# Stress-diffusion coupling. Application to interstitial diffusion

Lisa Germain, Sébastien Jegou, Laurent Barrallier

► To cite this version:

Lisa Germain, Sébastien Jegou, Laurent Barrallier. Stress-diffusion coupling. Application to interstitial diffusion. International Journal of Mechanical Sciences, 2024, 279, pp.109574. 10.1016/j.ijmecsci.2024.109574. hal-05340822

HAL Id: hal-05340822

https://hal.univ-brest.fr/hal-05340822v1

Submitted on 31 Oct 2025

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# Stress-diffusion coupling. Application to interstitial diffusion.

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Lisa GERMAINª, Sébastien JEGOUᵇ, Laurent BARRALLIERᵇ

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## Abstract

A numerical study of the stress-diffusion coupling theory based on the Cahn-Larché method is presented in the case of interstitial diffusion. Indeed, the interaction between diffusion and stress fields has been widely studied in the past. Most of the studies focused on the effect of chemical stress due to mass transport on diffusion. The majority of the studies have been focusing on the coupling between external mechanical and chemical stresses with diffusion. In this study, the effect of stress/strain on mobility and stress gradient mobility and driving forces are presented, based on the model of Voorhees and Larché. The analytical solutions are detailed, from general equations (interstitial diffusion, elastic material) to a macroscopic model (interstitial diffusion, isotropic elastic material, infinite dilute solution, 2D axisymmetric). Numerical simulations for different samples thickness and $\beta$ parameter helped to better understand and then experimentally identify the prevailing effect of stress on mobility or diffusion driving forces. These studies demonstrated the impact of constant compressive and tensile stress on diffusion kinetics through the mobility term of the equation, as well as the effects of negative and positive stress gradients via the Nernstian term. The results show that constant compressive stress and negative stress gradient hinder the diffusion. On the contrary, tensile stress and positive stress gradient promote the diffusion. This work also demonstrates the importance of coupling between mobility and Nernstian effect on the diffusion promotion.

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**Keywords:** diffusion, driving force, stress-diffusion coupling, residual stresses, interstitial alloy, nitriding

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<footnote>ªInstitut de Recherche Dupuy De Lôme, IRDL, Centre de Recherche Christian Huygens, Rue de Saint-Maudé, Lorient, 56100, France</footnote>

<footnote>ᵇArts et Métiers Institute of Technology, MSMP, 2 cours des Arts et Métiers, Aix-en-Provence, 13617, France</footnote>
1. Introduction

Diffusion is known through Fick's first law, demonstrated in 1855, which describes phenomenologically, and macroscopically, the proportionality of the flux of a concentration gradient through a diffusion coefficient (1). Researchers have demonstrated the failure of this first Fick equation in the case of solid state diffusion, thanks to the Pierre Curie's 1894 study, which showed that long-range elastic interactions resulting from compositional inhomogeneities give rise to non-local contributions (2). This stress-induced diffusion phenomenon is actually initiated by self-induced stress fields. In the absence of force, defects migrate randomly by jumping from one site to another through thermal agitation. A uniform stress cannot generate a flux of atoms, but induces local atomic jumps that allow a reorientation of the elastic distortion caused by the size effect (except in the case of cubic symmetry). Conversely, when a force is applied, a concentration gradient is created involving a diffusion that opposes the "forced" diffusion (3). The coupling between mechanical stresses (applied or residual) and chemical fields during a diffusion process within a so-called "deformable" and "reactive" solid has been a major research topic for many years (4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20). The transport of matter by diffusion under stress can lead to the progressive degradation of the microstructure of a material, as for example in the case of the oxidation of metals (diffusion of oxygen atoms (21; 22; 23)), or structural embrittlement by hydrogen (diffusion of hydrogen atoms (24; 25; 5)). But it can also be used to improve mechanical properties that are controlled by solid state diffusion like in the case of gas nitriding (26; 27; 28; 29; 30; 31; 32; 33; 34)). Early studies focused on the fact that matter transport in a solid was defined by chemical driving forces (in the absence of any other force), i.e. self-induced stresses (35; 4; 6). Subsequently, they have shown the effect of stresses, as activators (e.g. tension (36; 37; 38)), reducers (e.g. compression (36)) or seems with no effect (e.g. shear stress (5)). However, these effects have always been explained in relation to a driving force but not mobility although the effect of stress on mobility was introduced analytically by Cahn and Larché (6). Moreover numerical or

<footnote>*Corresponding author email: lisa.germain@univ-ubs.fr (Lisa GERMAIN)</footnote>
experimental studies have always taken into account an effective diffusion coefficient. The present work focuses on the decoupling of the stress/strain effect (mobility effect) and the stress gradient effect (mobility gradient and driving forces). The latter effect is already present in a diffusion process given the heterogeneities of the material (local) and induced by diffusion (long distance). The approach is based on a multi-scale analysis to predict the macroscopic coupling. This work is divided into two parts. The first is to understand the importance of self-induced stresses during a diffusion process, i.e. the difference in influence between mobility and driving force, and the second is to understand the influence of a stress gradient and a homogeneous stress on the diffusion kinetics.

In this paper, the complete analytical problems taking into account the mechanical re-equilibrium during the diffusion phenomenon, are demonstrated. For this purpose, some elements of mechanics and thermodynamics are first given in order to define the coupling equations between solid state atomic diffusion and stress. Calculations are carried out on different samples thickness in order to study the influence of mechanical equilibration and stress redistribution.

## 2. Elements of continuum mechanics and thermodynamics

### 2.1. *Fundamental statements of mechanics*

#### 2.1.1. *Stress-free strains*

In this case study, the theory of infinitesimal transformations will be taken into account, i.e. the hypothesis of small perturbations. The fundamental mechanical statements on the strain equations are detailed in Appendix A. In a general case, the stress-free strains, $\epsilon_{ij}^F$, are the sum of the plastic, thermal and volumetric strains. In the present work, thermal and plastic strains are neglected. Only the volumetric strains associated with the phase transformations and the solid solution during the diffusion process are taken into account (39). The stress-free strain defined by Cahn and Larché (40; 41; 39), in the case where a crystal lattice is reduced to a spherical inclusion in a homogeneous medium with anisotropic elastic properties (i.e. which does not correspond to the case of phase transformations), is given by :
$$ \varepsilon_{ij}^{\mathrm{F}}(c_{I}^{\phi}) = (\eta_{I}^{\phi})_{ij} \cdot c_{I}^{\phi} \qquad (1) $$

where $c_I^\phi$ is the concentration [-] of the element I in the phase $\phi$ and $(\eta_I^\phi)_{ij}$ is the chemical expansion tensor defined as $(\eta_I^\phi)_{ij} = \frac{\partial \varepsilon_{ij}^F}{\partial c_I^\phi}$.

In continuum mechanics, a tensor field of compatible strain in a plate is a unique tensor field that is obtained when the body is subjected to a single-valued continuous displacement field in order to guarantee the uniqueness of the solution (42; 43). Indeed, the residual stress field is statically admissible and the total strains $\varepsilon_{ij}^{\mathrm{tot}}$ must satisfy the six compatibility equations in the $(x, y, z)$ space:

$$ \left\{ \begin{array}{l} \varepsilon_{xx,yy}^{\mathrm{tot}} + \varepsilon_{yy,xx}^{\mathrm{tot}} - 2\varepsilon_{xy,xy}^{\mathrm{tot}} = 0 \\ \varepsilon_{yy,zz}^{\mathrm{tot}} + \varepsilon_{zz,yy}^{\mathrm{tot}} - 2\varepsilon_{yz,yz}^{\mathrm{tot}} = 0 \\ \varepsilon_{zz,xx}^{\mathrm{tot}} + \varepsilon_{xx,zz}^{\mathrm{tot}} - 2\varepsilon_{xz,xz}^{\mathrm{tot}} = 0 \\ \varepsilon_{xx,yz}^{\mathrm{tot}} + (\varepsilon_{yz,x}^{\mathrm{tot}} - \varepsilon_{xz,y}^{\mathrm{tot}} - \varepsilon_{xy,z}^{\mathrm{tot}})_{,x} = 0 \\ \varepsilon_{yy,xz}^{\mathrm{tot}} + (\varepsilon_{xz,y}^{\mathrm{tot}} - \varepsilon_{xy,z}^{\mathrm{tot}} - \varepsilon_{yz,x}^{\mathrm{tot}})_{,y} = 0 \\ \varepsilon_{zz,xy}^{\mathrm{tot}} + (\varepsilon_{xy,z}^{\mathrm{tot}} - \varepsilon_{yz,x}^{\mathrm{tot}} - \varepsilon_{zx,z}^{\mathrm{tot}})_{,z} = 0 \end{array} \right. \qquad (2) $$

### 2.1.2. Mechanical equilibrium

The equations of continuum mechanics Eq. 3, describes the variation of stresses in the absence of body forces ($f_i = 0$) on the solid. If the body is at static equilibrium, it must be satisfied at every point within the body. If the stress distribution is homogeneous, Eq. 3 is automatically satisfied since each derivative with respect to distance is identically equal to zero. These equations, which will be developed below, will follow Einstein's notation.

$$ \sigma_{ij,j} = 0 \quad \text{on} \quad \mathrm{V} \qquad (3) $$

In the case where the volume V element has a surface boundary $\partial V$, so that any surface force will also contribute to the equilibrium of forces, the boundary conditions of the stress distribution existing in the material necessary for equilibrium are given by Eq. 4:
$$ \sigma_{ij}n_j = g_i \quad \text{on} \quad \partial V \qquad (4) $$

Constitutive equations are described by the generalized Hooke's law relating stresses to strains as follows:

$$ \sigma_{ij} = C_{ijkl} (\varepsilon_{kl}^{\text{tot}} - \varepsilon_{kl}^{\text{F}}) \qquad (5) $$

The alternative form of the stress-strain relationship is:

$$ \varepsilon_{ij}^{\text{tot}} = \varepsilon_{ij}^{\text{F}} + S_{ijkl}\sigma_{kl} \qquad (6) $$

$$ S_{ijkl}C_{klmn} = \frac{1}{2}(\delta_{im}\delta_{jn} + \delta_{in}\delta_{jm}) \qquad (7) $$

## 2.2. Chemical potential applied to an interstitial solid solution in the case of an isotropic elastic material

In 1923, Gilbert Newton Lewis introduced the concept of chemical activity $a_I$ in the definition of chemical potential (44). At constant temperature and pressure, the chemical potential $\mu_I$ [J.mol⁻¹] of the element I is equal to:

$$ \mu_I(T, P, c_I^\phi) = \mu_I^0(T) + RT\ln(a_I) \qquad (8) $$

However, it is known that the chemical activity is function of the activity coefficient $\gamma_I$ and the concentration $c_I^\phi$ as $a_I = \gamma_I c_I^\phi$. So, in the case of an ideal solution, $\gamma_I = 1$ (45), and:

$$ \mu_I(T, P, c_I^\phi) = \mu_I^0(T) + RT \ln(c_I^\phi) \qquad (9) $$

In their study on the thermochemical equilibrium of multiphase solids under stress (46), Larché et al., have defined a new potential called "diffusion potential" M. Far from equilibrium, this potential is not constant throughout the solid and further diffusion can only occur
in directions determined by gradients in M (41; 4). They showed that the chemical potential of interstitials $\mu_I$ are constant and equal to the chemical potential $M_I^\Phi$ of the corresponding species (Eq. 10), the chemical potentials are assumed known at a hydrostatic pressure $P$, stress $\sigma_{ij}$ and composition $c_I^\phi$:

$$M_I^\Phi (P, \sigma_{ij}, c_I^\phi) = \mu_I (P, \sigma_{ij}, c_I^\phi) \quad (10)$$

In the case where the elastic constants are independent of composition, the diffusion potential $M_I^\phi (T, \sigma_{ij}, c_I^\phi)$ is given by:

$$M_I^\phi (T, \sigma_{ij}, c_I^\phi) = M_I^\phi (T, 0, c_I^\phi) - V_a \left( \frac{\partial \epsilon_{ij}^F}{\partial c_I^\phi} \right) \sigma_{ij} \quad (11)$$

Details of determining this equation are given in Appendix B. Eq. 11 is a simplification of Eq. B.7.

