Introduction:

Hydrogen in fusion

ITER

Plasma: mixture of Hydrogen (D-T) and Helium

Particle bombardment

Divertor

Why should we care?

T is rare

T is expensive

€£$

Material embrittlement

T is radioactive

☢

What's Tritium?

Protium

Deuterium

Tritium

Molar mass: 6.032 g/mol

What's Tritium?

Tritium

☢

800 \ \text{MWth} \approx 10^{20} \ \text{reactions/s} \approx 100 \ \text{kg/year}

Consumption in a single fusion reactor:

What's Tritium?

Tritium

☢

Half-life: 12 years

☢

World production:

CANDU reactors (Canadian Deuterium Uranium): 130 g per year
From cosmic rays: 200 g per year

We need to produce tritium!

World reserves:

$ \approx $ 100 kg

Cost: $30,000 per gram

Fuel self-sufficiency

Lithium is used to breed tritium

\mathrm{TBR} = \mathrm{\frac{tritium \ produced}{tritium \ consumed}}

🎯Goals

↗️Maximise TBR

↘️Minimise volume

Lithium is used to breed tritium

→ Li6 enrichment is an option

DT fusion neutrons

There are several concepts of breeding blanket

FLiBe

LiPb

Choose your breeder

Li in ceramics

There are several concepts of breeding blanket

FLiBe

LiPb

Self

Water

He+Water

Choose your breeder

Choose your coolant

Li in ceramics

There are several concepts of breeding blanket

FLiBe

LiPb

Self

Water

He+Water

Liquid

Segmented

Choose your breeder

Choose your coolant

Choose your geometry

Li in ceramics

Froio et al Fus Eng Design 2017

There are several concepts of breeding blanket

FLiBe

LiPb

Self

Water

He+Water

Liquid

Segmented

Choose your breeder

Choose your coolant

Choose your geometry

Li in ceramics

ARC's liquid immersion blanket

There are several concepts of breeding blanket

FLiBe

LiPb

Self

Water

He+Water

Liquid

Segmented

Choose your breeder

Choose your coolant

Choose your geometry

Li in ceramics

First Light Fusion

There are several concepts of breeding blanket

FLiBe

LiPb

Self

Water

He+Water

Liquid

Segmented

Choose your breeder

Choose your coolant

Choose your geometry

Li in ceramics

Water Cooled Lithium-Lead

Arena et al Energies 2023, 16, 2069

Segmented vs LIB

The breeding blanket is only one component of the fuel cycle

Need to understand how tritium behaves in each of these components
cf. Fuel Cycle Modelling

Meschini et al (submitted)

A startup inventory of T is required

Meschini et al (submitted)

The efficiency of the fuel cycle will influence the future fusion economy

Doubling time: time required for a reactor to produce enough tritium to start another reactor
Energy industry: $\tau_2 \approx $ 2-3 years
Constrains the required TBR and the startup inventory
Different fuel cycles can give different TBRr and startup inventories

P_\mathrm{max} = P_\mathrm{plant} \times 2^{t/\tau_2}

Safety

Tritium is a health hazard

☢

No external hazard: electron stopped by skin
Once ingested, can cause damage to internal tissues
Forms: HT, HTO, methane, titrides...
Biological half-life of HTO: 10 days
Biological half-life of OBT (Organically Bound Tritium): 40 days

Augustin Janssens. ‘Emerging Issues on Tritium and Low Energy Beta Emitters”’. en. In: (Nov. 2007), p. 100

Country	Water limit (Bq/L)
EU	100
USA	740
UK	100
Canada	7,000
Finland	30,000
Australia	76,103
Russia	7,700
WHO	10,000

Source: Standards and Guidelines for Tritium in Drinking Water (INFO-0766)

The content of tritium needs to be limited

Ferrero, et al (2022) A Preliminary CFD and Tritium Transport Analysis for ARC Blanket, Fusion Science and Technology, 78:8, 617-630

1. Keep inventory at a minimum

Tritium limit in the ITER vacuum vessel: 1 kg

Delaporte-Mathurin et al. 2021. “Fuel Retention in WEST and ITER Divertors Based on FESTIM Monoblock Simulations.” Nuclear Fusion 61 (12): 126001.

