Damien Regnault

Jean-Baptiste Rouquier

Eric Thierry

Stochastic

Éric Thierry

Stochastic Minority on Graphs

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Introduction

Cellular automata (CA) can be seen both as a model of computation with massive parallelism and as a model for complex systems in nature. They have been studied with various fields of applications like parallel/distributed computing, physics, biology or social sciences. Most of the work concerning CA assumes that their dynamics is deterministic and synchronous (all cells update simultaneously) and that the topology of the cells is very regular (the cells are the vertices of a graph which is usually a line or a 2D or 3D grid). Such assumptions can be questioned with regard to the applications and the real life constraints. Dynamics where those assumptions are perturbed have been far less studied and their analysis is very challenging. Here is a non-exhaustive list of related works about CA in literature, the models are stochastic CA since the perturbations are usually introduced as stochastic processes:

-Perturbation of the updating rule: resilience to random errors [START_REF] Gács | Reliable cellular automata with self-organization[END_REF][START_REF] Toom | Discrete local Markov systems[END_REF], mean field analysis of general Markovian rules [START_REF] Balister | Large deviations for mean fields models of probabilistic cellular automata[END_REF]. -Perturbation of synchronism: empirical studies about resilience to asynchronism [START_REF] Bersini | Asynchrony induces stability in cellular automata based models[END_REF][START_REF] Buvel | Structure in asynchronous cellular automata[END_REF][START_REF] Kanada | Asynchronous 1d cellular automata and the effects of fluctuation and randomness[END_REF][START_REF] Schönfisch | Synchronous and asynchronous updating in cellular automata[END_REF], mathematical analysis of some 1D CA under full asynchronism (only one random cell updates at each time step) or under α-asynchronism (each cell updates independently with probability α) [START_REF] Fatès | Fully asynchronous behavior of doublequiescent elementary cellular automata[END_REF][START_REF] Fatès | Asynchronous behaviour of double-quiescent elementary cellular automata[END_REF][START_REF] Fukś | Probabilistic cellular automata with conserved quantities[END_REF]. -Perturbation of the topology of cells: empirical studies [START_REF] Fatès | Perturbing the topology of the game of life increases its robustness to asynchrony[END_REF].

Many rules studied in literature cannot be extended to any topology of cells: rules for lines or grids of cells may refer to specific orientations like left/right or up/down in their definition. Nevertheless a large class of CA have a definition which is independent of the graph: the outer-totalistic CA. Their update rule only considers the state of the updated cell and the number of adjacent cells in each possible state (e.g. Game of Life). The threshold CA with states {0, 1} are a interesting sub-class where the new state of a cell only depends on its former state and on the comparison between some threshold and the number of 1 in the neighborhood. For example, consider a simplified frequency allocation problem with an interference graph. Each vertex is a station which has to choose between two possible frequencies, say 0 or 1, when emitting. Each edge represents a possible interference between two stations if they emit with the same frequency. To avoid interferences, suppose that stations frequently update their emission frequency by choosing the one which minimizes the number of potential interferences with its neighbors. This is the Minority rule and the evolution of the allocated frequencies corresponds to the dynamics of this threshold CA. The Minority rule may also arise in models from biology. Consider a set of cells sharing a simple gene regulatory networks where a gene exerts a feedback inhibiting its expression. The state of a cell is whether it expresses this gene or not. Assuming that each cell starts expressing the gene when less than half of its neighbors (including itself) express it, and that otherwise it stops expressing it, leads to the Minority rule [START_REF] Demongeot | Genetic regulation networks : circuits, regulons and attractors[END_REF]. Both those models are of course extreme simplifications of any real device or phenomena, but understanding such simple rules is an indispensable step towards the study of more involved CA.

In this paper, we choose to investigate how the topology of the cells acts upon the dynamics under asynchronous updates. We focus on Stochastic Minority where the Minority rule applies to two possible states {0, 1} and under full asynchronism (at each time step, only one random cell is updated with the uniform distribution). This simple rule already exhibits a surprisingly rich behavior as observed in [START_REF] Regnault | Progresses in the analysis of stochastic 2d cellular automata : A study of asynchronous 2d minority[END_REF][START_REF] Regnault | On the analysis of "simple" 2d stochastic cellular automata[END_REF] where it is studied for cells assembled into a torus. Minority is interesting because it is a typical CA with negative feedback. Some related stochastic models like Ising models or Hopfield nets have been studied under asynchronous dynamics (e.g. our model of asynchronism corresponds to the limit when temperature goes to 0 in the Ising model). These models are acknowledged to be harder to analyze when it comes to arbitrary graphs [START_REF] Istrail | Statistical mechanics, three-dimensionality and np-completeness : I. universality of intractability for the partition function of the ising model across non-planar lattices[END_REF][START_REF] Mccoy | The Two-Dimensional Ising Model[END_REF] or negative feedbacks [START_REF] Rojas | Neural Networks : A Systematic Introduction[END_REF]. Minority has been already well studied for arbitrary graphs under the synchronous regime [START_REF] Goles | Neural and automata networks, dynamical behavior and applications[END_REF] where it is shown that for any graph the dynamics eventually converges to a fixed point or a cycle of period two. As any other Markovian process on a finite number of configurations, any trajectory of Stochastic Minority ends in an attractor, i.e. a final strongly connected component of the transition graph. We study the structure of attractors and the average time to hit them, for several types of graphs: cycles (Section 3.1), cliques (Section 3.2), trees (Section 4). This hitting time can be seen as the end of the transitory part of the dynamics. The table below sums up these average hitting times: polynomial (Poly) or exponential (Exp) in the number of vertices, for the worst initial configuration. It can be compared to the worst hitting time under the synchronous dynamics. What is the graph parameter which enables to predict whether the average hitting time will be polynomial or exponential ? Does adding edges usually speed up or slow down the convergence ? The questions remain open, but our analyzes already disprove a simple link with the maximum degree or the mean degree, and they provide tools for further investigations about the influence of the topology on such CA dynamics.

The model

Topology

We consider in this part Stochastic Minority on arbitrary undirected graphs. Section 3.1, 3.2 and 4 will be respectively devoted to cycles, cliques and trees.

Definition 1 (Configuration). Let G = (V, E) be a finite undirected graph with vertices V and edges E. Q = {0, 1} is the set of states (0 stands for white and 1 stands for black). The vertices are also called cells and N := |V| denotes their number. The neighborhood N i of a cell i is the set of its adjacent cells (including itself ). A configuration is a function c : V → Q (c i denotes the state of cell i in configuration c).

Definition 2 (Stochastic Minority). We consider the following dynamics δ that associates with each configuration c a random configuration c obtained as follows: a cell i ∈ V is selected uniformly at random (we say that cell i is fired) and its state is updated to the minority state among its neighborhood (no change in case of equality), while all the other cells remain in their current state:

c i = 1 if j∈Ni c j < |Ni| 2 or j∈Ni c j = |Ni| 2 and c i = 1 0 if j∈Ni c j > |Ni|
2 or j∈Ni c j = |Ni| 2 and c i = 0 and c k = c k for all k = i. In a configuration, a cell is said active if its state changes in case the cell is fired. We denote by c t the random variable for the configuration obtained from an initial configuration c, after t steps of the dynamics: c 0 = c and c t = δ t (c) for all t 1.

Definition 3 (Attractors).

For the dynamics induced by δ, a set of configurations A is an attractor if for all c, c ∈ A, the time to reach c starting from c is finite almost surely. It is a strongly connected component with no arc leaving the component, in the transition graph where vertices are all the possible configurations and arcs (c, c

) satisfy P (δ(c) = c ) > 0.
The union of all attractors is denoted A and called the limit set.

Definition 4 (Convergence). We say that the dynamics δ converges from an initial configuration c 0 to an attractor A (resp. the limit set A) if the random variable T = min{t : c t ∈ A} (resp. T = min{t : c t ∈ A}) is almost surely finite.

Since we only consider finite graphs, the dynamics δ converges almost surely from any initial configuration to A. The variable T is a hitting time.

Energy and Particles

As in the Ising model [START_REF] Mccoy | The Two-Dimensional Ising Model[END_REF] or in Hopfield networks [START_REF] Rojas | Neural Networks : A Systematic Introduction[END_REF], we define a natural global parameter that one can consider to be the energy of the system since it counts the number of interactions between neighboring cells in the same state. This parameter will provide key insights on the evolution of the system. Definition 5 (Potential). The potential v i of cell i is the number of its neighbors (including itself ) in the same state as itself. If v i |Ni| 2 then the cell is in the minority state and is thus inactive; whereas, if

v i > |Ni| 2 then the cell is active. A configuration c is stable if and only if for all cell i ∈ V, v i |Ni| 2 . Definition 6 (Energy). The energy of configuration c is E(c) = i∈V (v i -1).
The energy of a configuration is always non-negative. More precisely, we have the following bounds which are direct consequences of the definitions.

Proposition 7 (Energy bounds).

The energy E satisfies 2|E|-2C max E 2|E|, where C max is the maximum number of edges in a cut of G.

As a consequence, computing the minimum energy for arbitrary graphs is NP-hard. There exists configurations of energy 0 if and only if G is bipartite. Those stable configurations are the 2-colorings of G.

Fact 8 (Energy is non-increasing) The energy is a non-increasing function of time and decreases each time a cell i with potential > |Ni| 2 fires.

Proof. When an active cell i of potential v i fires, its potential becomes |N i | -v i + 1, and the energy of the configuration becomes E + 2|N i | -4v i + 2.

Corollary 9. A configuration c belongs to the limit set if and only if there is no sequence of updates that would lead the energy to decrease, i.e. if and only if ∀t, P E(c t ) > E(δ(c t ))|c 0 = c = 0.

Proof. If the energy decreases when updating c to c , then c will never be reached again (because energy is non-increasing). Reciprocally, any update that keeps the energy constant is reversible: the fired cell can be fired again to get back to the previous configuration.

Remark 10. Since firing a cell of odd degree makes the energy decrease, such cells are inactive in the limit set.