The dependency of the chemical potential with the stress state was firstly developed from both Gibbs and Helmholtz free energies using a variational approach (46; 47). Li et al. have developed the definition of the chemical potential of a moving component in a stressed solid based on the Gibbs equation. The equality of such a potential ensures the diffusion equilibrium of the mobile component in a body subjected to an arbitrary stress (48).

### 3. Stress-diffusion coupling

The description proposed by Cahn and Larché is the most well known. They have defined a rigorous method based on the thermodynamics of irreversible processes. Larché and Cahn used the equation developed by Chen-Min Li to study the dependence of local diffusion stresses in a solid medium, where the stresses developed from inhomogeneities of composition (35). The approach is based on the thermodynamics of a multicomponent and purely elastic solid which reaches equilibrium under stress (non-hydrostatic) (40; 46; 41; 4). Equilibrium is possible because any relative movement of the atoms, whether by sliding or diffusion, is not taken into account. The solid is assumed composed of a network which is integrated with the solid and allows to define a displacement and strain. All the variables form a set
of state variables (strain, composition variables and entropy density). The hypothesis of infinitesimal strain is taken into account, implying that the thermodynamic fields which are expressed in densities will refer identically either to the current state of the crystal (eulerian description) or to the appropriate reference state (lagrangian description).

Other authors, Villani et al., have used the formalism of Le maître and Chaboche which is a method with a mechanical point of view that concluded on equivalent flux equation (14). These two formalisms start from Helmholtz's definition of free energy. In the present work, one method which links the two formalisms is proposed in order to have a complete description starting from the most general equation while remaining in the case of an elastic material and to progressively add hypotheses often used in studies: infinitely diluted solution, isotropic, induced and applied stress.

### 3.1. Coupling via solid state diffusion

Fick proposed the first Fick's law (1), in analogy with the thermal law by Fourier (49), that describes the diffusion of matter in a binary medium, assuming that the diffusion is isotropic in case of a stable state. Diffusion occurs then from regions of high concentration towards regions of low concentration, with a magnitude proportional to the concentration gradient (spatial derivative):

$$ (J_I^\phi)_i = -D_I^\phi \frac{dc_I^\phi}{dx_i} \qquad (12) $$

Where $(J_I^\phi)_i$ is the uniaxial flux of element $I$ in the phase $\phi$ and $D_I^\phi$ is the coefficient of diffusion of element $I$ in the phase $\phi$ described by an Arrhenius law (50; 51).

$$ D_I^\phi = D_0^\phi \exp \left( -\frac{\Delta H}{k_B T} \right) = D_0^\phi \exp \left( -\frac{E_A}{RT} \right) \qquad (13) $$

The Second Fick's (Eq. 14), is known as the law of mass conservation in a non-permanent equation. This equation applies only to strictly conservative species (or particles) (52). Eq. 14 predicts changes in concentration with respect to time in the form of a partial differential equation:
$$ \frac{\partial c_I^\phi}{\partial t} = \frac{\partial (J_I^\phi)_i}{\partial x_i} \quad \text{on} \quad \partial V \qquad (14) $$

Where $\frac{\partial c_I^\phi}{\partial t}$ is the gradient of concentration as a function of time and $\partial V$ is the surface of the volume $V$. The boundary conditions are given by:

$$ J_I^\phi = (J_I^\phi)_i \cdot n_i \quad \text{on} \quad V \qquad (15) $$

The Fick's law were showed non-validated in case of the inhomogeneity at a long distance (2). The stress-diffusion coupling shows the combined effect of a concentration gradient and a mechanically-induced driving force. More generally, when a force is applied, a concentration gradient is created involving diffusion which opposes "forced" diffusion (52).

$$ (J_I^\phi)_i = -(D_I^\phi)_{ij} \frac{\partial c_I^\phi}{\partial x_i} + \langle (v_I^\phi)_i \rangle c_I^\phi \qquad (16) $$

The Nernst-Einstein relationship gives $\langle (v_I^\phi)_i \rangle = \frac{F_j(D_I^\phi)_{ij}}{k_B T}$ with $F$ is the driving force (as the electrotransport, the thermotransport, the centrifugal force, the stress gradient, etc.). In the absence of force, defects migrate randomly by jumping from one site to another by thermal agitation.

$\langle (v_I^\phi)_i \rangle c_I^\phi$ corresponds to the Nernstian term i.e. the action of an external force, or driving force, under the effect of which the particles move with a certain average speed giving rise to a flow. In this article, $\langle (v_I^\phi)_i \rangle c_I^\phi$ corresponds to the Nernstian term i.e. the action of an external $F_j = \frac{d\sigma_{ij}}{dx_i}$ is focused on a driving force due to a stress gradient. It follows that Eq. 16 takes the form:

$$ (J_I^\phi)_i = -(D_I^\phi)_{ij}^{\text{eff}} \frac{\partial c_I^\phi}{\partial x_i} = -(D_I^\phi)_{ij} \left( 1 + \frac{\partial \sigma_{ij}}{\partial c_I^\phi} \frac{c_I^\phi}{k_B T} \right) \frac{\partial c_I^\phi}{\partial x_i} \qquad (17) $$

The presence of a gradient of stress induced by the composition is equivalent to an additional driving force. This term is a function of the state of stress and composition. It represents the stress-induced diffusion forces during diffusion of the species directly influ-
encing the diffusion velocity of the mechanically diffusing atom undergone by the structure.

If not considered, one experimentally determined a so-called effective diffusion coefficient $(D_I^\phi)_{ij}^{\text{eff}}$ rather than an intrinsic one as it will be defined below.

Cahn and Larché wrote a formalism based on the equation defined by Einstein (53) and Darken (54), showing that the flux is proportional to the gradient of the diffusion potential as presented in Eq. 18 (40; 41; 46). Eq. 18 comes from the thermodynamics of irreversible processes (TPI). Its demonstration is explained in detail in Appendix C.

$$ (J_I^\phi)_i = -B_{ij}^\phi \frac{\partial M_I^\phi}{\partial x_i} \quad i = 1, 2, 3 \qquad (18) $$

Larché and Cahn proposed a development regarding the B coefficient (4). It is a function of local temperature T, stress σ and composition $c_I^\phi$. Larché and Voorhees defined the coefficient $B_{ij}$ as a function of composition and stress in the case of a first-order stress (6):

$$ B_{ij} = \frac{c_I^\phi}{\Phi R T} (B_{ij}^0 + B_{ijkl}^1 \sigma_{kl}) \qquad (19) $$

Where:

$$ B_{ij}^0 = D_I^\phi \delta_{ij} \qquad (20) $$

$$ B_{ijkl}^1 = \beta \delta_{ij} \delta_{kl} + \gamma_I (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}) \qquad (21) $$

Larché and Voorhees proposed that the coefficients β and γ are simple scalars (6). Philibert mentioned γ is the thermodynamic activity coefficient which also intervenes in the chemical activity factor (see Eq. 8) and the coefficient β is a measure of the activation volume (9).

Thanks to simplifications, Eq. 19 becomes:

$$ B_{ij} = \frac{c_I^\phi}{\Phi R T} (D_I^\phi \delta_{ij} + \beta \sigma_{kk} \delta_{ij} + 2 \gamma_I \sigma_{ij}) \qquad (22) $$

The term $(\beta\sigma_{kk}\delta_{ij} + 2\gamma\sigma_{ij})$ shows the influence of the stresses on the mobility by means
of the hydrostatic stresses $\sigma_{kk}$ and the stress tensor $\sigma_{ij}$. The term $2\gamma_I\sigma_{ij}$ is equivalent to the effect of shear and does not seem to have any influence on the mobility (9; 6; 5). Furthermore, if the three components $\sigma_{kk}$ are equal, the well-known effect of hydrostatic pressure is found (10). Finally, it was noticed that $\beta\sigma_{kk}\delta_{ij}$ and $2\gamma_I\sigma_{ij}$ are homogeneous to a diffusion coefficient, $\beta$ and $\gamma$ are in [cm$^2$.s$^{-1}$.MPa$^{-1}$].

Li studied chemical stresses in elastic media of simple geometries (55). Chu et al., used the formalism of Li, and conducted an analysis of the effects of chemical stresses on diffusion (56). They found that chemical stresses improve mass transfer and that the effective diffusivity resulting from chemical stresses is proportional to the concentration.

Using Eqs. B.1, 11 and 22, and assuming no dependence of elastic properties with the composition change, the flux (Eq. 18) in general case is equal to:

$$ (J_I^\phi)_i = -\frac{c_I^\phi}{\Phi RT} (D_I^\phi \delta_{ij} + \beta\sigma_{kk}\delta_{ij} + 2\gamma_I\sigma_{ij}) \left( \frac{RT\Phi}{c_I^\phi} \frac{\partial c_I^\phi}{\partial x_i} - V_a\eta_{kl}\frac{\partial\sigma_{kl}}{\partial x_i} \right) \quad (23) $$

Some studies show that the diffusion could be affected by external stresses. In fact, stress can be decomposed as:

$$ \sigma_{ij}^{\text{total}} = \sigma_{ij}^{\text{residual}} + \sigma_{ij}^{\text{external}} \quad (24) $$

Where $\sigma_{ij}^{\text{residual}}$ is the residual stress tensor and $\sigma_{ij}^{\text{external}}$ is the external stress tensor. In fact, studies showed that the diffusion processes are affected by mechanical stresses, more precisely by crystal strains that cause them, through two possible origins (24; 57):

* residual, corresponding to a state of self-equilibrated stress, i.e. they occur in a volume of material without it being mechanically stressed (58; 42; 59);

* external, corresponding to a state of mechanical stress applied to the limits of the volume of material.

## 3.2. Coupling via composition fields

It is necessary to determine the coupling from a mechanical point of view as a function of the composition fields. As explained previously, at equilibrium, the diffusion potential $M$
is constant and satisfies the mass balance equation (Eqs. 14 and 15). The expression of the chemical equilibrium concentration is obtained as a function of the corresponding diffusion potential, $(M_I^\Phi)_{eq}$, from Eq. C.17, assuming that $c_I^\phi \ll 1$.

$$ (c_I^\Phi)_{eq} = \exp\left(\frac{V_a}{RT}(M_I^\Phi)_{eq} - \frac{E_f}{RT}\right) \exp\left(\frac{V_a}{RT}(\eta_I^\Phi)_{ij}\sigma_{ij}\right) \quad (25) $$

In the case of an elastic open-system, Cahn and Larché introduced a notion $\chi_I^\Phi$ as (4):

$$ \frac{1}{\chi_I^\Phi} = \rho\left(\frac{\partial M_I^\Phi}{\partial c_I^\Phi}\right)_{\sigma_{ij}=0} \quad (26) $$

Eqs. C.17 and 26, allow to simplified the chemical equilibrium concentration, as:

$$ (c_I^\Phi)_{eq} = (c_I^\Phi)_0 + \chi_I^\Phi (\eta_I^\Phi)_{ij} \sigma_{ij} \quad \text{avec} \quad \chi_I^\Phi = \frac{V_a}{RT} c_I^\phi \quad (27) $$

Where $(c_I^\Phi)_0 = \exp\left(\frac{V_a}{RT}(M_I^\Phi)_{eq} - \frac{E_f}{RT}\right)$.

