+05 s Elapsed time so far: 79.0 s
28.6 % 1.0e+05 s Elapsed time so far: 79.2 s
28.7 % 1.0e+05 s Elapsed time so far: 79.5 s
28.9 % 1.0e+05 s Elapsed time so far: 79.7 s
29.0 % 1.0e+05 s Elapsed time so far: 80.0 s
29.1 % 1.0e+05 s Elapsed time so far: 80.2 s
29.3 % 1.1e+05 s Elapsed time so far: 80.5 s
29.4 % 1.1e+05 s Elapsed time so far: 80.7 s
29.5 % 1.1e+05 s Elapsed time so far: 81.0 s
29.7 % 1.1e+05 s Elapsed time so far: 81.2 s
29.8 % 1.1e+05 s Elapsed time so far: 81.5 s
30.0 % 1.1e+05 s Elapsed time so far: 81.8 s
30.1 % 1.1e+05 s Elapsed time so far: 82.0 s
30.2 % 1.1e+05 s Elapsed time so far: 82.3 s
30.3 % 1.1e+05 s Elapsed time so far: 82.6 s
30.5 % 1.1e+05 s Elapsed time so far: 82.9 s
30.6 % 1.1e+05 s Elapsed time so far: 83.1 s
30.8 % 1.1e+05 s Elapsed time so far: 83.3 s
30.9 % 1.1e+05 s Elapsed time so far: 83.6 s
31.0 % 1.1e+05 s Elapsed time so far: 83.8 s
31.2 % 1.1e+05 s Elapsed time so far: 84.1 s
31.3 % 1.1e+05 s Elapsed time so far: 84.3 s
31.4 % 1.1e+05 s Elapsed time so far: 84.6 s
31.6 % 1.1e+05 s Elapsed time so far: 84.9 s
31.7 % 1.1e+05 s Elapsed time so far: 85.1 s
31.9 % 1.1e+05 s Elapsed time so far: 85.4 s
32.0 % 1.2e+05 s Elapsed time so far: 85.7 s
32.1 % 1.2e+05 s Elapsed time so far: 85.9 s
32.3 % 1.2e+05 s Elapsed time so far: 86.2 s
32.4 % 1.2e+05 s Elapsed time so far: 86.4 s
32.6 % 1.2e+05 s Elapsed time so far: 86.7 s
32.7 % 1.2e+05 s Elapsed time so far: 87.0 s
32.8 % 1.2e+05 s Elapsed time so far: 87.2 s
33.0 % 1.2e+05 s Elapsed time so far: 87.5 s
33.1 % 1.2e+05 s Elapsed time so far: 87.7 s
33.3 % 1.2e+05 s Elapsed time so far: 88.0 s
33.4 % 1.2e+05 s Elapsed time so far: 88.3 s
33.5 % 1.2e+05 s Elapsed time so far: 88.5 s
33.7 % 1.2e+05 s Elapsed time so far: 88.8 s
33.8 % 1.2e+05 s Elapsed time so far: 89.0 s
34.0 % 1.2e+05 s Elapsed time so far: 89.2 s
34.1 % 1.2e+05 s Elapsed time so far: 89.5 s
34.2 % 1.2e+05 s Elapsed time so far: 89.7 s
34.4 % 1.2e+05 s Elapsed time so far: 90.0 s
34.5 % 1.2e+05 s Elapsed time so far: 90.2 s
34.6 % 1.2e+05 s Elapsed time so far: 90.5 s
34.8 % 1.3e+05 s Elapsed time so far: 90.8 s
34.9 % 1.3e+05 s Elapsed time so far: 91.0 s
35.1 % 1.3e+05 s Elapsed time so far: 91.3 s
35.2 % 1.3e+05 s Elapsed time so far: 91.5 s
35.3 % 1.3e+05 s Elapsed time so far: 91.8 s
35.5 % 1.3e+05 s Elapsed time so far: 92.0 s
35.6 % 1.3e+05 s Elapsed time so far: 92.3 s
35.8 % 1.3e+05 s Elapsed time so far: 92.6 s
35.9 % 1.3e+05 s Elapsed time so far: 92.8 s
36.0 % 1.3e+05 s Elapsed time so far: 93.0 s
36.2 % 1.3e+05 s Elapsed time so far: 93.3 s
36.3 % 1.3e+05 s Elapsed time so far: 93.5 s
36.5 % 1.3e+05 s Elapsed time so far: 93.8 s
36.6 % 1.3e+05 s Elapsed time so far: 94.0 s
36.7 % 1.3e+05 s Elapsed time so far: 94.3 s
36.9 % 1.3e+05 s Elapsed time so far: 94.6 s
37.0 % 1.3e+05 s Elapsed time so far: 94.8 s
37.1 % 1.3e+05 s Elapsed time so far: 95.1 s
37.3 % 1.3e+05 s Elapsed time so far: 95.3 s