Definition 11. Let c be a configuration on G = (V, E), an edge {i, j} holds a particle if c i = c j . A configuration is fully characterized (up to the black/white symmetry) by its set of particles located at P ⊆ E. Note that the converse proposition "any subset P ⊆ E corresponds to a configuration" is true only on trees.

i i i i i i
Fig. 1. Transfers of particles (red diamonds) when firing cell i.

The energy of a configuration is clearly equal to twice its number of particles. With the particle point of view, when firing a cell i of degree deg(i) = |N i |-1, if the number of incident edges holding a particle is

deg(i)
2 , these particles disappear but new particles appear on the incident edges (if any) which had no particle (as illustrated on Fig. 1). Otherwise the particles do not move.

Switching between the coloring and the particle points of view may simplify the description of the configurations and the dynamics, e.g. when the energy is low and the dynamics comes to random walks of a few particles.

Regular graphs

In [START_REF] Regnault | Progresses in the analysis of stochastic 2d cellular automata : A study of asynchronous 2d minority[END_REF][START_REF] Regnault | On the analysis of "simple" 2d stochastic cellular automata[END_REF], Stochastic Minority has been studied for the torus with two different kinds of neighborhoods: the von Neumann neighborhood (4 neighbors/cell) and the Moore neighborhood (8 neighbors/cell). In both cases, the average hitting time of the limit set is conjectured to be polynomial. We investigate here two extreme cases of regular graphs: cycles and cliques.

Cycles

On cycles, the particle point of view is convenient and one can prove that Stochastic Minority behaves as random walks of annihilating particles on a discrete ring (see [START_REF] Fatès | Fully asynchronous behavior of doublequiescent elementary cellular automata[END_REF][START_REF] Grigoriev | Random walk of annihilating particles on the ring[END_REF]). On the right, the particles are the red diamonds.

Theorem 12. Stochastic Minority on cycles hits the limit set after O(N 3 ) steps on expectation. If N is even, there are two attractors of energy 0: the 2-colorings of the cycle. If N is odd, there is a single attractor: a cycle in the transition graph composed of all the configurations with only one particle (energy 2).

Cliques

We prove in this part that Stochastic Minority on cliques behaves as a coupon collector (see [START_REF] Grimmett | Probability and Random Process[END_REF]). In a clique, all cells have the same neighborhood N i = V and thus all black (resp. white) cells have the same potential.

Theorem 13. Stochastic Minority on cliques hits the limit set after O(N log N ) steps on expectation. If N is even, there are

N N 2 attractors, each one is a stable configuration of energy N 2
2 which is half black and half white. If N is odd there is only one attractor which is made of the 2

N N -1 2 configurations of energy N 2 +1
2 with a difference -1 or +1 between black and white cells.

Trees

First observations

Observations from simulations Stochastic Minority has a complex behavior on trees. There are similarities with the evolution on the torus [START_REF] Regnault | Progresses in the analysis of stochastic 2d cellular automata : A study of asynchronous 2d minority[END_REF][START_REF] Regnault | On the analysis of "simple" 2d stochastic cellular automata[END_REF]. The dynamics of Stochastic Minority on a random initial configuration is characterized by a fast initial energy drop. Afterwards, different regions colored by specific patterns emerge. Each region is a subtree with a 2coloration (corresponding to minimal energy 0 inside the subtree). The energy is located on the boundaries between the regions. These regions either merge or compete with each other. Finally either one region manages to extend to the whole configuration or the boundaries between the different regions stabilize: the dynamics has reach the limit set.

Configurations of minimum energy

As any bipartite graph, a tree admits exactly two configurations of energy 0 which are the two possible 2-colorings of the graph. Note that some trees admit stable configurations with non-null energy (consider an arbitrary tree all colored in black and add |N i | new leaves in white to each cell i).

The Dual Rule

We now introduce dual configurations and their dual rule to ease the study of the dynamics on trees. In this dual dynamics equivalent to Stochastic Minority, the stable configurations of minimum energy are the two configurations all black and all white, and the regions which compete are all white vs. all black subtrees. Definition 14 (Dual configurations). Consider a tree T and fix a root r in T. For any configuration c on T, its dual configuration ĉ is defined as ĉi = c i if h i is even and ĉi = 1-c 1 if h i is odd, where h i is the distance from r to i (see Fig. 2). The mapping c → ĉ is a bijection on the set of all the configurations: more precisely ĉ = c.

A equivalent definition consists in making a XOR with the 2-coloring of T such that r is in white (such a definition can be used for any bipartite graph, see e.g. [START_REF] Regnault | Progresses in the analysis of stochastic 2d cellular automata : A study of asynchronous 2d minority[END_REF]). Proposition 15 (Dual dynamics). Consider a sequence (c t ) for the Stochastic Minority dynamics δ and the sequence (ĉ t ) of the dual configurations, and define the dual dynamics δ as δ(ĉ) = δ(c) so that ĉt+1 = δ(ĉ t ). Then the dynamics δ is also a stochastic CA. It associates with each configuration ĉ a random configuration ĉ by updating one random cell i uniformly with the rule which selects the majority state in the neighborhood of i excluding itself (in case of equality its state changes):

ĉ i = 1 if j∈Ni {i} ĉj > |Ni|-1 2 or ( j∈Ni {i} ĉj = |Ni|-1 2 and ĉi = 0) 0 if j∈Ni {i} ĉj < |Ni|-1 2 or ( j∈Ni {i} ĉj = |Ni|-1 2 and ĉi = 1)
By construction, the dual sequences (c t ) and (ĉ t ) as well as their corresponding dynamics δ and δ are stochastically coupled (see [START_REF] Lindvall | Lectures on the coupling method[END_REF]) by firing the same random cell at each time step. then the cell is active. The dual energy E is the sum of the dual potentials over all the cells.

Given a configuration c and its dual ĉ, the potential of any cell i in c is equal to the dual potential of cell i in ĉ plus 1. Thus the dual energy of ĉ is exactly the energy of c.

From now on, we consider the dual dynamics.

5 Structure of the limit set on trees

An Algorithm for the limit set

In this section we describe an algorithm (algorithm 1) that checks if a configuration belongs to the limit set, and if not, gives a sequence of updates that leads to the limit set.

Fact 17 An attractor A decomposes the graph into three sets of vertices:

1. the vertices that are in the state 0 for every configuration of A; 2. the vertices that are in the state 1 for every configuration of A; 3. the vertices that can be either in the state 0 or 1, depending on the configuration in A.

Algorithm 1: Membership to the limit set: check that E(c ) = E(c).

Input: A configuration c. while There is an active cell i in state 0 do Fire i

1
while There is an active cell i in state 1 do Fire i b is now definitely white (case 1 of Fact 17), but was not identified as such at the end of phase 1. Thus, a third phase is necessary with this algorithm to identify b.

No other cell will be active during the execution of the algorithm.

Fig. 3. Three phases are necessary.

Structure

Section 5.1 has shown that in the limit set, some vertices are fixed while other are oscillating.

Let us now study more precisely the structure of the limit set.

We first pick an arbitrary node of degree 1 and set it as root r in the tree (it introduces "father" and "sons" relations). We assign a label to each cell to count the number of attractors and the size of the limit set. The number of attractors is the number of acceptable labelings. The set of labels we use is {( , 0), ( , 1), ( , 0), ( , 1)}. " " intuitively means "oscillating" while " " means "fixed" (like the recorder symbols play/stop). The second component is called the "preferred" state of the cell. Note that only vertices of even degree can have a label of the form ( , •).

Definition 20 (acceptable labeling). A labeling is acceptable if and only if, for each cell

The apparent asymmetry in case 2a (imposing α = 0) is there only to avoid double counts in Theorem 22, one could as well have defined acceptable labelings with α = 1.

Theorem 22 shows that a labeling corresponds to an attractor, and Theorem 23 details the meaning of a labeling, thus the structure of an attractor. Definition 21. The configuration snd(L) corresponding to a labeling L is the projection of the second component:

L(v) = (•, α) ⇒ snd(L)(v) = α.
Theorem 22. Given a tree, there is a bijection between attractors and acceptable labelings.

Theorem 23 (Structure of an attractor). Let L be an acceptable labeling. Then for every configuration c reachable by a sequence of updates from snd(L), for every cell v:

1. If L(v) = ( , α) then c(v) = α (
this is why " " intuitively means "fixed").

2. If L(v) = ( , α) (in this case deg(v) is even and thus v cannot be the root, which has degree 1) then (a) if the father has a label of the form ( , •), v is in the state appearing in majority among its neighbors (and can be in any state in case of equality); (b) otherwise, if v is in the state α, then its sons are also in their preferred states.

Moreover, if v is not in the state α, then all its sons that are not in their preferred state are in the same state.

6 Hitting time for trees

Trees of maximum degree 3

On trees where the degrees are at most 3, the dynamics ends by fixing the states of the cells of degree 1 and 3 (see Remark 10) and it may remain some isolated particles which oscillate on disjoint paths.

Theorem 24. Stochastic Minority on trees with degrees at most 3 hits the limit set in O(N 4 ) steps on expectation. The attractors of a tree T are in bijection with the matchings of the reduced tree T where each path of T has been replaced by an edge.

Trees of maximum degree 4

In this part, we introduce biased trees (Definition 27 and Figure 5) such that the dynamics ĉ converges in exponential time on this topology (Theorem 34). Vertices of biased trees have a degree at most 4. In fact, biased trees simulate biased random walks (Definition 25) which converge in exponential time. Biased trees are created from small trees called widgets (Definition 26 and Figure 4) arranged on a line. Except from the ends, this line of widgets is made of gates. According to the configuration, theses gates are either locked, unlocked or stable (Definition 28). On a correct configuration (Definition 29), the line of gates is split into two regions: all gates on the left side are stable and all gates on the right side are unstable (locked or unlocked). In a correct configuration, three different events may be triggered with the same probability 1/N (Fact 30 and Corollary 33):

the rightmost stable gate becomes an unlocked gate.

the leftmost unstable gate becomes stable if it is unlocked.

the leftmost unstable gate is switched from locked to unlocked or the contrary.

Thus stable gates tend to disappear. This dynamics will ultimately converge to the stable configuration ĉf (Definition 31). To reach this configuration all gates must be stable. Thus it takes exponential times to dynamics ĉ to converge on biased tree with an initial correct configuration.

