Enguerrand Prebet

On Up-to Context Techniques in the π-calculus

We present a variant of the theory of compatible functions on relations, due to Sangiorgi and Pous. We show that the up-to context proof technique for bisimulation is compatible in this setting for two subsets of the pi-calculus: the asynchronous pi-calculus and a pi-calculus with immediately available names.

Proving that two elements are bisimilar is usually done by relying on a relation that is a bisimulation and also contains the corresponding pair of elements. Up-to techniques provide a powerful way of simplifying such proofs, by requiring that a relation is only included in a bisimulation. One such example is the up-to context technique which allows us to remove contexts when playing along the bisimulation game. A general theory of those techniques is developed in [START_REF] Pous | Coinduction all the way up[END_REF], by focusing on the class of compatible functions on relations, that are both sound up-to techniques and have nice compositional properties.

In the π-calculus, up-to context is not a sound technique. In fact, bisimilarity is not even a congruence, due to the input prefix creating substitutions. However, in subcalculi like the Asynchronous π-calculus (Aπ), bisimilarity is closed by substitution making it a congruence. Thus, the question of soundness of the up-to context technique for this subcalculus arises again. It is known that up-to substitution is not compatible, and not even below the greatest compatible function (called the companion in [START_REF] Pous | Coinduction all the way up[END_REF]). Thus it seemed that even if up-to substitution is sound, it could not be used in conjunction with other techniques without having to redo the proofs all over again.

Intuitively, the reason why compatibility fails for up-to substitution is that compatibility assumes the knowledge about one step of transitions in the bisimulation game, while in the proof of the congruence for Aπ, the substitution is dealt with by looking at two successive transitions to deduce the behaviour of the next step of the program after substitution. More precisely, we need to look at two visible transitions to reason about an internal step. There is thus a distinction to be made between visible and internal steps which leads us to define the usual bisimulation function as the intersection of the two bisimulation functions represented by the diagrams below, with α ranging over visible actions.

P R Q P ′ R Q ′ α α P R Q P ′ R Q ′ τ τ
In this paper, we propose a new notion of compatibility for a bisimulation function defined as f ∩ g (above f would be the visible actions while g would be the silent ones). The key idea is to impose a stronger condition on f and a weaker condition on g while preserving soundness. This allows us to define a framework where standard up-to techniques, including full up-to context, are both compatible and sound.

We show this result for two subcalculi where bisimilarity is a congruence and similarly for weak bisimilarity.

We thank Damien Pous and Davide Sangiorgi for helpful discussions about this work.

Compatibility and Soundness

In this section, we present some standard results about compatibility and their usage to show the soundness of up-to techniques (Section 1.1). Then we introduce compatibility with a function that is a generalisation of compatibility (Section 1.2). This notion still provides a soundness result while keeping nice properties of compatible functions (like being composable).

Previous work

Here, we recall standard results for compatibility from [START_REF] Pous | Coinduction all the way up[END_REF].

Definition 1 (Compatibility). f is g-compatible if f • g ⊆ g • f Definition 2 (Soundness). f is g-sound via f ′ if f ′ is extensive and R ⊆ (g •f )(R) implies f ′ (R) ⊆ b(f ′ (R))
Compatible functions can be composed freely in a modular fashion.

Lemma 3. If f 1 , f 2 are g-compatible, then f 1 • f 2 is g-compatible.
If g is monotone, we also have that

f 1 ∪ f 2 is g-compatible.
Compatible functions are useful as they are sound up-to techniques.

Lemma 4. If f is monotone and g-compatible, then f is g-sound via f ω .

However, there are sound up-to techniques that are not exactly compatible. We can recover some of them using compatibility up-to.

Definition 5 (Compatible up-to). f is g-compatible up to f ′ when f ′ is expansive and f • g ⊆ g • f ′ • f .
Compatible functions up to f ′ can be related to compatible functions when f ′ is also compatible ensuring the soundness of such functions. Lemma 6. If f ′ is idempotent, monotone and expansive, g-compatible and f is g-compatible up to f ′ , then

f ′ • f is g-compatible. Proof. f • g ⊆ g • f ′ • f f ′ • f • g ⊆ f ′ • g • f ′ • f f ′ is monotone ⊆ g • f ′ • f ′ • f f ′ is g-compatible ⊆ g • f ′ • f f ′ is idempotent
In fact, compatible functions are a subset of compatible functions up to f ′ for any expansive f ′ .

Remark 7. If f ′ is expansive, g is monotone and f is g-compatible, then f is also g-compatible up to f ′ .

Compatibility with a function

Unfortunately, substitution is not a compatible function not even up to some compatible f ′ . To see why, we call b the bisimulation function associated to bisimilarity for the π-calculus (see Section 2.1). We need the following lemma where ⊤ is the universal relation: Being compatible up-to some compatible function implies being smaller than some other compatible function (f ⊆ f ′ • f in Lemma 5). So it is enough to show that substitution is not included in the companion t, which is the greatest compatible function. As t is compatible, we have that t • b 2 (⊤) ⊆ b 2 (⊤). Thus, Lemma 8 implies that sub(b 2 (⊤)) ⊆ t(b 2 (⊤)). This entails that up-to context is not compatible in the π-calculus.

Lemma 8. Taking notations from CCS, we have (a. c | c, a | c) ∈ b 2 (⊤) but ((a. c | c){ a /c}, (a | c){ a /c}) / ∈ b 2 (⊤)
This example is asynchronous and as we will see later, up-to substitution is sound for Aπ. Our goal is to adapt the notion of compatibility so that it captures up-to substitution.

The main idea is to proceed in two steps. We first focus on visible actions and establish compatibility with respect to the corresponding bisimulation function. Then, we exploit that result to prove a weaker version of compatibility on internal actions, which we define now.

Definition 9 (Compatibility with). We say that f is h-compatible with g (or

g, h-compatible) if f • (g ∩h) ⊆ h • f .
Intuitively, with the knowledge we have about g, we are able to prove a sort of compatibility result on h. The main use of "compatibility with" is to prove g ∩ h-soundness, in which case we do not need to prove exactly g, h-compatibility. For instance g-compatibility and g 2 , h-compatibility is sufficient (see Thereom 15).

In Section 2, we exploit this approach taking g as the bisimulation function restricted to visible actions and h the one restricted to internal actions.

"Compatibility with" can be linked back to standard compatibility as follows:

Lemma 10. If f is g ∩ h-compatible, then f is g, h-compatible and h, g-compatible. If f is monotone, g-compatible and g, h-compatible, then f is g ∩ h-compatible.
Proof.

• f • (g ∩ h) ⊆ (g ∩ h) • f ⊆ g • f (and h • f respectively)
• The following proof uses Lemma 11, presented below.

f • (g ∩ h) = f • (g ∩ (g ∩ h)) ⊆ f • g ∩ f • (g ∩ h) by Lemma 11 ⊆ g • f ∩ f • (g ∩ h) f is g-compatible ⊆ g • f ∩ h • f f is g, h-compatible = (g ∩ h) • f
If f is a compatible function for g ∩ h, like it is the case in Section 2, we cannot show directly that f is both g-compatible and g, h-compatible. Indeed, we only have the latter by Lemma 10. We show in Section 2 that in the case of the π-calculus, things can be handled smoothly.

Since we often use monotone functions and intersections, we rely on the following lemma:

Lemma 11. If f is monotone, then for any set A, B, f (A ∩ B) ⊆ f (A) ∩ f (B). Similarly, for any function g, h, f • (g ∩ h) ⊆ f • h ∩ f • g. Conversely, for any g, h, f • g ∪ f • h ⊆ f • (g ∪ h).
We can also build the composition and union of g, h-compatible functions under mild assumptions.

Lemma 12. If f 1 , f 2 are g-compatible, g, h-compatible and monontone, then

f 1 • f 2 is g, h-compatible.
Proof. By Lemma 10, f 1 and

f 2 are g ∩ h-compatible, so f 1 • f 2 is too, meaning that f 1 • f 2 is g, h- compatible. Lemma 13. If f 1 , f 2 , h are monotone and f 1 , f 2 are both g, h-compatible, then f 1 ∪ f 2 is g, h-compatible.
Proof.

(f 1 ∪ f 2 ) • (g ∩ h) = f 1 • (g ∩ h) ∪ f 2 • (g ∩ h) ⊆ h • f 1 ∪ h • f 2 f 1 , f 2 are g, h-compatible ⊆ h • (f 1 ∪ f 2 )
by Lemma 11

Corollary 14. If f, h are monotone, g-compatible and g, h-compatible, then f ω is g, h-compatible.
Here, we state the theorem that is the equivalent of Lemma 4 for compatibility with. Intuitively, if f is g-compatible, then we can use g any number of times to show the compatibility of f with respect to h (i.e f is g m , h-compatible), and this is enough to prove that f is a sound up-to technique for g ∩ h.

Theorem 15. If f, g, h are monotone and

f is both g-compatible, g m , h-compatible (with m ≥ 1) then f is g ∩ h-sound via f ′ω with f ′ = i≤m f i . Proof. First, notice that R ⊆ ((g ∩ h) • f )(R) implies R ⊆ (g • f )(R) and R ⊆ (h • f )(R).
On one side, we then have R ⊆

(g • f ) m (R), meaning R ⊆ (g m • f m )(R) by compatibility and so R ⊆ (g m • f ′ )(R). On the other side, we have that R ⊆ (h • f ′ )(R) as h is monotone and m ≥ 1. Therefore, R ⊆ ((g m ∩ h) • f ′ )(R). By Lemma 10, f is g m ∩ h-compatible, so f ′ is too. Thus f ′ is g m ∩ h-sound via f ′ω meaning f ′ω (R) ⊆ (g m ∩ h)(f ′ω (R)). Therefore we have f ′ω (R) ⊆ h(f ′ω (R)). As f is g-compatible, we can also prove that f ′ω (R) ⊆ g(f ′ω (R)).
By combining the two, we obtain that

f ′ω (R) ⊆ (g ∩ h)(f ′ω (R)).
To prove g ∩ h-soundness, we are thus interested in showing g-compatibility, and g m , h-compatibility. By showing a weakening of compatibility with, we are able to compose functions with different m.

