Alexandre Maurer
email: alexandre.maurer@lip6.fr

Sébastien Tixeuil
email: sebastien.tixeuil@lip6.fr

Parameterizable Byzantine Broadcast in Loosely Connected Networks

We consider the problem of reliably broadcasting information in a multihop asynchronous network, despite the presence of Byzantine failures: some nodes are malicious and behave arbitrarly. We focus on non-cryptographic solutions.

Most existing approaches give conditions for perfect reliable broadcast (all correct nodes deliver the good information), but require a highly connected network. A probabilistic approach was recently proposed for loosely connected networks: the Byzantine failures are randomly distributed, and the correct nodes deliver the good information with high probability. A first solution require the nodes to initially know their position on the network, which may be difficult or impossible in self-organizing or dynamic networks. A second solution relaxed this hypothesis but has much weaker Byzantine tolerance guarantees.

In this paper, we propose a parameterizable broadcast protocol that does not require nodes to have any knowledge about the network. We give a deterministic technique to compute a set of nodes that always deliver authentic information, for a given set of Byzantine failures. Then, we use this technique to experimentally evaluate our protocol, and show that it significantely outperforms previous solutions with the same hypotheses.

Introduction

As modern networks grow larger, they become more likely to fail. Indeed, some nodes can be subject to crashes, attacks, transient bit flips, etc. Many models for failures and attacks have been studied so far, but the most general one is the Byzantine model [START_REF] Lamport | The byzantine generals problem[END_REF]: we assume that the failing nodes have a totally arbitrary behavior. In other words, we must anticipate the most malicious strategies they could adopt. This encompasses all other possible types of failures, and has important security applications.

In this paper, we study the problem of reliably broadcasting information in a network despite the presence of Byzantine failures. This is a difficult problem, as a single Byzantine node, if not neutralized, can lie to the entire network. Our objective is to design broadcast protocols that prevent or limit the diffusion of malicious informations.

Related works. Many Byzantine-robust protocols are based on cryptography [START_REF] Castro | Practical byzantine fault tolerance[END_REF][START_REF] Vadim Drabkin | Efficient byzantine broadcast in wireless ad-hoc networks[END_REF]: the nodes use digital signatures or certificates. Therefore, the correct nodes can assess the validity of received informations and authenticate the sender across multiple hops. However, this approach may not be as general as we want, as the malicious nodes are supposed to ignore some cryptographic secrets: therefore, their behavior is not completely arbitrary. Besides, manipulating cryptographic operations requires important ressources, which may not always be available -for instance, in sensor networks. Finally, cryptography requires a trusted infrastructure that initially distributes public and private keys: if this initial infrastructure fails, the whole network fails. Yet, we want to design a solution where any component can fail. For these reasons, we focus on cryptography-free solutions.

Cryptography-free solutions have first been studied in completely connected networks [START_REF] Lamport | The byzantine generals problem[END_REF][START_REF] Attiya | Distributed Computing: Fundamentals, Simulations, and Advanced Topics[END_REF][START_REF] Malkhi | Diffusion without false rumors: on propagating updates in a Byzantine environment[END_REF][START_REF] Malkhi | Efficient update diffusion in byzantine environments[END_REF][START_REF] Minsky | Tolerating malicious gossip[END_REF]: a node can directly communicate with any other node, which implies the presence of a channel between each pair of nodes. Therefore, these approaches are hardly scalable, as the number of channels per node can be physically limited. We thus study solutions in multihop networks, where a node must rely on other nodes to broadcast informations.

A notable class of algorithms tolerates Byzantine failures with either space [START_REF] Masuzawa | Stabilizing link-coloration of arbitrary networks with unbounded byzantine faults[END_REF][START_REF] Nesterenko | Tolerance to unbounded byzantine faults[END_REF][START_REF] Sakurai | A self-stabilizing link-coloring protocol resilient to byzantine faults in tree networks[END_REF] or time [START_REF] Masuzawa | Bounding the impact of unbounded attacks in stabilization[END_REF][START_REF] Dubois | Maximum metric spanning tree made byzantine tolerant[END_REF][START_REF] Dubois | Bounding the impact of unbounded attacks in stabilization[END_REF][START_REF] Dubois | On byzantine containment properties of the min+1 protocol[END_REF][START_REF] Dubois | The impact of topology on byzantine containment in stabilization[END_REF] locality. Yet, the emphasis of space local algorithms is on containing the fault as close to its source as possible. This is only applicable to the problems where the information from remote nodes is unimportant (such as vertex coloring, link coloring or dining philosophers). Also, time local algorithms presented so far can hold at most one Byzantine node and are not able to mask the effect of Byzantine actions. Thus, the local containment approach is not applicable to reliable broadcast.

It has been shown that, for agreement in the presence of up to k Byzantine nodes, it is necessary and sufficient that the network is (2k + 1)-connected, and that the number of nodes in the system is at least 3k + 1 [START_REF] Dolev | The Byzantine generals strike again[END_REF]. Also, this solution assumes that the topology is known to every node, and that nodes are scheduled according to the synchronous execution model. Both requirements have been relaxed in [START_REF] Nesterenko | Discovering network topology in the presence of byzantine nodes[END_REF]: the topology is unknown and the scheduling is asynchronous. Yet, this solution retains 2k + 1 connectivity for reliable broadcast and k + 1 connectivity for detection (the nodes are aware of the presence of a Byzantine failure). In sparse networks such as a grid (where a node has at most four neighbors), both approaches can cope only with a single Byzantine node, independently of the size of the grid.

Another existing approach is based, not on connectivity, but on the fraction of Byzantine neighbors per node. Broadcast protocols have been proposed for nodes organized on a grid [START_REF] Koo | Broadcast in radio networks tolerating byzantine adversarial behavior[END_REF][START_REF] Bhandari | On reliable broadcast in a radio network[END_REF]. However, the wireless medium typically induces much more than four neighbors per node, otherwise the broadcast does not work. Both approaches are based on a local voting system, and perform correctly if every node has strictly less than a 1/4 fraction of Byzantine neighbors. This result was later generalized to other topologies [START_REF] Pelc | Broadcasting with locally bounded byzantine faults[END_REF], assuming that each node knows the global topology. Again, in loosely connected networks, this constraint on the proportion of Byzantine nodes in any neighborhood may be difficult to assess.

All aforementioned results rely on strong connectivity and Byzantine proportions assumptions in the network. In other words, tolerating more Byzantine failures requires to increase the connectivity, which can be a heavy constraint in a large network. To overcome this problem, a probabilistic approach for reliable broadcast has been proposed in [START_REF] Maurer | Limiting byzantine influence in multihop asynchronous networks[END_REF]. In this setting, the distribution of Byzantine failures is assumed to be random. This hypothesis is realistic if we consider that each node has a given probability to fail or to be corrupted by an adversary. Another possible use of such a setting is structured overlay networks that make use of distributed hash tables, where the identifier (and thus location) of a node joining the network is attributed randomly: Byzantine nodes joining the network are thus randomly located in the overlay. Also, as we initially accepted that some node may become Byzantine, we can accept that a small minority of correct nodes are fooled by the Byzantine nodes.

With these assumptions, the network can tolerate a number of Byzantine failures that largely Figure 1: Principle of the protocol exceeds its connectivity [START_REF] Maurer | Limiting byzantine influence in multihop asynchronous networks[END_REF]. This approach was recently generalized to tolerate a constant fraction of Byzantine nodes in a potentially infinite network with a bounded connectivity [START_REF] Maurer | A scalable byzantine grid[END_REF]. However, both solutions require a global view of the network: each node must know its position in the network, in order to join particular subsets called control zones. This may be difficult or impossible in many types of networks, such as self-organized wireless sensor networks or peer-to-peer overlays. This requirement was removed in [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF], at the cost of a much lesser number of tolerated Byzantine than [START_REF] Maurer | Limiting byzantine influence in multihop asynchronous networks[END_REF][START_REF] Maurer | A scalable byzantine grid[END_REF].

Our contribution. In this paper, we elaborate on [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF] and provide a cryptography-free, Byzantinerobust protocol adaptated to loosely connected network, that does not require the nodes to initially know their position on the network. Our proposal is parameterizable in such a way that Byzantine tolerance of [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF] is significantely improved (for example, our experiments show that four times more Byzantine nodes can be tolerated when 0.99 communication reliability probability is required).

The paper is organized as follows:

• In Section 2, we describe our protocol and give the algorithm executed by each correct node.

• In Section 3, we present a deterministic technique to determine a set of nodes that deliver the good information in any possible execution.

• In Section 4, we use this technique to experimentally evaluate the performances of our protocol.

Description of the Protocol

In this section, we provide an informal description of our protocol. Then, after stating the notations and hypotheses, we give the local algorithm that each correct node must execute.

Informal Description

The network is described by a set of processes, called nodes. Some pairs of nodes are linked by a channel, and can send messages to each other: we call them neighbors. The network is asynchronous: the nodes can send and receive messages at any time.

A particular node, called the source, wants to broadcast an information m to the rest of the network.

In the ideal case, the source would send m to its neighbors, which will transmit m to their own neighbors -and so forth, until every node receives m. Let us call this an unsecured broadcast. In our setting however, some nodes can be malicious (Byzantine) and broadcast false informations in the network. Of course, a correct node cannot know whether a neighbor is Byzantine or not. In the following, we say that a correct node delivers an information when it definitively considers that this information comes from the source.

To limit the diffusion of false informations, we introduce the following mechanism: to deliver an information, a node must receive it through several node-disjoint paths, each path having a maximal length. For instance, in Figure 1-a, the information must be received through . . .

• a first path of at most

H 1 = 3 hops • a second path of at most H 2 = 4 hops • a third path of at most H 3 = 2 hops
This specific setting of the protocol can be described by the triplet (H 1 , H 2 , H 3 ) = (3, 4, 2). To be more general, a setting of the protocol is described by a n-uplet (H 1 , . . . , H n ).