At equilibrium, where $\chi_I^\Phi$ is taken as a function of $c_I^\Phi = (c_I^\Phi)_0$:

$$ \varepsilon_{ij}^F (c_I^\Phi) = (\eta_I^\Phi)_{ij} c_I^\phi + \chi_I^\Phi (\eta_I^\Phi)_{ij} (\eta_I^\Phi)_{kl} \sigma_{kl} \quad (28) $$

According to Hooke's law (Eq. 5), the total strain tensor is written as:

$$ \varepsilon_{ij}^{tot} = (S_{ijkl} + \chi_I^\Phi (\eta_I^\Phi)_{ij} (\eta_I^\Phi)_{kl}) \sigma_{kl} + (\eta_I^\Phi)_{ij} c_I^\phi \quad (29) $$

In brackets, as according to Larché and Cahn (4), it is the open-system compliances, $S_{ijkl}^0$, related to the constant composition compliances $S_{ijkl}$.

# 4. Numerical implementation for isotropic material

## 4.1. Problem definition

The model used is based on a homogeneous and isotropic diffusion. It is assumed:

* the calculations are carried out in the case of a semi-infinite material (see Fig. 1). In this problem, the indices i and j will take the values x, y and z,
• the material is considered macroscopically elastic, homogeneous and isotropic,

• shear and cission along x and y are zero: $\epsilon_{xy}^{\text{tot}} = \epsilon_{yx}^{\text{tot}} = 0$, $\sigma_{xy} = \sigma_{yx} = 0, \forall z$.

<figure><img src="image_2.png" /><figcaption>Figure 1: Scheme of a semi-infinite body i.e. bounded in one direction (here along z) and unbounded in another.</figcaption></figure>

In the case of small strains, small displacements and isotropic material the elastic strains $\epsilon_{ij}^e$ is described as:

$$ \epsilon_{ij}^e = \frac{1+\nu}{E}\sigma_{ij} - \frac{\nu}{E}\sigma_{kk}\delta_{ij} \qquad (30) $$

Eq. 5, in the isotropic and elastic case, takes the form:

$$ \sigma_{ij} = 2\mu\epsilon_{ij}^e + \lambda\epsilon_{kk}^e\delta_{ij} \qquad (31) $$

Where $\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}$ and $\mu = \frac{E}{2(1+\nu)}$ are the Lamé's coefficient.

The compositional strain tensor in Eq. 1 takes the form:

$$ \eta_{ij} = \eta\delta_{ij} = \frac{\Delta V_I}{3V_a}\delta_{ij} \qquad (32) $$

Where $\Delta V_I$ is the partial molar volume of the element I and $V_a$ is the molar volume of the host lattice.
If the crystal is isotropic or has cubic symmetry, the compositional strain is given by Eq. 32. Based on thermodynamic theory, when a mobile solute in an ideal and dilute solid solution is subjected to stresses, the diffusion potential becomes:

$$M_I^{\phi}(T, \sigma_{ij}, c_I^{\phi}) = M_I^{\phi}(T, 0, c_I^{\phi}) - V_a \eta \sigma_{kk} \quad (33)$$

In the case of interstitial diffusion $\eta$ is positive, i.e. an increase in the concentration of the constituent increases the volume of the crystal. For a given composition, it follows that a compressive hydrostatic stress ($\sigma_{kk} < 0$) increases the diffusion potential, while a tensile stress reduces it. Since the diffusion potential $M_I^{\phi}$ is constant at equilibrium, either the stress is uniform, in which case the composition is uniform, or the stress is not uniform and correlatively the composition is not uniform.

In case of dilute solution i.e. $c_I^{\phi} \ll 1$ and isotropic material, the gradient of potential of diffusion becomes (4):

$$\frac{\partial M_I^{\phi}}{\partial x_i} = \frac{RT}{c_I^{\phi}} \frac{\partial c_I^{\phi}}{\partial x_i} - V_a \eta \frac{\partial \sigma_{kk}}{\partial x_i} \quad (34)$$

Eq. 34 was confirmed by Kirchheim (5). Under applied tensile stress, the change in chemical potential (here equal to the potential of diffusion) is proportional to the stress and the partial molar volume $V_a$ of hydrogen.

R. Kirchheim studied the effect of a torsion stress on wires, these experiments showed that the pure shear stress produced by torsional stress had no impact on diffusion kinetics relative to tensile stress and concluded on no change of the chemical potential. In addition, it was observed that torsion leads to a reorientation of point defects (5; 37).

From the definition of the thermodynamic factor given in Eq. B.1, in case of infinitely diluted solution $c_I^{\phi} \ll 1$, it is possible to write:

$$\frac{\partial \ln \gamma_I}{\partial c_I^{\phi}} \frac{\partial c_I^{\phi}}{\partial \ln c_I^{\phi}} = \frac{\dot{\gamma}_I}{\gamma_I} c_I^{\phi} \quad (35)$$

So the thermodynamic factor $\Phi$ is:
$$ \lim_{c_I^\phi \to 0^+} \Phi = \lim_{c_I^\phi \to 0^+} \left[ 1 + \frac{\partial \ln \gamma_I}{\partial \ln c_I^\phi} \right] = 1 \qquad (36) $$

In case of isotropic material, Eq. 23 becomes:

$$ \left(J_I^\phi\right)_i = -\frac{c_I^\phi}{RT} \left(D_I^\phi \delta_{ij} + \beta\sigma_{kk} \delta_{ij} + 2\gamma\sigma_{ij}\right) \left(\frac{RT}{c_I^\phi} \frac{\partial c_I^\phi}{\partial x_i} - V_a \eta \frac{\partial \sigma_{kk}}{\partial x_i}\right) \quad (37) $$

If Eq. 37 is written as Eq. 17, an effective diffusion coefficient can be defined as:

$$ D_{\text{eff}}^{\phi} = \left( D_{I}^{\phi} \delta_{ij} + \beta \sigma_{kk} \delta_{ij} + 2 \gamma \sigma_{ij} \right) \left( 1 - \frac{V_a \eta}{RT} c_{I}^{\phi} \frac{\partial \sigma_{kk}}{\partial c_{I}^{\phi}} \right) \quad (38) $$

In the case of isotropy material, the total strain $\varepsilon_{ij}^{\text{tot}}$, based on the generalised Hooke's law Eq. A.2 and Eqs. 1 to 30, can be written as:

$$ \varepsilon_{ij}^{\text{tot}} = \frac{1 + \nu}{E} \sigma_{ij} - \frac{\nu}{E} \sigma_{kk} \delta_{ij} + c_I^\phi \eta \delta_{ij} \qquad (39) $$

So:

$$ \varepsilon_{ij}^{\text{tot}} - \eta \delta_{ij} c_I^\phi = \frac{1 + \nu}{E} \sigma_{ij} - \frac{\nu}{E} \sigma_{kk} \delta_{ij} \qquad (40) $$

The previous equation becomes as a function of the strain:

$$ \sigma_{ij} = 2 \frac{E}{2(1+\nu)} \left( \varepsilon_{ij}^{\text{tot}} - c_I^\phi \eta_{ij} \right) + \frac{E\nu}{(1+\nu)(1-2\nu)} \left[ \varepsilon_{kk}^{\text{tot}} - c_I^\phi \eta_{kk} \right] \delta_{ij} \quad (41) $$

However, in an open-system case, $E$ and $\nu$ are respectively function of $E_0$ and $\nu_0$:

$$ \lambda = \frac{E_0 (\nu_0 + \chi \eta^2 E_0)}{(1 + \nu_0)(1 - 2\nu_0 - 3\chi \eta^2 E_0)} \qquad (42) $$

$$ \mu = \frac{E_0}{2(1 + \nu_0)} \qquad (43) $$

The equation of mechanical equilibrium Eq. 3 thus becomes in the case of an isotropic material:
$$ \mathrm{div} \left[ 2\mu (\varepsilon_{ij}^{\mathrm{tot}} - c_I^{\phi} \eta_{ij}) + \lambda [\varepsilon_{kk}^{\mathrm{tot}} - c_I^{\phi} \eta_{kk}] \cdot \delta_{ij} \right] = 0 \quad (44) $$

According to Eq. 29, the open-system compliances coefficient takes the form:

$$ S_{ijkl}^{0} = \left(-\frac{\nu}{E} + \chi\eta^{2}\right) \delta_{kl} + \frac{1}{2G} (\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) \quad (45) $$

The equation of mechanical equilibrium Eq. 14 thus becomes:

$$ \mathrm{div} \left[ S_{ijkl}^{0}^{-1} (\varepsilon_{kl}^{\mathrm{tot}} - \eta_{kl} c_{I}^{\phi}) \right] = 0 \quad (46) $$

This equation shows that the problem of diffusion/mechanical equilibrium is equivalent to solving the purely elastic problem for a "fictitious" coefficient $S_{ijkl}^{0}$ known. The elasticity constants for an open system are defined as follows:

$$ \nu_0 = \frac{\nu - \chi \eta^2 E}{1 + \chi \eta^2 E} \quad ; \quad E_0 = \frac{E}{1 + \chi \eta^2 E} \quad ; \quad G_0 = G \quad (47) $$

$$ E_0 = \frac{E}{1 + \chi \eta^2 E} \quad (48) $$

Where $\nu_0$ is the "fictitious" Poisson's coefficient, $E_0$ is the "fictitious" Young's modulus and $G = \frac{E}{2(1+\nu)}$ is the shear modulus. The fictitious tensor of compliance then becomes:

$$ S_{ijkl}^{0} = -\frac{\nu_0}{E_0} \delta_{ij} \delta_{kl} + \frac{1}{2G_0} (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}) \quad (49) $$

Eqs. 47 to 49 show that if the concentration, $c_I^{\phi}$, or the coupling parameter, $\eta$, are small; then, the open system constants reduce to the standard elastic constants.