Lemma 16. If f is monotone, is g, h-compatible, and g ′ ⊆ g, then f is g ′ , h-compatible. Proof. As f is monotone, f • (g ′ ∩ h) ⊆ f • (g ∩ h). So f • (g ′ ∩ h) ⊆ h • f . Lemma 17. If f 1 , f 2 are monotone, g-compatible, f 1 is g m , h-compatible, f 2 is g n , h-compatible with m ≥ n and g ⊆ id, then f 1 • f 2 and f 2 • f 1 are g m , h-compatible.
Proof. As g ⊆ id, so is g m-n meaning g m ⊆ g n . Thus, we have that f 2 is g m , h-compatible.

Also, f 1 , f 2 being g-compatible, they are also g m -compatible. Therefore, by Lemma 12,

f 1 • f 2 and f 2 • f 1 are g m , h-compatible.
Compatibility with can also be combined with compatibility up to (see Definition 5):

Definition 18. f is g, h-compatible up to f ′ when f ′ is expansive and f • (g ∩ h) ⊆ h • f ′ • f . Lemma 19. If f ′ is idempotent, monotone, g, h-compatible and f is g-compatible up to f ′ , g, h-compatible up to f ′ , then f ′ • f is g, h-compatible. Proof. f • (g ∩ h) = f • (g ∩ g ∩ h) ⊆ f • g ∩ f • (g ∩ h) by Lemma 11 ⊆ g • f ′ • f ∩ h • f ′ • f = (g ∩ h) • f ′ • f f ′ • f • (g ∩ h) ⊆ f ′ • (g ∩ h) • f ′ • f f ′ is monotone ⊆ h • f ′ • f ′ • f f ′ is g, h-compatible ⊆ h • f ′ • f f ′ is idempotent Lemma 20. If f ′ is idempotent, g, h-compatible up to f ′′ and f is g, h-compatible up to f ′ , then f ′ • f is g, h-compatible up to f ′′ . Proof. f • (g ∩ h) = f • (g ∩ g ∩ h) ⊆ f • g ∩ f • (g ∩ h) by Lemma 11 ⊆ g • f ′ • f ∩ h • f ′ • f = (g ∩ h) • f ′ • f f ′ • f • (g ∩ h) ⊆ f ′ • (g ∩ h) • f ′ • f f ′ is monotone ⊆ h • f ′′ • f ′ • f ′ • f f ′ is g, h-compatible up to f ′′ ⊆ h • f ′′ • f ′ • f f ′ is idempotent 2 Up-to context in the π-calculus
We apply the theory developed above to the π-calculus. We recall the syntax and operational semantics of the π-calculus in Figure 1 (symmetric transitions have omitted).

We now show the usage of compatibility with to prove the soundness of the up-to context techniques in subcalculi of the π-calculus. For that, we first show it using our framework on non-input contexts (Section 2.1). This result is not new, but along the way, we prove that these up-to context techniques are b α which will be required to compose it later on. Then, in Section 2.2, we isolate the key property (Definition 26) that is needed to have the soundness of up-to substitution, and show how it gives the soundness result for upto context. We end by giving two subcalculi, the Asynchronous π-calculus and a π-calculus with immediately available names, where this property holds, thus proving up-to context technique can be used for these calculi.

Up-to behavioural relations and evaluation contexts

We briefly recall the simulation (s, . . .

--→ P Rep !G | G µ -→ G ′ !G µ -→ G ′ Open P a b --→ P ′ (νb)P a(b) --→ P ′ if b = a Sum G µ -→ G ′ G + G ′′ µ -→ G ′ Res P µ -→ P ′ (νa)P µ -→ (νa)P if a / ∈ n(µ) Par P µ -→ P ′ P | Q µ -→ P ′ | Q if bn(µ) ∩ fn(Q) = ∅ Comm P a(b) --→ P ′ Q a b --→ Q ′ P | Q τ -→ P ′ | Q ′ Close P a(b) --→ P ′ Q a(b) --→ Q ′ P | Q τ -→ (νb)(P ′ | Q ′ ) if b / ∈ fn(P )
Q ′ s.t Q µ -→ Q ′ , P ′ R Q ′ } s α (R) def = {(P, Q) for all α, P ′ , P α -→ P ′ implies there exists Q ′ s.t Q α -→ Q ′ , P ′ R Q ′ } s τ (R) def = {(P, Q) for all P ′ , P τ -→ P ′ implies there exists Q ′ s.t Q τ -→ Q ′ , P ′ R Q ′ } b(R) def = s(R) ∩ s(R -1 ) b α (R) def = s α (R) ∩ s α (R -1 ) b τ (R) def = s τ (R) ∩ s τ (R -1 ) b def = id ∩ b b α (R) def = id ∩ b α b τ (R) def = id ∩ b τ
The variant b(R) only contains pairs that are also in R, thus corresponding to the notion of respectfulness. As all the up-to techniques we use are monotone, any results for b α (resp. b τ ) also holds with their variant b α (resp. b τ ). We note ∼ for the bisimilarity.

Remark 21.

• b = b α ∩ b τ , b = b α ∩ b τ .
• All functions are monotone.

• b, b α , b τ ⊆ id

We will now define some up-to techniques corresponding to evaluation contexts and prove their b αcompatibility and b τ -compatibility with b α .

F S (R) = SRS -1 refl(R) = {(P, P )} Lemma 22. F ∼ , id, refl are b α -compatible.
Evaluation contexts contain parallel composition and restriction.

res(R) = {((ν a)P, (ν a)Q) P R Q} pcomp(R) = {(P | P ′ , Q | Q ′ ) P R Q, P ′ R Q ′ } Lemma 23. res, pcomp are b α -compatible.
By direct application of Lemma 10 and existing results of b-compatibility [START_REF] Madiot | Bisimulations up-to: Beyond first-order transition systems[END_REF], we have that F ∼ , id, refl, res are b α , bτ -compatible, and pcomp is b α , bτ -compatible up to res.

Thus, we are able to take the union and compose while remaining sound according to Theorem 15.

Corollary 24.

(id ∪ F ∼ ∪ refl ∪ res ∪ pcomp) ω is b-sound.
Corollary 24 is not new, but we obtain it via b α -compatibility and b α , bτ -compatibility instead of bcompatibility. This is used below to compose those techniques with up-to substitution which is not bcompatible.

In fact, we can already go further and add the remaining non-input contexts.

tau(R) = {(τ . P, τ . Q) P R Q} out(R) = {(a b . P, a b . Q) P R Q} sum(R) = {(G 1 + G 2 , G ′ 1 + G ′ 2 ) G 1 R G ′ 1 , G 2 R G ′ 2 } rep(R) = {(!G, !G ′ ) G R G ′ } Lemma 25.
• tau, is b α -compatible.

• id ∪ out is bα -compatible.

• id ∪ sum is b α -compatible.

Up-to substitution and input for subcalculi of π

Next up, we can add substitution and input related contexts. The substitution makes use of Theorem 15 with m > 1.

The proof requires an additional property that is not true in general in the π-calculus.

Definition 26 (Aliased Communication Property). We say that a set of processes P satisfies the aliased communication property if for all processes P in P, we have the following properties:

• P a b --→ c b --→ P ′ implies P σ τ -→ P ′ σ for all σ s.t. aσ = cσ. • P a(b) --→ c b --→ P ′ implies P σ τ -→ (νb)P ′ σ for all σ s.t. aσ = cσ.
This property is for instance satisfied in the asynchronous π-calculus and used to show that bisimilarity on asynchronous π-terms is closed by substitution.

sub(R) = {(P σ, Qσ) P R Q} inp(R) = {(a(b). P, a(b). Q) P R Q} Lemma 27. • sub is b α -compatible and b 2 α , b τ -compatible up to F ≡ • res. • inp is b α , b τ -compatible.
• id ∪ inp is b α -compatible up to sub. Proof.

• We rely on [START_REF] Sangiorgi | The Pi-Calculus -a theory of mobile processes[END_REF]Lemma 1.4.13]:

Lemma 28.

′ -→ Q ′ , then Qσ α ′ σ --→ Q ′ σ.
Take a relation R and some processes P, Q such that (P, Q) ∈ b 2 α ∩ b τ (R). We want to show that for all σ, (P σ, Qσ)

∈ b τ • F ≡ • res • sub(R).
Take P σ τ -→ P ′ , by the lemma, we have three cases:

1. either P τ -→ P ′′ for some P ′′ with P ′′ σ = P ′ . Then, as (P,

Q) ∈ b τ (R), Q τ -→ Q ′ and P ′ R Q ′ . Thus, Qσ τ -→ Q ′ σ and (P ′ σ, Q ′ σ) ∈ sub(R) ⊆ F ≡ • res • sub(R). 2. or P a b --→ c b
--→ P ′′ for some P ′′ , a, b, c with aσ = cσ and P ′′ σ ≡ P ′ . Then, as (P, --→ P ′′ for some P ′′ , a, b, c with aσ = cσ and (νb)P ′′ σ ≡ P ′ . Then, as (P,

Q) ∈ b 2 α (R), Q a b --→ c b --→ Q ′′ and P ′′ R Q ′′ . Thus, by the Aliased Communication Property Qσ τ -→ Q ′′ σ and (P ′ , Q ′′ σ) ∈ F ≡ • sub(R) ⊆ F ≡ • res • sub(R).
Q) ∈ b 2 α (R), Q a(b) --→ c b --→ Q ′′ and P ′′ R Q ′′ . Thus, by the Aliased Communication Property Qσ τ -→ (νb)Q ′′ σ and (P ′ , Q ′′ σ) ∈ F ≡ • res • sub(R).
• Trivial (no transition)

• We can prove inp ⊆ b α • sub.
Then,

(id ∪ inp) • b α = b α ∪ inp • b α ⊆ b α ∪ b α • sub • b α ⊆ b α • (id ∪ sub • b α ) by Lemma 11 Thus we know that id ∪ sub • b α ⊆ sub ⊆ sub • (id ∪ inp).
On the other hand, we have (id

∪ inp) • b α ⊆ (id ∪ inp) ⊆ sub • (id ∪ inp). So (id ∪ inp) • b α ⊆ b α • sub • (id ∪ inp) ∩ sub • (id ∪ inp) = b α • sub • (id ∪ inp).
The property defined in Definition 26 is only used to show that sub is b 2 α , b τ -compatible up to res. However, because the compatibility of inp is shown up to sub, the soundness of the corresponding technique relies on the compatibility result for sub.

