Cristian Popi

Olivier Festor
email: festor@loria.fr

Flow Monitoring in Wireless MESH Networks

We present a dynamic and self-organized flow monitoring framework in Wireless Mesh Networks. An algorithmic mechanism that allows for an autonomic organization of the probes plane is investigated, with the goal of monitoring all the flows in the backbone of the mesh network accurately and robustly, while minimizing the overhead introduced by the monitoring architecture. The architecture of the system is presented, the probe organization protocol is explained and the performance of the monitoring framework is evaluated by simulation.

Introduction

Community, as well as operator based wireless mesh networks are gaining market shares at a fast pace in the area of Internet access. Easy deployment and the absence of any need of centralized coordination make them a promising wireless technology [START_REF] Akyildiz | Wireless mesh networks: a survey[END_REF]. The centerpiece of such a network is the wireless distribution system, named the backbone of the network. Such a backbone comprises a mesh of fixed nodes which play the role of either access points, or routers (when they relay packets for other routers), or both of them. The nodes in the backbone communicate in a multi-hop fashion over the wireless links established between them and based on technologies like WiFi or WiMax. The mesh structure provides robustness and availability: if one or multiple nodes fail, the packets will be rerouted on different paths by the "alive" nodes.

While these networks provide flexibility, their heterogeneous nature (for the case of community networks) comes with a range of challenges and difficulties; performance, interoperability and security being some of them. Second, mesh elements may operate under severe constraints such as reduced bandwidth, significant signal quality fluctuation due to environment conditions: obstacles, interferences, or hidden hosts. When designing a monitoring infrastructure for a wireless mesh network, one should therefore provide for situations like broken links, out of range nodes (weakening of the radio signal), or failed nodes. The solution should adapt to the new situation by choosing a different node for monitoring, or a different export path for the monitored information, for instance.

Flow monitoring is a well established approach to monitor the behavior of a network. On one hand, it is used for user monitoring, application and service profiling. On the other hand, certain flows may need to be treated differently from others (ie. separate queues in switches or routers) for traffic shaping, fair queueing, QoS; monitoring the flows in the network is therefore essential in order for such (traffic) handling decisions to be taken.

In this paper we investigate an algorithmic and protocol mechanism that allows for an autonomic organization of the probes plane, that scales well and is robust to node and link failure. The monitoring of the flows is done by the nodes in the mesh backbone. We make the assumption that flows are generated from and directed to mobile nodes (clients of the network) or to the other networks (Internet) via a node acting as a gateway. All the segments of a flow traverse the same path. The paths of flows going through the backbone, pass by one ore more mesh routers, which are all potentially probes. If more than one probe monitor the flow (case of MeshFlow in [START_REF] Huang | A flow-based network monitoring framework for wireless mesh networks[END_REF]), than the cost of monitoring increases. This becomes important especially in what concerns the network overhead at export time. This work looks into finding a mechanism for node self-organization such that all flows are monitored only once in their passage through the mesh backbone, while adapting to topology changes generated by loss of links or nodes. The solution relies on the knowledge of routing information for each flow on any mesh node, by requiring that each node distribute its routing table records to the other nodes in the backbone. For this, the underlying routing protocol must be a pro-active one, in which the routing tables are built based on topology knowledge and before any packet needs to be sent across the network (ie. OLSR [START_REF] Clausen | Optimized Link State Routing Protocol (OLSR)[END_REF], or DSDV [START_REF] Perkins | Highly Dynamic Destination-Sequenced Distance-Vector Routing (DSDV) for Mobile Computers[END_REF]).

The rest of the paper is structured as follows: section 2 introduces related work. Section 3 presents the architecture of the flow monitoring framework. The protocol for node self-organization is shown in section 4 and routing information distribution and the decision making process in section 5. Section 6 provides the evaluation of the proposed monitoring system, and the conclusions end the paper in section 7.

2 Related Work [START_REF] Kim | On accurate measurement of link quality in multi-hop wireless mesh networks[END_REF][START_REF] Natu | Probe station placement for robust monitoring of networks[END_REF] and [START_REF] Bejerano | Robust Monitoring of Link Delays and Faults in IP Networks[END_REF] all present active approaches towards monitoring the state of the network. In [START_REF] Kim | On accurate measurement of link quality in multi-hop wireless mesh networks[END_REF], K. Kim introduces a scheme for accurate measurement of link quality in a wireless mesh network. The proposed architecture, EAR (Efficient and Accurate link-quality monitoR) uses distributed and periodic measurement of unicast-based unidirectional data probes by dynamically choosing one of three schemes: passive, cooperative and active to measure link quality. In [START_REF] Natu | Probe station placement for robust monitoring of networks[END_REF], a greedy solution to probe station placement in a network is provided, to detect all node failures in the network. At each step of the algorithm, a node is added to the probe station set such that the set of shadow nodes (the set of the nodes in the networks that can not be reached by k independent paths from the probe stations, where k is the maximum number of node failures) is minimized. Like in our work, adaptability mechanisms are offered.

A self-configurable cluster-based monitoring system for a wireless mesh network is presented in [START_REF] Sailhan | Wireless mesh network monitoring: Design, implementation and experiments[END_REF]. Monitoring nodes are organized into a cluster-based hierarchical structure in a dynamical way; topological monitoring information, gathered from passively listening to OLSR (Optimized Link State Routing Protocol [START_REF] Clausen | Optimized Link State Routing Protocol (OLSR)[END_REF]) traffic is pushed up the hierarchical structure to a topology monitoring entity (which corresponds to a cluster head). In a flow monitoring scenario, flows would be monitored by all the nodes on their paths. We look rather at how we measure available data across the system by coordinating the participant nodes, such that flows are monitored by only one node.