Macroscopic strains and stresses can be expressed using the relations of continuum mechanics. The residual stresses determine a statically admissible field (S.A.) and satisfy the equilibrium equations Eq. 3 and that Eq. 4 is for $z = 0$, $\sigma_{ij}n_j = 0$ with
$$ 271 \quad n_j = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} : \begin{cases} \sigma_{xy} = 0 \\ \sigma_{yz} = 0 \\ \sigma_{zz} = 0 \end{cases}. $$

273 This leads to a plane state of cylindrical stress:

$$ \begin{bmatrix} \sigma_{xx} = \sigma & 0 & 0 \\ 0 & \sigma_{yy} = \sigma & 0 \\ 0 & 0 & 0 \end{bmatrix} \qquad (50) $$

$$ \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \epsilon_{zz} = \epsilon \end{bmatrix} \qquad (51) $$

274 Thanks to Eqs. 50 and 51, Eqs. 40 and 41 become:

$$ \begin{cases} \sigma = -2\mu\eta c_I^\phi + \lambda(\epsilon - 3\eta c_I^\phi) \\ 0 = 2\mu(\epsilon - \eta c_I^\phi) + \lambda(\epsilon - 3\eta c_I^\phi) \end{cases} \qquad (52) $$

$$ \begin{cases} \epsilon = \frac{2\mu+3}{2\mu+\lambda}\eta c_I^{\phi} \\ \sigma = -\frac{2\mu(2\mu+3)}{2\mu+\lambda}\eta c_I^{\phi} \end{cases} \qquad (53) $$

275 Considering a semi-infinite body implies that $(J_I^\phi)_i = (J_I^\phi)_z$ and $\gamma_I \sigma_{ij} = 0$ for any value of $\gamma_I$ (only the flux along $z$ is not equal to none). It follows that the flux is:

$$ \begin{align} (J_I^\phi)_z &= -\frac{c_I^\phi}{\Phi R T} (D_I^\phi + \beta \sigma_{kk}) \left( \frac{R T \Phi}{c_I^\phi} \frac{\partial c_I^\phi}{\partial z} - V_a \eta \frac{\partial \sigma_{kk}}{\partial z} \right) \tag{54} \\ &= - (D_I^\phi + \beta \sigma_{kk}) \frac{\partial c_I^\phi}{\partial z} + \frac{V_a \eta}{\Phi R T} (D_I^\phi + \beta \sigma_{kk}) c_I^\phi \frac{\partial \sigma_{kk}}{\partial z} \nonumber \end{align} $$

277 During diffusion, the mass conservation must be verified:
$$ \frac{\partial c_I^\phi}{\partial t} + \frac{\partial (J_I^\phi)_z}{\partial z} = 0 \qquad (55) $$

with

$$ \frac{\partial (J_I^\phi)_z}{\partial z} = \beta \frac{\partial \sigma_{kk}}{\partial z} \left( \frac{V_a \eta}{\Phi R T} c_I^\phi \frac{\partial \sigma_{kk}}{\partial z} - \frac{\partial c_I^\phi}{\partial z} \right) + (D_I^\phi + \beta \sigma_{kk}) \left( + \frac{V_a \eta}{\Phi R T} \left( \frac{\partial c_I^\phi}{\partial z} \frac{\partial \sigma_{kk}}{\partial z} + c_I^\phi \frac{\partial^2 \sigma_{kk}}{\partial z^2} \right) - \frac{\partial^2 c_I^\phi}{\partial z^2} \right) \qquad (56) $$

Finally, equation 55 can be simplified using $\sigma_{kk} = 2\sigma$ and considering a dilute solution $\Phi = 1$:

$$ \frac{\partial c_I^\phi}{\partial t} = 2\beta \frac{\partial \sigma}{\partial z} \left( \frac{\partial c_I^\phi}{\partial z} - 2 \frac{V_a \eta}{R T} c_I^\phi \frac{\partial \sigma}{\partial z} \right) + (D_I^\phi + 2\beta\sigma) \left( \frac{\partial^2 c_I^\phi}{\partial z^2} - 2 \frac{V_a \eta}{R T} \left( \frac{\partial c_I^\phi}{\partial z} \frac{\partial \sigma}{\partial z} + c_I^\phi \frac{\partial^2 \sigma}{\partial z^2} \right) \right) \qquad (57) $$

In contrast to the studies presented in the literature, Eq. 57 shows, in addition to the effect of a stress gradient (driving force) on the effect of diffusion kinetics, that mobility could be stress dependent according to the terms on stress-induced mobility ($D_I^\phi + 2\beta\sigma$) and stress-mobility $2\beta\frac{\partial\sigma}{\partial z}$.

## 4.2. Numerical solution

A finite difference model with an explicit Euler scheme has been written. Here, one quantity $q$ represents $c_I^\phi$, $\sigma$ or $\epsilon$. $q_z^t$ is the quantity at $t$ time and at $z$ depth and $q_{z+1}^{t+1}$ is the quantity at time $t+1$ and depth $z+1$.

The numerical solution of equation 57 is obtained using a finite difference schema and the explicit mode so that:
$$ \frac{c_z^t - c_z^{t-1}}{\Delta t} = 2\beta \frac{\sigma_z^t - \sigma_{z-1}^t}{\Delta z} \left( \frac{c_z^t - c_{z-1}^t}{\Delta z} - 2\frac{V_a \eta}{RT} c_z^t \frac{\sigma_z^t - \sigma_{z-1}^t}{\Delta z} \right) + \left( D_I^\phi + 2\beta \sigma_z^t \right) \left( \frac{c_{z+1}^t - 2c_z^t + c_{z-1}^t}{\Delta z^2} - 2\frac{V_a \eta}{RT} \frac{(c_z^t - c_{z-1}^t)(\sigma_z^t - \sigma_{z-1}^t) + c_z^t(\sigma_{z+1}^t - 2\sigma_z^t + \sigma_{z-1}^t)}{\Delta z^2} \right) \quad (58) $$

where $\Delta z$ and $\Delta t$ are respectively spatial and time step.

Equation 53 can be written as:

$$ \begin{aligned} \epsilon_z^t &= \frac{2\mu + 3}{2\mu + \lambda} \eta c_z^t \\ \sigma_z^t &= -\frac{2\mu(2\mu + 3)}{2\mu + \lambda} \eta c_z^t \end{aligned} \quad (59) $$

A convergent and stable solution is found if the following is verified (60):

$$ \Delta t = \alpha \frac{\Delta z^2}{D} \quad (60) $$

Where $\alpha \le 0.5$ for the stability.

The coupling is processed using experimentally known input data such as mechanical characteristics (Young's modulus $E$, Poisson's coefficient $\nu$), concentration data ($c_0$), time data ($t$), data related to the binary system under study ($V$, $V_a$, $\eta$) and sample thickness. Then, a time loop is performed in which the diffusion calculation is performed, then the mechanical calculation follows. Indeed, this code is a weak coupling, i.e. the calculations are performed one after the other.

For both cases, the steady-state concentration field and the mechanical equilibrium are determined by solving the coupled problems numerically using a python code solving the explicit method via an axisymmetric 2D model. The analytical solution of the equilibrium is directly adapted to the problem from Eqs. 37 and 44.
## 5. Interstitial diffusion of nitrogen in ferrite: results and discussions

Many experimental and numerical studies have focused on the influence of treatment parameters (e.g. gas nitriding with time t, temperature T and nitriding potential $K_N$) in order to understand the effects on diffusion kinetics, the evolution of residual stresses or also the effect of precipitation (not taken into account here) (61; 62; 63). The objective here is to understand the effect of stresses on the diffusion kinetics in order to obtain deeper depths and thus to reduce treatment times or temperatures.

### 5.1. Problem definition

Usually, the coefficients $\beta$ and $\gamma$ are not taken into account on the effective diffusion coefficient in the mobility part of the stress-diffusion coupling equation. As a result, a large number of papers published on this coupling have used an effective diffusion coefficient where the mobility is only a function of the stress-independent diffusion coefficient and where the stress only influences the diffusion potential gradient corresponding to the driving force. As presented earlier, Eq. 57 shows that mobility could be stress dependent. This paper aims to investigate the importance of the parameter $\beta$, the stress gradient and the chemical gradient/stress gradient distribution using a 2D axisymmetric model,

where the initial conditions are:

$$ \text{at} \quad t=0 \quad z>0 \quad c(z, 0) = c_0 \quad (61) $$

and the limits are:

$$ \forall t=0, \quad z=0 \quad c(0, t) = c_s \quad (62) $$

### 5.2. Residual stresses

The effect of chemical stress on the diffusion of mass transfer has been the subject of several studies in the literature, however the objective is to have information on the predomi-
nance between the effect of stress on the mobility part (see Eq. 54 with the ($D\delta_{ij} + \beta\cdot\sigma_{kk}\cdot\delta_{ij}$)) and the stress gradient (46; 35).

Fig. 2 shows more precisely the evolution of residual stresses. The diagram on the left shows that for the same composition profile, the mechanical equilibrium for two sample thicknesses will not be the same. Residual stresses, unlike applied stresses (without creep), evolve over time. At each moment the composition profile leads to stress-free strains, resulting in a local effect at the scale of the diffusion layer, which in turn plays a role in the global effect and this global effect in turn plays a role in the composition profile, as shown in the figure on the right.

<figure><img src="image_3.png" /><figcaption>Figure 2: (left) Schematic evolution of residual stresses for two sample thicknesses (semi-infinite and finite body) with the same diffusion profile of insertion atoms. (right) Evolution over time of residual stresses at local and global scales.</figcaption></figure>

Experimentally, it is known that as the treatment time increases, the stresses are initially intense and progressively decrease in intensity and spread out in depth, as schematically shown in Fig. 3. Indeed, the surface content is not constant and evolves over time until reaching a maximum which corresponds to the adsorption limit of the layer. However, it is difficult to understand what happens in the first few seconds of the diffusion. The boundary conditions of the flow, i.e. the Neumann conditions, could be used to observe these phenomena. However, in this work, the surface is assumed to be constant (Dirichlet boundary conditions) and its value does not exceed the solubility limit (no precipitation). Throughout
<table><caption>Table 1: Parameters values. $D_0$ value from Torchane (64), $\eta$ value from Cheng (65).</caption><thead><tr><th>D<sub>0</sub> [cm<sup>2</sup>.s<sup>-1</sup>]</th><th>c<sub>0</sub> [-]</th><th>η [-]</th><th>R [J/K/mol]</th><th>T [°C]</th></tr></thead><tbody><tr><td>4.9.10<sup>-8</sup></td><td>0.0025</td><td>2.64e<sup>-5</sup></td><td>8.314</td><td>520</td></tr><tr><td>E [GPa]</td><td>ν [-]</td><td>nb of step</td><td>Δz [μm]</td><td>Δt [s]</td></tr><tr><td>202</td><td>0,3</td><td>17640</td><td>1</td><td>(see Eq. 60)</td></tr></tbody></table>

this section, the surface concentration is fixed $c_0 = 2,5.10^{-3}$. This value corresponds to the solubility limit atomic percentage of nitrogen (N) in iron-alpha at the processing temperature 520 °C. This first section focuses on the evolution of the long range diffusion kinetics by playing on the following parameters: the influence of the $\beta$ coefficient, the treatment time and the thickness of the sample.

<figure><img src="image_4.png" /><figcaption>Figure 3: Schematic representation of experimental evolution of the concentration and residual stresses over time.</figcaption></figure>

Tab. 1 shows the parameters used in the residual stresses simulations corresponding to the mechanical characteristics of the material used, the diffusion data of the case of interstitial diffusion of nitrogen N in Fe-$\alpha$ and programming parameters.

### 5.2.1. Influence of $\beta$ parameter

The coefficient $\beta$, established by Larché and Voorhees (6), is a multiplier parameter of the stress amplitude allowing to observe the influence of the stresses on the mobility. This consideration of stresses by a degree 1 decomposition allows to observe that the amplitude and the sign of the stress can play an important role in the diffusion kinetics from an empirical point of view. The first study therefore investigates the effect of the parameter
$\beta$ on the diffusion kinetics of a semi-infinite plate (10 mm thick) treated for 1 hour. In this study, 3 values of $\beta$ cm$^2$.s$^{-1}$.MPa$^{-1}$ are analysed: 0, $10^{-11}$ and $10^{-10}$ cm$^2$.s$^{-1}$.MPa$^{-1}$ all compared to the Fick equation where the diffusion coefficient is equal to $D_0 = 4.9 \cdot 10^{-8}$ cm$^2$.s$^{-1}$. These selected values of $\beta$ were defined after several numerical simulations in order to compare only some interesting values.