Definition 25 (Biased random walks). Consider a sequence of random variables (X i ) i 0 defined on {0, . . . , n} such that X 0 = 0 and for all i 0:

-P (X i+1 = 1 | X i = 0) = 1 (reflecting barrier at 0). -P (X i+1 = n | X i = n) = 1 (absorbing barrier at n). -If 0 < x < n, P (X i+1 = x -1 | X i = x) + P (X i+1 = x + 1 | X i = x) = 1 and 0 < a < P (X i+1 = x + 1 | X i = x) < b < 1/2 for some a, b ∈ R + .
This sequence is called a biased random walk. Let

T = min{i|X i = n}, then θ(b) E [T ] θ(a) where θ(p) = 2p(1-p) (1-2p) 2 1-p p n -1 -n 1-2p .
Definition 26 (Widgets). A Widget W is a tree T = (V, E, b) where b ∈ V is called the bridge. We consider the three widgets described in figure 4.a: head, gate and tail and the three configurations ĉl , ĉu , ĉs for gates. Definition 27 (Biased trees). Let (W i ) 0 i n+1 be a finite sequence of widgets where

W i = (V i , E i , b i ).
From this sequence, we define the tree T = (V, E) where

V = ∪ n+1 i=0 V i and E = (∪ n+1 i=0 E i ) (∪ n i=0 b i b i+1
). Abusively we also denote by (W i ) 0 i n+1 the tree generated by this sequence. A biased tree of size n is a finite sequence of widgets (W i ) 0 i n+1 where W 0 is a head, for 1 i n, W i is a gate and W n+1 is a tail.

Definition 28 (Stable and unstable gates). Consider a biased tree (W i ) 0 i n+1 and a configuration ĉ. We denote by ĉWi , the restriction of ĉ to widget W i . We say that gate i is locked if ĉWi = ĉl , unlocked if ĉWi = ĉu and stable if ĉWi = ĉs . An unstable gate is a gate which is locked or unlocked.

Definition 29 (Correct configurations). Configuration ĉ is correct if cells of the head are black, the tail is white, and there exist j such that for all 1 i j gate i is stable and for all j < k n gate k is unstable. We say that configuration ĉ is on position j. We denote by P os(ĉ) the position of configuration ĉ. The position is unlocked if j = n or gate j + 1 is unlocked, the position is locked otherwise. Fact 30 (Active cells) Consider a correct configuration ĉ on position j. The active cells of ĉ are:

Head

-Cell b j if j = 0. -Cell b j+1 if j = n and gate W j+1 is unlocked. -Cell f i if j < i n. -Cell b n+1 if j = n.
Proof. Consider a correct configuration ĉ on position j. The only cells which may be active in ĉ are cells b i and f i for 1 i n and cell b n+1 . Cell b n+1 is active if and only if ĉ(b n ) = 1 that is to say gate W n is stable. For all 1 i n, cell f i is active if and only if the gate W i is unstable that is to say j < i n. For all 1 i n,

cell b i is inactive if ĉ(b i-1 ) = ĉ(b i ) = ĉ(b i+1
). Thus among cells (b i ) 1 i n , only cells b j and b j+1 may be active: cell b j is active and cell b j+1 is active if gate W j+1 is unlocked.

Definition 31 (Final configuration). The final configuration ĉf is the configurations where cells of the head are black, tail is black and every gate is stable. We say that ĉf is on position n + 1, P os(ĉ f ) = n + 1.

Lemma 32. Configuration ĉf is stable. Proof. Consider a correct configuration ĉ on position j and the configuration ĉ = δ(ĉ). If an inactive cell fires than ĉ = ĉ. Now consider that an active cell fires (see fact 30):

if j = 0 and cell b j fires: then gate W j becomes unlocked and ĉ is a correct configuration on unlocked position j -1. if j = n and cell b j+1 fires: then gate W j+1 becomes stable and ĉ is a correct configuration on position j + 1. if cell f i fires with j < i n: then gate W i becomes unlocked (resp. locked) in ĉ if it is locked (resp. unlocked) in ĉ. Configuration ĉ stays correct and on position j. if j = n and cell b n+1 fires: then ĉ = ĉf .

Theorem 34. On biased trees of size n (i.e. N = 8n + 4 vertices), starting from an correct configuration, Stochastic Minority converges almost surely to c f . Moreover the hitting time T of the limit set satisfies Θ(1.

5 n ) E [T ] Θ(n4 n ).

A Omitted proofs

Proof (proof of Theorem 12). The best way to analyze cycles is to study the movement of particles. Firing a cell incident to a particle either attract the particle on the next edge if it is free, or annihilate both particles on each side of the cell. Consequently, the dynamics comes to the analysis of random walks of annihilating particles on a discrete ring. If the number N of cells is even (resp. odd), any configuration has necessarily an even (resp. odd) number of particles. Due to the rules of particle movements, from any initial configuration, one can always destroy pairs of particles until it remains only one particle (if N is odd) or no particle (if N is even). Thus attractors are included in this set of configurations. As a matter of fact, all those configurations belongs to an attractor since it is not possible to decrease the energy from them. The exact structure of the attractors is clear if N is even (there are only two configurations of energy 0) and when N is odd, one can easily check that configurations with only one particle are organized as a cycle in the transition graph (defined by possible transitions over all the configurations).

To bound the expected hitting time of the limit set, associate with each configuration c t a weight X t which is the maximum distance between two consecutive particles (for a fixed orientation of the cycle, say for instance "clockwise") if there is at least two particles, or N if there is only one particle, or N +1 if there is no particle. For all t, X t ∈ {1, . . . , N +1} and c t belongs to the limit set if and only if X t = N or X t = N +1. Let ∆X t+1 = X t+1 -X t . One can check that E (∆X t+1 | c t = c ) 0 for any configuration c. Moreover E (∆X t+1 ) 2 c t = c 3/N for any configuration c. Using a classical Stopping Time theorem from martingales [START_REF] Grimmett | Probability and Random Process[END_REF], one gets the bound O(N 3 ) on E [T ] where T = min{t 0 | X t ∈ {N, N + 1}}, and thus on the expected hitting time of the limit set.

A careful look at the proof shows that this bound on the hitting time of the limit set also applies to the graphs which are paths.

Proof (proof of Theorem 13). Let n b be the number of black cells of a configuration. Since the neighborhood of a cell is V, the potential of a black cell is n b . The potential of a white cell is

N -n b . If n b > N +1 2 (resp. n b < N -1
2 ) then firing a black (resp. white) cell decreases the energy and the configuration is not in A. If a configuration is in A then N -1 N to fire a black cell. This kind of dynamics is known as coupon collector (see [START_REF] Grimmett | Probability and Random Process[END_REF]) and T = O(N log N ).

Proof (Proof of Proposition 18).

We first prove that the cells in state 0 at the end of phase 1 of the algorithm cannot switch to state 1, whatever the sequence of updates, i.e. there are in case 1 of Fact 17. Indeed, assume instead that there exists a cell i and sequence of configurations c 1 , c 2 , ..., c k such that c 1 is the configuration at the end of phase 1; each configuration is the result of firing one cell in the previous configuration; c 1 (i) = 0 and c k (i) = 1.

Let c be the configuration just before the first update of this sequence that fires an active cell j in state 0 (there exists one since at least i will be fired in this sequence). But j must have been already active in c 1 , since it had at least as many neighbors in state 1 as in c (this is a monotonicity argument). The algorithm thus would not have exited the first "while" loop, which is a contradiction.

Same holds for the cells in state 0 at the end of phase 3. Also, by symmetry, the cells in state 1 at the end of phase 2 are in case 2 of Fact 17.

The last thing to prove is that the remaining cells are in case 3. Indeed, those cells were white at the end of phase 2, which means they can be made white by a sequence of updates starting from the configuration c 1 (the configuration at the end of phase 1). By monotonicity, they can be made white by a sequence of updates starting from any configuration in the attractor.

were black at the end of phase 3, which means by a symmetric argument that they can be made black.

Let us now show that for any acceptable labeling L, snd(L) is in the limit set. As is the last paragraph, start from the configuration paint(L, 0), consider each node in a bottom-up order, update it if it is not in its preferred state. Updating a node makes it go to its preferred state, because it has an acceptable label. Then consider each node in a top-down order and update it if it is not in the state 1. Again, thanks to the definition of an acceptable labeling, updating a node makes it change its state and be in the state 1.

Symmetrically, there is a sequence of updates leading from paint(L, 1) back to paint(L, 0). This is a cycle, so the energy cannot decrease.

Moreover, there is no sequence of updates leading to change the state of a cell labeled ( , α): the first change of state of such a cell v would contradict the fact that it has more than deg(v)/2 neighbors with label ( , α) and thus in the state α.

Thus, the energy cannot decrease and the whole cycle, including snd(L), is in the limit set (see Proposition 9).

We have even shown that if we let A be attr • snd(L), then all vertices labeled ( , α) are in the state α in every configuration in A there are two configurations c, c in A such that, for every cell v labeled ( , •), c(v) = 0 and c (v) = 1.

In the definition of f , the value of α for vertices labeled ( , α) is entirely determined by the labeling of vertices labeled ( , •). Since L and f (attr • snd(L)) = L have the same set of vertices labeled ( , α) for α ∈ {0, 1}, they are equal. Finally, Section 5.1 has shown that for an attractor A, the configuration c in which all oscillating vertices are in the state 1 belongs to A. We have also shown that c := paint(f (A), 1) is in the limit set. But c assigns state 1 to vertices labeled ( , •), i.e. oscillating vertices, and assigns the state they have in all configuration of A to the remaining vertices. Thus, c = c , which means that the attractors attr • snd(f (A)) and A have a common element and are thus equal.

This concludes the proof by implying that f is a bijection.

Proof (Proof of Theorem 23). This is clearly true for snd(L). So it is sufficient to show recursively that if this is true for a configuration c, then it is true for the resulting configuration c after firing one cell. Any cell v labeled ( , α) has strictly more than deg(v)/2 in state α, so v is not active in c. This proves point 1.