Finally, if we aggregate all the results:

Theorem 29. If the aliased communication property holds, then (

F ∼ ∪ id ∪ refl ∪ sub ∪ res ∪ pcomp ∪ sum ∪ rep ∪ tau ∪ out ∪ inp) ω is b-sound.

Subcalculi satisfying the aliased communication property

We present two subcalculi satisfying the aliased communication property. The property does not hold in general because of processes like a. b. Thus, we look at Aπ, where outputs cannot guard processes, and processes with immediately available names, where dually inputs cannot be guarded.

Asynchronous π-calculus. The asynchronous π-calculus is defined by imposing that outputs no longer guard a process, meaning that there are forbidden in sums and in a b . P , we have P = 0.

Lemma 30. Aπ satisfies the Aliased Communication Property.

Proof. This is the direct application of Lemma 5.3.2 (3) and ( 4) in [START_REF] Sangiorgi | The Pi-Calculus -a theory of mobile processes[END_REF].

Immediately available names. Immediately available names may only be used in input as soon as the name is created. This is a weaker notion than linear receptiveness or uniform receptiveness [START_REF] Sangiorgi | The name discipline of uniform receptiveness[END_REF] which impose that exactly one input (resp. replicated input) must be accessible. This discipline is formalised by the following typing rules where Γ is the set of name that can be used as input.

∅ ⊢ P Γ ⊢ τ . P, a b . P Γ ⊢ G Γ ⊢ !G ∅ ⊢ P a ∈ Γ Γ ⊢ a(b). P Γ, a ⊢ P Γ ⊢ (νa)P Γ ⊢ P Γ ⊢ Q Γ ⊢ P | Q ∅ ⊢ G ∅ ⊢ G ′ ∅ ⊢ G + G ′
Note that because of the typing rule for sum, inputs are forbidden in sums. Typable processes form a subcalculus of the π-calculus. Indeed, the set of typable processes is closed by transitions as expressed by the lemma below.

Lemma 31 (Subject Reduction). If Γ ⊢ P and P µ -→ P ′ , then Γ ∪ bn(µ) ⊢ P ′ . Lemma 32. The set of typable processes satisfies the Aliased Communication Property.

As a consequence of Theorem 29, the up-to context techniques is sound for both subcalculi.

The weak case

We show how these results can be adapted to the weak case. The weak arrows are defined as usual:

= ⇒ def = τ -→ * , α = ⇒ def = = ⇒ α -→= ⇒, α = ⇒ def = α = ⇒ and τ = ⇒ def = = ⇒.
We define the simulation functions for the weak case ws, ws α , ws τ , the corresponding bisimulation functions wb, wb α , wb τ , and their variant wb, wb α , wb τ follow as expected.

ws(R) def = {(P, Q) for all µ, P ′ , P µ -→ P ′ implies there exists Q ′ s.t Q µ = ⇒ Q ′ , P ′ R Q ′ } ws α (R) def = {(P, Q) for all α, P ′ , P α -→ P ′ implies there exists Q ′ s.t Q α = ⇒ Q ′ , P ′ R Q ′ } ws τ (R) def = {(P, Q) for all P ′ , P τ -→ P ′ implies there exists Q ′ s.t Q = ⇒ Q ′ , P ′ R Q ′ } Weak bisimilarity is noted ≈ a .
Most results true in the strong case also hold in the weak case. We give details about those whose statement or proof need to be adapted.

First, it is known that up-to weak bisimilarity is not a sound technique. However, we can still use up-to strong bisimilarity but also use the expansion preorder . Take s ′ (R) def = {(P, Q) for all µ, P ′ , P µ -→ P ′ implies there exists

Q ′ s.t Q µ -→ Q ′ , P ′ R Q ′ } where α -→ def = α -→ and τ -→ def = τ -→ = . Then is the largest relation R such that R ⊆ ws(R) ∩ s ′ (R -1 ). Thus, if P a b = == ⇒ c b = == ⇒ P ′ , then P ⇒ a b --→ c b
--→⇒ P ′ so Aπ satisfies the weak aliased communication property.

For immediately available names, the reasoning is reversed. As inputs are immediately available, we can show that they can be preponed: --→⇒ P ′ and so we can conclude.

We can notice a symmetry between Lemmas 37 and 38, the former delays outputs while the latter anticipates inputs.

This shows that both calculi are also a congruence for the weak bisimilarity and that the up-to context technique is sound.

Remark 39. Note that, in the (weak) aliased communication property, we quantify over all names a, c, being the subject of the output and input respectively. If we impose that the property holds for only some names, for instance if only a subset of names are asynchronous or immediately available, then we have the soundness of up-to substitution restricted to those asynchronous names (resp. immediately available names), and up-to input that only carry asynchronous names (resp. immediately available names).

A Language with lookahead

Theorem 15 is sufficient to derive result with operator enabling after more than transitions.

Consider the language P ::= op(P ) a. P 0 with the following semantic a. P It is known that up-to-bisimilarity-and-context is unsound. However, one could tweak this language to make it sound.

A.1 Two prefixes

A first way would be to consider two prefixes a and b, with a. P

A.2 Second version

First, we need to show a variation of Theorem 15. Here, we add the condition that g ⊆ id along with a rather technical condition (⋆) (stated to remain as general as possible), which is always verified in practice (for instance, if f is expansive). It allows us to show the soundness when we add an extra h in front of g m , i.e in the case where f is h • g m , h-compatible.

Theorem 40. If f, g, h are monotone, g ⊆ id and f is both g-compatible, h • g m , h-compatible (with m ≥ 1) and verifies the following: ∀n ∈ N, f n • (

i≤mn+1 f i ) ⊆ i∈N f i (⋆) then f is g ∩ h-sound via f ω . Proof. Note that if R ⊆ ((g ∩ h) • f )(R) then R ⊆ (g • f )(R) and R ⊆ (h • f )(R).
We already know that f is g-sound via f ω so f ω (R) ⊆ g(f ω (R)). As g ⊆ id, we have h • g m ∩ h = h • g m . Thus f • h • g m ⊆ h • f . Now, let's show that the same inclusion holds for h. We prove by induction on n that

f n • h • g mn ⊆ h • f n (△)
For n = 0, trivial. For n ≥ 0,

f n+1 • (h • g m(n+1) ) = f n • f • h • g m • g mn ⊆ f n • h • f • g mn f is g m , h-compatible ⊆ f n • h • g mn • f f is g-compatible ⊆ h • f n • f by induction
By monotonicity of g and f , we can prove by a simple induction that R ⊆ (g • f )(R) implies R ⊆ (g • f ) mn (R). Then, we have R ⊆ (h

• f )(R) ⊆ (h • f ) • (g • f ) mn (R)
So R ⊆ (h • g mn • f mn+1 )(R) by compatibility and if we note f ω mn+1 = i≤mn+1 f i , as g and h are monotone we have R ⊆ (h • g mn • f ω mn+1 )(R).

Proof.

  First, by definition (c, 0) ∈ ⊤ so (a. c, a), (c | c, c) ∈ b(⊤). Therefore, (a. c | c, a | c) ∈ b 2 (⊤). Then, we have (a. a | a){ a /c} = a. a | a τ -→ a and the only transition that the second process can do is (a | a){ a /c} = a | a τ -→ 0. Thus, as a a -→ but 0 a -→, we have (a, 0) / ∈ b(⊤), meaning that (a. a | a, a | a) / ∈ b 2 (⊤).

  ) and bisimulation functions b, b and introduce their weaker versions b α , b τ that only impose conditions on visible and internal actions respectively. P, Q ::= !G P | Q (νa)P G G, G ′ ::= 0 a b . P a(b). P τ . P G + G ′ α ::= a b a(b) a b µ ::= α τ Inp a(b). P a c --→ P { c /b} Out a b . P a b

Figure 1 :

 1 Figure 1: Syntax and Early Labelled Transition System of the full π-calculus

•

  id ∪ rep is b α -compatible up-to pcomp ∪ id Similarly, using Lemma 10 and existing results, we have that id ∪ tau and id ∪ out are b α , b τ -compatible, id ∪ sum is b α , b τ -compatible and id ∪ rep is b α , b τ -compatible up to pcomp ∪ id.

1 .

 1 If P σ α -→ P ′ then P α ′-→ P ′′ for some α ′ , P ′′ with α ′ σ = α and P ′′ σ = P ′ . 2. If P σ τ -→ P ′ then (a) P τ -→ P ′′ for some P ′′ with P ′′ σ = P ′ , or (P ′′ for some P ′′ , a, b, c with aσ = cσ and P ′′ σ ≡ P ′ , or (P ′′ for some P ′′ , a, b, c with aσ = cσ and (νb)P ′′ σ ≡ P ′ . b α -compatibility follows from the lemma and that if Q

  α

  ′ . One could define simply b a , b b two bisimulations using actions a and b respectively. Then with op 1 (R) = {(op 1 (P ), op 1 (Q)) P R Q}, we have op 1 • b b ⊆ b b • op 1 (there cannot be any b transition) and op 1 • (b 2 b ∩ b a ) ⊆ b a • op 1 .