37.4 % 1.3e+05 s Elapsed time so far: 95.6 s
37.6 % 1.4e+05 s Elapsed time so far: 95.8 s
37.7 % 1.4e+05 s Elapsed time so far: 96.1 s
37.8 % 1.4e+05 s Elapsed time so far: 96.3 s
38.0 % 1.4e+05 s Elapsed time so far: 96.6 s
38.1 % 1.4e+05 s Elapsed time so far: 96.8 s
38.3 % 1.4e+05 s Elapsed time so far: 97.0 s
38.4 % 1.4e+05 s Elapsed time so far: 97.3 s
38.5 % 1.4e+05 s Elapsed time so far: 97.5 s
38.7 % 1.4e+05 s Elapsed time so far: 97.8 s
38.8 % 1.4e+05 s Elapsed time so far: 98.0 s
39.0 % 1.4e+05 s Elapsed time so far: 98.3 s
39.1 % 1.4e+05 s Elapsed time so far: 98.6 s
39.2 % 1.4e+05 s Elapsed time so far: 98.8 s
39.4 % 1.4e+05 s Elapsed time so far: 99.1 s
39.5 % 1.4e+05 s Elapsed time so far: 99.3 s
39.6 % 1.4e+05 s Elapsed time so far: 99.6 s
39.8 % 1.4e+05 s Elapsed time so far: 99.9 s
39.9 % 1.4e+05 s Elapsed time so far: 100.1 s
40.1 % 1.4e+05 s Elapsed time so far: 100.4 s
40.2 % 1.4e+05 s Elapsed time so far: 100.6 s
40.4 % 1.5e+05 s Elapsed time so far: 100.9 s
40.5 % 1.5e+05 s Elapsed time so far: 101.1 s
40.6 % 1.5e+05 s Elapsed time so far: 101.3 s
40.8 % 1.5e+05 s Elapsed time so far: 101.6 s
40.9 % 1.5e+05 s Elapsed time so far: 101.9 s
41.0 % 1.5e+05 s Elapsed time so far: 102.1 s
41.2 % 1.5e+05 s Elapsed time so far: 102.4 s
41.3 % 1.5e+05 s Elapsed time so far: 102.6 s
41.5 % 1.5e+05 s Elapsed time so far: 102.9 s
41.6 % 1.5e+05 s Elapsed time so far: 103.1 s
41.7 % 1.5e+05 s Elapsed time so far: 103.3 s
41.9 % 1.5e+05 s Elapsed time so far: 103.6 s
42.0 % 1.5e+05 s Elapsed time so far: 103.8 s
42.1 % 1.5e+05 s Elapsed time so far: 104.1 s
42.3 % 1.5e+05 s Elapsed time so far: 104.3 s
42.4 % 1.5e+05 s Elapsed time so far: 104.6 s
42.6 % 1.5e+05 s Elapsed time so far: 104.9 s
42.7 % 1.5e+05 s Elapsed time so far: 105.0 s
42.9 % 1.5e+05 s Elapsed time so far: 105.2 s
43.0 % 1.5e+05 s Elapsed time so far: 105.5 s
43.1 % 1.6e+05 s Elapsed time so far: 105.8 s
43.3 % 1.6e+05 s Elapsed time so far: 106.0 s
43.4 % 1.6e+05 s Elapsed time so far: 106.3 s
43.5 % 1.6e+05 s Elapsed time so far: 106.6 s
43.7 % 1.6e+05 s Elapsed time so far: 106.8 s
43.8 % 1.6e+05 s Elapsed time so far: 107.1 s
44.0 % 1.6e+05 s Elapsed time so far: 107.3 s
44.1 % 1.6e+05 s Elapsed time so far: 107.6 s
44.2 % 1.6e+05 s Elapsed time so far: 107.8 s
44.4 % 1.6e+05 s Elapsed time so far: 108.1 s
44.5 % 1.6e+05 s Elapsed time so far: 108.4 s
44.7 % 1.6e+05 s Elapsed time so far: 108.6 s
44.8 % 1.6e+05 s Elapsed time so far: 108.9 s
44.9 % 1.6e+05 s Elapsed time so far: 109.1 s
45.1 % 1.6e+05 s Elapsed time so far: 109.4 s
45.2 % 1.6e+05 s Elapsed time so far: 109.6 s
45.4 % 1.6e+05 s Elapsed time so far: 109.9 s
45.5 % 1.6e+05 s Elapsed time so far: 110.1 s
45.6 % 1.6e+05 s Elapsed time so far: 110.4 s
45.8 % 1.6e+05 s Elapsed time so far: 110.7 s
45.9 % 1.7e+05 s Elapsed time so far: 110.9 s
46.0 % 1.7e+05 s Elapsed time so far: 111.1 s
46.2 % 1.7e+05 s Elapsed time so far: 111.4 s
46.3 % 1.7e+05 s Elapsed time so far: 111.6 s