The content of tritium needs to be limited

1. Keep inventory at a minimum

2. Reduce inventory

Heating components help releasing their tritium content (cf. Basics of H transport)

Delaporte-Mathurin et al. 2023. “3D effects on hydrogen transport in ITER-like monoblocks” submitted

The content of tritium needs to be limited

1. Keep inventory at a minimum

2. Reduce inventory

3. Avoid contamination of coolants

Metal

Tritiated environment

"Clean" environment

Permeation

The content of tritium needs to be limited

1. Keep inventory at a minimum

2. Reduce inventory

3. Avoid contamination of coolants

Metal

Tritiated environment

"Clean" environment

Permeation

Permeation barrier

The content of tritium needs to be limited

1. Keep inventory at a minimum

2. Reduce inventory

3. Avoid contamination of coolants

Ceramics are promising candidates:

oxides
carbides
nitrides

Permeation barriers are caracterised by their PRF (Permeation Reduction Factor)

\mathrm{PRF} = \frac{\mathrm{uncoated \ flux}}{\mathrm{coated \ flux}}

Target for breeding blankets PRF ≈ 100-1000

Luo et al Surface and Coatings Technology 2020

Component embrittlement

Agglomeration of hydrogen can lead to blistering

Accumulation of H at defects:
- Voids
- Reaction with impurities (formation of methane)
- grain boundaries
Creation of a filled cavity
Growth of the cavity without bursting

Kuznetsov, Alexey S. et al. “Hydrogen-induced blistering of Mo/Si multilayers: Uptake and distribution.” Thin Solid Films 545 (2013): 571-579.

Cavities can lead to Hydrogen Induced Cracking

Review of HIC by Sofronis

https://doi.org/10.1016/j.jngse.2022.104547

Take aways

H transport

Safety

Fusion Economy

Materials

Minimising inventories
Limiting permeation
Protecting personel

Fuel cycle
Tritium breeding
Losses

Embrittlement
H induced cracks

Question time!

Fundamentals of hydrogen transport in materials

Outline

Sources of H transport in fusion (2 mins)
Basics of H transport (10 mins)
- Surface effects (Sievert vs Henry, recombination, dissociation,)
- Bulk diffusion (Fick, Soret...)
- Interfaces
- Trapping (intrinsic, extrinsic)
- Isotopic exchange

What are the sources of tritium in a tokamak?

1. Direct implantation from the plasma

2. Neutron capture from Lithium

3. Neutron capture by Helium-3

Hydrogen transport in metals

Diffusion

Even more particles:

continuity approximation

1 particle:

Random walk

Many particles

Diffusion

\varphi_\mathrm{diffusion} = - D \ \nabla c

$ \varphi_\mathrm{diffusion} $: diffusion flux

$ D $: diffusion coefficient

$ c $: diffusive hydrogen concentration

Fick's first law of diffusion

Diffusion

\varphi_\mathrm{diffusion} = - D \ \nabla c

$ \varphi_\mathrm{diffusion} $: diffusion flux

$ D $: diffusion coefficient

$ c $: diffusive hydrogen concentration

$ S$: source term

Fick's 1st law of diffusion

\frac{\partial c}{\partial t} = - \nabla \cdot \varphi_\mathrm{diffusion} + S

Fick's 2nd law of diffusion

Diffusion

\varphi_\mathrm{diffusion} = - D \ \left( \nabla c + \frac{Q\ c}{R \ T^2} \ \nabla{T} - \frac{c \ V_H}{R \ T} \nabla \sigma \right)

$ \varphi_\mathrm{diffusion} $: diffusion flux

$ D $: diffusion coefficient

$ c $: diffusive hydrogen concentration

$ S$: source term

Soret effect (or thermophoresis)

Stress assisted diffusion

Particle implantation

Ziegler et al. 2010. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 268 (11): 1818–23. https://doi.org/10.1016/j.nimb.2010.02.091.

Implantation range

Implantation range & width and reflection coefficient can be computed with SRIM, SDTRIM...

Mutzke et al, SDTrimSP Version 6.00 2019

Particle implantation

S = (1-r) \ \Gamma_\mathrm{incident} \ f(x)

$\Gamma_\mathrm{incident} $: incident flux (particle/m2/s)

$ f(x) $: Gaussian distribution (/m)

$r $: reflection coefficient

Transport in liquids

Advection + diffusion

Carrying H in the liquid flow

Normal diffusion process

Requires to do some Computational Fluid Dynamics

Transport in liquids

Advection + diffusion

Carrying H in the liquid flow

Normal diffusion process

Requires to do some Computational Fluid Dynamics

Correlations

Transport in liquids

Heat transfer analogy

H transport

q = h \ (T_\infty - T_0)

\varphi = k \ (c_\infty - c_0)

$T_\infty$

$T_0$

$c_\infty$

$c_0$

Transport in liquids

Heat transfer analogy

H transport

q = h \ (T_\infty - T_0)

\varphi = k \ (c_\infty - c_0)

$ h $: heat transfer coefficient (W/K/m2)