The underlying idea is as follows: if the Byzantine nodes are sufficiently spaced, they will never manage to broadcast false informations. Indeed, to deliver a false information in setting (3, 4, 2), a node must be located at . . .

• at most H 1 = 3 hops from a first Byzantine node

• at most H 2 = 4 hops from a second Byzantine node

• at most H 3 = 2 hops from a third Byzantine node This critical configuration is illustrated in Figure 1-b. However, if (for instance) the third Byzantine node is located at more than 2 hops, the false information will never be delivered. This is illustrated in Figure 1-c. This intuitive idea is demonstrated further in Theorem 1.

This protocol is a generalization of the protocol proposed in [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF], which corresponds to the settings (1, H), H being a parameter. The condition for safety was the following: the minimal distance between two Byzantine nodes is at least H + 2 hops. Here, the general condition for safety is more complex to evaluate. However, we show in this paper that the performance of [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF] can be significantely improved using this more general approach (see Section 4).

Detailed description

Let us precise the notations and hypotheses, and give the local algorithm executed by each correct node.

Notations and Hypotheses Let (G, E) be a non-oriented graph representing the topology of the network. G denotes the nodes of the network. E denotes the neighborhood relationship. A node can only send messages to its neighbors. Some nodes are correct and follow the protocol described thereafter. We consider that all other nodes are totally unpredictable (or Byzantine) and may exhibit an arbitrary behavior. We exclude the case where the source is Byzantine, which obviously cannot be correct.

We consider an asynchronous message passing network: any message sent is eventually received, but it can be at any time. We assume that, in an infinite execution, any process is activated inifinitely often. However, we make no hypothesis on the order of activation of the processes. Finally, we assume local topology knowledge: when a node receives a message from a neighbor p, it knows that p is the author of the message. Therefore, a Byzantine node cannot lie about its identity to its direct neighbors. This model is referred to as the "oral" model in the literature.

Preliminaries The setting of the protocol is described by a n-tuple of integers (H 1 , . . . , H n ), known by all correct nodes. These values, once settled, should be considered as an inherent part of the protocol: they are "hard-coded" with the rest of the algorithm. Therefore, they are not concerned by any broadcast or agreement problem. Let H = max i∈{1,...,n} H i .

The messages exchanged in the protocol are tuples of the form (m, S):

• m is the information broadcast by the source, or pretending to be it.

• S is a set containing the identifiers of nodes already visited by the message. It ensures that a node is never visited twice, and that the paths are actually node-disjoint.

Each correct node holds a dynamic set Rec, where the messages (m, S) received are recorded. In the following, the set Rec of a node v is referred to as v.Rec.

Finally, we say that a node multicasts a message when it sends it to all its neighbors.

Local execution of the protocol Initially, the source multicasts (m, ø). Then, each correct node executes the following algorithm:

1. When a message of type (m, S) is received from a neighbor q:

(a) If q is the source, deliver m and multicast (m, ø) (b) If q / ∈ S and card(S) < H:

• Add (m, S ∪ {q}) to the set Rec • Multicast (m, S ∪ {q}) 2.
When there exists (m, S 1 , . . . , S n ) such that:

(a) ∀i ∈ {1, . . . , n}, we both have (m, S i ) ∈ Rec and card

(S i ) ≤ H i (b) ∀(i, j) ∈ {1, . . . , n} 2 , S i ∩ S j = ø
Then, deliver m and multicast (m, ø).

Protocol Properties

In this section, we adopt the point of view of an omniscient external observer, knowing the positions of Byzantine nodes. These positions are, of course, unknown by the correct nodes: this is just a global view of the system.

In 3.1, we give a condition on the placement of Byzantine nodes to guarantee the safety of the network -that is, no correct node can deliver a false information. Then, in 3.2, we give a methodology to determine a reliable node set -that is, a set of nodes that always deliver the good information, in any possible execution. This methodology is used in the next section to evaluate the performances of the protocol, without actually simulating it.

In addition, 3.3, we show that the bounds of our properties are tight. At last, in 3.4, we evaluate the message complexity of our protocol.

Network Safety

Let us give a general condition on the placement of Byzantine nodes, ensuring that no correct node ever delivers a false information. The following theorem is the demonstration of the intuitive idea exposed in 2.1, proving the correctness of our algorithm. Notice that is does not necessarily ensure that the correct nodes actually deliver the good information: this aspect is studied further in 3.2.

Fisrt, let us give some preliminary definitions.

Definition 1 (Path). A N-hops path is a sequence of distinct nodes (p 0 , . . . , p N ) such that, ∀i ∈ {1, . . . , N }, p i and p i-1 are neighbors. We say that this path connects p 0 and p N . This path is correct if all its nodes are correct.

Definition 2 (Disjoint paths). Two path (p 0 , . . . , p N ) and (p 0 , . . . , p

N ) are disjoint if {p 1 , . . . , p N -1 }∩ {p 1, . . . , p N -1 } = ø.
This definition is introduced for the needs of the following theorems. Note that we do not claim that disjoint paths are strictly equivalent to node-disjoint paths.

Theorem 1 (Network Safety). For a given correct node u, let Critical(u) be the following proposition: There exists . . .

• n distinct Byzantine nodes (b 1 , . . . , b n )

• n disjoint paths (X 1 , . . . , X n ) such that, ∀i ∈ {1, . . . , n}, X i is a path of at most H i hops connecting u and b i .

If, for every correct node u, Critical(u) is false, then no correct node can deliver a false information.

Proof. The proof is by contradiction. Let us suppose the opposite: for every correct node u, Critical(u) is false, yet at least one correct node delivers a false information. Let v be the first correct node to deliver a false information. In the following, we show that Critical(v) is necessarily true, contradicting the previous statement. This contradiction achieves the proof.

Let m = m be the information delivered by v. As v is correct and delivers m , according to the protocol: there exists (S 1 , . . . , S n ) such that, ∀i ∈ {1, . . . , n}, (m , S i ) ∈ v.Rec and card(S i ) ≤ H i .

Let us consider a given index i ∈ {1, . . . , n}, and let q 0 = v. Let P k be the following proposition: There exists a path (q 1 , . . . , q k ), with {q 1 , . . . , q k } ⊆ S i , such that q k-1 received (m , S i -{q 1 , . . . , q k }) from q k ∈ S i -{q 1 , . . . , q k-1 }. In our notations, {q 1 , . . . , q k-1 } = ø for k = 1.

• First, let us show that P 1 is true. According to the protocol, the statement (m , S i ) ∈ v.Rec implies that v received (m, S i -{q 1 }) from a node q 1 ∈ S i . It is actually possible, as card(S i -{q 1 }) ≤ H i -1 < H. So P 1 is true.

• Now, let us suppose that P k is true, for k < card(S i ). So q k sent (m , S i -{q 1 , . . . , q k }) to q k-1 . Let us suppose that q k is correct. Then, according to the protocol, it implies that q k received (m, S i -{q 1 , . . . , q k+1 }) from a node q k+1 ∈ S i -{q 1 , . . . , q k }. It is actually possible, as card(S i -{q 1 , . . . , q k+1 }) ≤ H i -k -1 < H. So either P k+1 is true, either q k is Byzantine.

Therefore, by induction:

• Either there exists an index k ∈ {1, . . . , card(S i ) -1} and a path (q 1 , . . . , q k ) such that q k is Byzantine. Let b i = q k , and let X i be the path (v, q 1 , . . . , q k ).

• Either P card(S i ) is true, and q card(S i ) sent (m , S i -{q 1 , . . . , q card(S i ) }) = (m , ø) to q card(S i -1) . The node q card(S i ) cannot be the source, as m = m. Also, it cannot be a correct node, as v is the first correct node to deliver m . So q card(S i ) is necessarily Byzantine. Let b i = q card(S i ) , and let X i be the path (v, q 1 , . . . , q card(S i ) ).

In both cases, the path X i connects v and b i with at most card(S i ) ≤ H i hops.

According to Definition 2, for (i, j) ∈ {1, . . . , n} 2 , the path X i and X j are disjoint if (S i -{b i }) ∩ (S j -{b j }) = ø. As v is correct and delivers m , according to the protocol: ∀(i, j) ∈ {1, . . . , n} 2 , S i ∩ S j = ø. Therefore, the paths (X 1 , . . . , X n ) are disjoint, and the nodes (b 1 , . . . , b n ) are distinct. Thus, Critical(v) is true. Thus, the result.

Also, as Critical(u) requires n distinct Byzantine nodes, a sufficient condition for safety is that the number of Byzantine nodes does not exceeds n -1.

Network Reliability

Here, we suppose that the conditions of Theorem 1 are satisfied: no false information can be delivered by a correct node. We now give a methodology to construct a reliable node set.

Definition 3 (Reliable node set). For a given source and a given set of Byzantine nodes, a reliable node set is a set of correct nodes that always deliver the good information, in any possible execution.

The following theorem explains how to construct such a set node by node: for a given reliable node set R and a given correct node v, we can determine if v can be added to R (R := R ∪ {v}) -and so forth, until we obtain the largest possible reliable node set. To initialize the construction of R, simply notice that the source and its correct neighbors form a reliable node set, according to the protocol.

Theorem 2 (Construction of a reliable node set). Let us suppose that the conditions of Theorem 1 are satisfied. Let R be a reliable node set, and let v / ∈ R be a correct node. If there exists . . .

• n distinct nodes (r 1 , . . . , r n ) of R

• n disjoint correct paths (X 1 , . . . , X n ) such that, ∀i ∈ {1, . . . , n}, X i is a path of at most H i hops connecting v and r i .

Then, R ∪ {v} is also a reliable node set.

Proof. According to Theorem 1, no correct node can deliver a false information. So, if a correct node delivers an information, it is necessary the good information.