46.5 % 1.7e+05 s Elapsed time so far: 111.9 s
46.6 % 1.7e+05 s Elapsed time so far: 112.1 s
46.7 % 1.7e+05 s Elapsed time so far: 112.4 s
46.9 % 1.7e+05 s Elapsed time so far: 112.7 s
47.0 % 1.7e+05 s Elapsed time so far: 112.9 s
47.2 % 1.7e+05 s Elapsed time so far: 113.2 s
47.3 % 1.7e+05 s Elapsed time so far: 113.4 s
47.4 % 1.7e+05 s Elapsed time so far: 113.7 s
47.6 % 1.7e+05 s Elapsed time so far: 114.0 s
47.7 % 1.7e+05 s Elapsed time so far: 114.2 s
47.9 % 1.7e+05 s Elapsed time so far: 114.5 s
47.9 % 1.7e+05 s Elapsed time so far: 114.7 s
48.1 % 1.7e+05 s Elapsed time so far: 115.1 s
48.1 % 1.7e+05 s Elapsed time so far: 115.4 s
48.3 % 1.7e+05 s Elapsed time so far: 115.8 s
48.4 % 1.7e+05 s Elapsed time so far: 116.1 s
48.5 % 1.7e+05 s Elapsed time so far: 116.4 s
48.7 % 1.8e+05 s Elapsed time so far: 116.7 s
48.8 % 1.8e+05 s Elapsed time so far: 116.9 s
49.0 % 1.8e+05 s Elapsed time so far: 117.1 s
49.1 % 1.8e+05 s Elapsed time so far: 117.3 s
49.2 % 1.8e+05 s Elapsed time so far: 117.6 s
49.4 % 1.8e+05 s Elapsed time so far: 117.8 s
49.5 % 1.8e+05 s Elapsed time so far: 118.1 s
49.6 % 1.8e+05 s Elapsed time so far: 118.4 s
49.8 % 1.8e+05 s Elapsed time so far: 118.6 s
49.9 % 1.8e+05 s Elapsed time so far: 118.9 s
50.1 % 1.8e+05 s Elapsed time so far: 119.1 s
50.2 % 1.8e+05 s Elapsed time so far: 119.4 s
50.3 % 1.8e+05 s Elapsed time so far: 119.6 s
50.5 % 1.8e+05 s Elapsed time so far: 119.9 s
50.6 % 1.8e+05 s Elapsed time so far: 120.1 s
50.8 % 1.8e+05 s Elapsed time so far: 120.4 s
50.9 % 1.8e+05 s Elapsed time so far: 120.7 s
51.0 % 1.8e+05 s Elapsed time so far: 121.1 s
51.2 % 1.8e+05 s Elapsed time so far: 121.4 s
51.3 % 1.8e+05 s Elapsed time so far: 121.7 s
51.5 % 1.9e+05 s Elapsed time so far: 122.0 s
51.6 % 1.9e+05 s Elapsed time so far: 122.2 s
51.7 % 1.9e+05 s Elapsed time so far: 122.5 s
51.9 % 1.9e+05 s Elapsed time so far: 122.7 s
52.0 % 1.9e+05 s Elapsed time so far: 123.0 s
52.1 % 1.9e+05 s Elapsed time so far: 123.2 s
52.3 % 1.9e+05 s Elapsed time so far: 123.5 s
52.4 % 1.9e+05 s Elapsed time so far: 123.7 s
52.6 % 1.9e+05 s Elapsed time so far: 124.0 s
52.7 % 1.9e+05 s Elapsed time so far: 124.2 s
52.9 % 1.9e+05 s Elapsed time so far: 124.5 s
53.0 % 1.9e+05 s Elapsed time so far: 124.7 s
53.1 % 1.9e+05 s Elapsed time so far: 125.0 s
53.3 % 1.9e+05 s Elapsed time so far: 125.2 s
53.4 % 1.9e+05 s Elapsed time so far: 125.5 s
53.5 % 1.9e+05 s Elapsed time so far: 125.8 s
53.7 % 1.9e+05 s Elapsed time so far: 126.0 s
53.8 % 1.9e+05 s Elapsed time so far: 126.3 s
54.0 % 1.9e+05 s Elapsed time so far: 126.6 s
54.1 % 1.9e+05 s Elapsed time so far: 126.8 s
54.2 % 2.0e+05 s Elapsed time so far: 127.1 s
54.4 % 2.0e+05 s Elapsed time so far: 127.3 s
54.5 % 2.0e+05 s Elapsed time so far: 127.6 s
54.6 % 2.0e+05 s Elapsed time so far: 127.8 s
54.8 % 2.0e+05 s Elapsed time so far: 128.1 s
54.9 % 2.0e+05 s Elapsed time so far: 128.3 s