Lemma 33. F ∼ , F is wb α -compatible.

The aliased communication property needs also to be changed to use weak arrows so that the proof of substitution goes without trouble.

Definition 34 (Weak Aliased Communication Property). We say that a set of processes P satisfies the weak aliased communication property if for all process P in P, we have the following properties:

= == ⇒ P ′ implies P σ = ⇒ P ′ σ for all σ s.t. σ(a) = σ(c). Weak bisimilarity is not a congruence for sum. Indeed, we have τ . a ≈ a a but τ . a+b ≈ a a+b. Congruence is usually recovered by considering non-degenerate contexts, that is, contexts where the hole is not directly under a sum operator. Therefore, we want to prove the soundness of up-to non-degenerate contexts, and thus we use the up-to guarded sum technique instead of the previous up-to sum technique:

Proof. We prove

The proof for replication needs also to be changed. We show instead that id ∪ rep is b α , b τ -compatible up to F ∼ • (pcomp ∪ id). Intuitvely, the problem is similar to the case of the sum, but because we have the law !G ∼ !G | G, it does not break soundness.

The other proofs can be carried out without any modification, and we can then conclude with the soundness of the whole up-to technique:

We can now show that both subcalculi also satisfy the weak aliased communication property. In asynchronous π-calculus, we know that outputs, being asynchronous, may always be postponed as expressed below.

In the end, we have

This

We may also take P This approach may require a bit more redundancy to prove compatibility results, but it does make statements easier to read. Here, we aim to decompose b using more than 2 functions, so we will use this incremental approach.

This Lemma allows us to chain Lemma 10 into one compatible function. For simplicity, we will assume f is expansive.

and f is g 1 -compatible and for some (m i ) i≤n with m i ≥ 1, g mi i , g i+1 -compatible, then f is g n -sound via f ω . Proof. First, we define h i with h 1 = g 1 and h i+1 = g i+1 ∩ h mi i . We show by induction on i that f is h i -compatible.

When i = 1, this is true by assumption. For i = 2, as f is h 1 -compatible, it is also h m1 1 -compatible. Thus, by Lemma 10, f is g 2 ∩ h m1 1 -compatible. For i ≥ 2, as f is g i ∩h mi-1 i-1 -compatible and g mi i , g i+1 -compatible, by Lemma 41, f is g i+1 ∩(g i ∩h mi-1 i-1 ) micompatible.

Next, we will show by induction that R ⊆

A.4 Unverified approaches

Having integers n ≥ 1 as prefixes, i.e n. P


Beatrice R Bricchi

Luca Mascaretti

Simona Garattoni

Matteo Mazza

Matteo Ghidelli

Alberto Naldoni
email: alberto.naldoni@upol.cz

Andrea Li Bassi
email: andrea.libassi@polimi.it

Nanoporous titanium (oxy)nitride films as broadband solar absorbers

Broadband absorption of solar light is a key aspect in many applications that involve an efficient conversion of solar energy to heat. Titanium nitride (TiN)-based materials, in form of periodic arrays of nanostructures or multilayers, can promote significant heat generation upon illumination thanks to their efficient light absorption and refractory character. In this work, pulsed laser deposition was chosen as a synthesis technique to tune the properties of TiN films from metallic bulk-like TiN to oxynitride hierarchical films featuring a nanoparticle-assembled tree-like morphology. The nanoporous hierarchical films exhibit strong broadband solar absorption (~ 90% from the UV to the near infrared range) and could generate temperatures of ~475 °C under moderate light concentration (17 Suns). The high heat generation achieved by the tree-like films is ascribed to their peculiar morphology, structure and composition, which is favorable for light trapping and heat dissipation . These properties pave the way for the implementation of such films as solar absorber structures.

Introduction

The abundant and widespread availability of solar energy makes it a potential source for various sustainable energy conversion technologies. Solar-thermal conversion, for example, aims at transforming photons coming from the Sun into heat that could be used for residential, commercial or industrial applications. [START_REF] Weinstein | Concentrating Solar Power[END_REF] An ideal solar absorber material should exhibit a unitary and omnidirectional absorption in the 250-2500 nm range of the electromagnetic spectrum, which can be achieved, for example, by metal-insulator-metal (MIM) multilayers. [START_REF] Deng | Broadband Perfect Absorber Based on One Ultrathin Layer of Refractory Metal[END_REF][START_REF] Kajtár | Theoretical Model of Homogeneous Metal-Insulator-Metal Perfect Multi-Band Absorbers for the Visible Spectrum[END_REF] Additionally, broadband absorbers must convert light energy to heat without undergoing thermal degradation, which typically requires the use of refractory metals and oxides [START_REF] Deng | Broadband Perfect Absorber Based on One Ultrathin Layer of Refractory Metal[END_REF][START_REF] Chirumamilla | Multilayer Tungsten-Alumina-Based Broadband Light Absorbers for High-Temperature Applications[END_REF] or carbon-based components. [START_REF] Lenert | A Nanophotonic Solar Thermophotovoltaic Device[END_REF] Titanium nitride (TiN) is a well-known refractory material employed in the complementary metal oxide semiconductor (CMOS) technology. [START_REF] Patsalas | Conductive Nitrides: Growth Principles, Optical and Electronic Properties, and Their Perspectives in Photonics and Plasmonics[END_REF] TiN in the form of nanoparticles (NPs) or nanostructures exhibits similar plasmonic properties as Au, but it offers several advantages compared to the latter, including the compatibility with semiconductor technology, [START_REF] Patsalas | Conductive Nitrides: Growth Principles, Optical and Electronic Properties, and Their Perspectives in Photonics and Plasmonics[END_REF] high thermal stability, [START_REF] Li | Refractory Plasmonics with Titanium Nitride: Broadband Metamaterial Absorber[END_REF][START_REF] Krekeler | Unprecedented Thermal Stability of Plasmonic Titanium Nitride Films up to 1400 °C[END_REF] and tunability of its plasmonic resonance by controlling its stoichiometry or crystalline quality. [START_REF] Patsalas | Conductive Nitrides: Growth Principles, Optical and Electronic Properties, and Their Perspectives in Photonics and Plasmonics[END_REF] A further advantage of TiN is that its absorption peak can be extended towards the near-infrared (NIR) range and its optical losses can lead to higher photothermal heating compared to Au. 9,10 Therefore, it is not surprising that this material has been considered in thermoplasmonics, [START_REF] Baffou | Thermo-Plasmonics: Using Metallic Nanostructures as Nano-Sources of Heat[END_REF][START_REF] Baffou | Applications and Challenges of Thermoplasmonics[END_REF] in which light-to-heat conversion mediated by plasmonic structures is exploited for a wide range of applications, [START_REF] Naldoni | Applying Plasmonics to a Sustainable Future[END_REF] including solar steam generation, [START_REF] Neumann | Solar Vapor Generation Enabled by Nanoparticles[END_REF] optical trapping [START_REF] Ndukaife | Long-Range and Rapid Transport of Individual Nano-Objects by a Hybrid Electrothermoplasmonic Nanotweezer[END_REF] and photothermal catalysis. [START_REF] Naldoni | Solar Thermoplasmonic Nanofurnace for High-Temperature Heterogeneous Catalysis[END_REF][START_REF] Zhou | Light-Driven Methane Dry Reforming with Single Atomic Site Antenna-Reactor Plasmonic Photocatalysts[END_REF] Broadband solar absorbers based on TiN can be realized by coupling multiple TiN nanostructures in different arrangements. For example, ordered arrays or periodic metamaterials made of nm-sized units, such as nanocavities, [START_REF] Naldoni | Solar Thermoplasmonic Nanofurnace for High-Temperature Heterogeneous Catalysis[END_REF][START_REF] Mascaretti | Solar Steam Generation on Scalable Ultrathin Thermoplasmonic TiN Nanocavity Arrays[END_REF] nanotubes [START_REF] Ishii | Extreme Thermal Anisotropy in High-Aspect-Ratio Titanium Nitride Nanostructures for Efficient Photothermal Heating[END_REF] and hollow squares, [START_REF] Li | Refractory Plasmonics with Titanium Nitride: Broadband Metamaterial Absorber[END_REF] as well as non-periodic structures, such as nanodonuts [START_REF] Le | Titanium Nitride Nanodonuts Synthesized from Natural Ilmenite Ore as a Novel and Efficient Thermoplasmonic Material[END_REF] and nanopillars [START_REF] Chirumamilla | Large-Area Ultrabroadband Absorber for Solar Thermophotovoltaics Based on 3D Titanium Nitride Nanopillars[END_REF] , have been reported. These approaches can prevent thermal stresses in MIM multilayers due to the different thermal expansion coefficients in the materials. Apart from optimizing the thermal stability of the solar absorber, the achievement of a full absorption with a simple fabrication method is also challenging. [START_REF] Chirumamilla | Large-Area Ultrabroadband Absorber for Solar Thermophotovoltaics Based on 3D Titanium Nitride Nanopillars[END_REF] A simple approach in this regard is represented by porous films made of NP assemblies grown on a flat substrate, in which broadband absorption can be achieved as a result of coupling among individual localized surface plasmon resonances (LSPRs) of the NPs and lighttrapping effects (i.e., multiple reflections/scattering) promoted by the nm-scale porosity. [START_REF] Yan | Plasmonic Near-Touching Titanium Oxide Nanoparticles to Realize Solar Energy Harvesting and Effective Local Heating[END_REF][START_REF] Li | Solution-Processed All-Ceramic Plasmonic Metamaterials for Efficient Solar-Thermal Conversion over 100-727[END_REF] Compared to standard NP-assembled materials, TiN thin films can be synthesized by vacuum-phase methods including magnetron sputtering, [START_REF] Patsalas | Electronic, and Transport Properties of Nanocrystalline Titanium Nitride Thin Films[END_REF][START_REF] Reddy | Temperature-Dependent Optical Properties of Plasmonic Titanium Nitride Thin Films[END_REF][START_REF] Mascaretti | Controlling the Plasmonic Properties of Titanium Nitride Thin Films by Radiofrequency Substrate Biasing in Magnetron Sputtering[END_REF] glancing angle deposition, [START_REF] Khan | Gold-Nanoparticle-Decorated Titanium Nitride Electrodes Prepared by Glancing-Angle Deposition for Sensing Applications[END_REF][START_REF] Jen | Obliquely Deposited Titanium Nitride Nanorod Arrays as Surface-Enhanced Raman Scattering Substrates[END_REF] atomic layer deposition (ALD), [START_REF] Langereis | Situ Spectroscopic Ellipsometry Study on the Growth of Ultrathin TiN Films by Plasma-Assisted Atomic Layer Deposition[END_REF][START_REF] Van Bui | On the Difference between Optically and Electrically Determined Resistivity of Ultra-Thin Titanium Nitride Films[END_REF] and pulsed laser deposition (PLD). [START_REF] Sugavaneshwar | Fabrication of Highly Metallic TiN Films by Pulsed Laser Deposition Method for Plasmonic Applications[END_REF][START_REF] Perego | Hierarchical TiN Nanostructured Thin Film Electrode for Highly Stable PEM Fuel Cells[END_REF] The latter, in particular, allows depositing virtually any material with high tunability of morphology and structure: compact layers, [START_REF] Sugavaneshwar | Fabrication of Highly Metallic TiN Films by Pulsed Laser Deposition Method for Plasmonic Applications[END_REF][START_REF] Kölbach | Elucidating the Pulsed Laser Deposition Process of BiVO4 Photoelectrodes for Solar Water Splitting[END_REF] surface-supported NPs, [START_REF] Misra | Tailorable Au Nanoparticles Embedded in Epitaxial TiO2 Thin Films for Tunable Optical Properties[END_REF][START_REF] Ghidelli | Engineering Plasmonic Nanostructured Surfaces by Pulsed Laser Deposition[END_REF] hierarchical tree-like films, [START_REF] Perego | Hierarchical TiN Nanostructured Thin Film Electrode for Highly Stable PEM Fuel Cells[END_REF][START_REF] Mahjouri-Samani | Nonequilibrium Synthesis of TiO2 Nanoparticle "Building Blocks" for Crystal Growth by Sequential Attachment in Pulsed Laser Deposition[END_REF][START_REF] Noh | Aligned Photoelectrodes with Large Surface Area Prepared by Pulsed Laser Deposition[END_REF] and ultra-porous foams [START_REF] Mascaretti | Syngas Evolution from CO2 Electroreduction by Porous Au Nanostructures[END_REF][START_REF] Maffini | Growth Dynamics of Pulsed Laser Deposited Nanofoams[END_REF] can be obtained without the need of substrate heating. The latter is an important feature in terms of process energy utilization and usage of flexible substrates. In this work, we exploit the versatility of PLD to deposit TiN/TiOxNy nanostructured films in controlled atmosphere (from vacuum up to 100 Pa of N2/H2) to tune the morphology, structure and optical absorption, with the aim to realize broadband solar absorbers. Tree-like films deposited at 50 and 100 Pa, in particular, exhibited the highest optical absorption over the entire UV-visible-NIR range.