Proof of Theorem 22 has shown that snd(L) is in the limit set. Thus, any cell v has always at least deg(v)/2 neighbors in the same state as v (otherwise, firing it would decrease the energy). This proves point 2a. Now, consider a cell v in case 2b. There are only two cases to consider:

v is in its preferred state and a son v of v is fired. But then v must also have been in its preferred state α, and same for all its sons. Which means v has one more son labeled (•, α) than sons labeled (•, 1 -α), and is thus inactive. v not in its preferred state and is fired. Then v has as many neighbors in each state (because we are in the limit set), all its sons that are not in their preferred state are in the same state (recursion hypothesis), and v has one more son labeled (•, α) than sons labeled (•, 1 -α) (definition of an acceptable labeling). Which implies that all the sons of v are actually in their preferred state.

Proof (proof of Theorem 24). We study here the movements of particles in the initial tree T.

One can divide T into its induced subgraphs which are paths, denoted P 1 ,...,P . Those paths link the cells of odd degree (1 or 3). The reduced tree T is obtained by replacing each path P j by an edge. Consider a configuration c on T which belongs to an attractor. There can not be two particles on the same path P j , otherwise a sequence of updates could lead to the collision of these two particles and thus to an energy decrease. In the same way, there can not be two particles on two paths P j and P k which share a common extremity. This extremity would necessarily be a cell of 3, then a sequence of updates could position the two particles on the edges incident to this cell. Firing this cell at that time would decreases the number of particles by at least 1 and lead to an energy decrease. Consequently, in configurations of the limit set, there is at most one particle by path P j and there can not be two particles on two incident paths. Reciprocally, one can easily check that such configurations always belong to the limit set. Moreover in such configurations, once a particle is stuck on a path P j , it can not leave it any longer. One can deduce from all those remarks that the attractors are in bijection with the matchings of T (the matching indicates where the isolated cells are located in T).

To prove the bound on the expected hitting time of the limit set, we find a bound on the time until at least one particle disappears. Consider a configuration where there exist two particles on a same path P j of length n. One can suppose that these two particles follow a random walk on this path with reflecting barriers at each extremity, unless they collide with another particle (leading to the loss of two particles in the tree) or unless one of the two particles leaves the path (leading to the loss of at least one particle in the tree since leaving necessarily involves a cell of degree 3 fired with two incident particles). Thus under the condition that they have not disappeared before, a bound on the expected time elapsed until they collide can be derived from classical studies of random walks with reflecting barriers [START_REF] Grimmett | Probability and Random Process[END_REF]: this expected time is bounded by O(n 3 ), and consequently it is also a bound on the time until at least one particle disappear.

Consider a configuration where there exist two particles on two paths P j and P k (of respective length n and m) sharing a common extremity: the cell i. With the same reasoning, assuming that the two particles have not led to the removal of another particle means that they follow random walks on their respective paths with reflecting barriers at the extremities. Then they can only disappear by being both incident to cell i when cell i is fired. By analyzing the two-dimensional random walk corresponding to the evolution of the respective distances to cell i, this event occurs after at most O(max(n, m) 3 ) steps on expectation, as proved in Section C using standard tools for multi-dimensional random walks.

Finally, from any configuration which does not belong to the limit set, at least one particle disappear within O(N 3 ) steps on expectation. Since the number of particles in any initial configuration on a tree is bounded by N , the expected time to hit the limit set is bounded by O(N 4 ).

Proof (proof of Theorem 34). Consider a biased tree of size n, an initial correct configuration ĉ0 on position 0 and the sequence (ĉ t ) t 0 . Dynamics δ converges almost surely from initial configuration ĉ0 and c T = c f . We define the sequence of random variable (t i ) i 0 as t 0 = 0 and t i+1 = min{t > t i |P os(ĉ ti+1 ) = P os(ĉ ti ) or P os(ĉ ti+1 ) = n+1}. Consider the sequence of random variable (X i ) i 0 such that X i = P os(ĉ ti ). According to corollary 33, |X i-1 -X i | = 1.

Consider a configuration ĉt on locked position n > j > 0 then firing cell b j leads to a configuration on position j -1 and firing cell f j leads to a configuration on unlocked position j. Firing other cells does not affect the position of the configuration. Consider a configuration ĉt on unlocked position n > j > 0 then firing cell b j leads to a configuration on position j -1, firing cell f j leads to a configuration on locked position j and firing cell b j+1 leads to a configuration on position j + 1. Firing other cells does not affect the position of the configuration. A cell has a probability 1/N to fire where N = 4 + 8n. Thus, the evolution of a configuration on position 0 < j < n can be summarized as:

Unlocked Position j Position j+1 Locked Position j Position j-1 1/N 1/N 1/N 1/N 1/N (N-3)/N (N-2)/N
A basic analysis yields that:

-if 1 x n then P (X i+1 = x + 1 | X i = x) = 1 -P (X i+1 = x -1 | X i = x) and 1/5 P (x i+1 = x + 1 | X i = x) 2/5. -P (X i+1 = 1 | X i = 0) = 1. -P (X i+1 = n + 1 | X i = n + 1) = 1.
Thus the behavior of (X i ) i 0 is as described in definition 25. We define the random variable T = min{i|X i = n} which corresponds to the first time when all gates are stable, then Θ(

3 2 n ) E [T ]
Θ(4 n ) on expectation (see Def. 25). We call c f -1 the correct configuration on position n (i.e. all gates are stable). Then c

T = c f , c T -1 = c f -1 and P (c t+1 = c f |c t = c f -1 ) = 1/2. Thus, E [T ] = Θ(E [t T ]). By definition, t T = T i=1 (t i -t i-1 ) = ∞ i=1 [(t i -t i-1 )1 ti<T ].
Since there are at most 2 cells which may modify the position of a correct configuration, we have 1

E [t i+1 -t i-1 ] Θ(n). Thus ∞ i=1 (1 ti<T ) E [t T ] Θ( ∞ i=1 (n1 ti<T )). We conclude that Θ( 3 2
n ) E [T ] Θ(n4 n ).

B Biased random walk

Consider a sequence of random variable (X i ) i 0 defined on {0, . . . , n} such that X 0 = 0 and for all i 0:

-P (X i+1 = 1 | X i = 0) = 1 (reflecting barrier at 0).

-P (X i+1 = n | X i = n) = 1 (absorbing barrier at n).

-If 0 < x < n, P (X i+1 = x -1 | X i = x) + P (X i+1 = x + 1 | X i = x) = 1 and 0 < a < P (X i+1 = x + 1 | X i = x) < b < 1/2 for some a, b ∈ R + . This theorem is a direct consequence of classical analyzes of simple random walks on {0, . . . , n} where for all 0 < x < n, P (X i+1 = x + 1 | X i = x) = p and P (X i+1 = x -1 | X i = x) = q, with p + q = 1.If P (X i+1 = 1 | X i = 0) = 1 and P (X i+1 = n | X i = n) = 1, then for all 0 k n, the expectation of T k is ex-

actly E [T k ] = 2p(1-p) (1-2p) 2 1-p p n -1-p p k -n-k 1-2p
. This formula is obtained by solving the following system of equations:

     E [T k ] = p(1 + E [T k+1 ]) + (1 -p)(1 + E [T k-1 ]) for all 0 < k < n E [T n ] = 0 E [T 0 ] = 1 + E [T 1 ]
C Bound on the hitting time of a 2D finite Markov Chain

C.1 Background on Markov chain theory

We recall only the necessary background on Markov chains to get a bound on the hitting time of a 2D finite Markov chain. For a gentle introduction and proofs, we refer for instance to Chapter 7 of [START_REF] Levin | Markov Chains and Mixing Times[END_REF].

Let (X t ) t∈N be a Markov chain. We

C.2 Application to our case

If a neighbor does not exists (because the node is on the border), the edge points to the node itself. Each edge has the same probability 1 4 . An invariant probability is the uniform probability π : x → 1 nm . Our Markov chain is invariant with respect to this probability. So we can use the definitions of section C.1: That is, a flow on a single path from a to b. E(θ) (n + m)4nm. We conclude with the commute time identity that the average hitting time is O(n 3 ), assuming wlog that n m.

Fig. 2 .

 2 Fig.2. A configuration c and its dual configuration ĉ (with regard to root r).

2 while 3 Output:

 23 There is an active cell i in state 0 do Fire i The current configuration c . Proposition 18. The configuration c returned by algorithm 1 is in the limit set.Corollary 19. The input configuration c is in the limit set if and only if the energy has not decreased during execution of the algorithm.Three phases are necessary, as shown on Fig.3. configuration, phase 1 of the algorithm makes the cell a become black. Then, phase 2 makes its left neighbor b become white, as well as a.

  v, if the cell has label 1. ( , α) then it has strictly more than deg(v)/2 neighbors with label ( , α); 2. ( , α) then (a) if the father has a label of the form ( , β), then α = 0 and v has one more son labeled (•, 1 -β) than sons labeled (•, β); (b) otherwise, i.e. the father has a label of the form ( , •), v has one more son labeled (•, α) than sons labeled (•, 1 -α).

  -The 3 widgets used in the construction of a biased tree. Gray denotes the fact that cell state is not represented 4.b -The three configurations ĉl ,ĉu and ĉs.

Fig. 4 .

 4 Fig. 4. Widgets used in the construction of a biased tree.

Fig. 5 .

 5 Fig. 5. A biased tree and a correct configuration on unlocked position 2.

Proof.

  Consider the correct configuration ĉ on position n. According to fact 30, only cells b n and b n+1 are active. If cell b n+1 fires, these two cells become inactive and no other cell becomes active. Firing cell b n+1 leads to configuration ĉf . Thus ĉf is stable. Corollary 33. Consider a correct configuration ĉ, then configuration ĉ = δ(ĉ) is either correct or ĉf . Moreover |P os(ĉ ) -P os(ĉ)| 1.