55.1 % 2.0e+05 s Elapsed time so far: 128.6 s
55.2 % 2.0e+05 s Elapsed time so far: 128.9 s
55.4 % 2.0e+05 s Elapsed time so far: 129.1 s
55.5 % 2.0e+05 s Elapsed time so far: 129.2 s
55.6 % 2.0e+05 s Elapsed time so far: 129.5 s
55.8 % 2.0e+05 s Elapsed time so far: 129.8 s
55.9 % 2.0e+05 s Elapsed time so far: 130.0 s
56.0 % 2.0e+05 s Elapsed time so far: 130.3 s
56.2 % 2.0e+05 s Elapsed time so far: 130.5 s
56.3 % 2.0e+05 s Elapsed time so far: 130.8 s
56.5 % 2.0e+05 s Elapsed time so far: 131.0 s
56.6 % 2.0e+05 s Elapsed time so far: 131.3 s
56.7 % 2.0e+05 s Elapsed time so far: 131.6 s
56.9 % 2.0e+05 s Elapsed time so far: 131.8 s
57.0 % 2.1e+05 s Elapsed time so far: 132.1 s
57.2 % 2.1e+05 s Elapsed time so far: 132.3 s
57.3 % 2.1e+05 s Elapsed time so far: 132.6 s
57.4 % 2.1e+05 s Elapsed time so far: 132.8 s
57.6 % 2.1e+05 s Elapsed time so far: 133.1 s
57.7 % 2.1e+05 s Elapsed time so far: 133.4 s
57.9 % 2.1e+05 s Elapsed time so far: 133.6 s
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58.1 % 2.1e+05 s Elapsed time so far: 134.1 s
58.3 % 2.1e+05 s Elapsed time so far: 134.4 s
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59.0 % 2.1e+05 s Elapsed time so far: 135.7 s
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59.2 % 2.1e+05 s Elapsed time so far: 136.2 s
59.4 % 2.1e+05 s Elapsed time so far: 136.4 s
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59.7 % 2.1e+05 s Elapsed time so far: 137.0 s
59.8 % 2.2e+05 s Elapsed time so far: 137.2 s
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60.2 % 2.2e+05 s Elapsed time so far: 138.0 s
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60.9 % 2.2e+05 s Elapsed time so far: 139.0 s
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62.4 % 2.2e+05 s Elapsed time so far: 141.1 s
62.6 % 2.3e+05 s Elapsed time so far: 141.3 s
62.7 % 2.3e+05 s Elapsed time so far: 141.5 s
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63.1 % 2.3e+05 s Elapsed time so far: 142.1 s
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64.5 % 2.3e+05 s Elapsed time so far: 144.0 s
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64.9 % 2.3e+05 s Elapsed time so far: 144.6 s
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65.2 % 2.3e+05 s Elapsed time so far: 145.0 s
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66.2 % 2.4e+05 s Elapsed time so far: 146.3 s
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67.0 % 2.4e+05 s Elapsed time so far: 147.4 s
67.2 % 2.4e+05 s Elapsed time so far: 147.6 s
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68.0 % 2.4e+05 s Elapsed time so far: 148.7 s
68.1 % 2.4e+05 s Elapsed time so far: 149.0 s
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100.0 % 3.6e+05 s Elapsed time so far: 192.6 s
Comparison with experimental data and MHIMS: D content#
The results produced by FESTIM are in good agreement with the experimental data and correlate perfectly with MHIMS.