In the case of $\beta = 0$, i.e. with only the effect of a residual stress gradient ($\frac{\partial\sigma}{\partial x} \neq 0$), it is observed that the effect of a positive stress gradient shows an acceleration of the long range diffusion kinetics. However, as the value of $\beta$ increases, the diffusion slows down. More precisely, it is interesting to note that from a certain value of $\beta$, the diffusion is slowed down compared to the analytical Fick profile. The $\beta = 10^{-11} \text{cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$ curve shows a diffusion which remains accelerated compared to the Fick curve but which is slowed down compared to the curve taking into account only the stress gradient. In the $\beta = 10^{-10} \text{cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$ curve, the diffusion is considerably slowed down compared to the Fick curve. As a consequence, the generation of compressive stress is lower and stress gradient are steeper.

This study shows the competition between the mobility gradient and the mechanical driving force (stress gradient alone without effect on mobility). Fig. 4b) illustrates the fact that when $\beta$ reaches a particular value, mobility takes over the influence of the stress gradient. This study shows that the effect of stress on mobility is predominant when $(\beta\sigma_{kk} = 0.1 \times D_0)$.

Fig. 5a) shows the comparative curves for the case $\beta = 10^{-10} \text{cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$ with and without taking into account the mechanical driving force given by $\frac{\partial\sigma_{kk}}{\partial z}$ in Eq. 54. This comparison reveals that the coupling of the stress gradient with mobility leads to a slowdown 3 times greater than the profile that only takes mobility into account (see the green and red arrows in Fig. 5b)).

### 5.2.2. Time evolution

The temporal effect of residual stresses on the diffusion kinetics (Eq. 54) is presented in Fig. 6a). The numerical simulations are performed on a semi-infinite plate (10 mm thick)
<figure><img src="image_5.png" /><figcaption>Figure 4: a) Evolution of concentration and residuals stresses profiles as a function of depth for a 10 mm thick sample at 520 °C for 1 h. Comparison of the Fick profile and three measures of the activation volume $β$ values of the coefficient: 0, $10^{-11}$ and $10^{-10}$ cm$^2$.s$^{-1}$.MPa$^{-1}$. b) Inset of Fig. 4a).</figcaption></figure>

which is subjected to three treatment times (1 h, 2 h and 3 h). As the treatment time increases, the diffusion depth increases and compressive residual stresses are generated and spread over deeper depth. A positive stress gradient induces accelerated diffusion whereas the combination with the effect of the stress on the mobility for $β = 10^{-10}$ cm$^2$.s$^{-1}$.MPa$^{-1}$ promotes slower diffusion. The evolution of residual stresses with time is characterized by a lower stress gradient with time, meaning a more spread stress field within the depth below the surface. Indeed, Fig. 6b) shows the evolution of the gradients of concentration and stress with space.

The effect of the stress gradient is more important at the beginning of the diffusion process. Furthermore, the combined effect of mechanical driving force and mobility ($β > 0$) leads to the evolution of a higher stress gradient at the beginning of the diffusion layer and a lower one at a greater depth compared to the mechanical driving force alone ($β = 0$). However, at the beginning of the diffusion and at a depth of less than 200 µm, the effect of mobility compared to the effect of the driving force alone leads to a deceleration of
<figure><img src="image_6.png" /><figcaption>Figure 5: a) Comparison of the concentration profiles (relative to the Fick profile) and residual stress as a function of depth for the case ($\beta > 0$ and $\frac{\partial\sigma}{\partial x} = 0$) and the case ($\beta > 0$ and $\frac{\partial\sigma}{\partial x} \neq 0$) for a 10 mm thick sample at 520 °C for 1 h for $\beta = 10^{-10} \text{ cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$. b) Inset of Fig. 5a).</figcaption></figure>

the kinetics at the beginning of the treatment and to an increase as time increases. This deceleration effect of the kinetics (mobility effect) at the beginning of the treatment leads to a general deceleration of the final concentration profile as observed in Fig. 6a).

### 5.2.3. Thickness sample evolution

Figs. 7 show the influence of stress on mobility through 3 sample thicknesses (2, 5 and 10 mm). The result shows that in the case of $\beta = 0$ the concentration profiles are almost identical for the three samples. In the case of $\beta = 10^{-10} \text{ cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$, the concentration profiles are characterized by a sluggish diffusion. The higher sample thickness, the slower diffusion. Surface stresses are highest for $\beta = 10^{-10} \text{ cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$ and large thickness. The stresses from 50 µm are stronger for $\beta = 0$ and large thickness. The stress gradients are stronger for $\beta = 10^{-10} \text{ cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$, and high thickness.

### 5.3. Controlled loading study

f Within the framework of this study and the conditions chosen, it was shown that in order to observe a significant influence of the stress gradient (in the case where there are only residual stresses, i.e. a positive gradient and negative stresses), it is necessary to use
<figure><img src="image_7.png" /><figcaption>Figure 6: a) Evolution of concentration and residual stress profiles in function of depth; b) Evolution of $\frac{\partial c}{\partial x}$ and $\frac{\partial \sigma}{\partial x}$ in function of depth; for a 10 mm thick sample, for 3 treatment times (1h, 2h and 3h). Comparison between the Fick's law and two measures of the activation volume $\beta$ values: 0 and $10^{-10} \text{ cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$.</figcaption></figure>

a thin sample with a short treatment time and a large stress amplitude. Throughout this subsection, numerical simulations for a 1 mm thick sample treated for 1 h with a constant surface concentration of $c_0 = 0.001$ will be presented. This value of surface concentration is arbitrary because the expansion of the α-iron lattice due to stress-free strains is not taken into account ($\eta = 0$). Only the case of a fixed stress throughout the treatment is taken into account.

A study of time-constant homogeneous or linear loading, i.e. without any source of creep, was carried out. The objective of this study is to understand what happens at the level of long range stress field. The work focused on the impact of stress fields that can be easily obtained experimentally, such as a tension/compression test ($\sigma \neq 0$ and $\frac{\partial \sigma}{\partial z} = 0$) or a bending test ($\sigma \neq 0$ and $\frac{\partial \sigma}{\partial z} \neq 0$). Analytically from Eq. 57, it can be seen that a homogeneous stress will only have an effect on diffusivity ($\beta > 0$), because $\frac{\partial \sigma}{\partial z} = 0$. Conversely, a stress gradient ($\frac{\partial \sigma}{\partial z} \neq 0$) will have an effect on diffusivity but also on driving forces.
<figure><img src="image_8.png" /><figcaption>Figure 7: a) Evolution of concentration and residual stress profiles for three sample thicknesses (2 mm, 5 mm and 10 mm) in function of two measures of the activation volume β values (0 and $10^{-10}$ cm$^2$.s$^{-1}$.MPa$^{-1}$). b) Inset of Fig. 7a).</figcaption></figure>

### 5.3.1. Homogeneous stress

The study on homogeneous stress, meaning constant stress as a function of the depth below the surface, will be divided into two parts. The first is the study of a tensile stress of 1 GPa evolving through 5 β values: 0, $10^{-13}$, $10^{-12}$, $10^{-11}$ and $10^{-10}$ cm$^2$.s$^{-1}$.MPa$^{-1}$. The second is the study of a tensile and compressive stress (|1| GPa) for a given value of β.

In the case of these homogeneous stresses, i.e. $\frac{\partial\sigma}{\partial z} = \frac{\partial^2\sigma}{\partial z^2} = 0$, Eq. 57 takes the following form:

$$ \frac{\partial c_I^\phi}{\partial t} = (D_I^\phi + 2\beta\sigma) \frac{\partial^2 c_I^\phi}{\partial z^2} \qquad (63) $$

Fig. 8a) shows the evolution of the concentration profile for a tensile stress of 1 GPa with different values of β. This magnitude of stress cannot be reached in pure iron (having a very low elasticity limit) but this value is used to help setting up future experiments. Fig. 8a) shows that a tensile homogeneous stress accelerates the diffusion kinetics. When $\beta = 10^{-12}$ cm$^2$.s$^{-1}$.MPa$^{-1}$, the acceleration becomes considerable, i.e. when the orders of magnitude is closed to $\beta\sigma_{kk} = 0, 1D_0$ in agreement with the previous results in Fig. 4.
Fig. 8b) shows the influence of a tensile and compressive stress (|1| GPa) for $\beta = 10^{-11}$ cm$^2$.s$^{-1}$.MPa$^{-1}$. It shows that a compressive stress decreases the diffusion kinetics, and inversely for a tensile stress. This is in agreement with the definition of a stress-dependent diffusion coefficient ($D_I^\phi + 2\beta\sigma$). Fig. 8b) shows the depths affected for three concentration values: $4 \cdot 10^{-4}$, $2 \cdot 10^{-4}$ and $5 \cdot 10^{-5}$. This diagram illustrates that the same level of stress in tension and compression does not result in symmetrical diffusion. A compressive stress decreases the diffusion kinetics more significantly than a tensile stress promote diffusion using the depths of the three concentration points, a compressive stress produces 20 % less kinetics than a tensile stress.

<figure><img src="image_9.png" /></figure>

Figure 8: a) Evolution of concentration profiles as a function of five values of $\beta$ (0, $10^{-13}$, $10^{-12}$, $10^{-11}$ and $10^{-10}$ cm$^2$.s$^{-1}$.MPa$^{-1}$) subjected to a homogeneous tensile test (1 GPa); b) Evolution of concentration profiles for two mechanical tests (pure tensile at 1 GPa and pure compressive at -1 GPa) as a function of the measure of the activation volume $\beta = 10^{-11}$ cm$^2$.s$^{-1}$.MPa$^{-1}$; for a 1 mm thick sample, at 520 °C during a 1 h treatment.

### 5.3.2. Applied stress gradient

The study of an applied stress gradient will be divided into two parts. In Fig. 9a), the study of two tensile stress gradients of 2 and 4 GPa.mm$^{-1}$ with 2 $\beta$ values : 0 and $10^{-13}$
cm².s⁻¹.MPa⁻¹ is given. Fig. 9b) gives the effect of a tensile and compressive stress gradient (|4| GPa.mm⁻¹ ) for a given β = 10⁻¹³ cm².s⁻¹.MPa⁻¹.

In the case of these linear stress gradients, i.e. $\frac{\partial^2 \sigma}{\partial z^2} = 0$, Eq. 57 takes the following form:

$$ \frac{\partial c_I^{\phi}}{\partial t} = 2 \left( \beta (1 - \sigma) - D_I^{\phi} \right) \frac{V_a \eta}{RT} \frac{\partial c_I^{\phi}}{\partial z} \frac{\partial \sigma}{\partial z} + \left( D_I^{\phi} + 2\beta\sigma \right) \frac{\partial^2 c_I^{\phi}}{\partial z^2} \quad (64) $$

Eq. 64 shows that in the case of a linear gradient, the amplitude of the stress will also play a role on the kinetics. In other words, there will be a coupled effect between the stress amplitude and the stress gradient.

Fig. 9a) confirms the previous observations that when the parameter $\beta = 0$, the greater the tensile gradient, the faster the diffusion kinetics are and that when $\beta$ increases a slowing down of the kinetics is observed. On the contrary, Fig. 9a) shows that when $\beta = 0$, the larger the compressive gradient, the slower the diffusion kinetics and when $\beta$ increases, an acceleration of the kinetics is observed.