- 2 N 2 ( 1 2

 221 If N is even then all cells have potential N 2 and these C N configurations are stable configurations. -If N is odd we call C b (resp. C w ) the set of configurations where n b = N +1 resp. n b = N -). White (resp. black) cells of a configuration in C b (resp. C w ) are inactive and black (resp. white) cells are active, firing one of them leads to a configuration of C w (resp. C b ). Thus from any configuration of C w ∪ C b , there is no sequence of updates that causes a drop of energy and C w ∪ C b = A. Now, we prove that A is made of only one attractor. Consider two configurations c and c , we define the distance from c to c as d(c, c ) = |{i|c i = c i }|. Consider two configurations c and c such that c, c ∈ C b and c = c . Since c and c have the same number of black cells then there exist i and j such that c i = 1, c i = 0, c j = 0 and c j = 1. Firing cells c i and afterward c j in c leads to a configuration c and d(c , c ) = d(c, c ) -2. Thus from any configuration c of C b ∪ C w there is a sequence of updates which leads to any configuration c of C b ∪ C w . The set C b ∪ C w is an attractor of size 2C N -1 2 N . Now consider a configuration where n b > N +1 2 . As long as the configuration does not become a configuration of A, white cells are inactive. When n b -N +1 2 black cells have fired the configuration is in A. At each time step there is a probability n b

Theorem 35 .

 35 For all 0 k n, let T k := min{i|X i = k} be the hitting time of position k.Then θ k (b) E [T k ] θ k (a)where θ k (p) = 2p(1

  note τ b the hitting time of b, i.e. the first time the Markov chain is in state b: τ b := min {t | X t = b } If (X, P ) is a Markov chain reversible with respect to the probability π, the conductance of an unoriented edge (x, y) is c(x, y) := π(x)P (x, y) = π(y)P (y, x)Let a and b be two distinguished vertices, representing source and sink. We use the following potential, or voltage, of a vertex:V (x) := P (τ a < τ b | X 0 = x )Clearly V (a) = 1 and V (b) = 0. Now, define the current flow on oriented edges asI(x, y) := c(x, y) (V (x) -V(y)) and I := x I(a, x) The effective resistance between a and b is R(a, b) := V (a) -V (b) I Theorem 36 (Commute time identity). E (τ b | X 0 = a ) + E (τ a | X 0 = b ) = cR(a, b) In our case, c = 1, so R(a, b) is an upper bound for the average hitting time E (τ b | X 0 = a ). Here is how one can bound R(a, b). A flow from a to b is a function on oriented edges which is antisymmetric: θ(x, y) = -θ(y, x) and which obeys Kirchhoff's node law: ∀v / ∈ {a, b} x θ(x, v) = 0 This is just the requirement "flow in equals flow out". The strength of a flow is θ := x θ(a, x) Theorem 37 (Thomson's Principle). For any finite connected graph, R(a, b) = inf {E(θ) | θ a unit flow from a to b } where E(θ) := x,y (θ(x, y)) 2 c(x, y) .

  to Thomson's principle, it is sufficient to construct a flow from a to b to get an upper bound on R(a, b). If a = (i, j) and b = (n, m), we consider the trivial (and far from optimalk, j), (k + 1, j)) := 1 if i k < n θ((n, k), (n, k + 1)) := 1 if j < m θ(x, y) := 0 elsewhere.

  Definition 16 (Dual potential & Energy). The dual potential vi of cell i is the number of its neighbors (excluding itself ) in a different state than itself. If vi < |Ni|-1

		2	then the cell
	is in the majority state and is thus inactive; whereas, if vi	|Ni|-1 2

Proof (Proof of theorem 22). Let attr(c) be the attractor in which the configuration c lies. The function attr is naturally defined only on the limit set.

We define a function f that maps an attractor A to an acceptable labeling L. We then conclude the proof by showing that for every attractor A, attr • snd(f (A)) = A and for every acceptable labeling L, f (attr • snd(L)) = L.

To define f , let A be an attractor, we construct L = f (A). For all vertices that have the same state α in any configuration of A, define L(v) := ( , α). All leaves are now labeled. We define the labeling of the remaining vertices inductively (in a bottom-up order). Each remaining node has an even degree and an odd number of sons. Considering a cell v having all its sons labeled:

if the father is already labeled (thus having a label of the form ( , •)), define L(v) := ( , 0); otherwise, the father is not already labeled, let α be the majority state among the preferred states of the sons and define L(v) := ( , α). Note that v can change its state in A, so deg(v) is even, so v has an odd number of sons and the majority state is well defined.

Let us show that the labeling we have just defined is acceptable. Let paint(L, α) be the following configuration (intuitively "set to state α as many cells as possible"):

Section 5.1 has shown that, given the way we construct L, this configuration is in A.

1. Consider a cell v labeled ( , 0) and the configuration paint(L, 1). All the neighbors of v not labeled ( , 0) are in the state 1. So, there are necessarily more than deg(v)/2 neighbors labeled ( , 0) (and thus in the state 0), otherwise, updating v would make it change its state, contradicting the definition of L(v) = ( , •). By symmetry, point 1 of the definition of acceptable labelings is fulfilled. 2. Now consider a cell labeled ( , α).

(a) If the father has a label of the form ( , •) then L(v) has been defined as ( , 0). So, point 2a is fulfilled. (b) We consider the remaining nodes (those with a father labeled ( , •)) in a bottom-up order and prove inductively that for each one, point 2b is fulfilled. Assume for now that there is a configuration c of A in which all the nodes of the subtree of v are in their preferred state, except v which is in state 1 -α. Then if v had more than one more son labeled (•, α), updating v would make the energy decrease, contradicting the fact that c is in the limit set. So, point 2b is fulfilled. Such a configuration c can be constructed inductively: start from the configuration paint(L, 1 -α), consider each node in the subtree of v in a bottom-up order, update it if it is not in its preferred state. Updating a node makes it go to its preferred state, because it has an acceptable label.


A Akhatova

F Robaut

M Verdier

M Yescas

F Roch

C Tassin

Hugo P Van

H P Van Landeghem
email: hugo.van-landeghem@grenoble-inp.fr

Microstructural and mechanical investigation of the near fusion

Keywords: Dissimilar Metal Weld, Gas Tungsten Arc Welding, Low-Alloy Steel, Nickel-based filler metal, Hardness, Phase transformation

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Microstructural and mechanical investigation of the near fusion

boundary region in thermally aged 18MND5 / Alloy 52 narrow-gap dissimilar metal weld

I. Introduction

Dissimilar metal welds (DMWs), such as the joints between the reactor pressure vessel nozzles and safe-ends to the main coolant pipes are key elements in the design and performance of nuclear power plants. High-quality welds are commonly performed by gas tungsten arc welding (GTAW) with Ni-based filler alloys, with or without buttering layers. In these joints, large variations in chemical composition occur at fusion boundaries between the base and filler metals. In particular, the carbon composition changes abruptly from the carbon rich low-alloy steel (LAS) to the carbon lean filler metal. The resulting composition gradient at the LAS/filler metal fusion boundary leads to the diffusion of carbon when it has sufficient mobility. This transfer of carbon is one of the key aspects of the durability of such DMWs [START_REF] Brentrup | Fabrication and Characterization of Graded Transition Joints for Welding Dissimilar Alloys[END_REF].

Carbon migration from the heat-affected zone of the steel into the weld metal during post-weld heat treatment (PWHT) is particularly strong with high-Cr filler alloys, such as Alloy 52 [START_REF] Hänninen | Dissimilar metal weld joints and their performance in nuclear power plant and oil refinery conditions[END_REF], [START_REF] Mouginot | Characterization of a Ni-base NG-DMW weld of modern PWR[END_REF]. The driving force for carbon migration is the difference in carbon chemical potential between the ferritic LAS and the austenitic nickel-based alloy caused by the variation in chromium and other carbide formers (Nb, Mo) content, as well as the change in structure, across the weld. Experimental results from heat-treated dissimilar metal joints demonstrated the accumulation of carbon close to the fusion boundary on the weld side. It can lead to a carbon build-up of up to about 1.5 wt% C and the formation of Cr-rich precipitates [START_REF] Dupont | Microstructural evolution and high temperature failure of ferritic to austenitic dissimilar welds[END_REF], [START_REF] Huang | Carbon migration in 5Cr-0.5Mo/21Cr-12Ni dissimilar metal welds[END_REF] in the first few tens of microns of the weld against the fusion boundary. Accordingly, many authors have reported an increase in the hardness in weld metal next to the fusion boundary [START_REF] Chen | Microstructural and hardness investigations on a dissimilar metal weld between low alloy steel and Alloy 82 weld metal[END_REF]- [START_REF] Wang | Microstructures and microhardness at fusion boundary of 316 stainless steel/Inconel 182 dissimilar welding[END_REF]. For instance, Nevasmaa et al. [START_REF] Nevasmaa | Fracture mechanical characterisation of ferrite-austenite dissimilar metal welds (DMWs) for elevated temperature service in view of metallurgical mismatch[END_REF] compared two configurations of ferrite-austenite DMWs: one made using a beveled V-groove and Alloy 82/182 filler metal with a buttering layer and one narrow-gap weld using Alloy 52 as a filler metal with no buttering. They found that the applied PWHTs induced different effects on the strength mismatch and fracture behavior of DMWs. In the mock-up with buttering layer, the PWHT was shown to reduce the strength contrast in comparison to the as-welded state, whereas PWHT of narrow-gap GTAW sample accentuated this contrast and resulted in inferior fracture resistance, compared to the as-welded state.

Similarly, the LAS / Alloy 52 fusion boundary was found to be significantly affected by carbon diffusion during aging close to the operation temperature. Sarikka et al. [START_REF] Sarikka | Microstructural Characterization of Alloy 52 Narrow-Gap Dissimilar Metal Weld After Aging[END_REF] reported that the hardness peak found on the weld side following PWTH tends to level out after 5 000 h and even further after 10 000 h at 400°C. They explained this evolution by the diffusion of carbon from the near-fusion-boundary region deeper into the weld metal. Ahonen et al. [START_REF] Ahonen | Effect of Thermal Aging on Fracture Mechanical Properties and Crack Propagation Behavior of Alloy 52 Narrow-Gap Dissimilar Metal Weld[END_REF] found that such aging did lower the fracture resistance of the overall joint but it remained well within specifications. Ahonen et al. [START_REF] Ahonen | Fracture mechanical and microstructural characterization of narrow-gap safe-end dissimilar metal weld[END_REF] believe this drop in mechanical properties is mostly owed to phosphorous segregation.