Let us consider a given index i ∈ {1, . . . , n}. Let X i = (q 0 , . . . , q M ), with q 0 = r i and q M = v.

By definition, we have M ≤ Hi ≤ H. Let us prove the following property P k by induction, ∀k ∈ {1, . . . , M }:

The node q k eventually receives (m, {q 0 , . . . , q k-2 }) from q k-1 . In our notations, {q 0 , . . . , q k-2 } = ø for k = 1.

• First, let us show that P 1 is true. As R is a reliable node set, according to Definition 3, the node r i ∈ R eventually delivers the good information. According to the protocol, it implies that r i also multicasts (m, ø). So q 1 eventully receives (m, ø) from q 0 , and P 1 is true.

• Now, let us suppose that P k is true for k ≤ M . As q k-1 / ∈ {q 0 , . . . , q k-2 }, and card({q 0 , . . . , q k-2 }) < M ≤ H, q k eventually multicasts {q 0 , . . . , q k-1 }. So q k+1 eventually receives {q 0 , . . . , q k-1 } from q k , and P k+1 is true. So P M is true and the node q M = v eventually receives (m, {q 0 , . . . , q M -2 }) from q M -1 . As q M -1 / ∈ {q 0 , . . . , q M -2 } and card({q 0 , . . . , q M -2 }) < M ≤ H, v eventually adds (m, {q 0 , . . . , q M -1 }) to the set Rec. Let S i = {q 0 , . . . , q M -1 }.

So, ∀i ∈ {1, . . . , n}, we have (m, S i ) ∈ v.Rec and card(S i ) < H i . Besides, as the paths (X 1 , . . . , X n ) are disjoint, according to Definition 2: ∀(i, j) ∈ {1, . . . , n} 2 , we have (S i -{r i }) ∩ (S j -{r j }) = ø. Thus, as the nodes (r 1 , . . . , r n ) are distinct, we have S i ∩ S j = ø. Therefore, according to the protocol, v eventually delivers m, and R ∪ {v} is a reliable node set.

Bounds Tightness

Let us show that the condition for safety (Theorem 1) is tight, and that the methodology to construct a reliable node set (Theorem 2) is optimal in a safe network.

Theorem 3 (Bounds tightness for Theorem 1). If the condition of Theorem 1 is not satisfied, it is strictly impossible to guarantee network safety.

Proof. Let us suppose the opposite: the condition of Theorem 1 is not satisfied, yet the network is safe -that is, no correct node can deliver a false information. As the condition of Theorem 1 is not satisfied, there exists at least one correct node u such that Critical(u) is true: There exists . . .

• n distinct Byzantine nodes (b 1 , . . . , b n )
• n disjoint paths (X 1 , . . . , X n ) such that, ∀i ∈ {1, . . . , n}, X i is a path of at most H i hops connecting u and b i .

Thus, it is possible that the Byzantine nodes (b 1 , . . . , b n ) unanimously multicast (m , ø), with m = m. If so, with a reasoning similar to the proof of Theorem 2, we show that u eventually delivers the false information m . Therefore, the network cannot be safe. This contradiction completes the proof.

Theorem 4 (Bounds tightness for Theorem 2). Let us suppose that the network is safe. Then, the set constructed with Theorem 2 is the largest possible reliable node set.

Proof. Let us suppose the opposite: the set R constructed with Theorem 2 is not the largest possible reliable node set. Let R be the largest possible reliable node set.

Let there be an execution where all the nodes of R delivered the good information, but no other correct node delivered the good information so far. Such an execution is possible, as the construction of R with Theorem 2 does not require that any node u / ∈ R delivers the good information.

Let v be the first node of R -R that delivers the good information in the following of the execution. Then, by a reasoning similar to the proof of Theorem 1, we show that there must exist . . .

• n distinct nodes (u 1 , . . . , u n )

• n disjoint correct paths (X 1 , . . . , X n ) such that, ∀i ∈ {1, . . . , n}, X i is a path of at most H i hops connecting v and r i .

. . . such that the nodes (u 1 , . . . , u n ) have previously delivered the good information. As the only correct nodes that have previously delivered the good information are the nodes of R, we have {u 1 , . . . , u n } ⊆ R, and the condition of Theorem 2 is verified for R and v. So R ∪ {v} is a reliable node set, and R is not the largest reliable node set that can be constructed with Theorem 2. This contradiction completes the proof.

Message complexity

At last, let us evaluate the message complexity of our protocol. We only consider the case where all nodes are correct, as the Byzantine nodes can send as many messages as they want.

Let |G| be the number of nodes, and let d be the maximal degree of the network -that is, the maximal number of neighbors for a single node. According to our protocol, each node u multicasts (m, ø) at most once. Then, each neighbor of u multicasts (m, {u}), which makes at most 1 + d messages. This process is repeated n times, which makes at most 1

+ d + d 2 + • • • + d n = o(d n ) messages. So, o(d n |G|
) messages are sent by the protocol.

Therefore, if we consider that d and n are bounded, o(|G|) messages are sent, and the message complexity is the same as an unsecured broadcast protocol (see 2.1).

Experimental Evaluation

In this section, we experimentally compare the performances of different settings of our protocol. We detail our motivations, then describe the methodology and comment on the results. Finally, we provide a quantitative comparison with existing protocols, and show the improvement.

Motivation

We would like to evaluate the performances of different settings of our protocol, and also compare them with other existing protocols.

As our protocol is designed for loosely connected networks, we choose grid-shaped networks for experimentation (see Figure 2), where each node has at most 4 neighbors. Very few Byzantinerobust protocols exist for such sparse networks. We also compare the performances between grid and torus networks (a grid without border), to show the influence of border effects.

In order to quantify these performances, we assume a random uniform distribution of Byzantine failures. Our metric is the probability that a node delivers the good information. We choose this probabilistic model for two reasons:

• As stated in the introduction, it is difficult or impossible to achieve perfect reliable broadcast is sparse networks. Therefore, we release deterministic guarantees, and aim at probabilistic guarantees -for instance, a node must deliver the good information with a probability of 0.99.

• This random model has realistic applications: in a network, each component has a weak yet positive probability to misbehave. Besides, some Byzantine agents can join a peer-topeer overlay at any moment, or be introduced in a collection of wireless sensors before their deployment on the field.

Simulating a distributed protocol in the presence of Byzantine failures is a far too difficult problem. Indeed, it would implie to make hypotheses on the order of activation of processes, on the order of Figure 2: A 7 × 7 grid network reception of messages, and above all, on the behavior of Byzantine nodes: how can we guarantee that they actually adopt the worst possible behavior? To bypass these difficulties, we do not directly simulate the protocol, but use the reliability properties of Section 3 to evaluate its performances without adding more hypotheses.

Methodology

We perform our evaluation on torus and grid networks.

Definition 4 (Torus and grid). A N × N torus (resp. grid) network is a network such that:

• Each node has a unique identifier (i, j) with 1 ≤ i ≤ N and 1 ≤ j ≤ N .

• Two nodes (i 1 , j 1 ) and (i 2 , j 2 ) are neighbors if and only if one of these two conditions is satisfied:

i 1 = i 2 and |j 1 -j 2 | = 1 or N (resp. 1).

j 1 = j 2 and |i 1 -i 2 | = 1 or N (resp. 1).

These topological identifiers (i, j) are not to mismatch with the identifiers used in the protocol: according to our hypotheses, the nodes do not know their position on the grid.

An example of grid network is given in Fig 2 . A torus network can be seen as a grid network with a continuity between the left-right and up-down extremities.

We assume an uniform distribution of Byzantine failures: each node has the same probability λ to be Byzantine. We call this probability Byzantine rate. We would like to evaluate, for a given Byzantine rate λ, the probability P (λ) for a node to deliver the good information. For this purpose, we use a Monte-Carlo method:

• We generate several random distributions of Byzantine nodes, with the Byzantine rate λ.

• For each distribution, we randomly choose a source node s and a node v. Then, we use Theorem 1 and Theorem 2 to construct a reliable node set (see Definition 3). If v is in the reliable node set, the simulation is a success -else, it is a failure.

• With a large number of simulations, the fraction of successes approximates P (λ).

Il the following, we compare the performances of 4 different settings of our protocol:

• Setting A: (1, 2) • Setting B: (1, 2 , 5)
Probability of safety The probability to meet safety conditions increases with the complexity of the setting (2 paths for setting A, 3 paths for settings B and C, 4 paths for setting D). Indeed, as we use more paths, it becomes less likely to have a critical configuration of Byzantine nodes (see Figure 1-b).

This probability decreases with the size of the network: for a given Byzantine rate, a larger network has a greater frequency of critical configurations. Besides, the disparities between the different settings also increase with the size: the results on Figure 3 are more dispersed than on Figure 4.

At last, the results are slightly better for the grid topology: as the grid is less connected than the torus, a critical configuration is less likely to occur.

Border effects When the Byzantine rate equals zero (all nodes are correct), the probability to deliver the good information equals 1 for the torus topology, but not for the grid topology. Indeed, in the torus, each node has 4 neighbors, which is sufficient for the 2, 3 or 4 node-disjoint paths required by our settings. However, in the case of the grid, some border nodes have only 3 or 2 neighbors (see Figure 2). Therefore:

• The nodes with 2 neighbors cannot deliver any information with settings B, C and D.

• The nodes with 3 neighbors cannot deliver any information with setting D.

Also, we could exclude these problematic border nodes from our statistics, by dividing the probabilities by P (0). If we do so, the results are roughly equivalent for the torus and the grid: the grid offers better chances to meet safety conditions, but the construction of a reliable node set becomes slightly more difficult.