Fig. 9b) shows that the magnitude of the stress gradient in tension and compression does not lead to symmetrical diffusion. Using the depths of the three concentration points, compressive stress produces lower stress kinetics than tensile stress. This result shows a similar trend to that observed in the case of homogeneous stress. However, this study showed that for a tensile gradient with compressive stresses, the slowing down effect is accentuated. This is due to the stress-dependent mobility.

## 6. Discussion

Industrially, effective diffusion depths, specific to the diffusing atoms (e.g. interstitial nitrogen, oxygen and hydrogen atoms), are used and defined to determine the depth of a diffusion layer. This depth corresponds to the long-range diffusion and research is constantly being carried out to increase it in order to have an increasingly resistant material (e.g. resistance to fatigue, wear, etc.). In addition, thermochemical treatments are expensive due to the processing time and high temperatures. Therefore, many researchers have been interested in coupling mechanical pre-treatment with thermochemical treatment to improve
<figure><img src="image_10.png" /><figcaption>Figure 9: a) Evolution of concentration profiles as a function of two values of $\beta$ (0 and $10^{-13} \text{ cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$) subjected to two bending tests (2 GPa.mm$^{-1}$ and 4 GPa.mm$^{-1}$); b) Evolution of concentration profiles for a bending loading (4 GPa.mm$^{-1}$) as a function of the measure of the activation volume $\beta = 10^{-13} \text{ cm}^2 \cdot \text{s}^{-1} \cdot \text{MPa}^{-1}$; for a 1 mm thick sample, at 520 °C during a 1 h treatment.</figcaption></figure>

the mechanical properties by increasing the diffusion depths in shorter time. The results of these studies have been based on the predominance of plastic strain phenomena (e.g. nanocrystallisation), i.e. short-range diffusion due to the creation of diffusion short circuits. Nanostructured surfaces induce an increase in grain boundary and dislocation densities which have a significant influence on the diffusion kinetics (66; 30; 67). Nanostructures offer faster diffusion paths for interstitial atoms at the material surface. However, these mechanical pretreatments also cause residual stress gradients (i.e. elastic strains). The effect of stresses, whether in the form of gradients or homogeneous stresses, has revealed that the sign of the stress dominates the diffusion kinetics: positive stresses increase the diffusion kinetics, conversely for negative stresses (5; 22; 68; 34). These observations were mainly hypothesised by molecular dynamics, as schematically shown in Fig. 10, based on the work of Araki et al. (22). This figure shows how potential energies are affected by homogeneous stresses. A
tensile stress results in a larger lattice distance, which leads to a larger atomic jump, and vice versa in the case of a homogeneous compressive stress. Although this difference in lattice distance seems to be more favourable in the case of compressive stress (higher jumping statistics), this diffusion enhancement in the case of tensile stress has been explained by potential energy differences through higher energy gradients on the tensile stress side. But it is also attributed to the elastic recovery force, i.e. the elastic strain energy, caused by the tensile stress, and not to the lattice strain itself. In other words, the recovery forces caused by the tensile stress would attract the inserted atom to another free position, while the repulsive force caused by the compressive stress could inhibit the migration of the atom.

<figure><img src="image_11.png" /><figcaption>Figure 10: Schematic representation of the effect of homogeneous stresses (tension/compression) on jump distances and potential energies from the work of (22).</figcaption></figure>

Numerically, studies of stress-diffusion couplings have largely been studied in the case of the effect of stress by a driving force and never by an effect on mobility. The effect of stress on mobility was introduced analytically by Larché and Voorhees (6). However, numerical or experimental studies have always taken into account an effective diffusion coefficient according to the complexity of applying stress or studying in-situ residual stress during a diffusion phenomena. Therefore, in this paper, the study focused on the decoupling between stress gradient and stress-dependent mobility on long distance diffusion.

The investigation of the evolution of kinetics in the case of the generation of stresses with time (i.e. residual stresses), allowed to observe more precisely the effects and to draw conclusions on the predominance between the influence of a gradient of stress ($\frac{\partial\sigma}{\partial x}$) and that of the stress on mobility ($D_0 + \beta\sigma_{kk}$). Tab. 2 summarises the evolution of surface stresses $\sigma_{\text{surface}}$, stress gradients $\frac{\partial\sigma}{\partial x}$ (surface and depth) and concentration gradients $\frac{\partial c}{\partial x}$ (surface and depth) as a function of four factors (time $t$, sample thickness $e$, stress amplitude and $\beta$)
parameter). For a better understanding, Fig. 11 shows the schematic representation of the evolution of gradient between surface and in the depth.

<table><caption>Table 2: Summary table of the evolution observed as a function of four factors (time *t*, sample thickness *e*, stress amplitude and parameter *β*).</caption><thead><tr><th colspan="2">Parameter evolution</th><th>|σ<sub>surface</sub>|</th><th>(∂σ/∂x)<sub>surface</sub></th><th>(∂σ/∂x)<sub>in depth</sub></th><th>(∂c/∂x)<sub>surface</sub></th><th>(∂c/∂x)<sub>in depth</sub></th></tr></thead><tbody><tr><td>Time t</td><td>↓</td><td>↑</td><td>↑</td><td>↓</td><td>↑</td><td>↓</td></tr><tr><td>Thickness e</td><td>↓</td><td>↓</td><td>↑</td><td>↓</td><td>↓</td><td>↓</td></tr><tr><td>Stress amplitude</td><td>↑</td><td>↑</td><td>↑</td><td>↑</td><td>↑</td><td>↑</td></tr><tr><td>β parameter</td><td>↑</td><td>↑</td><td>↑</td><td>↓</td><td>↑</td><td>↓</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Figure 11: Schematisation of the terms: surface stress, surface gradient and in-depth gradient.</figcaption></figure>

A study on the long-range stress field was studied in the case of a homogeneous stress and a stress gradient. The results showed that similar stress amplitudes (negative/positive) did not show symmetrical phenomena with respect to the Fick profile (without coupling with stress). These observations were expected in view of the known Einstein relation: $z^2 = 2D_0t$. Indeed, it shows that the depth *z* is propositional to the root of the diffusion coefficient $D_0$ for a given time *t* and temperature *T*. By modifying this equation by adding the mobility stress part, the equation shows an even more non-linear behaviour: $z^2 = 2(D_0 + \beta\sigma_{kk})t$. The simulations showed that in the case of a homogeneous stress, a compressive stress slows down diffusion by 20 % as compared to the increase in tensile stress. In the case of a stress gradient, a positive gradient in a negative stress domain shows 3 times slower diffusion than in the case of a negative gradient in a positive stress domain. The study on
stress gradients shows that if one consider a nitrided sample **under bending**, there are two
possible observations to conclude on the effect of stresses on mobility:

• Negligible effect on mobility ($\beta = 0$): the compression side ($\sigma < 0$) due to the positive gradient ($\frac{\partial\sigma}{\partial x} > 0$) accelerates the diffusion and vice versa for the side in tension,

• Mobility effect: the compression side can slow down the diffusion and vice versa for the tension side, so the $\beta$ parameter should be quantified/identified.

These results highlight and support the findings of Guillot's 2018 study (34). A sample subjected to a bending test was nitrided at 520 °C for 50 h with $K_N = 4.37 \text{ atm}^{-1/2}$. The study focused on a stress gradient of +/- 300 MPa to remain within the elastic limit of the material. XRD analysis revealed a decrease in the gradient of around 100 MPa across the nitrided layer. Fig. 12 shows the nitrogen concentration profile for both sides of the sample and compares it to a reference sample. The results show an acceleration of the kinetics on the tensile side ($\sigma > 0$ and $\frac{\partial\sigma}{\partial z} < 0$) and a slowing down on the compressive side ($\sigma < 0$ and $\frac{\partial\sigma}{\partial z} > 0$). This indicates that, under this stress condition, the impact of stress on mobility prevails over the gradient effect. Numerical simulations corroborate these results: homogeneous positive stress accelerates diffusion, while homogeneous negative stress slows it down. Similarly, positive stress gradients accelerate diffusion, while negative gradients hinder it, in a symmetrical relationship with the Fick curve. Taking these two effects into account is crucial for analysing stress-diffusion coupling. In scenarios such as simple bending, the effects become asymmetric, with significant variations depending on the magnitude of the stress and the gradient.

When these two effects are combined, it is crucial to consider the mobility effect when analyzing the coupling between stress and diffusion. For example, in the case of simple bending, the effects are no longer symmetrical and, depending on the values of the stresses and gradients, the effects can be significant.

The numerical simulations were applied to the binary system Fe-N. This system shows that the interstitial diffusion of a binary system, in the case of the hypothesis of an infinitely
<figure><img src="image_13.png" /><figcaption>Figure 12: Nitrogen composition profiles of reference and nitrided samples under bending loading, stress gradient of +/- 300 MPa, at 520 °C for 50 h and a nitriding potential $K_N = 4.37 \text{ atm}^{-1/2}$ plotted from B. Guillot data (34).</figcaption></figure>

diluted solution (i.e. a concentration lower than the solubility limit of the insertion atom in the master matrix), did not create high residual stresses due to the low stress-free strains. For the binary system Fe-N, $\eta = 2.6 \cdot 10^{-5}$ with a nitrogen solubility limit in the ferritic lattice of 0.1 w.% (65). There are other examples of binary systems that could be applied in this study, such as the Pd-H binary system, where $\eta = 2.8 \cdot 10^{-4}$ with a hydrogen solubility limit in the paladium of 0.0015 w.% (69) or the Nb-O binary system, where $\eta = 3.5 \cdot 10^{-3}$ with an oxygen solubility limit in nobium of 1.7 w.% (70). These three examples show low expansion coefficients with more or less low solubility limits which imply low residual stress values. This numerical work does, however, provide a basis for a qualitative understanding of the effects potentially observed experimentally and thus allow conclusions to be drawn on the effect of stress or mechanical driving force dependent mobility. This effect is, however, difficult to decouple experimentally. Indeed, studies have shown that in the case of pure nitrided iron studying the composite layer (i.e. the precipitation of iron nitrides), the surface stresses are also of low values (71), knowing that precipitation leads to higher free stresses than the diffusion of nitrogen in solid solution (72). Industrially, the materials used are strongly or weakly alloyed materials leading to precipitation and co-diffusion phenomena. Indeed, in an application such as gas nitriding of a low-alloyed steel, a much higher residual stress
field (linked to precipitation phenomena) is observed than in pure iron, with surface stresses ranging from -500 to -1000 MPa. However, the simulation of such a treatment would lead to complex interactions between diffusion and precipitation kinetics or to the influence of stresses that could impact on the precipitation kinetics.