$ k $: mass transport coefficient (m/s)

Nu = a\ Re^b \ Pr^c

Sh = a\ Re^b \ Sc^c

Nu = \frac{h}{\lambda/L} = \frac{\mathrm{Convective \ rate}}{\mathrm{Conduction \ rate}}

Sh = \frac{k}{D/L} = \frac{\mathrm{Convective \ rate}}{\mathrm{Diffusion \ rate}}

Surface effects

H2 molecules

Metal lattice

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

Dissociation coefficient (H/m2/s/Pa)

Partial pressure of H (Pa)

Adsorbed H

Metal lattice

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^2

Metal lattice

Recombination coefficient (m4/s)

Concentration (H/m3)

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^2

Metal lattice

\varphi_\mathrm{net} = \varphi_\mathrm{dissociation} - \varphi_\mathrm{recombination}

Waelbroeck model

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^2

Metal lattice

At equilibrium:

\varphi_\mathrm{net} = 0

c = \sqrt{\frac{K_\mathrm{d}}{K_\mathrm{r}}} \ \sqrt{P}

\varphi_\mathrm{net} = \varphi_\mathrm{dissociation} - \varphi_\mathrm{recombination}

c = S \ \sqrt{P}

Sievert's law of solubility

Surface effects

\varphi_\mathrm{dissociation} = K_\mathrm{d} \ P

\varphi_\mathrm{recombination} = K_\mathrm{r} \ c^\textcolor{black}{1}

Non-metallic liquid

At equilibrium:

\varphi_\mathrm{net} = 0

c = \textcolor{black}{\frac{K_\mathrm{d}}{K_\mathrm{r}} \ P}

\varphi_\mathrm{net} = \varphi_\mathrm{dissociation} - \varphi_\mathrm{recombination}

c = K_H \ P

Henry's law of solubility

Interfaces

Material 1

Material 2

Interfaces

Partial pressure and flux are continuous

Material 1

Material 2

Interfaces

Material 1

Material 2

Case 1:

Metal-Metal

- D_1 \nabla c_1 = - D_2 \nabla c_2

P = \left( \frac{c}{K_S} \right) ^2

\left( \frac{c_1}{K_{S,1}} \right)^2 =\left( \frac{c_2}{K_{S,2}} \right)^2

Sievert's law

Interfaces

Material 1

Material 2

Case 2:

Non metal-non metal

- D_1 \nabla c_1 = - D_2 \nabla c_2

P = \frac{c}{K_H}

\frac{c_1}{K_{H,1}} = \frac{c_2}{K_{H,2}}

Henry's law

Interfaces

Material 1

Material 2

Case 3:

Metal-Non metal

- D_1 \nabla c_1 = - D_2 \nabla c_2

P = \left( \frac{c}{K_S} \right) ^2

\left( \frac{c_1}{K_{S,1}} \right)^2 = \frac{c_2}{K_{H,2}}

Sievert's law

P = \frac{c}{K_H}

Henry's law

Interfaces

Material 1

Material 2

Steady state concentration profile

different diffusivities $\rightarrow$ different concentration gradients
different solubilities $ \rightarrow $ concentration discontinuity

$x$

$c$

Permeation barriers are low solubility, low duffisivity

Metal

Tritiated environment

"Clean" environment

Permeation

Permeation barrier

Permeation barriers are low solubility, low duffisivity

Pressure $P$

High gradient = high flux

Low gradient = low flux

Pressure $P$

Trapping

Trap = anything binding to H

vacancy
grain boundary
impurity
chemical reaction
...

Trapping

Potential energy

Distance

Diffusion barrier

Energy barrier = activation energy

Trap binding energy

Trapping energy

Common assumption:

$ E_k = E_D $

Trapping

\mathrm{H} + [\ \ \ ]_\mathrm{trap} \ \substack{p \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k} \ [\mathrm{H}]_\mathrm{trap}

\frac{\partial c_\mathrm{t}}{\partial t} = -\frac{\partial c_\mathrm{m}}{\partial t} = k \ c_\mathrm{m} \ n_\mathrm{free \ trap} - p c_\mathrm{t}

c_\mathrm{t}

Since $n_\mathrm{trap} = n_\mathrm{free \ trap} + c_\mathrm{t} $

c_\mathrm{m}

n_\mathrm{free \ trap}

Trapping

\mathrm{H} + [\ \ \ ]_\mathrm{trap} \ \substack{p \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k} \ [\mathrm{H}]_\mathrm{trap}