Therefore, investigating the evolution of the microstructure and the mechanical properties during aging near the LAS / Alloy 52 fusion boundary in narrow-gap DMWs is critical to understand the long-term behavior and performance of such joints in nuclear power plants and is the aim of the present study. Particular attention was paid to the prominent role of carbon distribution in the near-fusion-boundary region and its evolution during aging treatments, using both experimental measurements and thermodynamic calculations. Its impact on local mechanical properties was assessed by nano-indentation.

II. Experimental

A.

Materials and processing

The DMW investigated in this paper is an narrow-gap GTAW weld, with Alloy 52 filler metal, between a 18MND5 LAS and a Z2CND18-12 austenitic stainless steel with controlled nitrogen.

The chemical compositions of the 18MND5 LAS, Alloy 52 filler metal and the average composition of the weld far from the fusion boundary used for this work are listed in Table 1.

Table 1. Chemical composition of base materials (in wt%) [START_REF] Joly | Fracture Toughness in the Ductile-Brittle Transition and Thermal Ageing Behavior of Decarburized Heat Affected Zone of Alloy 52 Dissimilar Metal Welds of Nuclear Components[END_REF]. For weld metal, Cr, Ni, Fe and Mn contents were measured by electron probe micro-anaysis and other elements contents estimated by dilution. To reduce internal stresses developed during welding the mock-up was subjected to the following PWHT: the whole mock-up was first heated up to 610°C and treated for 6h15 min, then cooled down to 350°C at 40°C/h followed by air cooling to room temperature. It was later subjected to a second similar treatment of 10 h for a total PWHT duration of 16 h at 610°C.

Materials

Subsequently, coupons cut from the mock-up were aged in air in the temperature range from 350°C to 450°C for various times between 5 000 and 50 000 hours as listed in Table 2.

B. Sample preparation and etching

Cross-sections of the 18MND5 / Alloy 52 fusion boundary were cut perpendicularly to the weld beads and polished with SiC abrasive paper down to 4000 grit, followed by polishing using diamond pastes of 3 μm and 1 μm size. In the final step, the samples were polished using an aluminum oxide polishing paste of 0.1 µm grit size and cleaned in ethanol in an ultrasonic cleaner. The ferritic side of the samples was etched using Nital (3% HNO3 in methanol), the weld side was electrochemically etched in a 10 % chromic acid aqueous solution (10 g CrO3 in 100 ml H2O) under a 3 V potential at room temperature. The microstructure of the weld mockup was observed using an Olympus BX60M light microscope.

Once regions of interest (ROIs) were identified and marked, etching was removed by grinding and polishing again, starting back from 1200 grit SiC to 1 µm diamond. Samples were further finished with 0.1 µm alumina suspension prior to EPMA and nanoindentation analyses.

C. Scanning electron microscopy and energy dispersive spectroscopy

For the pre-selection of comparable ROIs in all samples, evolutions of Fe, Ni and Cr contents in the partially mixed zone of the weld were first semi-quantitatively determined by energy dispersive spectroscopy (EDS). EDS measurements were performed in a JEOL JSM 6400 SEM equipped with a BRUKER AXS SDD detector with a profile step size of 1 µm.

D. Electron probe micro-analysis

Quantitative analysis of Fe, Ni, Cr, Mn and C concentration across the fusion boundary was carried by wavelength dispersive spectroscopy (WDS). Analyses were performed using a Cameca SX50 electron microprobe analyzer (EPMA). Measurements were made every 20 µm in the homogeneous parts of the base and weld metals far from the fusion boundary, and every 4 µm in the near-fusion-boundary region. The measurement was done at 15 kV with beam current of 900 nA for C and Mn and 150 nA for Cr, Ni, Fe. A liquid nitrogen cold trap and a low-pressure jet of oxygen onto the specimen were used to reduce carbon contamination.

Substitutional element contents were measured with LiF monochromators (2d spacing = 0.4 nm). The classic quantitative analytical procedure was taken into account using the ratio of measured intensities of the K line from the sample and the pure metallic standards. Background intensity was measured and deducted from the peak intensity.

Carbon concentration was measured with a tungsten/silicon multilayer monochromator (2d spacing = 9.5 nm -crystal PC2-). The procedure was based on ISO standard 16592:2012, which postulates a linear relationship between the intensity of the C Kα peak and the carbon content in the probed volume (in wt%). The samples used as references were electrolytic iron and various fully martensitic steels with carbon contents varying from 0.0006 to 0.99 wt%.

E. Nano-indentation

Nano-indentation tests were performed to map hardness across the fusion boundary with a high spatial resolution (typically 2 µm pitch), using an MTS XP machine equipped with a diamond Berkovich indenter. Two mapping patterns were used. The first is a coarse matrix (2 × 28 points, 25 µm pitch) with a final indentation depth of 400 nm. The second is a finer matrix (5 × 25 points, 2.5 µm pitch) with a 100 nm final depth, centered over the fusion boundary. Hardness and elastic modulus were continuously determined along the indentation depth using the standard Oliver and Pharr method [START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF] and superimposed 2 nm oscillations at 45 Hz.

F. Transmission electron microscopy

Transmission electron microscopy (TEM) specimens were prepared using focused ion beam (FIB) lift-out in a dual beam microscope to control their locations within the determined ROIs. Before observation, the specimens were cleaned in a Gatan Solarus plasma cleaner for 4 min in O2/H2 mixture followed by for 4 min in Ar/H2. The specimens were observed in a JEOL

2100F TEM operated at 200 kV.

G. Thermodynamic calculations

Thermodynamic calculations were performed using the DICTRA (DIffusion Controlled

III. Results

A.

Evolution of composition and microstructure through the weld The first weld bead in contact with the LAS was taken as a starting point for their numbering. steel grade. Close to the fusion boundary, it is replaced by ferrite with a grain size of a few tens of micrometers, as a result of carbon diffusion from the LAS side to the weld side that notably leads to the dissolution of cementite, as illustrated in Figure 3b. The main part of the weld exhibits a cellular / dendritic structure with large grains elongated, millimeter-sized in the solidification direction, perpendicular to the solid-liquid interface, and a few hundreds of micrometer wide in the transversal section. A white layer where no microstructure is revealed by etching can be observed along the fusion boundary on the weld side: it corresponds to a thin layer where the local solidification conditions lead to plane front growth [START_REF] Mas | Growth Morphologies and Primary Solidification Modes in a Dissimilar Weld between a Low-Alloy Steel and an Austenitic Stainless Steel[END_REF]. Dark-etched bands on the LAS side are related to the carbon micro-segregation zones and are also known as ghost lines [START_REF] Ahonen | Fracture mechanical and microstructural characterization of narrow-gap safe-end dissimilar metal weld[END_REF]. Micro-segregation is a known phenomenon that takes place during the casting of large ingots [START_REF] Pickering | The Consequences of Macroscopic Segregation on the Transformation Behaviour of a Pressure-Vessel Steel[END_REF]. It is a material heterogeneity in the form of a chemical gradient that deviates from the nominal composition and may locally exceed specification limits. Microprobe analysis has shown dark-etched bands to be enriched in solute elements such as C, Cr, Ni and Mn, as detailed in Table 3.

A similar result was reported by Naudin in 18MND5 LAS [START_REF] Naudin | Modélisation de la ténacité de l'acier de cuve REP en présence de zones de ségrégation[END_REF] where it was found that A-segregations were enriched in solute elements and impurities as shown in Table 3. The similarity between average A-segregations and micro-segregations was already noted by Andrieu [START_REF] Andrieu | Mécanismes et modélisation multi-échelle de la rupture fragile trans-et intergranulaire des aciers pour réacteurs à eau sous pression, en lien avec le vieillissement thermique[END_REF] who considered them to behave similarly in terms of phase transformations and mechanical properties. Carbon concentration in segregated zone was found to be significantly greater than bulk content, about 1.8 times higher on average. Consequently, the influence of segregated zones on the carbon concentration in the weld will be examined in the present article. TEM observation of the microstructure at the carbon peak revealed an abundant precipitation of nano-sized elongated particles, as illustrated by Figure 5. These particles were found in all samples in varying amounts. Electron diffraction confirmed the presence of a secondary phase by showing a lattice of dimmer diffracted beams in addition to those of austenite. They can be indexed as M23C6, a chromium rich carbide with a FCC structure in a cube-on-cube orientation relationship with the matrix and a lattice parameter (≈ 10.6 Å) approximately 3 times as large as that of the matrix (≈ 3.6 Å) [START_REF] Lewis | Precipitation of M23C6 in austenitic steels[END_REF].

B. Evolution of carbon distribution and hardness during aging

Table 4 summarizes the fusion boundary characteristic data: white layer and partially mixed zone widths, corresponding measured peak values of carbon and hardness in the weld.

The extent of the white layer is determined visually and that of the partially mixed zone is chosen as the distance after which Ni content is greater than 98 % of its plateau value. The profile index corresponds to the number of the welding bead, N and S indicate non-segregated and segregated zones, and the last number is used to separate two profiles close together in the same welding bead. The collection of measurements found in Table 4 highlights the strong heterogeneity that occurs along the same fusion boundary, which is common in narrow-gap GTAW DMWs. To finely characterize the microstructure in the near-fusion-boundary region, local scale characterization techniques were used such as EPMA or nano-indentation and, in an effort to capture this heterogeneity, multiple measurements were distributed along the fusion boundary. In what follows, this aspect must be kept in mind when assessing the agreement across different results.