All the investigated films were tested by non-contact thermal measurements under moderate solar irradiation. We show thatthe TiOxNy film deposited at 100 Pa produced the highest temperature of ~ 475 °C under 17 Suns, which was ascribed to its broadband optical absorption, unlike metallic-like TiN films deposited at pressures lower than 20 Pa.Our results open the way to the utilization of TiN-based broadband solar absorbers with controlled functional properties fabricated by PLD in solar-thermal devices.

Experimental methods

Samples preparation

PLD was performed in a vacuum chamber equipped with mass flow controllers to tune the partial gas pressure. Ablation was performed with a ns-pulsed laser (Nd:YAG, 2 nd harmonic, λ =532 nm) with pulse duration in the 5-7 ns range and repetition rate of 10 Hz. The laser pulses were focused on the target material through a viewport and the target of stoichiometric TiN (99.9% purity, Mateck Gmbh) was mounted on a roto-translational manipulator ensuring a uniform ablation. The laser fluence on the target was set at 6.5 J cm -2 . Depositions were performed at room temperature and in vacuum (~10 -3 Pa) or N2/H2 (95/5%) background gas mixture at the overall pressure equal to10, 20, 50, and 100 Pa. Si (100), soda-lime glass and Ti plate substrates were cleaned in ultrasonic bath with isopropanol and mounted on a rotating sample holder placed head-on the target at a fixed distance of 50 mm. The deposition time was set at 2 hours.

Material characterization

The thin films morphology was evaluated by means of a field emission scanning electron microscopy (SEM, Zeiss SUPRA 40) on samples grown on silicon. The SEM microscope is equipped with an Oxford Instruments Si(Li) detector for energy-dispersive X-ray spectroscopy (EDX), which was employed to estimate the atomic percentage (at.%) of Ti, N and O in the films, by employing an accelerating voltage of 10 kV. The chemical composition of selected films was further analyzed by EDX mapping in a highresolution transmission electron microscope (HRTEM, Titan G2) operated in scanning TEM mode (STEM) using a Super-X system with four silicon drift detectors (Bruker). STEM images were taken with a high-angle annular dark-field imaging (HAADF) detector (Fischione, model 3000). TEM lamellae were prepared with a FEI Helios Focused Ion Beam/SEM (Thermo Scientific). X-ray photoelectron spectroscopy (XPS) analysis was performed by a PHI 5000 VersaProbe II XPS system (Physical Electronics) with a monochromatized Al Kα source (15 kV, 50 W) and a photon energy of 1486.7 eV.

The crystalline structure of the films was investigated by X-ray diffraction (XRD) using a high-resolution X-ray powder diffractometer (PANalytical X'Pert Pro MPD) with Co Kα radiation (λ = 0.1789 nm). The measurements were performed in Bragg-Brentano geometry in a 2θ range of 22°-100° with a step size of 0.033°. Further qualitative information about stoichiometry/composition of the films was gained by Raman spectroscopy using a Renishaw InVia micro-Raman spectrometer equipped with a diode-pumped solid-state laser (λ = 660 nm, incident power on the sample of 0.94 mW, spectral resolution ~ 3 cm -1 ).

The optical characterization of the films in the spectral range 250-2000 nm was evaluated by transmittance (T) and reflectance (R) spectra on samples deposited on soda-lime glass using a PerkinElmer Lambda 1050 spectrophotometer equipped with an integrating sphere (150 mm diameter).

In the spectral range 1330-25000 nm, transmittance and reflectance spectra on samples deposited on silicon were acquired by a vacuum Fourier-transform infrared (FTIR) Vertex 80v spectrophotometer.

Infrared thermal imaging

The heat generation produced by the films under irradiation was evaluated by measuring the temperature of the Ti substrate (thickness 0.125 mm) by a FLIR X6580sc infrared camera. The thermal camera was placed on the backside of the samples that were irradiated from the front side with a 1000 W solar simulator (Sciencetech A4 Lightline C250) equipped with an AM 1.5G filter and an aspheric condenser lens (ACL25416U, Thorlabs). The samples were kept in a custom-made vacuum cell equipped with a sapphire viewport on the front side (420GSG040-saphir, Pfeiffer Vacuum GmbH) and a CaF2 viewport (VPCH512, Thorlabs) on the backside in a pure Ar atmosphere. The temperatures measured by the thermal camera were corrected by the spectral emissivity of the Ti plates measured by FTIR spectroscopy at room temperature and averaged in the sensitivity range of the thermal camera (2500-5500 nm).

Results and discussion

TiN films with different morphologies were synthetized by varying the background pressure during the deposition from vacuum up to 100 Pa of N2/H2 to favor Ti-N bonds formation during film growth. SEM cross-section and top-view micrographs are reported in Figure 1. The TiN film deposited in vacuum shows a compact and dense columnar structure (Figure 1a) and a smooth surface (Figure 1b). For the samples deposited at 10 and 20 Pa (Figures 1c-1f), the columns composing the film exhibit a slight deviation from a perfectly vertical growth direction with a consequent increase of porosity. A further increase of the background pressure during the deposition leads to the growth of a hierarchical nanoparticle assembly structure (Figures 1g and1h), clearly visible for the film deposited at 100 Pa (Figures 1i-1l). As a consequence of the increase of film porosity with background pressure during deposition, the thickness of the films for a deposition duration of 2 h increased from 2.4 µm for the vacuum-deposited film up to 3, 4.5, 6, and 8.5 µm for the ones deposited at 10, 20, 50, and 100 Pa, respectively.