Optimal setting There seems to exist an optimal complexity for the setting of the protocol. Indeed, as we increase the number of paths, we increase the probability to meet safety conditions (according to Theorem 1), but we also make the construction of a reliable node set more difficult (according to Theorem 2). Therefore, there is a compromise to find between these two tendancies. This is illustred on the 10 × 10 torus, where setting D (the most complex setting) offer the best safety probability, but also the worst communication probability. Thus, setting C is the best setting for this network.

Besides, the size of the network seems to have an influence on this optimum. Indeed, on the 50 × 50 torus, setting D now offers the best communication probability for the Byzantine rates λ > 0.005. However, this is not the case for smaller Byzantine rates. Indeed, the neighbors of a Byzantine node cannot deliver the good information with setting D, as they have at most 3 correct neighbors (4 correct node-disjoint paths are required). Therefore, setting C achieves the best performances in both networks.

Comparison with existing solutions

At last, let us provide a quantitative comparison with existing solutions.

According to our hypotheses, there is no initial topology knowledge: the nodes do not know a priori their position on the grid. Therefore, our protocol must be compared with other Byzantine-robust protocols that respect this constraint, yet still work on such sparse networks. To our knowledge, the only protocols meeting these two conditions are Explorer [START_REF] Nesterenko | Discovering network topology in the presence of byzantine nodes[END_REF] and the previous protocol [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF].

Let us suppose that we want to guarantee a communication probability P (λ) of at least 0.99 on a 50 × 50 torus. Then, we can tolerate a Byzantine rate λ of:

• 4 × 10 -6 with an unsecured broadcast (see 2.1)

• 5 × 10 -5 with Explorer [START_REF] Nesterenko | Discovering network topology in the presence of byzantine nodes[END_REF] • 5 × 10 -4 with the previous protocol [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF] • 2 × 10 -3 with our protocol (improvement of factor 4)

These performances are to compare with the Control Zones protocol [START_REF] Maurer | Limiting byzantine influence in multihop asynchronous networks[END_REF], where the nodes are required to know their position on the grid. With this additionnal hypothesis, we can tolerate a Byzantine rate of 8 × 10 -3 . As we can see, releasing topology knowledge still has a cost in term of performances.

All these solutions are represented in Figure 5, on a logarithmic scale.

Conclusion

In this paper, we proposed a paremetrizable approach for Byzantine Broadcast in loosely connected networks, assuming a minimal number of hypotheses. We showed that, by manipulating protocol parameters, we can optimize performance in the presence of random Byzantine failures. However, not using topology knowledge still has a cost in term of Byzantine tolerance.

To go further, it would be interesting to make experimentations on real-life topologies, such as sensor networks, robot networks or peer-to-peer overlays. Also, an motivating open challenge is to obtain theoretical probabilistic guarantees with global network parameters -diameter, node degree, connectivity -in order to determine the optimal setting automatically.

• Setting D: (

According to Theorem 1, we have interest to use the smaller parameters (H 1 , . . . , H n ) for our settings. However, if we substract 1 to any parameter of the aforementionned settings, the construction of a reliable node set becomes impossible on grid and torus topologies. Therefore, the choice of these 4 settings corresponds to a certain optimum.

In particular, setting A, of the form (1, H), is the optimal setting for the previous protocol [START_REF] Maurer | On byzantine broadcast in loosely connected networks[END_REF] on grid and torus topologies. Our experiments show that it is outperformed by other settings.

Results

The experimental results are presented in Figure 3 and4.

• We ran simulations on two networks: a "small" network of 10 × 10 nodes (Figure 3), and a "large" network of 50 × 50 nodes (Figure 5).

• For both networks, we consider the torus topology (left column) and the grid topology (right column).

• For each case, we represented the probability to meet the safety conditions (no correct node can deliver a false information), then the probability P (λ) to deliver the good information.

Let us comment on these results. We study the three following aspects: the probability of safety, the border effects and the existence of an optimal setting.


Koji Hirano
email: hirano.koji@nsc.co.jp

Remy Fabbro

Maryse Muller

Experimental determination of temperature threshold for melt surface deformation during laser interaction on iron at atmospheric pressure

Keywords: Laser processing, Evaporation, Temperature measurement PACS: 42.62.-b, 81.20.-n, 47.80.Fg

Recoil pressure is the principal driving force of molten metal in laser processing in the intensity range of 10 -1 ~ 10 2 MW/cm 2 . It is thus essential to estimate the recoil pressure in order to describe physical processes or to carry out numerical simulations. However, there exists no quantitative estimation of the recoil pressure near the boiling temperature (T v ), which is particularly important in welding process. In this study we experimentally investigated the recoil pressure of pure iron around T v . The main interest was to determine the threshold surface temperature to start deformation of melt surface. Using camera-based temperature measurement with accurate evaluation of emissivity from experiment, it was shown that the surface temperature has to reach T v to initiate the melt surface deformation. This result provides the first experimental evidence for the frequently used assumption that a deep keyhole welding requires surface temperature over T v . It is indicated also that, in normal gas-assisted laser cutting process, the recoil pressure hardly contributes to material ejection when the surface temperature is lower than T v , as opposed to commonly believed presumption.

Introduction

A number of laser metal processing such as welding, drilling, grooving utilise the intensity range of 10 -1 ~ 10 2 MW/cm 2 , which are easily obtained by multi-kW class fibre, disk, Nd:YAG and CO 2 lasers. In this intensity range, materials are driven or removed in the form of liquid, while the fraction of vaporised material is limited. Understanding of hydrodynamics of molten material is thus essential to consider process efficiency or final quality achieved. Recoil pressure, which is sometimes called evaporation pressure or ablation pressure, is generated as a counter momentum to evaporation from the surface. It is widely accepted that the recoil pressure can be a dominant driving force of the molten material in various laser metal processing [START_REF] Von Allmen | Laser-Beam Interactions with Materials[END_REF][START_REF] Semak | The role of recoil pressure in energy balance during laser materials processing[END_REF], even though the evaporation itself contributes little to the mass removal.

It is well known that the recoil pressure exerted on the surface increases with the surface temperature T s . One may then pose a question: what is the minimum surface temperature to initiate the driving process of molten material? This question is very important, since this temperature determines limit of energy efficiency of various processes. Although there exist a few experimental measurements of the recoil pressure for several metal species [START_REF] Kuznetsov | Recoil momentum at a solid surface during developed laser ablation Quantum[END_REF][START_REF] Batanov | Evaporation of metallic targets caused by intense optical radiation[END_REF][START_REF] Apostol | Recoil impulse received by metal targets as a result of interaction with microsecond CO 2 laser radiation[END_REF][START_REF] Golovin | Some features of the evaporation of magnesium and lead by quasi-cw laser radiation[END_REF][START_REF] Lee | Analysis of recoil force during Nd:YAG laser ablation of silicon[END_REF], these are limited to the ablation regime with high laser intensity (typically > 1 MW/cm 2 for iron) well above the threshold of the ejection process. It should be noted also that in these studies, the recoil pressure was determined only as a function of laser intensity and it could not be related to the surface temperature. Consequently our understanding of the threshold temperature is not at all conclusive. Concerning a deep keyhole welding, for example, not a few studies have assumed that the keyhole surface temperature is equal to T v [START_REF] Dowden | The flow of heat and the motion of the weld pool in penetration welding with a laser[END_REF][START_REF] Kaplan | A model of deep penetration laser welding based on calculation of the keyhole profile[END_REF][START_REF] T J Colla | Heat transport in melt flowing past the keyhole in deep penetration welding[END_REF][START_REF] Solana | A mathematical model for penetration laser welding as a free-boundary problem[END_REF][START_REF] Zhao | Macroporosity free aluminum alloy weldments through numerical simulation of keyhole mode laser welding[END_REF][START_REF] Guen | 3D heat transfer model of hybrid laser Nd:Yag-MAG welding of S355 steel and experimental validation[END_REF], implying that this is the minimal temperature in order to carry out the keyhole welding process. In the well-known paper by Semak and Matsunawa [START_REF] Semak | The role of recoil pressure in energy balance during laser materials processing[END_REF], however, this assumption was denied. They claimed that the keyhole can be obtained at a temperature well below T v . This kind of debates is caused by the lack of information on the temperature dependence of recoil pressure, especially in the region near T v .

As for theoretical description of the recoil pressure, the expressions proposed by Anisimov [START_REF] Anisimov | Vaporization of Metal Absorbing Laser Radiation Sov[END_REF] and Knight [START_REF] Knight | Theoretical Modeling of Rapid Surface Vaporization with Back Pressure[END_REF] have often been used for numerical studies of laser welding, drilling and cutting [START_REF] Semak | The role of recoil pressure in energy balance during laser materials processing[END_REF][START_REF] Ki | J Modeling of Laser Keyhole Welding: Part I. Mathematical Modeling, Numerical Methodology, Role of Recoil Pressure, Multiple Reflections, and Free Surface Evolution[END_REF][START_REF] Pang | A three-dimensional sharp interface model for self-consistent keyhole and weld pool dynamics in deep penetration laser welding[END_REF]. As explained in detail in the next section, an important assumption in their analyses is that the surrounding ambient gas does not touch the evaporating surface and thus does not contribute as a partial pressure to the total pressure at the evaporating surface. In this case, the total pressure is composed uniquely of the recoil pressure of metal evaporation. This assumption should be valid for the temperature range well above the boiling temperature T v , where strong metal vapour jet emerging from the surface can completely expel the surrounding gas atoms. However, in the temperature range near T v , where only weak evaporation takes place, the assumption is possibly not appropriate and the contribution of ambient gas may have to be considered to evaluate the total pressure exerted on the evaporating surface. But the problem is that, there exists no theoretical model that can take into account this contribution of the surrounding gas atoms to the total pressure at the surface in the case of laser processing under atmospheric ambient pressure. Experimental studies are thus required to clarify the temperature dependence of the total pressure at the surface.