Figure 7. Analysis of post-weld heat-treated specimen E3: (a) light micrograph with profile locations, (b) carbon concentration profiles across the fusion boundary (steel on the left and weld on the right), (c) nano-hardness profile across the fusion boundary (steel on the left and weld on the right)

Figure 7a shows the locations of three ROIs in the PWHT sample E3. All three carbon composition profiles shown in Figure 7b were recorded in non-segregated zones in the LAS and have similar widths of white layer and partially mixed zone as seen in Table 4. On the LAS side, carbon concentration is low compared to the bulk content of the material. That is related to the proximity with non-segregated bands, which are carbon-lean compared to segregated bands as mentioned above. The nano-hardness profile recorded in ROI 18N1 is presented in Figure 7c. It reveals that hardness is very homogeneous in the LAS up until a few tens of microns from the fusion boundary, where it starts dropping. Past the fusion boundary, the hardness increases higher than the hardness level recorded further in the weld. The extent of this zone of elevated hardness matches that of the zone of elevated carbon content.

In Figure 8, similar analyses are presented for sample B4, aged 5 000 h at 400°C following PWHT. It can observed here that the ROI near a micro-segregation in the LAS, noted S, show a higher peak carbon content than regions away from them, noted N. This observation can be generalized to the other samples as can be seen from Table 4. The overall size of the carbonenriched zone remains similar (≈ 60 µm) to the PWHT state in all ROIs. In terms of hardness, it is slightly higher on the LAS side in micro-segregations, which is expected given the slightly higher carbon content. On the weld side, the hardness peak is also higher near microsegregations, showing here again a strong correlation between hardness and local carbon content. This correlation is also seen in the other samples listed in Table 4.

Figure 8: Analysis of specimen B4, aged for 5 000 h at 400°C following post-weld heattreatment: (a) micrograph with profile locations, (b) carbon concentration profiles across the fusion boundary (steel on the left and weld on the right), (c) nanohardness profiles across the fusion boundary (steel on the left and weld on the right).

C. Modelling chemical evolution across dissimilar interfaces

DICTRA simulation was carried out based on experimental data from profile 18N1 recorded on PWHT sample E3 and shown in Figure 7. It passes through a non-segregated zone.

As shown in Table 3, it was found that the average concentration of carbon in segregated and non-segregated zones 0.36 ± 0.07 and 0.17 ± 0.05 wt%, respectively. To investigate the effect of LAS carbon content on the evolution of the peak carbon content after aging, a carbon profile with higher carbon concentration (0.3 wt% at the maximum) was assumed on the LAS side to model segregated, carbon-rich bands, as plotted in Figure 1b.

The diffusion simulations consisted in isothermal holds corresponding to the experimental aging treatments, i.e. 350°C during 50 000 h, 400°C during 30 000 h, and 450°C

during 5 000 h. The simulation results in Figure 9 show that carbon is predicted to diffuse from the LAS side to the weld side, since the weld has both a higher chromium concentration and an fcc structure, which result in a lower carbon activity. This diffusion has resulted in minor increase of carbon in the carbon depleted zone (CDZ) just before the fusion boundary, and a carbon peak increase of ≈ 22 %, ≈ 33 % and ≈ 45 % after aging at 350°C during 50 000 h, at 400°C during 30 000 h, and at 450°C during 5 000 h, respectively. This increase takes place by the formation of a new carbon peak closer to the fusion boundary than the original one. The position of this peak changed across the different simulation going from 7 µm after 50 000 h at 350°C to 11 µm after 5 000 h at 450°C. The effect of carbon content in the steel was investigated by simulating the thermal treatment at 450°C during 5 000 h with two different starting carbon profiles on the LAS side as illustrated by Figure 10. As expected, the carbon peak is higher in the austenite adjacent to segregated zone. However, the difference is not as pronounced as in the experimental S and N profiles.

IV. Discussion

A.

Representativeness of ROIs

It is evident from the analytical data presented in this paper that the local composition profiles across the interface vary greatly along the fusion boundary. Since it is desirable to maintain high spatial resolution perpendicularly to the fusion boundary, it is challenging from a practical standpoint to acquire a representative dataset. This limitation imposes to select a restricted number of ROIs in each sample; therefore, it is important to select comparable ROIs across all samples. Consequently, it would be desirable to be able to select such ROIs on a microstructural feature. For this reason, the first criterion utilized was the size and morphology of the white layer. The study focused on regions where the boundaries of the fusion boundary and the white layer are parallel because it represents the larger fraction of the fusion boundary length and because these regions are less prone to abrupt changes in the out-of-plane direction.

However, it can be observed in Table 4 that the size of the white layer, chosen here to be around 15 µm, does not correlate well with the size of the partially mixed zone, which is a direct characteristic of the local composition profile. This requires additional measurements after the initial ROI selection step.

Consequently, EDS substitutional composition profiles were recorded across all ROIs to ensure that they feature comparable local substitutional concentration fields. Since those fields will not be affected by thermal aging, the evolution of the microstructure, interstitial content, and mechanical properties can be investigated as a function of their environment of substitutional elements.

B. Carbon mass transport

There exists a gradient of carbon chemical potential across the 18MND5 / alloy 52 fusion boundary that leads to the diffusion of the carbon contained in the LAS to the weld side of the fusion boundary when its mobility is sufficient. This carbon potential gradient related to the composition gradient of all solutes, in particular chromium, which has a strong attractive interaction with carbon. Given the high solute contents on the weld side, this carbon transfer also results in an intense precipitation of carbides on the weld side, mostly chromium rich M23C6 as evidenced by Figure 5 and Figure 6. The carbon atoms involved in such a precipitation reaction no longer participate to diffusion, leading to interconnected diffusion and precipitation kinetics.

Carbon analysis shows that carbon diffusion from the LAS into the weld occurs already during welding and PWHT, the maximum carbon peak value measured in the weld after PWHT being 0.79 wt%, measured in a zone adjacent to non-segregated band in LAS. The width of the carburized zone is about 60 µm.

Given the data shown in Table 4, it can be seen that the position of the carbon peak is closely related to its aging conditions in all ROIs. The carbon peak is, in all aged conditions, closer to the fusion boundary than right after PWHT. However, the carburized depth in the weld seems to be little affected, if at all, as seen in Figure 8. Additionally, it seems that there exists a shoulder after the peak in most ROIs of aged samples whereas this feature is absent in the PWHT sample. This is consistent with the carbon transport mechanism revealed by the numerical simulation. It can be seen Figure 9 that during aging, the modification of the carbon profile occurs by creation of a new carbon content peak, higher than the one obtained after PWHT and closer to the fusion boundary. In the rest of the carburized region, the profile remains almost identical. The distance from the fusion boundary of the peak formed during aging was systematically underestimated. This is also possibly due to the assumption of the model used here that assumes that phase transformations are instantaneous and thus that precipitate fractions are always at equilibrium. Carbon captured in carbides is no longer mobile and is thus trapped much closer to the fusion boundary than if part of it remained longer in solid solution due to finite reaction kinetics. This suggests that precipitation kinetics should be accounted for in order to quantitatively model carbon transport in this DMW.

The maximum values reached at the carbon concentration peaks are quite scattered. One possible explanation for this disagreement is the local variations of carbon in the LAS. It can be noticed from Table 4 that ROIs adjacent to dark segregation bands present a larger carbon peak. The larger carbon content in those bands on the LAS leads to a steeper chemical potential gradient across the fusion boundary and the partially mixed zone, and a larger mass transfer during aging. Results from the simulation showed a reasonable agreement with the peak content found in non-segregated ROIs. Nevertheless, simply assuming a base carbon level in the LAS as the segregated content [START_REF] Naudin | Modélisation de la ténacité de l'acier de cuve REP en présence de zones de ségrégation[END_REF] did not reproduce the peak content found in segregated ROIs. It is thought that the original peak of the PWHT condition plays a role as well. This peak is likely to be higher in segregated regions but no measurement in such regions could be successfully carried out during the present study, thus this hypothesis remains to be confirmed.

It should also be reminded here that the gradient of carbon chemical potential is not only affected by the local carbon concentration but also by other elements as well and in particular by chromium. Figure 11 shows the Cr profiles across the fusion boundary for ROIs that are all adjacent to non-segregated bands in the LAS, where carbon content is lower. From these profiles, it can be observed that shallower Cr gradients are associated to lower carbon peaks.

Chromium is known to decrease the chemical potential of carbon and thus, a shallower Cr content gradient leads to a shallower C chemical potential gradient. This, in turn, results in a lower mass transfer and smaller concentration peak after identical aging.

C. Evolution of microstructure and mechanical properties

Nano-hardness measurements revealed substantial variations of the mechanical properties across the fusion boundary in all conditions. On the LAS side, far from the fusion boundary, a hardness of about 4 GPa was recorded and is consistent with tempered bainite.

Within 150 µm from the fusion boundary, the value gradually drops to about 3 GPa due to carbon depletion. Past the fusion boundary into the weld, the hardness increases sharply to a peak whose maximum and location depend on the aging condition. Further away, it decreases to between 4 and 5 GPa

The data set in Table 4 reveals a close correlation between peak carbon concentration and peak hardness. In fact, Figure 7 and Figure 8 show that the shape of hardness profiles is closely similar to the carbon concentration profile.

Peak hardness after aging can be extremely high, close to 10 GPa compared to a base line of about 4.5 GPa in the weld metal. It is expected that carbon diffusion hardens the austenitic weld metal through solid solution strengthening, but that sole mechanism seems insufficient for such elevated hardness levels. Those hardness levels can be explained by the presence of the nano-precipitates shown in Figure 5. Those precipitates were shown to be M23C6 carbides, the presence of which is predicted by thermodynamic calculations throughout the PMZ. Those calculations also predict the presence of M7C3 carbides in the first few microns past the fusion boundary, which was not observed here.

V. Conclusion

In summary, the experimental results highlight the challenge of fine scale characterization of the near-fusion-boundary region in narrow-gap DMWs. The inhomogeneity inherent to the welding process calls for careful selection of multiple ROIs in the same welded joint. This work demonstrates the importance in this process of white layer and partially mixed zone thicknesses as well as the adjacency to segregated/non-segregated zone on the LAS side.

The following conclusions can be drawn from the study.

1. During welding and PWHT, carbon diffuses from the steel heat-affected zone and leads to the carbon enrichment in a narrow zone on the weld side of the fusion boundary. EPMA measurements showed the peak value of carbon to be 0.6 ± 0.1 wt% following PWHT.

Subsequent thermal aging in the 350°C-450°C range increases the peak value of carbon up to 1.93 wt% (400°C, 5 000 h).