A morphological transition from compact to nonporous films by increasing the background gas pressure is typical of the PLD process. [START_REF] Mahjouri-Samani | Nonequilibrium Synthesis of TiO2 Nanoparticle "Building Blocks" for Crystal Growth by Sequential Attachment in Pulsed Laser Deposition[END_REF][START_REF] Geohegan | Dynamics of Laser Ablation Plume Penetration through Low Pressure Background Gases[END_REF] Upon the interaction of the focused pulsed laser on the target material in a controlled atmosphere, i.e., laser ablation, a plasma plume is generated and expands from the target toward the substrate. In low-pressure conditions, the ablated species possess high kinetic energy and lead to a compact or bulk-like growth. In high-pressure conditions, the background gas molecules and the ablated species undergo collisions, which decrease the kinetic energy of the latter and lead to a clusterassembled growth regime.. [START_REF] Mascaretti | Syngas Evolution from CO2 Electroreduction by Porous Au Nanostructures[END_REF][START_REF] Maffini | Growth Dynamics of Pulsed Laser Deposited Nanofoams[END_REF] The peculiar hierarchical nanoparticle assembly (tree-like) morphology observed for the films deposited at 50 and 100 Pa (Figures 1g-1j) has been frequently observed for various oxides [START_REF] Mahjouri-Samani | Nonequilibrium Synthesis of TiO2 Nanoparticle "Building Blocks" for Crystal Growth by Sequential Attachment in Pulsed Laser Deposition[END_REF][START_REF] Noh | Aligned Photoelectrodes with Large Surface Area Prepared by Pulsed Laser Deposition[END_REF][START_REF] Gondoni | Enhancing Light Harvesting by Hierarchical Functionally Graded Transparent Conducting Al-Doped ZnO Nano-and Mesoarchitectures[END_REF] as well as for TiN. [START_REF] Perego | Hierarchical TiN Nanostructured Thin Film Electrode for Highly Stable PEM Fuel Cells[END_REF] In the latter, similarly to the present case, by increasing the background gas deposition pressure a morphological evolution featuring a porosity and thickness increase in parallel with a density decrease was observed. However, a transition from compact to treelike films was observed at 15-30 Pa, i.e., lower than in this case (50-100 Pa), which could be ascribed to the use of a different laser wavelength (i.e., λ = 280 nm) and a lower fluence (i.e., 3.4 J cm -2 ). The structural properties were investigated by XRD in Bragg-Brentano geometry, which is sensitive to the preferential orientation of crystalline domains along the film growth direction. The diffractograms for all the films exhibited the characteristic peaks of the cubic phase of TiN (Fm-3m space group), but a remarkable change of texture and shift of the diffraction angles were found by increasing the background pressure during deposition. The vacuum-deposited film exhibited the (111) and (222) reflections at lower diffraction angles than the corresponding counterparts in bulk TiN. On the contrary, the film deposited at 10 Pa exhibited also the (200), which was the most intense in this case, (220) and (311) reflections (the latter slightly visible), all of them at higher diffraction angles than in the case of bulk TiN. By further increasing the deposition background pressure, all the peaks decreased in intensity and shifted to higher angles, while an amorphous background emerged at P > 20 Pa (Figure 2a). The peak shift effect can be better highlighted by evaluating the lattice constant a using the Bragg's law for cubic crystal system from the (111) reflections (vacuum, 10 Pa and 20 Pa) and ( 200) reflections (10-100 Pa), as shown in Figure 2b. Taking as reference value for bulk TiN aTiN = 4.2380 Å, the vacuum-deposited film showed a > aTiN.

For all the other films, conversely, a < aTiN and the lattice constant decreased with the deposition pressure. Interestingly, for the films deposited at P > 20 Pa the lattice constant was lower than that of cubic titanium monoxide, γ-TiO (aTiO = 4.182 Å), which has the same rock-salt crystal structure as TiN.

Furthermore, the average domain size (τ) along the film growth direction was evaluated through the broadening of the diffraction peaks by using the Scherrer equation on the (111) reflection for the films deposited in vacuum and 10-20 Pa, and on the (200) reflection for films deposited at 10-100 Pa (Figure 2c). For the film deposited in vacuum, τ ~18 nm; this value increases reaching a maximum for the film deposited at 10 Pa (τ ~ 46 nm for the (111) reflection and τ ~32 nm for (200) reflection), and finally decreases with the background pressure down to τ ~5 nm for both the films deposited at 50 and 100 Pa.

XRD data could be interpreted in terms of the growth regimes induced by the increase of deposition background gas pressure. The features of the film deposited in vacuum (a > aTiN) are consistent with two effects: on the one hand, a nitrogen sub-stoichiometry induced by non-stoichiometric transfer from the target and, therefore, nitrogen loss; [START_REF] Schou | Physical Aspects of the Pulsed Laser Deposition Technique: The Stoichiometric Transfer of Material from Target to Film[END_REF][START_REF] Canulescu | Nonstoichiometric Transfer during Laser Ablation of Metal Alloys[END_REF] on the other hand, in-plane compressive stresses due to highly energetic particles having a peening effect on the growing film. [START_REF] Patsalas | Electronic, and Transport Properties of Nanocrystalline Titanium Nitride Thin Films[END_REF] The residual stresses indeed were retrieved by evaluating the macro-strain as εmacro = |a-aref|/aref = 0.015, that is consistent with a compact film with compressive stress deposited by physical vapor deposition. [START_REF] Mascaretti | Controlling the Plasmonic Properties of Titanium Nitride Thin Films by Radiofrequency Substrate Biasing in Magnetron Sputtering[END_REF][START_REF] Patsalas | Surface Kinetics and Subplantation Phenomena Affecting the Texture, Morphology, Stress, and Growth Evolution of Titanium Nitride Films[END_REF] By ablating the target in the presence of a N2/H2 background gas, instead, the species in the plume slowed down because of the collisions with the gas molecules. As a consequence, at 10 Pa a trade-off between in-plume cluster nucleation and sufficiently large kinetic energy of the ablated species probably promoted a good crystallization with small residual stresses (εmacro = 0.006) and N incorporation in the growing film. A further pressure increase in the chamber promoted a less directional ablation plume and a low kinetic energy of the clusters formed in the plume. Therefore, a progressive decrease of the average domain size in parallel with amorphization of the film are expected. [START_REF] Mahjouri-Samani | Nonequilibrium Synthesis of TiO2 Nanoparticle "Building Blocks" for Crystal Growth by Sequential Attachment in Pulsed Laser Deposition[END_REF][START_REF] Leichtweiss | Amorphous and Highly Nonstoichiometric Titania (TiOx) Thin Films Close to Metallike Conductivity[END_REF] Another effect coming into play was a partial oxidation of the films due to residual impurities in the background gas. Previous studies, indeed, reported a decrease of the lattice parameter (down to 4.16 Å) as well as of the average domain size (2.8 nm) by decreasing the flow rate of N2 and increasing that of O2 during magnetron sputtering experiments, thus producing titanium oxynitrides, i.e., TiOxNy. [START_REF] Trenczek-Zajac | Structural and Electrical Properties of Magnetron Sputtered Ti(ON) Thin Films: The Case of TiN Doped in Situ with Oxygen[END_REF] For such materials, a lower lattice parameter compared to both standard TiN and TiO materials could be explained by the presence of ion vacancies. The data presented in Figures 1 and2 show the key role of the N2/H2 deposition pressure in controlling not only the morphology, but also the structure and the composition of the films. Since the latter itself deeply affects the functional properties of titanium nitride-based materials, a further compositional characterization was addressed by different techniques (Figure 3). EDX microanalysis was carried out to estimate the atomic content (% at.) of Ti, N and O in the films deposited at different pressure (Figure 3a) and the nitrogen to titanium ratio (N/Ti) was also considered (Figure 3b) to discern among substoichiometric (N/Ti < 1), stoichiometric (N/Ti = 1) and over-stoichiometric (N/Ti > 1) films. For the film deposited in vacuum, the ratio N/Ti was ~ 0.4 (Ti ~ 47% at., N ~ 18% at.) with a high amount of oxygen (32% at.). For the film deposited at 10 Pa, oxygen was not detected and a condition of slight overstoichiometry was found (N/Ti ~ 1.1). The film deposited at 20 Pa was characterized by a substantial amount of oxygen (O ~ 33% at.) and a decrease of N/Ti to ~ 0.8. By further increasing the deposition pressure, the N/Ti ratio significantly decreased down to ~ 0.5 and the oxygen content increased up to ~ 50% at. It should be taken into account that EDX measurements were also performed on the TiN target, finding N/Ti = 0.8 and 13% at. O (Figure S1). Despite the intrinsic under-estimation of light atoms, such as nitrogen and oxygen, of the EDX technique, the compositional analysis confirmed the indirect observations from XRD (Figure 2). The relatively high oxygen content in the vacuum-deposited film and in the target could be simply ascribed to the formation of a native thin oxide surface layer due to air exposure. [START_REF] Milošev | Comparison of TiN, ZrN and CrN Hard Nitride Coatings: Electrochemical and Thermal Oxidation[END_REF][START_REF] Ernsberger | Low Temperature Oxidation Behavior of Reactively Sputtered TiN by X-ray Photoelectron Spectroscopy and Contact Resistance Measurements[END_REF] The same effect would also hold for the case of porous films; additionally, oxygen was likely included during the growth due to the deposition at high background gas pressures. More information in this regard was gained by Raman spectroscopy (Figure 3c). The vacuum-deposited film exhibited evident acoustic Raman bands (~ 200-300 cm -1 ) without optical bands, which was ascribed to nitrogen vacancies and, therefore, under-stoichiometry (see Note S1 for more details on the interpretation of Raman spectra in TiN). The Raman spectra for the films deposited at 10 and 20 Pa, instead, were consistent with nearly-stoichiometric TiN. By further increasing the deposition pressure, the Raman spectra became broader, which could be explained by an increasing degree of amorphization and oxidation due to residual oxygen molecules in the chamber saturating the nitrogen vacancies. [START_REF] Trenczek-Zajac | Structural and Electrical Properties of Magnetron Sputtered Ti(ON) Thin Films: The Case of TiN Doped in Situ with Oxygen[END_REF] In other words, the films deposited at 50 and 100 Pa were partially oxidized through the whole film thickness. To confirm this claim, further analysis was performed by STEM-EDX mapping on lamellae prepared from a film deposited at 100 Pa (Figures 3d-3g). The maps show that Ti, N and O are present all along the tree-like structure. The measured elemental composition was Ti = 43.4% at., N = 23.5% at., and O = 33.1% at., with N/Ti = 0.54 in agreement with EDX microanalysis. Further information on the composition of this film was gained by XPS. The spectra revealed a rather low nitrogen amount (~ 10% at.), while the binding energies of the peaks in the Ti 2p and O 1s regions were comparable to reduced titanium dioxide (see details in Figure S2 and Table S1). Therefore, the XPS data overall suggested an extensive surface oxidation for the film deposited at 100 Pa. It is possible to conclude that, while the films deposited at P ≤ 20 Pa are consistent with titanium nitride materials with different stoichiometries, i.e., TiNx, and limited (surface) oxidation, those deposited at P ≥ 50 Pa correspond to titanium oxynitride materials, i.e., TiOxNy. The morphological, structural and compositional evolution of the films with the increase of deposition pressure discussed above was accompanied by a change in the optical properties (Figure 4). Absorptance spectra were retrieved by transmittance (T(λ)) and reflectance (R(λ)) measurements (Figure S3) according to the formula ( ) 1 ( )