Basu and DebRoy [START_REF] Basu | Liquid metal expulsion during laser irradiation[END_REF] and later He et al. [START_REF] He | Liquid metal expulsion during laser spot welding of 304 stainless steel[END_REF] examined melt expulsion during laser spot welding of metals both experimentally and numerically. Their results indicated that ejection of spatters is initiated after melt surface temperature exceeds T v . This hints that the total pressure at the surface becomes large enough to eject the melt layer only in the evaporation regime. The result, however, does not provide us precise information of the temperature dependence of the total pressure near T v , because it is difficult to correlate the total pressure with the threshold of the spatter ejection. It should also be pointed out that the temperature distribution in their study was given by numerical simulations.

The present study investigates the total pressure exerted on the surface for the surface temperature range near T v with experimental measurement of the temperature for the first time. Evolutions of surface shape and surface temperature are monitored with a high speed video camera during single impact irradiation on a pure iron sample. Special attention is paid to determination of the threshold temperature at which the surface starts to be deformed due to the pressure. This point should be contrasted with the above-mentioned works [START_REF] Basu | Liquid metal expulsion during laser irradiation[END_REF][START_REF] He | Liquid metal expulsion during laser spot welding of 304 stainless steel[END_REF], where the melt ejection, which occurs after the deformation, was analysed. Our focus on the small initial melt surface deformation allows us to evaluate the corresponding total pressure more precisely and to determine the threshold temperature for surface deformation in a more fundamental sense. Difficulty of the temperature measurement should be one of the factors that have prevented this kind of fundamental investigation on the pressure at the surface. In this work, thermal radiation detected by the camera is processed using emissivity that is obtained experimentally, as reported in [START_REF] Kraus | Optical spectral radiometric method for measurement of weld-pool surface temperatures[END_REF]. This accurate determination of emissivity enables us to reduce substantially error in the temperature measurement.

The outline of this paper is briefly summarised as follows. In the next section 2, theoretical aspects of the recoil pressure and the total pressure at the surface are described. The experimental methods of the surface shape and temperature measurements are explained in section 3, and the results are presented in section 4. In section 5 we discuss the consequences of the results on our understandings of physical processes during laser welding and cutting.

Theoretical background

It is schematically shown in figure 1 how the pressure applied on the surface deforms and ejects the molten material during a single laser pulse irradiation. We call the pressure measured at the laser beam centre the "total pressure" P tot in this paper. It is called "total", because in general it is expressed as the sum of two partial pressures:

g r tot p p P   (1)
where the first contribution p r comes from the conventional recoil pressure as a counter force of metal evaporation from the surface and the latter p g is the partial pressure of surrounding gas.

The value that is important to drive the melt is the pressure difference P between P tot at the centre and the pressure at the periphery of the melt layer. The latter can be given by the ambient pressure P amb , because the periphery is normally out of the laser beam and so the evaporation does not occur. Thus

amb tot P P P    (2)
The ambient pressure P amb can vary but is usually the atmospheric pressure (1.01 x 10 5 Pa).

Using (1) and (2), one obtains

amb g r P p p P     (3)
Please note that p g is the partial pressure of surrounding gas measured locally at the target surface. Therefore, p g can be different from P amb and the two pressures do not generally cancel to each other.

The pressure difference P is utilised to counteract the surface tension yielded from a deformation and to eject the material. Let us consider the case of figure 1, where the central part is depressed due to P and the melt in the centre is redistributed to the periphery as an extruded rim [START_REF] Semak | Effect of surface tension on melt pool dynamics during laser pulse interaction[END_REF]. The curvatures of the melt deformation in the central and rim parts are expressed by R and r m , respectively. As shown in figure 1, R and r m are defined to be positive when the centres of the curvatures are situated above and below the melt surface, respectively. (In conditions of this study, both R and r m become positive at any time.) When the melt is ejected from the rim with the velocity V m , the pressure balance can be written using the Bernoulli equation.

2

2 2 m m m V r R P        (4)
Please note that the factor 2 in front of (/R) comes from the fact that the deformation at the centre consists of the two same curvatures along two orthogonal directions on the sample surface.

In the following, we will examine dependence of p r and p g in P (eq.( 3)) on the surface temperature T s .

The recoil pressure p r is a monotonously increasing function of the surface temperature T s . Analytic expressions of p r have been proposed by a number of authors. In a general form, the pressure can be approximated as [START_REF] Samokhin | First-order phase transitions induced by laser radiation in absorbing condensed matter[END_REF]  

s sat R r T p p 2 1    (5)
where p sat is the saturated pressure at T = T s . It can be obtained from the well-known Clausius-Clapeyron's relation:

                   s v v B v sat T T T k H p p 1 exp 0 (6)
where T v is the vaporisation temperature under the pressure p 0 and v v mL H   [START_REF] Lee | Analysis of recoil force during Nd:YAG laser ablation of silicon[END_REF] is the enthalpy of phase transition from liquid to vapour per atom (m: mass per atom, L v : latent heat of vaporisation, L v = 6.1 x 10 6 J/kg for iron). The coefficient  R represents the fraction of re-condensing particles to evaporating ones. By its definition,  R is related to the mass flow balance. The total mass flux from melt layer to gas phase can be estimated from modified Langmuir expression [START_REF] Samokhin | First-order phase transitions induced by laser radiation in absorbing condensed matter[END_REF][START_REF] Bellot | Evaporation of Fe and Cr from Induction-stirred Austenitic Stainless Steel. Influence of the Inert Gas Pressure[END_REF]:

    s sat s B R v m T p T k m V 2 / 1 2 1              (8)
Here k B is the Boltzmann constant.  R is influenced by gas flow of evaporating particles above the surface. It depends on T s , P amb , three-dimensional surface profile around the considered surface point, and so on. When T s ≈ T v it is expected that the evaporation is not so strong and  R ≈ 1 (so p r ≈ p sat ). With the increase of T s ,  R decreases while the upward velocity of metal vapour flow increases until it reaches the sound velocity. In this limit,  R becomes 0.18 [START_REF] Anisimov | Vaporization of Metal Absorbing Laser Radiation Sov[END_REF][START_REF] Knight | Theoretical Modeling of Rapid Surface Vaporization with Back Pressure[END_REF][START_REF] Samokhin | First-order phase transitions induced by laser radiation in absorbing condensed matter[END_REF]. According to eq.( 5), p r = 0.6p sat in this regime. The equations ( 5) and [START_REF] Dowden | The flow of heat and the motion of the weld pool in penetration welding with a laser[END_REF] assume that atoms which recombine to the surface have the same temperature T s as the evaporating ones. This assumption is valid only in weak evaporation regime where  R ≈ 0, but comparison with rigorous expressions in [START_REF] Knight | Theoretical Modeling of Rapid Surface Vaporization with Back Pressure[END_REF], which take into account temperature decrease of the recombining atoms, shows that the error of p r caused by the approximate expression [START_REF] Apostol | Recoil impulse received by metal targets as a result of interaction with microsecond CO 2 laser radiation[END_REF] is less than 10%. Now let us consider p g in eq. [START_REF] Kuznetsov | Recoil momentum at a solid surface during developed laser ablation Quantum[END_REF]. It is considered that, when evaporation starts, p g becomes smaller than P amb , because evaporating metal particles reduces the number of surrounding gas atoms that impinge onto the surface. The only exception is processing under vacuum, where p g = P amb = 0 at any T s . In most cases, where a metal target is placed under atmospheric pressure of 1.01 x 10 5 Pa, the estimation of P is rather complicated, due to the fact that p g and P amb do not cancel to each other. This point is important for the following discussions. The characteristics of p g and resulting P can be estimated for the two extreme cases of low and high temperatures with respect to T v . First let us consider the case where T s is so small that p r ≈ 0 (see eqs.( 5) and ( 6)). In this case, there are very few evaporating metal particles. It can thus be assumed that p g = P amb = 1.01 x 10 5 Pa. From eq.( 3) one obtains P = p r ≈ 0, which means that there is no driving force of molten material. On the other hand, if T s reaches the strong evaporation limit of  R = 0.18, the intense metal vapour flow prevents the surrounding gas atoms from reaching the surface. Thus p g ≈ 0 and consequently P ≈ p r -P amb = 0.6p sat -P amb . In figure 2, P in this limit is shown with the curve a for the case of iron. The problem is that we do not have enough information on P in the intermediate temperature range near T v , where with the increase of T s , p g decreases from P amb to zero and p r gains from zero depending on the magnitude of the unknown parameter  R .

There exists no theoretical model that can predict this variation of P around T v in the case of processing under ambient pressure. The first analytical model by Anisimov [START_REF] Anisimov | Vaporization of Metal Absorbing Laser Radiation Sov[END_REF] was limited to the case of vaporisation into vacuum. Knight proposed another model [START_REF] Knight | Theoretical Modeling of Rapid Surface Vaporization with Back Pressure[END_REF], which can take into account the existence of ambient gas. It was assumed, however, that ambient gas is completely screened out from the evaporating surface by metal vapour that covers totally the surface, which corresponds to the case of p g = 0. As mentioned above, this assumption should be valid in strong evaporation regime, but its validity is not conclusive for weak evaporation in the vicinity of T v . The consideration of mixing of the ambient gas and the metal vapour is still a challenging problem even with modern numerical techniques and there are very few works. Bellot and co-workers [START_REF] Bellot | Evaporation of Fe and Cr from Induction-stirred Austenitic Stainless Steel. Influence of the Inert Gas Pressure[END_REF] numerically solved the Boltzmann equation to study the influence of surrounding gas pressure and geometrical scale factor for the evaporation problem of Cr from iron liquid. Unfortunately, the partial pressure of the surrounding gas they used, was limited approximately under 100 Pa. Due to the lack of theoretical model, experiments are indispensable to estimate the pressure P in the temperature region near T v .