## 7. Conclusion

The stress-diffusion coupling applied to interstitial diffusion could be described analytically from the equations of thermodynamics to understand the known equations of the Cahn-Larché formalism. The driving forces (i.e. the gradient of the diffusion potential) were decomposed and presented from the general equation to the most simplified equation in a particular case (infinitely dilute solution, isotropic elastic material, interstitial diffusion). The mobility was also decomposed with two additional terms showing the influence of stresses. The results and discussion were based on two studies: a first study on the residual stresses (stress-free strains) in the case of a semi-infinite body and on the applied stresses in the form of homogeneous/constant stresses and gradients (positive/negative). Numerical simulations showed that the effect of stress on mobility was significant when $\beta\sigma_{kk} = 0.1D_0$. This factor was obtained for both sources of stress studied: residual (i.e. time dependent) and applied (i.e. time independent). The numerical simulations correlated with the experimental results from the bibliography allowed to understand which effect is predominant (stress-dependent mobility or mechanical driving force). These observations open another opportunity for coupled simulations of thermochemical treatments. In the case of gas nitriding, for example, there may be antagonistic effects in terms of mechanical driving forces and mobility due to difficulties in controlling the diffusion kinetics. For example, the precipitation kinetics may be complicated by a decrease in the gradient during the transformation, which may imply a more sensitive stress effect at the beginning of the transformation than at the end. Jegou et al. showed that the maximum compressive stress decreases with increasing processing time, due to the transformation of carbides to nitrides dominating the precipitation of nitrides directly from the solid solution substitute as at the beginning of the nitriding treatment (73).
Furthermore, from an analytical point of view, the shear components are involved in Eq. 597
37 via the term $(\gamma_I \sigma_{ij})$. Previous experimental studies, presented by Kirchheim (5), have 598
shown that shear does not seem to have any effect on the diffusion phenomena. This is why 599
in this study the parameter $\gamma_I$ was not studied. However, it would be interesting to carry 600
out numerical simulations of shear stresses on a 2D numerical model. 601

Experimentally, it is difficult to implement the superposition of different types of stresses 602
by applied stresses, the study of in-situ stresses during diffusion, or the separation of the 603
different origins of diffusion shorts. On the other hand, the correlation between numerical 604
and experimental studies is an interesting perspective in understanding thermochemical 605
treatments. This correlation could allow to qualitatively identify which effect dominates 606
experimentally and to develop an inverse model to quantify the experimental $\beta$ parameter. 607
# Appendix A. Strain equations

In the case of small perturbations, the state of strain is then represented by the linearized strain tensor $\epsilon_{ij}^{\text{tot}}$ from $v_{i,j}$ displacement, named Cauchy's strain tensor (or small strain tensor):

$$ \epsilon_{ij}^{\text{tot}} = \frac{1}{2} (v_{i,j} + v_{j,i}) \qquad (A.1) $$

The total strain is the sum between elastic strains $\epsilon^e$, which are compatible strains and the stress-free strains (eigenstrains) $\epsilon^F$, which are incompatible strains (74).

$$ \epsilon_{ij}^{\text{tot}} = \epsilon_{ij}^{e} + \epsilon_{ij}^{F} \qquad (A.2) $$

# Appendix B. Chemical potential applied to interstitial solid solution

Thanks to Eqs. 8 and 10, the gradient of the potential diffusion is equal to:

$$ \frac{\partial M_I^{\phi}}{\partial c_I^{\phi}} = RT \frac{\partial \ln(a_I)}{\partial c_I^{\phi}} = RT \frac{\partial \ln(\gamma_I c_I^{\phi})}{\partial c_I^{\phi}} = \frac{RT}{c_I^{\phi}} \left[ 1 + \frac{\partial \ln(\gamma_I)}{\partial \ln(c_I^{\phi})} \right] \qquad (B.1) $$

Where $\phi = 1 + \frac{\partial \ln(\gamma_I)}{\partial \ln(c_I^{\phi})}$ is the thermodynamic factor.

To couple the stress with the diffusion phenomena, Larché and Cahn used the Maxwell's equation as (40):

$$ V_a \left( \frac{\partial \epsilon_{ij}}{\partial c_I^{\phi}} \right)_{T, \sigma_{mn}} = - \left( \frac{\partial M_I^{\phi}}{\partial \sigma_{ij}} \right)_{T, c_I^{\phi}, \sigma_{mn} \neq i, j} \qquad (B.2) $$

So Eq. 6, the derivation with respect to the composition of element I gives:

$$ \left( \frac{\partial \epsilon_{mn}}{\partial c_I^{\phi}} \right)_{T, \sigma_{ij}} = \left( \frac{\partial \epsilon_{mn}^F}{\partial c_I^{\phi}} \right)_{T, \sigma_{ij}} + \left( \frac{\partial S_{mnij}}{\partial c_I^{\phi}} \right)_{T, \sigma_{ij}} \sigma_{ij} \qquad (B.3) $$

Eq. B.3 becomes:
$$ \rho_a \left( \frac{\partial M_I^\phi}{\partial \sigma_{ij}} \right)_{T, c_I^\phi, \sigma_{mn} \neq i, j} = - \left( \frac{\partial \varepsilon_{ij}^F}{\partial c_I^\phi} \right)_{T, \sigma_{mn}} - \left( \frac{\partial \mathcal{S}_{ijkl}}{\partial c_I^\phi} \right)_{T, \sigma_{mn}} \sigma_{kl} \quad (B.4) $$

Where $\rho_a = \frac{1}{V_a}$ is the molar density of the host lattice.

The diffusion potential on the volume under stress $\sigma_{ij}$, at the same composition and temperature, is obtained by integrating Eq. B.4 as (4):

$$ \int_{M_I^\phi(T, P, c_I^\phi)}^{M_I^\phi(T, \sigma_{ij}, c_I^\phi)} \rho_a dM_I^\phi = - \int_{-P\delta_{ij}}^{\sigma_{ij}} \left( \frac{\partial \varepsilon_{ij}^F}{\partial c_I^\phi} \right) d\sigma_{ij} - \int_{-P\delta_{ij}}^{\sigma_{ij}} \left( \frac{\partial \mathcal{S}_{ijkl}}{\partial c_I^\phi} \right) \sigma_{kl} d\sigma_{ij} \quad (B.5) $$

The composition derivative is independent of the stress state, the potential of diffusion becomes:

$$ M_I^\phi(T, \sigma_{ij}, c_I^\phi) = M_I^\phi(T, P, c_I^\phi) - V_a \left( \frac{\partial \varepsilon_{ij}^F}{\partial c_I^\phi} \right) (\sigma_{ij} + P\delta_{ij}) - \frac{V_a}{2} \left( \frac{\partial \mathcal{S}_{ijkl}}{\partial c_I^\phi} \right) (\sigma_{ij}\sigma_{kl} + P^2\delta_{ij}\delta_{kl}) \quad (B.6) $$

Eq. B.6 gives the stress dependence of the diffusion potential when the small strain approximation is valid. The stress $\sigma_{ij}$ is only a function of the real composition $c_I^\phi$ and the real strain $\varepsilon_{ij}$, regardless of the path chosen to reach that state. The stress-free strains $\varepsilon_{ij}^F$ are produced by a change of composition from 0 to $c_I^\phi$ at zero stress ($\sigma_{ij} = 0$). According to the small strain assumption, a change in stress from 0 to $\sigma_{ij}$ at constant composition $c_I^\phi$ implies that in the reference state, the pressure around the system is equal to 0. Eq. B.6 then is simplified as (40; 41):

$$ M_I^\phi(T, \sigma_{ij}, c_I^\phi) = M_I^\phi(T, 0, c_I^\phi) - V_a \left( \frac{\partial \varepsilon_{ij}^F}{\partial c_I^\phi} \right) \sigma_{ij} - \frac{V_a}{2} \left( \frac{\partial \mathcal{S}_{ijkl}}{\partial c_I^\phi} \right) \sigma_{ij} \sigma_{kl} \quad (B.7) $$

Where $M_I^\phi(T, 0, c_I^\phi) = \mu_I(T, 0, c_I^\phi)$ is the diffusion potential at zero stress.
## Appendix C. Thermodynamics of Irreversible Processes (TIP)

According to the physical description of atomic transport in a solid, the diffusion process must be detailed using the TIP theory. From the Helmholtz free energy $F$. It is possible to determine the useful work that can be provided by a closed thermodynamic system, at constant temperature $T$, during a reversible transformation:

$$F = U - TS \qquad (C.1)$$

It is then possible to introduce a new variable named the free energy density, in the case of elastic and isothermal study (75):

$$f_v(\varepsilon^e, c_I^\phi) = u_v - Ts_v \qquad (C.2)$$

In case of an isothermal study, the total derivative of $f_v$ is given by:

$$\frac{df_v}{dt}(\varepsilon^e, c_I^\phi) = \frac{du_v}{dt} - \frac{ds_v}{dt}T \qquad (C.3)$$

The internal energy and entropy can be defined according to thermodynamic laws (76; 77; 78). In continuum mechanics, the Clausius-Duhem inequality is often used to express the second law of thermodynamics to determine whether the constitutive relationship of a material is thermodynamically permissible. This inequality makes it possible to assert the irreversibility of natural processes, particularly with regard to energy dissipation. The objective is now to determine the Clausius-Duhem inequality in terms of the entropy. In fact, the entropy balance describes how the entropy of a body changes as a result of any modification of the system. The thermodynamic state of a crystal is by definition heterogeneous since composition, strain and entropy are functions of position and time. This implies that the entropy $S$ or the internal energy $U$ of the system must be expressed as the integrale of their density over the volume $V$ of the system. In keeping with the equilibrium principle, the perturbation in the local entropy density must be accomplished holding the total entropy of the crystal, $S$, fixed:
$$ \delta S = \int_V \delta \frac{ds_v}{dt} \ dV = 0 \qquad (C.4) $$

$$ S = \int_V \frac{ds_v}{dt} \ dV \qquad (C.5) $$

From equilibrium thermodynamics of homogeneous systems it is known that the entropy flux is the quotient of the diffusion flux $\vec{j}$ in the case of an open system and a similar relationship is postulated between the specific heat production $r$ and the entropy production. With these assumptions, the global formulation of the Clausius-Duhem inequality is derived from the second law of thermodynamics (79):

$$ \frac{d}{dt} \int_V s_v dV \geq \int_S \frac{r}{T} \rho dS - \int_S \frac{M_I^\Phi (J_I^\phi)_i}{T} n_i dS \qquad (C.6) $$

All the variables are functions of a material point at $x$ at time $t$.

The total entropy density $s_v$ is known as the difference between the bulk entropy and the surface entropy. In our case, the system is adiabatic i.e. the surface entropy will be not taking into account. From the hypotheses, the derivative of the total entropy density is:

$$ \frac{ds_v}{dt} \geq -\mathrm{div} J_s \qquad (C.7) $$

Eq. C.7 shows that the local entropy production is positive if $J_s$ is defined as (80):

$$ J_s = -\frac{1}{T} \left( \sum_{I=1}^{n} M_I^{\phi} (J_I^{\phi})_i \right) \qquad (C.8) $$

So Eq. C.7 combined with Eq. C.8 take the form of an inequality as:

$$ T \frac{\partial s_v}{\partial t} - \frac{\partial (M_I^{\phi} (J_I^{\phi})_i)}{\partial x_i} \geq 0 \qquad (C.9) $$

Through the same development as entropy $S$, the first principle of thermodynamic, in the case where $r = 0$, in terms of density, is given as:
$$ \frac{du_v}{dt} = \sigma_{ij} \frac{\partial \varepsilon_{ij}^{\text{tot}}}{\partial t} \qquad (C.10) $$

The volumic dissipation takes the form:

$$ \mathcal{D}_v = \frac{du_v}{dt} - T \frac{ds_v}{dt} - \frac{df_v}{dt} = \sigma_{ij} \frac{\partial \varepsilon_{ij}^{\text{tot}}}{\partial t} - \frac{\partial \left( M_I^{\phi} (J_I^{\phi})_i \right)}{\partial x_i} - \frac{df_v}{dt} \geq 0 \qquad (C.11) $$

From Eq. C.2, the free energy density is considered to have a mechanical part and a chemical part (14):

$$ f_v \left( \varepsilon_{ij}^e, c_I^\phi \right) = f_v^{\text{mech}} \left( \varepsilon_{ij}^e, c_I^\phi \right) + f_v^{\text{chem}} \left( c_I^\phi \right) \qquad (C.12) $$

Where

$$ f_v^{\text{mech}} \left( \varepsilon_{ij}^e, c_I^\phi \right) = \frac{1}{2} \varepsilon_{ij}^e : C_{ijkl} \left( c_I^\phi \right) : \varepsilon_{ij}^e \qquad (C.13) $$

and

$$ f_v^{\text{chem}} \left( c_I^\phi \right) = E^f c_I^\phi + W R T \qquad (C.14) $$

Where W is the number of possible paths for an interstitial atom. It is written in the form of a binomial coefficient. In a case of an ideal solution and for a monoatomic crystal i.e. an infinite and regular stack of identical atoms for interstitial atoms W = $c_I^\phi \ln(c_I^\phi) + (1 - c_I^\phi) \ln(1 - c_I^\phi)$ and in a infinite solution W = $c_I^\phi \ln(c_I^\phi)$, E^f is the enthalpy of formation of one mole for the species considered.