( ) A T R λ λ λ = - -
(see Note S2 for details on the measurements in the different wavelength ranges and on data treatment). Figures 4a and4b show the optical spectra in the ultravioletvisible-near infrared (UV-vis-NIR, i.e., 250-2000 nm) range compared to the standard solar irradiance and in the medium-infrared (MIR, i.e., 1330-25000 nm) range, respectively. The absorptance monotonically increased and the overall behavior of the films dramatically changed by increasing the deposition pressure. In particular, the film deposited in vacuum showed zero transmittance (Figures S3a andS3b) and a well-defined reflectance minimum (Figures S3c andS3d) or absorptance maximum in the UV region of the electromagnetic spectrum (i.e., at 318 nm, Figure 4a). This behavior is very similar to that of bulk TiN, which exhibits a well-defined interband transition threshold at ~ 500 nm and a low absorption/high reflectance due to intraband transitions at longer wavelengths. [START_REF] Patsalas | Conductive Nitrides: Growth Principles, Optical and Electronic Properties, and Their Perspectives in Photonics and Plasmonics[END_REF] For the films deposited at 10 and 20 Pa the absorption peak redshifted to 516 and 658 nm, respectively, and broadened, thus leading to a higher absorptance in the full UV-MIR range than the film deposited in vacuum.

Additionally, these films exhibited a non-zero transmittance in the MIR range (maximum ~ 10% at ~10 µm, Figure S3b). Since the absorption maximum (or the reflectance minimum) position is associated to the plasma frequency of the film, its redshift can be explained with a change of charge carrier density, which implies a change of stoichiometry or composition. [START_REF] Meng | Characterization of Titanium Nitride Films Prepared by d.c. Reactive Magnetron Sputtering at Different Nitrogen Pressures[END_REF][START_REF] Braic | Titanium Oxynitride Thin Films with Tunable Double Epsilon-Near-Zero Behavior for Nanophotonic Applications[END_REF] In this case, according to EDX analysis (Figures 2a-2b), a higher nitrogen or oxygen content contributed to the redshift of the absorption peak, thus decreasing the metallic character of the films. Moreover, the broadening of the absorption could be associated to a transition from a smooth film surface to a nanostructured one (Figure 1), which can promote light scattering and light trapping phenomena. This effect became more evident for the films deposited at 50 and 100 Pa, which exhibited a broadband absorption in the whole investigated range (Figures 4a and4b). These films exhibited a non-zero transmittance at λ > 530 nm (Figure S3a) which increased up to ~ 60% at MIR wavelengths, then abruptly decreased to zero (Figure S3b). As a result, the absorptance exhibited broad maxima in the visible (λ ~ 500 nm), in the NIR (λ ~ 1000 nm) and in the MIR (λ ~ 13000 nm) ranges of the electromagnetic spectrum, with an absolute minimum at ~ 10000 nm.

Various effects could explain such an optical behavior. The main contribution is likely due to oxygen incorporation in the films, thus featuring a TiOxNy composition, which is supported by EDX, Raman and XPS data (Figures 3 andS2). TiOxNy materials show indeed a non-zero transmittance [START_REF] Trenczek-Zajac | Structural and Electrical Properties of Magnetron Sputtered Ti(ON) Thin Films: The Case of TiN Doped in Situ with Oxygen[END_REF][START_REF] Quijorna | TiN Dielectric Contrasts Obtained by Ion Implantation of O2+; Structural, Optical and Electrical Properties[END_REF] and a more extended energy range for interband transitions involving the additional O 2p orbitals [START_REF] Yamaguchi | Black Titanium Oxynitride Thin Films Prepared by Nitrogen Plasma-Assisted Pulsed Laser Deposition for Flat-Panel Displays[END_REF] . On the other hand, broadband absorption was also shown for highly substoichiometric TiOx [START_REF] Yan | Plasmonic Near-Touching Titanium Oxide Nanoparticles to Realize Solar Energy Harvesting and Effective Local Heating[END_REF] and commercial TiN [START_REF] Li | Solution-Processed All-Ceramic Plasmonic Metamaterials for Efficient Solar-Thermal Conversion over 100-727[END_REF] NPs assemblies and for TiN nanopillars. [START_REF] Chirumamilla | Large-Area Ultrabroadband Absorber for Solar Thermophotovoltaics Based on 3D Titanium Nitride Nanopillars[END_REF] In all these cases, a superposition of individual localized surface plasmon resonances (LSPRs) of the individual units, i.e., plasmon hybridization, [START_REF] Prodan | A Hybridization Model for the Plasmon Response of Complex Nanostructures[END_REF][START_REF] Lassiter | Close Encounters between Two Nanoshells[END_REF] increased and broadened the overall absorption of the film. Yan et al. further showed that the porous nanostructure promoted light trapping due to multiple reflections and scattering of light as well as a reduced reflectance at the air-solid interface due to an effective graded refractive index layer. [START_REF] Yan | Plasmonic Near-Touching Titanium Oxide Nanoparticles to Realize Solar Energy Harvesting and Effective Local Heating[END_REF] Similarly, in the present case, an increase of light scattering ability of the films with the morphological transition from compact to porous was assessed by retrieving the haze factor, i.e., the ratio between the diffuse and total components of the transmittance (Note S2), which increased with the deposition pressure for the films deposited at 50 and 100 Pa (Figure S3e).

To evaluate the overall performance of the films as solar absorbers, the spectrally averaged solar absorptance was calculated according to the formula 57

2000 280 ( ) ( ) s solar A S d I λ λ λ α = ∫ (1)
where Ss(λ) is the spectral solar irradiance (AM 1.5G), Isolar is the total irradiance, and the calculation is performed in the investigated wavelengths range, i.e., 280-2000 nm. Figure 4c shows that this quantity monotonically increased with the deposition pressure. In particular, α abruptly increased from ~ 0.29

for the film prepared in vacuum to ~ 0.61 for the film deposited at 10 Pa, and then it reached the maximum value of ~ 0.89 at the deposition pressure of 100 Pa, thus confirming the trend discussed above. The α value found for the film deposited at 100 Pa is comparable to other solar absorbers reported in the literature, such as a 2D Ta photonic crystal (α = 0.864), [START_REF] Rinnerbauer | Metallic Photonic Crystal Absorber-Emitter for Efficient Spectral Control in High-Temperature Solar Thermophotovoltaics[END_REF] a Ti/Al2O3/Ta plasmonic metamaterial (α = 0.913), [START_REF] Li | Scalable, and High-Temperature Selective Solar Absorbers Based on Hybrid-Strategy Plasmonic Metamaterials[END_REF] TiN nanopillars (α = 0.94), [START_REF] Chirumamilla | Large-Area Ultrabroadband Absorber for Solar Thermophotovoltaics Based on 3D Titanium Nitride Nanopillars[END_REF] and TiN/TiN NPs/SiO2 ceramic layer (α = 0.95). [START_REF] Li | Solution-Processed All-Ceramic Plasmonic Metamaterials for Efficient Solar-Thermal Conversion over 100-727[END_REF] Hence, the data presented in Figure 4 highlight the possibility of achieving a broadband solar absorber behavior for the tree-like TiOxNy films deposited at high pressures. Compared to broadband absorbers made of complex TiN metamaterial structures, [START_REF] Li | Refractory Plasmonics with Titanium Nitride: Broadband Metamaterial Absorber[END_REF][START_REF] Gao | Ultraviolet to near Infrared Titanium Nitride Broadband Plasmonic Absorber[END_REF] the hierarchical nanoporous TiOxNy films could represent a potential alternative thanks to their relatively simple preparation process. The performance of the TiN/TiNxOy films for solar-thermal conversion applications was evaluated by non-contact thermal measurements under solar irradiation (Figure 5). The temperature reached under solar-simulated light from 1.3 to 17 Suns (1 Sun = 100 mW cm -2 ) was measured with an infrared thermal camera pointing the back surface of the films deposited on titanium substrates (see Figure S4 for the experimental details and Figure S5 for a discussion on the substrate effect). The samples were kept in a home-made vacuum cell under inert Ar atmosphere to prevent surface oxidation and to allow multiple experiments under different irradiation conditions for extended periods of time. [START_REF] Naldoni | Solar Thermoplasmonic Nanofurnace for High-Temperature Heterogeneous Catalysis[END_REF][START_REF] Guler | Plasmonics on the Slope of Enlightenment: The Role of Transition Metal Nitrides[END_REF][START_REF] Goya | Thermal Oxidation of TiN Nanocylinder Arrays: Effects of Insulator Coatings by Atomic Layer Deposition[END_REF] Figure 5a shows the temperature profiles during time under the maximum irradiation conditions (i.e., 17 Suns) for all the investigated films compared with an uncoated Ti substrate (see Figure S6 for the results for all the films under all irradiation conditions). It is possible to observe that the films heated up very quickly (i.e., in less than 10 s), contrarily to the bare Ti plate, and they reached a steady-state temperature after ~ 20 s.