It should be pointed out that little attention has been paid to the contribution of ambient gas on P, which has been proposed as in eq.( 3). In a number of papers dealing with recoil pressure during laser material processing, the following type of expressions has been used.

r p P   (9)
This should be a good approximation, when T s is so high that p g = 0 and p r >> P amb . However, in order to describe lower temperature regime, such as the regime of a deep penetration keyhole welding, the validity of eq.( 9) must be questioned, since it implicitly assumes that p g = P amb . As already mentioned, in the case of processing under atmospheric pressure, p g must decrease from P amb , when evaporation starts.

Methods of experiments

Schematic of the experimental setup is shown in figure 3. We investigated evolutions of surface shape and temperature during a single impact irradiation. The position of the laser beam was fixed during each irradiation. A disk laser (Trumpf, TruDisk10002) was utilised as the laser beam source. A non-polarised laser beam with the wavelength  L of 1030 nm, which was transmitted through an optical fibre with the core diameter of 600 m, was imaged on a workpiece with a combination of a collimator lens (f = 200 mm) and a focusing lens (f = 560 mm). The obtained focus diameter was 1.6 mm and the focus position was always set at the sample surface. The beam profile at the focus position was analysed with a commercial CMOS camera and the top-hat distribution of the intensity was confirmed. The samples were pure iron (Goodfellow FE000480; purity 99.95%). Their surfaces were polished with sand papers (# 2400) and cleaned with ethanol before the laser irradiation. Argon gas was provided as shielding gas to avoid undesirable oxidation. Different configurations were used to observe the evolutions of surface shape and surface temperature, with the above-mentioned processing conditions fixed.

As for the observation of the surface shape, we illuminated the surface with a halogen lamp and observed a reflected image with a high speed video camera (Photron, FASTCAM/APX-RS). As shown in figure 3(a), the angles of the illumination and the observation are set at 45 degrees. The spatial resolution per pixel was 6.0 m x 8.5 m. We placed a standard glass filter of the type KG3 in front of the camera lens to block the laser beam reflected from the irradiation spot. Therefore the whole visible spectrum region was utilised. The acquisition rate was set at 20 kHz and the shutter speed was 2 s.

Two-dimensional distribution of surface temperature was monitored using a camera-based technique, which was reported for example in [START_REF] Kraus | Optical spectral radiometric method for measurement of weld-pool surface temperatures[END_REF]. Thermal radiation from the surface was detected by a high speed video camera. Based on the Planck's law, the surface temperature was calculated from the measured radiation level and the emissivity that was obtained experimentally.

Thermal radiation level was measured with a CMOS sensor of the above-mentioned high speed video camera. We used the same optical system, observation angle (45 degrees) and acquisition rate as the observation of the surface shape, which allowed us the high spatial resolution (6.0 m x 8.5 m) and sampling rate (20 kHz). The differences from the surface shape observation were: we did not use the lamp for illumination; we selected a near infrared (NIR) spectrum band as specified below; and we used an adapted shutter speed (50 s).

According to the Planck's radiation theory, the grey scale level G measured at a pixel of the CMOS sensor is generally expressed as

    T B T K G , 0    (10)
K is a constant, which depends on characteristics of the observation system, for example, numerical aperture of the lens system, transmittance of the thermal radiation from the laser spot to the sensor surface, shutter speed and the sensibility spectrum of the sensor. (  ) is the emissivity of the surface at the central wavelength   of the spectrum band selected. B(T) represents the temperature dependence of the black body radiation [START_REF] Solana | A mathematical model for penetration laser welding as a free-boundary problem[END_REF] where is the width (FWHM) of the selected band, and h and c are the Planck constant and the speed of light, respectively. B(T) can easily be calculated from eq.( 11), so that once the constant K and  are obtained, the surface temperature T can be determined from the grey scale level G.

                 2 2 5 2 0 0 1 exp 1 2        T k hc hc d T B B
The emissivity  is a function of wavelength and of temperature, which is then a function of time. Based on the Kirchhoff's law, the temporal evolution of  was estimated from that of the absorptivity A, which was evaluated for the process laser wavelength  L (1030 nm), assuming that

        t T A t T L , , 0     (12)
It can be mentioned here that this correspondence of  and A has been recently utilised for monitoring of keyhole front profile during laser welding [START_REF] Weberpals | Utilization of Quantitative Measurement Categories for Process Monitoring Proc ICALEO[END_REF]. Considering that in depends on the wavelength, it is better to use   near  L and small  for the NIR radiation observation by the video camera. We selected a spectrum band of   0.9 m and = 0.07 m using a band pass filter.

The setup for the measurement of A is shown in figure 3(b). The sample was placed just under an integrating sphere for the measurement. More details of the absorptivity A variation with operating parameters will be discussed in a future publication. For the present study, the evolution of A was measured using exactly the same laser power and duration as those of the NIR radiation observation. The incidence angle was set at 10 degrees in order to capture the first reflection from the sample within the sphere. The angle was different from that of the NIR observation (45 degrees), but the influence is negligibly small. According to the theoretical Fresnel formula and the complex refraction index of iron, the difference of  (or A) between 10 degrees and 45 degrees is very small for a non-polarised beam; less than 1% [START_REF] Dausinger | Energy Coupling Efficiency In Laser Surface Treatment[END_REF].

The constant K of this observation system was calibrated at the melting points of Ta (3290 K) and Nb (2750 K) through the same scheme of the evaluations of G and A as the measurement for pure iron. As is deduced from eq.( 10), systematic error in a measured temperature can come from errors in K and . The error in the determination of K in the calibration processes with Ta and Nb was about +/-7 % (= K/K). This leads to the error of +/-50 K in T s if it is evaluated at T s ~ T v . The experimental measurement of A allowed us to minimise the error of  within +/-3 %, which corresponds to the error of +/-50 K in T s . By adding the two contributions, the total error in T s is estimated to be +/-100 K.

The present method of temperature measurement applies only to the case where the deformation is so small that one observes only a component of NIR light that is emitted from the surface and directly transferred to the video camera. When the deformation becomes large, for example, as in the case of a keyhole, reflected component, which is emitted from another point, can be overlapped to the observed light in addition to the direct emission. The method in this study is based on a presumption that there is no such contribution from these reflections. In order to measure temperature also for the case where single or multiple reflections occur, it would be necessary to make the observation of the NIR image with the same angle as the incidence angle of the absorptivity measurement, for example, by installing the camera coaxially to the laser beam axis.

While a laser pulse is irradiated on the sample surface the surface temperature T s is continuously elevated or stagnated. The surface shape strongly depends on P (= P tot -P amb ), which is a function of T s as explained above. Although the evolution of T s is a dynamic phenomenon, it is considered that P tot is dependent only on the instantaneous value of T s , independent from any dynamical factor. This assumption is justified because the time needed for equilibrating P tot for each instance is estimated to be as small as the order of 20 s, if one estimates from (L/v s ) [START_REF] Kaplan | Unbounded keyhole collapse and bubble formation during pulsed laser interaction with liquid zinc[END_REF], where L is the characteristic relaxation length of the uprising vapour (≲ 10 mm) and v s is the sound velocity (≈ 500 m/s). As shown later, the characteristic time of the evolution of T s is much longer in our experimental conditions.

Apart from the observations of deformation and temperature, in order to estimate the recombination rate  R in the eqs.( 5) and ( 8), mass loss by evaporation was evaluated by measuring weight of samples before and after irradiation using a commercial balance (METTLER TOLEDO AT261). The precision of each weight measurement was +/-0.02 mg.

Using the high speed video camera we confirmed that there was no droplet detached from the surfaces during the laser beam irradiation, which would have caused overestimation of the evaporation rate.

Results

Evolution of melt surface shape

Depending on the level of T s , and thus on the level of P tot , we can observe the melt pool evolution during the pulse irradiation in three stages: (i) flat, (ii) deformation, and (iii) key-hole.

In the following, two typical conditions are shown. One is P = 2 kW, t = 80 ms, for which the system stagnates in the stage (ii). In the other condition of P = 4 kW, t = 25 ms, the system gets into the final keyhole stage (iii).

The evolution of surface shape observed for P = 2 kW and t = 80 ms is shown in figure 4(a). Please note that information on surface profile tends to be amplified with the illumination by the halogen lamp. From the observed video images, it was confirmed that melting starts at time t = 1.7 ms from the centre of the laser beam spot. The melted region spread with the elapse of time after that. In this first stage the liquid surface stays almost flat but is never stationary. As shown in the image of t = 7 ms, ring patterns, such that we observe when a raindrop plunges into water surface, were frequently observed. The detection of this phenomenon was made possible owing to the fast shutter speed of the camera (2 s).

The second regime sets in at t = 17 ms, when one can confirm the initiation of concave deformation in the centre. The diameter d and the depth h of the deformation area gradually increase with time until the end of the pulse. The depth h was estimated from small displacement of the centre of the circular deformed region that can be observed due to the inclined angle of visualisation of the high speed video camera (Figure 3(a)). Due to the inclination angle of 45°, h can be estimated by the displacement distance d p of the centre measured in the plane of the sample surface (refer to an example at t = 80 ms in figure 4(a)). At t = 60 ms, the diameter d of the deformed region was 1.1 mm +/-0.2 mm and the depth h was 80 m +/-20 m.

Let us estimate the pressure difference P from the values d and h measured at t = 60 ms. Using eq.( 4) with the ejection velocity V m = 0, the pressure difference P is estimated from the following equation

m r R P      2 (13)
The curvature R of the liquid surface at the bottom can be estimated from the geometrical formula R = ((d/2) 2 + h 2 )/2h, which gives R = 2 +/-1 mm. In order to estimate the curvature r m of the rim, we assume that the rim extruded above the sample surface is represented by a part of a torus which is cut at the sample surface, as in ref. [START_REF] Semak | Effect of surface tension on melt pool dynamics during laser pulse interaction[END_REF]. This extruded volume should be provided from the depressed volume in the central part. This volume balance can be approximated as

                     2 2 2 2 1 2 3 1 2 w d wh h d m   (14)
The width w is measured to be 0.5 +/-0.1 mm from the image at t = 60 ms. Substituting this value to eq.( 14), one obtains h m ≈ 0.02 +/-0.01 mm. Then r m is estimated to be 1.5 +/-1 mm from the geometrical formula r m = ((w/2) 2 + h m 2 )/2h m . Using ≈ 1.5 Pa m, we finally obtain P ≈ 3 +/-2 x 10 3 Pa. (We neglect temperature dependence of .)