2. The average carbon concentrations in segregated and non-segregated zones in steel are 0.36 ± 0.07 and 0.17 ± 0.05 wt%, respectively. The carbon peak values near fusion boundaries adjacent to segregated zones are on average ≈ 30 % higher than in zones near nonsegregated bands.

3. The carbon peak value and its position relative to the fusion boundary coincide well with the nanohardness peak. This correspondence confirms that the high hardness in a narrow zone on the weld side of the fusion boundary is a result of the increased carbon content in the weld and the carbide precipitation it leads to.

4. The position of the carbon peak with respect to the fusion boundary seems to be mostly controlled by the aging condition. This causes the carbon peak value to be lower in wider partially mixed zone for a given aging condition, since the shallower Cr concentration gradient and the associated shallower carbon chemical potential gradient lead to a smaller carbon transfer to the weld.

5. Thermodynamic calculations showed that the amount of carbon transferred across the fusion boundary increases with aging temperature and time, in a given substitutional composition gradient.

6. Thermodynamic calculations revealed the mechanism involved in the evolution of the carbon profile during aging, with the formation of a new peak close to the fusion boundary that progresses and overtakes the initial peak formed during welding and PWHT. Such calculations showed a reasonable ability to predict the peak value. Its distance from the fusion boundary was however, systematically underestimated.

7. Thermodynamic calculations predicted the presence of chromium rich M23C6 carbides throughout the PMZ, which was experimentally confirmed by electron diffraction in TEM. They also predict the presence of M7C3 carbides close to the fusion boundary, which was not observed here.

8. The effect of carbon content in the steel was investigated by simulation of thermal aging at 450°C during 5 000 h. The predicted carbon peak value did increase with the carbon content in LAS but not as much as experimentally observed. Further work is still necessary to quantify the effect of dark bands on carbon transfer across the fusion boundary, starting from the as-welded state.

VI. Acknowledgement

VII. Data availability

The raw and processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

VIII. Bibliography

  TRAnsformations) package in Thermo-Calc® software (Solna, Sweden)) together with the TCFE9 and MOB2 databases. This tool allows the prediction of atomic diffusion in one dimension resulting from chemical potential gradients. The DMW was treated using the cell concept in DICTRA, each material (LAS and weld) being placed in a separate cell. In the calculations, the program iterates until it detects that the chemical potentials at the cell boundary conform to the mass balance. The phases selected for the computations were the matrix phases (BCC ferrite for the LAS and FCC austenite for the weld) and several carbides (cementite M3C, M23C6, M7C3, M2C and M3C2). Elements considered were C, Cr, Fe, Ni, Mn, Mo, Si and input composition profiles were those measured by EPMA (C, Cr, Fe, Ni, Mn) on the PWHT sample or calculated by dilution (Mo, Si) based on the Ni content. Thermal aging treatments were reduced to their holding temperature and time, neglecting the heating and cooling parts.The WDS-measured profiles of C, Cr, Fe, Ni and Mn were fitted in order to the get smooth profile with geometrical series of step (initial step = 1 µm). The experimental profiles with their fitting curves are presented in Figure1. Fe, Cr, Ni and Mn concentration profiles are fitted according to,𝐹(𝑥 𝑖 ) = 𝐹 0 -𝐴 * 𝑒𝑟𝑓(𝐵 * (𝐶 -𝑥 𝑖 ))(1)Where 𝑥 𝑖 is a distance, A, B, C are the fit parameters and 𝐹 0 is the fixed concentration of atoms at the left end of the couple (in the LAS). C profile was fitted with a set of exponential functions at the intervals [-800:0], [0:16],[16:23], [23:400] µm.

Figure 1 :

 1 Figure 1: Experimental concentration profiles of (a) iron, nickel, chromium and (b) carbon concentration as a function of the distance from the interface (610°C, 16 h) given by symbols. The lines represent the corresponding fitted profiles used as the input of the simulation.

Figure 2

 2 Figure2shows a macrograph of the cross-section of the mock-up after PWHT, revealing the different zones of DMW, including the heat-affected zone of the LAS about 2.5 mm wide.

Figure 2 :

 2 Figure 2: Optical macrograph of the dissimilar metal weld mock-up after post-weld heat treatment. Numbering of weld beads was started (N=0) at the first bead in contact with LAS, as shown by the dotted line.

Figure 3

 3 Figure 3 shows optical micrographs of the weld joint microstructure after PWHT. The LAS side far from the fusion boundary displays the typical bainitic microstructure of 18MND5

Figure 3 :

 3 Figure 3: a) Optical micrograph of 18MND5/Inconel 52 fusion boundary cross section in a sample aged for 5 000 h at 400°C following post-weld heat treatment. b) Close-up of the near-fusionboundary region at the center of a).

  Figure 4 shows a distinct intermediate zone with substitutional content gradients, characteristic of a partially mixed zone. From the LAS to the weld metal, Fe, Ni and Cr contents change abruptly across the fusion boundary, then more gradually in the partially mixed zone, before they finally become constant in the fully mixed zone. It is important to note that this abrupt change was used throughout this study to determine the position of the fusion boundary in composition profiles. Given the 4 µm step used for those profiles, the distances to the fusion boundary are given ± 2 µm. Regarding the overall aspect of carbon profiles, carbon content is almost constant far from the fusion boundary, up until within about 100 µm, where it starts decreasing due to the carbon transport occurring from the LAS to weld during the various thermal treatments, including PWHT. The carbon transferred into the metal is then accumulated in a region right after the fusion boundary. The size of this region and the amount of carbon involved depend on the total thermal paths, as detailed further. Past this carbon-rich region, the weld metal shows very low carbon content. It sometimes spikes over a single point in the profiles, which is due to the presence of sparse micron-sized titanium carbides, likely formed during solidification given their extremely low solubility.

Figure 4 :

 4 Figure 4: Composition profiles measured across the fusion boundary between the 18MND5 low alloy steel (left side) and weld metal (right side) in sample A4, aged for 5 000 h at 450°C following postweld heat treatment.

Figure 5 :

 5 Figure 5: Scanning electron transmission micrograph of region 25S1 in C8 at the carbon concentration peak, 16µm away from the fusion boundary. The horizontal variations in contrast are an artifact of focused ion beam preparation of the specimens.

Figure 6 :

 6 Figure 6: Selected area electron diffraction patterns recorded from the same specimen as Figure 5 in a) [001]γ, b) [011]γ and c) [112]γ zone axes. The large, intense reflections are produced by the austenitic matrix. The smaller, dimmer ones match the crystal structure of M23C6 carbide in cubeon-cube orientation relationship with austenite.

Figure 9 :

 9 Figure 9: Simulated carbon concentration profiles after (a) 50 000 h at 350°C, (b) 30 000 h at 400°C and (c) 5 000 h at 450°C.

Figure 10 :

 10 Figure 10: Simulated carbon concentration profiles near (a) non-segregated and (b) segregated zones after 5 000 h at 450°C.

Figure 11 :

 11 Figure 11: Cr profiles across the fusion boundary (steel in the left and weld in the right) adjacent to the non-segregated zone of LAS in (a) sample A4 (450°C, 5 000 h), (b) sample B4 (400°C, 5 000 h).

  Part of this work was performed with the support of the Center of Excellence of Multifunctional Architectured Materials "CEMAM" No. AN-10-LABX-44-01 funded by the "Investment for the Future" Program operated by the National Research Agency (ANR).

Table 2 .

 2 Artificial aging conditions

	Sample	Temperature, °C	Aging time, h
	E3	-	0
	A4	450	5 000
	B4	400	5 000
	B7	400	30 000
	C8	350	50 000

Table 3 .

 3 Chemical composition of segregated and non-segregated zones in 18MND5 measured by EPMA in sample A4, compared to composition measurements in segregated zones in 18MND5 reported by Naudin[START_REF] Naudin | Modélisation de la ténacité de l'acier de cuve REP en présence de zones de ségrégation[END_REF].As usually observed in DMWs and consistently with the chemical compositions given in Table1, a composition gradient exists between the LAS and the weld metal.

	Source	wt%	C	Cr	Ni	Mn
		Bulk composition of LAS	0.19	0.17	0.72	1.46
	Present work	Average concentration in segregated zone	0.36 ± 0.07	0.20 ± 0.01	0.82 ± 0.05	1.72 ± 0.06
		Average concentration in non-segregated zone	0.17 ± 0.05	0.17 ± 0.02	0.72 ± 0.03	1.48 ± 0.06
		Bulk composition of LAS	0.16	0.19	0.70	1.35
	Naudin [20]	Average concentration in segregated zone	0.25	0.22	0.86	1.88
		Maximum concentration in segregated zone	0.40	0.4	1.0	2.3

Table 4 .

 4 Summary of fusion boundary characteristics

	Sample	Profile	White Layer size, µm	Partially Mixed Zone size, µm	Carbon peak Value, wt% Position, µm	Hardness peak Value, Position, GPa µm
	E3	17N	12	91	0.79	22.5		-
	Post-Weld Heat-	18N1	15	88	0.67	20	6.5±0.3		18
	Treated (PWHT)	18N2	15	93	0.66	20		-
	C8	24N	18	92	0.72	18	6.8±0.2		11
	PWHT + 350°C,	25S1	17	56	0.88	16	8.6±0.5		12
	50 000 h	25S2	19	85	1.31	14		-
	A4 PWHT + 450°C, 5000 h	15N 15S 16N 16S	17 17 14 14	84 62 52 50	0.60 0.82 1.21 1.43	13 13 13.5 9	7.2±0.4 8.1±0.1	--	17.5 15
	B4	22S	14	86	1.93	8	9.9±0.1		9
	PWHT + 400°C,	23N	16	67	1.1	9	5.7±0.1		6-26
	5000 h	24N	14	54	1.37	8		-
	B7	20S	18	45	1.16	18.5	8.1±0.9		20
	PWHT + 400°C, 30 000 h	20N	16	41	1.34	19	7.7±0.2		20


Luis Suárez

David Espes

Frédéric Cuppens

Philippe Bertin

Cao-Thanh Phan

Philippe Le Parc

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