The steady-state temperature value measured at the end of the experiment, hereinafter labeled as Tmax (maximum temperature) is shown for all the films under all the investigated solar intensities in Figure 5b. The generated temperature increased with the deposition pressure up to the maximum value of ~ 475 °C under 17 Suns irradiation for the film deposited at 100 Pa, which is a result comparable to the temperature generated by periodic TiN cylindrical nanocavities under the same moderately concentrated solar condition. [START_REF] Naldoni | Solar Thermoplasmonic Nanofurnace for High-Temperature Heterogeneous Catalysis[END_REF] This outcome is even more interesting considering that similar temperatures were reached by metallic nanoparticles array and TiN periodic nanostructures (i.e., nanotubes and trench) under laser irradiation with 10 6 -fold and 10 4 -fold greater power densities, respectively. [START_REF] Ishii | Extreme Thermal Anisotropy in High-Aspect-Ratio Titanium Nitride Nanostructures for Efficient Photothermal Heating[END_REF][START_REF] Mutlu | Thermoplasmonic Ignition of Metal Nanoparticles[END_REF] Notably, the films did not undergo any degradation upon solar irradiation. Raman spectroscopy experiments (Figure S7), indeed, revealed only minor changes in the spectra compared to the pristine samples. While the experiments shown in Figures 6a and6b were carried out in inert atmosphere, scale-up in environmental conditions could be realized by including a capping layer on top of the films, i.e., Al2O3 or Si3N4 by ALD. [START_REF] Guler | Plasmonics on the Slope of Enlightenment: The Role of Transition Metal Nitrides[END_REF][START_REF] Goya | Thermal Oxidation of TiN Nanocylinder Arrays: Effects of Insulator Coatings by Atomic Layer Deposition[END_REF] To characterize the solar-thermal performance of the films, the spectrally averaged emissivity at each irradiation condition in correspondence of Tmax was calculated as 57 25000

0 max 1330 max 25000 max 1330 ( , ) ( , ) ( ) ( , ) b b T E T d T E T d λ ε λ λ λ ε λ λ = ∫ ∫ (2)
In Equation ( 2), all the quantities are referred to the NIR-MIR range (i.e., the range investigated by FTIR measurements) because it is the typical range for thermal emission corresponding to surface temperatures of few hundreds of °C.

B hc E T hc k T π λ λ λ   =       -     (3)
where h is the Planck constant, c is the speed of light, and kB is the Boltzmann constant. Figure 5c shows the values for max ( ) T ε for all films under all irradiation conditions compared to the uncoated Ti substrate.

The film deposited in vacuum exhibited a very low emittance (0.12-0.14) due to its higly metallic and reflective properties (Figure S3d). For all the other films, the emittance values fell in the interval 0. 4b) and, therefore, lower emittance (Figure 5c). The performance of the hierarchical TiNxOy films at moderate temperature conditions is therefore lower than that of spectrally selective solar absorbers, [START_REF] Li | Solution-Processed All-Ceramic Plasmonic Metamaterials for Efficient Solar-Thermal Conversion over 100-727[END_REF][START_REF] Rinnerbauer | Metallic Photonic Crystal Absorber-Emitter for Efficient Spectral Control in High-Temperature Solar Thermophotovoltaics[END_REF] which is not surprising since this work did not address a control of the spectral emissivity of the TiN/TiNxOy films. In fact, the present results could be exploited to develop more advanced films limiting their emissivity in the MIR range. For example, compact/porous multi-layer structures could be realized in the same PLD process, [START_REF] Li | Solution-Processed All-Ceramic Plasmonic Metamaterials for Efficient Solar-Thermal Conversion over 100-727[END_REF][START_REF] Passoni | Self-Assembled Hierarchical Nanostructures for High-Efficiency Porous Photonic Crystals[END_REF] or anti-reflection coatings could further be included. [START_REF] Guler | Plasmonics on the Slope of Enlightenment: The Role of Transition Metal Nitrides[END_REF][START_REF] Goya | Thermal Oxidation of TiN Nanocylinder Arrays: Effects of Insulator Coatings by Atomic Layer Deposition[END_REF] More complex architectures, such as nanopatterned surfaces, [START_REF] Guler | Plasmonic Titanium Nitride Nanostructures via Nitridation of Nanopatterned Titanium Dioxide[END_REF] could also be designed thanks to the possibility of depositing TiNxOy at room temperature on plastic substrates.

Figure 1 .

 1 Figure 1. SEM cross-sectional (first row) and top views (second row) images of the films deposited (a,b) in vacuum, (c,d) at 10 Pa, (e,f) at 20 Pa, (g,h) at 50 Pa, and (i, j) at 100 Pa of N2/H2; (k) cross-sectional and (l) top view magnifications of the film deposited at 100 Pa.

47

 47

Figure 2 .

 2 Figure 2. (a) XRD patterns of TiN films deposited in vacuum on Si substrate (main reflection marked with *) and at 10, 20, 50, and 100 Pa N2/H2 on glass substrates (vertical dashed lines: reference TiN peaks). (b) Lattice constant a and (c) average domain size (τ) evaluated for the (111) and (200) XRD reflections. The horizontal dashed lines in (b) correspond to the lattice constants for bulk TiN (4.238 Å)

Figure 3 .

 3 Figure 3. (a) Atomic percentage of Ti, N and O by EDX microanalysis and (b) N to Ti ratio (N/Ti) of the atomic percentage for film deposited from vacuum to 100 Pa N2/H2 (c) Raman spectra of the TiN films deposited in vacuum and at 10-100 Pa. (d) High-angle annular dark-field STEM (HAADF-STEM) image and corresponding EDX mapping of Ti (e), N (f) and O (g) for the film deposited at 100 Pa.

Figure 4 .

 4 Figure 4. (a) Optical absorptance of the TiN films compared to the spectral solar irradiance (ASTM G173-03 AM 1.5 Global). (b) Optical absorptance in the MIR range retrieved by FTIR spectroscopy. (c) Spectrally averaged absorptance in the 280-2000 nm wavelength range. For all panels, the color legend is reported in (c).

=

  is the spectral emissivity according to the Kirchhoff's law of thermal radiation evaluated at room temperature (see a related discussion on the systematic errors introduced by instrumental limitations of FTIR spectroscopy in Note S2)

  film deposited at 50 Pa. The moderate difference in emittance values for the films deposited at 10-100 Pa (Figure5c) compared to the dramatic increase of the spectrally averaged absorptance with the deposition pressure (Figure4c) is due to the transparency window in the NIR range (up to ~ 10 µm) for the porous films (FigureS3b).Taking into account the values of α (Figure4c) σ = 5.670367×10 -8 W m -2 K -4 is the Stefan-Boltzmann constant, Tamb = 25 °C is the ambient temperature during the experiments, and C is the solar concentration factor(1.3-17). Figure5dshows that the samples deposited at pressures ≥ 20 Pa outperformed the bare Ti substrate. The maximum thermal transfer efficiency was found for the film deposited at 100 Pa at 1.3 Suns (ηth ~ 79% at Tmax ~ 54.9 °C), and the efficiency values decreased with the irradiation intensity for all the investigated films. This is because the radiative losses are very limited at room temperature, but they follow a ~ T[START_REF] Chirumamilla | Multilayer Tungsten-Alumina-Based Broadband Light Absorbers for High-Temperature Applications[END_REF] dependence, thus making the emittance contribution increasingly more relevant at increasing temperatures. It is possible to note that under 17 Suns illumination, the film deposited at 50 Pa outperformed the one deposited at 100 Pa (ηth ~ 35% at Tmax ~ 462 °C vs. ηth ~ 22% at Tmax ~ 475 °C) because of its lower absorptance in the NIR-MIR range (Figure

Figure 5 .

 5 Figure 5. Solar-thermal performance of the investigated films. (a) Temperature profiles under 17 Suns as a function of time. (b) Maximum temperature (steady-state value at the end of each experiment) measured as a function of the solar power (1.3-17 Suns). (c) Spectrally averaged emittance and (d) thermal transfer efficiency as a function of the maximum temperature under irradiation. All panels report also the data for an uncoated titanium substrate.

Acknowledgments L. M. and A. N. acknowledge support of the Ministry of Education, Youth and Sports of the Czech Republic through the Operational Programme Research, Development and Education -European Regional Development Fund, project no. CZ.02.1.01/0.0/0.0/15_003/0000416, and the Czech Science Foundation (GACR) through the project no. 20-17636S. The authors also thank Ondřej Tomanec for TEM and EDS measurements, Seyyedmohammadhossein Hejazi for XRD measurements, Veronika Šedajová for XPS measurements, and Filip Münz for his aid with FTIR measurements. The authors acknowledge CzechNanoLab Research Infrastructure supported by MEYS CR (LM2018110).

Conclusions

Titanium (oxy)nitride films of tunable morphology, structure and composition were deposited by pulsed laser deposition at room temperature. By increasing the deposition background pressure from vacuum to 100 Pa, the films properties changed from bulk-like TiN to hierarchical nanoparticle assembly of TiOxNy.

In particular, the hierarchical nanoporous films exhibited an ultra-fine nanocrystalline structure with a high degree of oxygen incorporation promoted by the high background gas (N2/H2) pressure during deposition. The light absorption of the films increased with the deposition pressure, thus allowing a broadband absorptance from the UV to the IR wavelength range. The films were studied as perspective candidates for solar-thermal applications by measuring the temperature produced under solar-simulated irradiation with moderate light concentration, which revealed the superior performance of the porous TiOxNy films (maximum temperature ~ 475 °C under 17 Suns) and resistance to oxidation. Further performance optimization could be addressed by simple design strategies thanks to the flexibility of pulsed laser deposition, such as by realizing compact/porous multilayers or by including anti-reflection oxide layers.

Associated content

The Supporting Information is available free of charge at… Content of the SI: EDX microanalysis of the TiN target; high-resolution XPS spectra for the film deposited at 100 Pa; optical properties of the TiN films in the UV-vis-NIR and in the MIR ranges; details on the thermal camera measurements; top view SEM images of an uncoated Ti substrate and of all the investigated films on Ti substrates; temperature profiles under solar irradiation; Raman spectra before and after solar irradiation for all the investigated films; details on XPS peak fitting for the film deposited at 100 Pa; details on Raman spectroscopy experiments; details on optical spectroscopy experiments.

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Notes

The authors declare no competing financial interest.