Images observed with P = 4 kW and t = 25 ms are shown in figure 5. Melting starts at t = 0.4 ms, and the second deformation stage begins at t ≈ 4 ms. The concaved surface is modulated with small wavy structure as one can confirm in the image of t = 6.5 ms. Unlike the case of P = 2 kW, the evolution of the surface profile is not stationary, and finally at t ≈ 15 ms, a keyhole is produced.

Evolution of melt surface temperature

Because the method of temperature measurement cannot be applied to the case of strong deformation, we present only the result for the condition of P = 2 kW and t = 80 ms. Observed NIR images are shown in figure 4(b). Results of the absorption measurement and the calculated surface temperature T s are plotted in figure 6 as a function of time. The grey scale level G was evaluated as an average in a small circle at the beam centre with the diameter of 0.2 mm. Variation of the temperature within the circle was typically +/-10 K. Considering that this variation can generally be caused by real temperature variation and/or noise of each pixel, it can be said that the value (+/-10 K) defines the maximal value of the temperature variation.

As shown in figure 6, the temperature T s in the central region increases and reaches T s = 3250 K at t = 17 ms and stagnates after that at about T s = 3270 K. This characteristic rising time of T s can be explained by the following 1D analytical solution of heat conduction problem for surface heating of a semi-infinite body with constant intensity [START_REF] Carslaw | Conduction of Heat in Solids 2[END_REF]. The temperature increase at the centre on the surface is given by

 t K AI T T L a s 2   (15)
where I L is the incident laser intensity (0.1 MW/cm 2 ), K is the heat conductivity (80 W/m K),  is the thermal diffusivity (2 x 10 -5 m 2 /s), and T a is the ambient temperature (300 K). This 1D equation neglects three-dimensional thermal diffusion, which is estimated to start at  ~ 10 ms based on the relation 2() 1/2 ≈ (D/2). We neglect the latent heats of melting and evaporation in this discussion. The assumption of no evaporation is justified, since it was experimentally confirmed that evaporated mass is negligibly small until t = 17 ms, as is shown later. Using the experimentally obtained absorptivity (A ≈ 0.35), the rising time to T s = 3250 K is estimated to be 17 ms, which shows a good agreement with the experiment in spite of the approximations.

The observation of surface shape showed that the surface deformation starts at t = 17 ms. Thus the threshold surface temperature for the melt surface deformation is determined to be 3250 K +/-100 K. It can be concluded that the surface deformation starts just above T v (3134 K).

Whereas T s at the centre stagnates, the 2D temperature field continues to evolve even after t = 17 ms. Further analysis showed that the region with the maximum temperature (= 3270 K) slightly spreads with time until t ≈ 60 ms. The diameter of this region stays constant at about 0.5 mm for t = 60 ~ 80 ms. The temperature distribution was quasi-stationary during this period.

Evaporation rate

Mass loss by evaporation measured for the case of P = 2 kW and t = 80 ms is shown in figure 7. Comparison with the evolution of T s suggests that the mass loss by the evaporation only starts after T s surpasses T v . The mass loss rises roughly proportionally to the pulse duration after this moment. The mass evaporation rate in the quasi-stationary regime (t = 60 ~ 80 ms) can be determined as m  = 1.4 x 10 -5 kg/s from the gradient of the plots. Based on the modified Langmuir eq.( 8), the recombination rate  R was calculated from the following relation  dS

T p T k m m s sat s B R            2 / 1 2 1    (16)
with the expression of p sat taken from eq.( 6) and the integrated area defined by the beam diameter. For T s , the temperature distribution at t = 60 ms was used. The obtained recombination rate  R was 0.9. Now let us discuss about the stagnation of T s at the centre observed after t = 17 ms. It is considered that the stagnation of T s at the centre is caused by increase of three-dimensional thermal diffusion and reduction of heat flux into the sample due to evaporation heat loss. If we consider surface heating of a semi-infinite body with a circular beam that has the intensity I abs and the diameter D, the analytical solution predicts that the temperature increase at the centre stagnates at (I abs D/2K) because of the heat conduction. Taking also into account the reduction of the incoming energy flux due to evaporation loss, the temperature increase in the stationary regime is estimated from

  DK L m AP T T v L a s      2 (17)
This equation gives T s = 3.5 x 10 3 K, which is close to the stagnation level observed experimentally.

Discussion

Temperature dependence of P

As explained above, the pressure difference P at t = 60 ms is estimated to be P ≈ 3 +/-2 x 10 3 Pa, from the result of the surface shape observation. The measurement of surface temperature shows that T s is quasi stationary at 3270 K around t = 60 ms. These results add an experimental point in figure 2, which shows the dependence of P on T s for the case of processing under P amb = 1.01 x 10 5 Pa. The important conclusion is that the melt surface deformation or the melt ejection due to the recoil pressure is obtained only above T v , where P becomes large enough. This finding allows us to understand better physical processes during laser welding and cutting, as discussed below in the sections 5.2 and 5.3.

The recombination rate determined from the mass loss measurement ( R = 0.9) may seem to be high, but similar results can be found in the past works. Mundra and DebRoy [START_REF] Mundra | Calculation of Weld Metal Composition Change in High-Power Conduction Mode Carbon Dioxide Laser-Welded Stainless Steels[END_REF] measured mass evaporation rate during conduction mode laser welding of stainless steel. It was reported that the evaporation rate of iron was only 15% of the value predicted from the Langmuir equation at T s ≈ 3200 K, which is about the same as the temperature in our case (3270 K), although their temperature was not measured experimentally but estimated from a numerical simulation. The rate  R in their work was 0.85, which is very close to our result. As for numerical investigations of  R , Bellot et al. [START_REF] Bellot | Evaporation of Fe and Cr from Induction-stirred Austenitic Stainless Steel. Influence of the Inert Gas Pressure[END_REF] studied the dependence of  R on ambient Ar pressure for evaporation of Cr from liquid iron surface in near vacuum condition. It was shown that, when the Ar pressure becomes comparable to the saturated pressure of Cr,  R for Cr evaporation rises up and approaches 1. These previous results support the rather high value of  R obtained in this study.

It must be mentioned, however, that the experimentally obtained P and  R at T s = 3270 K are not completely consistent with the theoretical prediction. Indeed, using  R = 0.9 obtained at T s = 3270 K, p r is calculated to be 1.6 x 10 5 Pa from eq.( 5). Considering that P amb =1.0 x 10 5 Pa and p g  0, the right hand side of eq.( 3) is equal to or larger than 6 x 10 4 Pa, which is larger than P (= 3 x 10 3 Pa). The errors in the experimental measurements of T s and m  might have caused this inconsistency. Nevertheless, the fact that the term (p r -P amb ) in the right hand side of the eq.( 3) is already larger than P suggests that p g ≈ 0. This means that the shielding Ar atoms are expelled from the surface of the beam spot already at T s = 3270 K. This indicates that the effect of the ambient atmospheric pressure at T s ~ T v is not the direct contribution as the partial pressure to the total pressure at the surface, but the confinement of metal vapours, which results in the high value of  R (= 0.9) compared with the vacuum case ( R = 0.18). Now let us discuss about the expression of P as a function of T s . Assuming that  R = 0.9 and p g =0, the pressure difference P in the range of T s ~ T v may be expressed as (0.95p sat (T s ) -P amb ), which is very close to the curve b (p sat (T s ) -P amb ) in figure 2. As already mentioned in section 2, when T s increases from T v , it is predicted that  R decreases down to 0.18 and correspondingly P approaches (0.6p sat (T s ) -P amb ), which is shown with the curve a in figure 2. Further investigation on the dependence of  R on T s is out of the scope of this work, it may be worthwhile to mention that the asymptotic convergence at high temperature depends more or less on geometry around the considered surface point. Aden et al. [START_REF] Aden | Laser-induced vaporization of a metal surface[END_REF][START_REF] Aden | Laser-induced vaporization of metal as a Riemann problem[END_REF] theoretically analysed metal vapour expansion dynamics in 3D geometry. It was shown that with the increase of the laser intensity and that of T s ,  R approaches ~ 0.2. It was also revealed that, when the beam diameter becomes larger,  R reaches this limit at higher laser intensity and that this delay is most pronounced at the extreme of 1D case. Similar effect of evaporating surface size on the magnitude of  R was also reported in [START_REF] Bellot | Evaporation of Fe and Cr from Induction-stirred Austenitic Stainless Steel. Influence of the Inert Gas Pressure[END_REF]. It was shown that, when the area of the evaporating surface becomes small, one needs higher ambient pressure of Ar to restrict the evaporating mass of Cr, that is, to keep  R at a high level. Apart from this surface size effect, we point out that the global geometry can also influence  R . For example, in the case of a deep keyhole welding, it is considered that  R effectively increases, since metal atoms vaporised from a keyhole wall can be absorbed again to the other side of the wall surface. As a result, P will reach the strong evaporation limit of (0.6p sat (T s ) -P amb ) at higher T s compared with an open geometry.