Show code cell source
import plotly.graph_objects as go
import plotly.express as px
from scipy.interpolate import interp1d
def RMSPE(x_sim, x_exp):
error = np.sqrt(np.mean((x_sim - x_exp) ** 2)) / np.mean(x_exp)
return error
retention = (
np.array(derived_quantities[0].data)
+ np.array(derived_quantities[1].data)
+ np.array(derived_quantities[2].data)
)
t = np.array(derived_quantities.t)
exp_ret = np.loadtxt("./reference_data/exp_ret.csv", delimiter=",", skiprows=1)
interp_ret = interp1d(t / 3600, retention, fill_value="extrapolate")
error = RMSPE(
interp_ret(
exp_ret[:, 0],
),
exp_ret[:, 1],
)
print(f"RMSPE between FESTIM and experimental data is {error*100:.2f}%")
MHIMS_ret = np.loadtxt("./reference_data/MHIMS_ret.csv", delimiter=",", skiprows=1)
fig = go.Figure()
fig.add_trace(
go.Scatter(
x=t / 3600,
y=retention / 1e20,
mode="lines",
line=dict(width=4, color=px.colors.qualitative.Plotly[1]),
name="FESTIM",
)
)
fig.add_trace(
go.Scatter(
x=MHIMS_ret[::100, 0],
y=MHIMS_ret[::100, 1] / 1e20,
mode="markers",
marker_symbol="square",
marker=dict(size=10, color=px.colors.qualitative.Plotly[2], opacity=0.6),
name="MHIMS",
)
)
fig.add_trace(
go.Scatter(
x=exp_ret[:, 0],
y=exp_ret[:, 1] / 1e20,
mode="markers",
marker=dict(size=10, color=px.colors.qualitative.Plotly[0], opacity=0.6),
name="Exp.",
)
)
fig.update_yaxes(
title_text="D inventory, 10<sup>20</sup> m<sup>-2</sup>",
range=[0, 5],
tick0=0,
dtick=1,
)
fig.update_xaxes(title_text="Time, h", range=[0, 100], tick0=0, dtick=10)
fig.update_layout(template="simple_white", height=600)
# The writing-reading block below is needed to avoid the issue with compatibility
# of Plotly plots and dollarmath syntax extension in Jupyter Book
# For mode details, see https://github.com/jupyter-book/jupyter-book/issues/1528
fig.write_html("./markelj_comparison_ret.html")
from IPython.display import HTML, display
display(HTML("./markelj_comparison_ret.html"))
RMSPE between FESTIM and experimental data is 7.30%
Comparison with experimental data: D depth distribution#
FESTIM reproduces well the experimental NRA measurements.