\frac{\partial c_\mathrm{t}}{\partial t} = -\frac{\partial c_\mathrm{m}}{\partial t} = k \ c_\mathrm{m} \ (n_\mathrm{trap,0} - c_\mathrm{t}) - p c_\mathrm{t}

c_\mathrm{m}

n_\mathrm{free \ trap}

c_\mathrm{t}

Total concentration of traps

Trapping

\frac{\partial c_\mathrm{t}}{\partial t} = -\frac{\partial c_\mathrm{m}}{\partial t} = k \ c_\mathrm{m} \ (n_\mathrm{trap,0} - c_\mathrm{t}) - p c_\mathrm{t}

With diffusion

and

1 trap

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \left(D\nabla c_\mathrm{m} \right) + S - \frac{\partial c_\mathrm{t}}{\partial t}

\frac{\partial c_\mathrm{t}}{\partial t} = k \ c_\mathrm{m} \ (n_\mathrm{trap,0} - c_\mathrm{t}) - p c_\mathrm{t}

Trapping

3 traps

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \left(D\nabla c_\mathrm{m} \right) + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}

\frac{\partial c_\mathrm{t, 1}}{\partial t} = k_1 \ c_\mathrm{m} \ (n_\mathrm{trap,1} - c_\mathrm{t,1}) - p_1 c_\mathrm{t,1}

\frac{\partial c_\mathrm{t, 2}}{\partial t} = k_2 \ c_\mathrm{m} \ (n_\mathrm{trap,2} - c_\mathrm{t,2}) - p_2 c_\mathrm{t,2}

\frac{\partial c_\mathrm{t, 3}}{\partial t} = k_3 \ c_\mathrm{m} \ (n_\mathrm{trap,3} - c_\mathrm{t,3}) - p_3 c_\mathrm{t,3}

McNabb & Foster model

Trapping

Other models assume traps can hold more than one H

\mathrm{H} + [\ \ \ ]_\mathrm{trap} \ \substack{p_1 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_1} \ [\mathrm{H}_1]_\mathrm{trap}

\mathrm{H} + [\mathrm{H}_1]_\mathrm{trap} \ \substack{p_2 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_2} \ [\mathrm{H}_2]_\mathrm{trap}

\mathrm{H} + [\mathrm{H}_2]_\mathrm{trap} \ \substack{p_3 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_3} \ [\mathrm{H}_3]_\mathrm{trap}

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot \left(D\nabla c_\mathrm{m} \right) + S - \sum{k_{i} \ c_\mathrm{m} \ c_{\mathrm{t},i-1} - p_i c_{\mathrm{t},i}}

\frac{\partial c_{\mathrm{t},i}}{\partial t} = k_{i} \ c_\mathrm{m} \ c_{\mathrm{t},i-1} - p_i c_{\mathrm{t},i} - k_{i+1} \ c_\mathrm{m} \ c_{\mathrm{t},i} + p_{i+1} c_{\mathrm{t},i+1}

Many of these processes are thermally activated

Recombination

Dissociation

Absorption

Trapping

Detrapping

Diffusion

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{k_B \ T}\right)}

Pre-exponential factor

Activation energy (eV/H)

Temperature (K)

Boltzmann constant (eV/H/K)

k_B = 8.617\times 10^{-5} \ \mathrm{eV/H/K}

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{R \ T}\right)}

Pre-exponential factor

Activation energy (J/mol)

Temperature (K)

Gas constant (J/mol/K)

k_B = 8.617\times 10^{-5} \ \mathrm{eV/H/K}

R = 8.314 \ \mathrm{eV/H/K}

\frac{E_D [eV/H]}{k_B} = \frac{E_D [J/mol]}{R}

Conversion:

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{R \ T}\right)}

$ 1/T $ (1/K)

E_X > 0

E_X < 0

Arrhenius law

X = X_0 \ \exp{\left(\frac{-E_X}{R \ T}\right)}

\log{(X)} = \log{(X_0)} \ - \frac{E_X}{k_B \ T}

\log{(X)} = \log{(X_0)} \ - \frac{E_X}{k_B} \ \frac{1}{T}

Intercept

+ Slope

Y =

Arrhenius law

Arrhenius parameters:

Diffusivity
Solubility
Permeability
Recombination coeff.
Dissociation coeff.
Trapping rate
Detrapping rate

Check out the HTM project!

github.com/RemDelaporteMathurin/h-transport-materials

Take aways

Key processes of H transport can be modelled mathematically
These processes are thermally activated
Next up:
- How to model these processes at the atomistic scale → Duc Nguyen's lecture
- at the component and reactor scale → R Delaporte-Mathurin's lecture
- How to characterise them experimentally → Thomas Fuerst's lecture
- Numerical modelling hands on workshop