As for practical interests such as boundary conditions of pressure for numerical simulations of melt pool dynamics driven by the recoil pressure, the variation of  R from 1 to 0.18 with increase of T s is possibly not so important, because the temperature difference between the curves a and b for a certain P level is only ~ 100 K, as shown in figure 2. Therefore, it can be sufficient to use the curve a (0.6p sat (T s ) -P amb ) with extrapolation down to T s ~ T v , if one accepts the corresponding error. The measured point (P = 3 x 10 3 Pa at T s = 3270 K), which lies near the curve a, supports the validity of this approximation. It should be emphasised that, in order to conduct accurate simulations, what is much more important than the variation of  R is to really take into account the ambient atmospheric pressure P amb , as discussed below.

Welding

The present result allows us to correct our common understandings on some processes in which T s becomes comparable to T v . First let us discuss consequence on a deep penetration keyhole welding. The minimum pressure difference P to carry out a keyhole welding can be estimated from the following relation.

2 2 m m k V d P      (18)
where d k is the diameter of the keyhole. For example, if we take d k = 1 mm and consider the limit of low velocity (V m ≈ 0), P = 3 x 10 3 Pa. According to figure 2, it is predicted that T s must be at least in the order of T v to keep the keyhole open. In theoretical analyses of a deep penetration keyhole welding, it has often been assumed that the temperature T s of the keyhole surface is equal to T v . The present result provides the first experimental support to this assumption.

We have to point out, however, that this assumption has been sometimes doubted in the past. For example, Semak and Matsunawa [START_REF] Semak | The role of recoil pressure in energy balance during laser materials processing[END_REF] denied this assumption and claimed that the recoil pressure can trigger lateral ejection at a lower temperature even near the melting temperature. Using the Bernoulli type equation (the equation number [START_REF] T J Colla | Heat transport in melt flowing past the keyhole in deep penetration welding[END_REF]

in [2]) 2 2 m m r V p   (19)
they claimed that a meaningful ejection velocity can be obtained even for T s < T v . As already discussed in section 2, the equation ( 19) is based on the implicit assumption that p g = P amb . This assumption is not correct according to the present result which suggests that p g ≈ 0 at the threshold temperature for melt surface deformation.

It is predicted that the threshold temperature for keyhole welding strongly depends on the ambient pressure. In figure 2, we plot P = p r for the extreme case of processing under vacuum, assuming that  R = 0.18 (the curve c in figure 2). P can take non-zero value even when T s < T v . For example, the above-discussed deep keyhole welding with the diameter of 1 mm, which requires P = 3 x 10 3 Pa, will be realised at T s ≈ 2600 K. This decrease of the process temperature must be one of the reasons why the penetration depth achieved in a deep penetration welding under vacuum is larger than the case under the atmospheric pressure for the same laser condition [START_REF] Katayama | Effect of vacuum on penetration and defects in laser welding[END_REF][START_REF] Mueller | Laser Beam Welding in Sub-Atmospheric Pressure[END_REF].

The same discussion applies also to boundary condition of pressure in numerical simulations of a keyhole welding process, which have already been reported by several groups [START_REF] Ki | J Modeling of Laser Keyhole Welding: Part I. Mathematical Modeling, Numerical Methodology, Role of Recoil Pressure, Multiple Reflections, and Free Surface Evolution[END_REF][START_REF] Pang | A three-dimensional sharp interface model for self-consistent keyhole and weld pool dynamics in deep penetration laser welding[END_REF][START_REF] Geiger | A 3D transient model of keyhole and melt pool dynamics in laser beam welding applied to the joining of zinc coated sheets[END_REF]. The result of our study highlights the importance of imposing the ambient atmospheric pressure P amb to non-evaporating surfaces, although this boundary condition was not clearly described in these papers. If one omits P amb for the non-evaporating surfaces, the calculation will represent the case of welding under vacuum, and it will predict that surface temperature threshold for keyhole regime can be lower than T v .

Cutting

We have to correct our understanding also for laser cutting process. In some of the analytical models for laser cutting, the recoil pressure P recoil have been taken into account as an important mechanism of melt ejection in lateral direction from the kerf front to the side region [START_REF] Petring | Anwendungsorientierte Modellierung des Laserstrahl-schneidens zur rechnergestützten Prozeßoptimierung[END_REF][START_REF] Mas | Steady-state laser cutting modelling[END_REF][START_REF] Quintero | Theoretical analysis of material removal mechanisms in pulsed laser fusion cutting of ceramics[END_REF]. It has been shown that the fraction of the lateral ejection can be the same order as that of the longitudinal ejection by downward assist-gas force, even when the surface temperature T s is well below T v [START_REF] Petring | Anwendungsorientierte Modellierung des Laserstrahl-schneidens zur rechnergestützten Prozeßoptimierung[END_REF][START_REF] Mas | Modelisation physique du procede de decoupe de metaux par laser[END_REF]. Our present result shows, however, that the lateral ejection by the recoil pressure cannot occur when T s < T v . A recent experimental investigation of temperature distribution on the kerf front [START_REF] Onuseit | Space-resolved Spectrometric Measurements of the Cutting Front[END_REF] indicates that T s reaches T v only for relatively high cutting speed near the process limit and this is restricted in a bottom part of the kerf. The absence of lateral ejection suggests that the downward melt flow by assist gas should be the only principal mechanism of melt ejection from the kerf. This was supported by our recent observation [START_REF] Hirano | Experimental investigation of hydrodynamics of melt layer during laser cutting of steel[END_REF] that the melt flows downwards with little lateral velocity component except for a high velocity range where T s is possibly comparable to T v . In the lowest velocity range, we observed interaction between melt accumulations in the central and side parts of the kerf front [START_REF] Hirano | Experimental investigation of hydrodynamics of melt layer during laser cutting of steel[END_REF]. It should be noted, however, that this interaction does not originate from the recoil pressure but from surface tension. Of course, the lateral ejection induced by the recoil pressure can be important when the surface temperature T s becomes higher than T v . For instance, in remote laser cutting of steel, where assist-gas is not utilised, the ejection by the recoil pressure should be the principal mechanism of melt removal from kerfs. Also in the case of normal gas assisted cutting, it is reported that the lateral ejection occurs depending on the surface temperature in pulsed laser cutting of ceramics [START_REF] Quintero | Theoretical analysis of material removal mechanisms in pulsed laser fusion cutting of ceramics[END_REF].

Future prospects

Whereas only pure iron samples were utilised in this study, it will be interesting to investigate the recoil pressure using practical materials with impurity elements. It is well acknowledged that volatile elements, which have lower boiling temperature than matrices, can cause process instabilities and defects. Fundamental investigations of the effect of additional elements on the characteristics of the recoil pressure will serve to understand the mechanisms of these problems.

We demonstrated a fast 2D measurement of melt surface temperature up to the range near T v . This technique can also be applied to real time process monitoring or to development of numerical models, although at the present stage the technique is restricted to relatively flat surface, for example, melt surface around a keyhole. An important finding in this study is that the emissivity is almost independent of T s up to ~ T v . Consequently, even without measurement of the emissivity, one can estimate T s with reasonable accuracy from thermal radiation intensity detected by a 2D sensor.

Conclusion

In this work we studied recoil pressure of iron around its boiling temperature T v during laser processing under atmospheric pressure. Single laser pulse irradiation on iron was experimentally investigated with a specific aim to determine the threshold temperature of the melt surface deformation. The camera-based 2D temperature measurement technique, where thermal radiation intensity was processed by emissitivity obtained from experiments, was applied for the first time to investigate the fast evolution of melt surface temperature T s up to the evaporation regime. It was revealed that the deformation starts at T s = 3250 K +/-100 K, which is just above T v . Furthermore, the experimental determination of the recombination rate  R in the evaporation process clarified the role of ambient atmospheric pressure on the total pressure exerted on the surface. That is, at T s ~ T v , ambient gas atoms do not contribute to the total pressure at the surface by direct impingement, but they confine metal vapours, which results in the increase of the recombination rate  R . The determination of the threshold temperature for melt surface deformation leads to some important conclusions for physical mechanisms in practical laser processing. It provides the first experimental support for the assumption that the temperature on the keyhole surface during deep penetration laser welding must be at least T v . As for laser cutting, it is suggested that lateral melt ejection process should be negligible as far as T s does not reach T v .

Figure 1

 1 Figure 1 Pressure induced deformation of melt layer during irradiation of single laser pulse.

Figure 2

 2 Figure 2 Pressure difference P as function of the surface temperature T s . The two curves a and b show theoretical equations in the case of processing of iron under P amb = 1.01 x 10 5 Pa (a: P = 0.6p sat (T s ) -P amb , b: P = p sat (T s ) -P amb ). The curve a corresponds to the asymptotic curve in the limit of low  R . The curve c shows the theoretical curve under vacuum conditions (P = 0.6p sat (T s )). The level of P = 3 x 10 3 Pa corresponds to the minimal pressure difference needed to keep open a keyhole with the diameter of 1 mm. The rectangular dot (■) is the experimental point obtained in this work (see the sections 3-5 for details).

Figure 3

 3 Photo diode

Figure 4

 4 Figure 4 Temporal evolutions of surface shape (a) and NIR emission (b) observed with the high speed video camera in the case of P = 2 kW and t = 80ms. In the image at t = 80 ms of (a), it is observed that the centre of the circular deformed region (cross mark) is displaced by d p from the initial level of the central line on the surface (dotted line). This value was used to estimate the depression depth h at the centre (h ≈ d p ).

Figure 5

 5 Figure 5 Temporal evolution of surface shape for P = 4 kW and t = 25 ms. The scale of the images is the same as figure 4.

Figure 6

 6 Figure 6Results of measurements of absorptivity and surface temperature in the case of P = 2 kW and t = 80ms.

Figure 7

 7 Figure 7Results of measurement of evaporated mass in the case of P = 2 kW and t = 80ms.

2

 2

Acknowledgements

The authors would like to thank S. Kojima (PHOTRON LIMITED) and M. Sugiura (Nippon Steel Corporation) for useful advice about the temperature measurement.