Show code cell source
fig = go.Figure()
FESTIM_profiles = np.genfromtxt("./FESTIM_sim.txt", names=True, delimiter=",")
NRA_exp = np.loadtxt("./reference_data/exp_NRA.csv", delimiter=",", skiprows=1)
NRA_sto = np.loadtxt("./reference_data/sto_NRA.csv", delimiter=",", skiprows=1)
def create_slider(fig):
steps = []
for i in range(0, len(fig.data), 2):
step = dict(
method="update",
args=[
{"visible": [False] * len(fig.data)},
{"title": "Time: " + f"{export_times[int(i/2)]/3600:.2f}"},
], # layout attribute
label=f"{export_times[int(i/2)]/3600:.2f} h",
)
step["args"][0]["visible"][i] = True # Toggle i'th trace to "visible"
step["args"][0]["visible"][i + 1] = True # Toggle i+1'th trace to "visible"
steps.append(step)
sliders = [
dict(active=0, currentvalue={"prefix": "Time: "}, pad={"t": 50}, steps=steps)
]
return sliders
for i, t in enumerate(export_times):
x = FESTIM_profiles["x"]
y = FESTIM_profiles[f"t{t:.2e}s".replace(".", "").replace("+", "")]
# order y by x
x, y = zip(*sorted(zip(x, y)))
color = px.colors.qualitative.Plotly[i]
fig.add_trace(
go.Scatter(
x=np.array(x) / 1e-6,
y=np.array(y) / rho_W * 100,
mode="lines",
line=dict(width=3.5, color=px.colors.qualitative.Plotly[i]),
name="FESTIM",
visible=False,
)
)
if i <= 4:
fig.add_trace(
go.Scatter(
x=NRA_exp[:, 0],
y=NRA_exp[:, i + 1],
mode="lines",
line=dict(
width=3.5, color=px.colors.qualitative.Plotly[i], dash="dash"
),
name="NRA",
visible=False,
)
)
else:
fig.add_trace(
go.Scatter(
x=NRA_sto[:, 0],
y=NRA_sto[:, i - 4],
mode="lines",
line=dict(
width=3.5, color=px.colors.qualitative.Plotly[i], dash="dash"
),
name="NRA",
visible=False,
)
)
fig.data[0].visible = True
fig.data[1].visible = True
fig.update_yaxes(title_text="D concentration, at.%", range=[0, 0.5], tick0=0, dtick=0.1)
fig.update_xaxes(title_text="Depth, µm", range=[0, 5], tick0=0, dtick=1)
fig.update_layout(template="simple_white", sliders=create_slider(fig), height=600)
# The writing-reading block below is needed to avoid the issue with compatibility
# of Plotly plots and dollarmath syntax extension in Jupyter Book
# For mode details, see https://github.com/jupyter-book/jupyter-book/issues/1528
fig.write_html("./markelj_profiles_exp.html")
display(HTML("./markelj_profiles_exp.html"))
Comparison with MHIMS: D depth distribution#
FESTIM agrees with MHIMS. Slight differences are due to the use of the precise value of Fernandez’s diffusivity pre-factor. FESTIM uses the value from HTM divided by \(\sqrt{2}\): \(1.93 \times 10^{-7} / \sqrt{2}\), whereas the presented diffusivity pre-factor in [26] is \(1.9 \times 10^{-7} / \sqrt{2}\).
Show code cell source
fig = go.Figure()
FESTIM_profiles = np.genfromtxt("./FESTIM_sim.txt", names=True, delimiter=",")
MHIMS_profiles = np.loadtxt(
"./reference_data/MHIMS_profiles.csv", delimiter=",", skiprows=1
)
for i, t in enumerate(export_times):
x = FESTIM_profiles["x"]
y = FESTIM_profiles[f"t{t:.2e}s".replace(".", "").replace("+", "")]
# order y by x
x, y = zip(*sorted(zip(x, y)))
color = px.colors.qualitative.Plotly[i]
fig.add_trace(
go.Scatter(
x=np.array(x) / 1e-6,
y=np.array(y) / rho_W * 100,
mode="lines",
line=dict(width=3.5, color=px.colors.qualitative.Plotly[i]),
name="FESTIM",
visible=False,
)
)
fig.add_trace(
go.Scatter(
x=MHIMS_profiles[:, 0],
y=MHIMS_profiles[:, i + 1],
mode="markers",
marker=dict(size=10, color=px.colors.qualitative.Plotly[i], opacity=0.4),
name="MHIMS",
visible=False,
)
)
fig.data[0].visible = True
fig.data[1].visible = True
fig.update_yaxes(title_text="D concentration, at.%", range=[0, 0.5], tick0=0, dtick=0.1)
fig.update_xaxes(title_text="Depth, µm", range=[0, 5], tick0=0, dtick=1)
fig.update_layout(template="simple_white", sliders=create_slider(fig), height=600)
# The writing-reading block below is needed to avoid the issue with compatibility
# of Plotly plots and dollarmath syntax extension in Jupyter Book
# For mode details, see https://github.com/jupyter-book/jupyter-book/issues/1528
fig.write_html("./markelj_profiles_MHIMS.html")
display(HTML("./markelj_profiles_MHIMS.html"))