Thanks to Eq. A.2 and Eqs. C.11 to C.14, the volumic dissipation $\mathcal{D}_v$ becomes (75; 14):

$$ \mathcal{D}_v = \left( \sigma_{ij} - \frac{\partial f_v}{\partial \varepsilon_{ij}^e} \right) \frac{\partial \varepsilon_{ij}^e}{\partial t} + \left( M_I^\phi + \sigma_{ij} \frac{\partial \varepsilon_{ij}^{\text{tot}}}{\partial c_I^\phi} - \frac{\partial f_v}{\partial c_I^\phi} \right) \frac{\partial c_I^\phi}{\partial t} - \left( J_I^\phi \right)_i \frac{\partial M_I^\phi}{\partial x_i} \geq 0 \qquad (C.15) $$

Since the Clausius-Duhem inequality holds regardless of any particular $\dot{\varepsilon}_{ij}^e$ and according
to Coleman-Noll's argument, the state laws for stress $\sigma_{ij}$ and diffusion potential $M_I^\phi$ are (81):

$$ \sigma_{ij} = \frac{\partial f_v}{\partial \varepsilon_{ij}^e} \qquad (C.16) $$

$$ M_I^\phi = \frac{\partial f_v}{\partial c_I^\phi} - V_a \sigma_{ij} (\eta_I^\phi)_{ij} \qquad (C.17) $$

From Eqs. C.15 to C.17, the inequality dissipation is simplified:

$$ \mathcal{D}_v = - \left( J_I^\phi \right)_i \frac{\partial M_I^\phi}{\partial x_i} \ge 0 \qquad (C.18) $$

To ensure positivity of $\mathcal{D}_v$, the existence of a convex dissipation potential $\Omega(\frac{\partial M_I^\phi}{\partial x_i})$ is used (75):

$$ \left( J_I^\phi \right)_i = - \frac{\partial \Omega^{\text{chem}}}{\partial \left( \frac{\partial M_I^\phi}{\partial x_i} \right)} \qquad (C.19) $$

$$ \Omega^{\text{chem}} = \frac{1}{2} B_{ij} \frac{\partial M_I^\phi}{\partial x_i} \frac{\partial M_I^\phi}{\partial x_i} \qquad (C.20) $$

These two equations give Eq. 18.
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1 Graphical Abstract

2 Stress-diffusion coupling. Application to interstitial diffusion.

3 Lisa GERMAIN, Sébastien JEGOU, Laurent BARRALLIER
<table>
  <tr>
    <td>E<sub>A</sub></td>
    <td>activation energy</td>
  </tr>
  <tr>
    <td>γ<sub>I</sub></td>
    <td>activity coefficient</td>
  </tr>
  <tr>
    <td>⟨v⟩</td>
    <td>average speed</td>
  </tr>
  <tr>
    <td>f<sub>i</sub></td>
    <td>body force</td>
  </tr>
  <tr>
    <td>k<sub>B</sub></td>
    <td>Bolztmann's constant</td>
  </tr>
  <tr>
    <td>σ<sub>ij</sub></td>
    <td>Cauchy's stress tensor</td>
  </tr>
  <tr>
    <td>a<sub>I</sub> = γ<sub>I</sub>c<sub>I</sub><sup>φ</sup></td>
    <td>chemical activity</td>
  </tr>
  <tr>
    <td>η<sub>ij</sub></td>
    <td>chemical expansion tensor</td>
  </tr>
  <tr>
    <td>μ<sub>I</sub></td>
    <td>chemical potential</td>
  </tr>
  <tr>
    <td>S<sub>ijkl</sub> = C<sub>ijkl</sub><sup>-1</sup></td>
    <td>compliance tensor</td>
  </tr>
  <tr>
    <td>c<sub>I</sub><sup>φ</sup></td>
    <td>concentration of the element I of the phase φ</td>
  </tr>
  <tr>
    <td>Ω</td>
    <td>convex dissipation potential</td>
  </tr>
  <tr>
    <td>D<sub>I</sub><sup>φ</sup></td>
    <td>diffusion coefficient</td>
  </tr>
  <tr>
    <td>M<sub>I</sub><sup>φ</sup></td>
    <td>diffusion potential</td>
  </tr>
  <tr>
    <td>D<sub>0</sub><sup>φ</sup></td>
    <td>diffusivity of the element</td>
  </tr>
  <tr>
    <td>v<sub>i,j</sub></td>
    <td>displacement</td>
  </tr>
  <tr>
    <td>D<sub>v</sub></td>
    <td>dissipation</td>
  </tr>
  <tr>
    <td>D<sub>eff</sub><sup>φ</sup></td>
    <td>effective diffusion coefficient</td>
  </tr>
  <tr>
    <td>ε<sup>e</sup></td>
    <td>elastic strain</td>
  </tr>
  <tr>
    <td>I</td>
    <td>element</td>
  </tr>
  <tr>
    <td>ΔH</td>
    <td>enthalpy</td>
  </tr>
  <tr>
    <td>E<sub>f</sub></td>
    <td>Enthalpy of formation</td>
  </tr>
  <tr>
    <td>S</td>
    <td>entropy</td>
  </tr>
  <tr>
    <td>s<sub>v</sub> = S/V<sub>a</sub></td>
    <td>entropy densities per unit volume of the reference configuration</td>
  </tr>
  <tr>
    <td>J<sub>s</sub></td>
    <td>entropy flux due to mass and energy diffusion</td>
  </tr>
  <tr>
    <td>J<sub>I</sub><sup>φ</sup></td>
    <td>flux of the element I of the phase φ</td>
  </tr>
  <tr>
    <td>f<sub>v</sub></td>
    <td>free energy density</td>
  </tr>
  <tr>
    <td>F</td>
    <td>Helmholtz free energy</td>
  </tr>
  <tr>
    <td>P = -1/3σ<sub>kk</sub></td>
    <td>hydrostatic pressure of a fluid</td>
  </tr>
  <tr>
    <td>σ<sub>kk</sub></td>
    <td>hydrostatic stress</td>
  </tr>
  <tr>
    <td>U</td>
    <td>internal energy</td>
  </tr>
  <tr>
    <td>u<sub>v</sub> = U/V<sub>a</sub></td>
    <td>internal energy densities per unit volume of the reference configuration</td>
  </tr>
  <tr>
    <td>δ<sub>ij</sub></td>
    <td>Kronecker's symbol</td>
  </tr>
  <tr>
    <td>λ ; μ</td>
    <td>Lamé's coefficients</td>
  </tr>
</table>
<table>
  <tr>
    <td>ρ</td>
    <td>mass density</td>
  </tr>
  <tr>
    <td>λ ; μ</td>
    <td>Lamé's coefficients</td>
  </tr>
  <tr>
    <td>ρ</td>
    <td>mass density</td>
  </tr>
  <tr>
    <td>β</td>
    <td>measure of the activation volume</td>
  </tr>
  <tr>
    <td>B<sub>ij</sub></td>
    <td>Mobility coefficient</td>
  </tr>
  <tr>
    <td>ρ<sub>a</sub> = 1/V<sub>a</sub></td>
    <td>molar density of the host lattice</td>
  </tr>
  <tr>
    <td>V<sub>a</sub></td>
    <td>molar volume of the host lattice</td>
  </tr>
  <tr>
    <td>W</td>
    <td>number of possible paths for an atom</td>
  </tr>
  <tr>
    <td>ΔV<sub>I</sub></td>
    <td>partial molar volume of the element I</td>
  </tr>
  <tr>
    <td>φ</td>
    <td>phase</td>
  </tr>
  <tr>
    <td>ν</td>
    <td>Poisson's coefficient</td>
  </tr>
  <tr>
    <td>r</td>
    <td>specific heat production</td>
  </tr>
  <tr>
    <td>C<sub>ijkl</sub></td>
    <td>stiffness tensor</td>
  </tr>
  <tr>
    <td>ε̇<sub>ij</sub></td>
    <td>strain rate</td>
  </tr>
  <tr>
    <td>ε<sup>F</sup></td>
    <td>stress-free strain</td>
  </tr>
  <tr>
    <td>∂V</td>
    <td>surface</td>
  </tr>
  <tr>
    <td>g<sub>i</sub></td>
    <td>surface force density in ∂V</td>
  </tr>
  <tr>
    <td>T</td>
    <td>temperature</td>
  </tr>
  <tr>
    <td>Φ = 1 + (∂lnγ<sub>I</sub>/∂lnc<sub>I</sub><sup>φ</sup>)</td>
    <td>thermodynamic coefficient</td>
  </tr>
  <tr>
    <td>t</td>
    <td>time</td>
  </tr>
  <tr>
    <td>ε<sup>tot</sup></td>
    <td>total strain</td>
  </tr>
  <tr>
    <td>n<sub>j</sub></td>
    <td>unit vector in direction j̄ on the boundary</td>
  </tr>
  <tr>
    <td>R</td>
    <td>universal gas constant</td>
  </tr>
  <tr>
    <td>V</td>
    <td>Volume</td>
  </tr>
  <tr>
    <td>E</td>
    <td>Young's modulus</td>
  </tr>
  <tr>
    <td>χ<sub>I</sub><sup>Φ</sup> = V<sub>a</sub>/RT c<sub>I</sub><sup>φ</sup></td>
    <td>Notion for chemical-mechanical coupling</td>
  </tr>
</table>
Initial host lattice

<figure><img src="image_14.png" /></figure>

Interstitial diffusion + residual stress

Local effect
(Stress free strains/ Eigenstrains $\epsilon_{ij}^F$)

<figure><img src="image_15.png" /></figure>

Long-distance effect
(elastic strains $\epsilon_{ij}^e$)

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

Hydrostatic pressure

<figure><img src="image_18.png" /></figure>

$\beta = 0$
$\beta = 1.3 \times 10^{-5}$

<figure><img src="image_19.png" /></figure>

a) [•] Concentration
b) [•] Hydrostatic pressure

Depth [µm]

Depth [µm]

<figure><img src="image_20.png" /></figure>

Interstitial diffusion + mechanical stress

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

Stress [MPa]

<figure><img src="image_23.png" /></figure>
5 Highlights

6 Stress-diffusion coupling. Application to interstitial diffusion.

7 Lisa GERMAIN, Sébastien JEGOU, Laurent BARRALLIER

8 • Role of stress on interstitial diffusion is studied thank to Cahn and Larché theory

9 • Constant compressive (tensile) stress hinders (promotes) diffusion (Mobility part)

10 • Negative (positive) gradient of stress hinders (promotes) diffusion (Nernstian part)

11 • Simulation can identify the prevailing effect of stress on mobility or driving force.