In [START_REF] Suh | Locating network monitors: Complexity, heuristics and coverage[END_REF] the problem of sampling packets in a cost-effective manner is proposed, by solving the two conflicting optimization objectives: maximization of the fraction of IP flows sampled vs. minimization of the total monitoring cost. The solutions of the posed problems determine the minimal number of monitors and their optimal locations under different sets of constraints. In [START_REF] Chaudet | Optimal positioning of active and passive monitoring devices[END_REF] Chaudet et al. also found that in terms of cost (of deployment and running of probes), it may be worthwhile to monitor only a part of the traffic (ie. 95%) (this reduces the number of probes needed to almost half in the examples presented in their simulations). These last two approaches are theoretical and they require a priori knowledge about the topology of the network, and the traffic flowing through the network. We aim to build a system that automatically and dynamically organizes the monitoring probes plane to capture at a minimum cost a high percentage of the traffic passing the network without knowing the flow graph in advance.

[8] proposes a monitoring architecture for IP flows in a wireless mesh network. MeshFlow records are being created on every mesh router in the path of a packet. By aggregation of these records a complete transportation path of packets can be deduced. Our proposal minimizes the export overhead by choosing one probe only on the path of a flow to monitor it.

In [START_REF] Gonzalez | A-GAP: An adaptive protocol for continuous network monitoring with accuracy objectives[END_REF], Gonzalez et al. build a network monitoring scheme, A-GAP, with accuracy objectives. To reduce the overhead of monitored information between the monitoring nodes and the management station the authors introduce a filter scheme by which a monitoring node does not send updates for small variations of its state or partial aggregate state computed on it. The filter is dynamically computed on each node based on a discrete-time Markov chains stochastic model. The results lead to the conclusion that accepting small errors in monitoring accuracy, overhead reduction is gained.

Architecture

The routers are all possible probes, that is they all have flow counting capabilities. In order for the probes to make decisions on which one monitors a flow, a global vision of the routing entries of all the nodes in the backbone is required on every node. This allows a probe that sees a flow passing through its interface to trace the flow's path. A prerequisite for monitoring a flow is that a probe P that sees a flow F on one of its interfaces has to know the flow's entry and exit points in the backbone, as well as the next hop towards the exit point for each node on the path of the flow. In accordance with this, the functional architecture of the monitoring system is presented in figure 1. The routing plane builds up the routing table of the mesh nodes (with the help of a pro-active routing protocol). It then provides the routing table entries of all nodes to the monitoring overlay, which uses this information to organize the nodes into monitoring or non-monitoring probes. Two components come into the decision making process when organizing the nodes for monitoring: the routing information received from the routing plane to locally build the path of a flow on a node, and the metrics that allow to differentiate between nodes located on the path of the flow. These metrics are distance (in number of hops) of the node from the collector (to which the node is configured to send flow records), connectivity degree and up link quality. Nodes with better distance, higher connectivity or up link quality are the ones elected to monitor the flow. The building blocks of a node's architecture are: the neighbor discovery service, the Multi Point Relay (MPR) manager, the global routing service and the flow monitoring service.

The neighbor discovery service gives information about the one hop and the two-hop neighbors. It relies on periodically polling neighbor nodes via Hello messages; this enables the tracking of alive nodes and of changes in the topology.

The Multi Point Relay manager computes, locally, for each node, the set of Multi Point Relay nodes (MPR) which are used to flood management messages from a node into the network. The MPRs (an OLSR concept [START_REF] Clausen | Optimized Link State Routing Protocol (OLSR)[END_REF]) selected by a node are the 1-hop neighbors of this node that allow it to reach all of its symmetric 2-hop neighbors. The use of MPRs to flood information into the network is an optimization of a classical flooding algorithm. This service relies on the underlying neighbor discovery service and is used, in our monitoring architecture, by the global routing table service for flooding local routing table information.

The global routing table service makes a view of the routing tables of all nodes available to any node by optimized flooding of messages. The messages flooded into the network by a node k contain tuples of this form: {N extHop i (k), N ode ID List}, with i = 1, N k , where N k stands for the number of entries in k's routing table, and N ode ID List is a list containing all the destinations that are reached from k via a next hop node N extHop i (k). Apart from routing information, other information local to a node is spread into the network: the distance (in number of hops) from k to the collector (the node, or station which collects all flow export messages sent by probes, and which can be reached independently by all probes by configuring them with the IP address of the collector), the connectivity degree of k (which is the number of direct neighbors it has), and the up link quality (the quality in bandwidth of the path from k to the collector).

The flow monitoring service takes decisions per flow on whether a node monitors the flow, based on the information provided by the global routing table service. The source and destination of the flow are extracted from packet headers to deduce the entry and exit points in the backbone. From the routing pieces of information of all nodes in the backbone available locally on each node, the entire path of the flow is constructed. A comparison of the metrics of all the nodes on the flow's path (which are provided locally as well) with the node's own metrics, gives the node the means for deciding if it monitors the flow. The flow monitoring service also handles the flow cache management and flow exportation. We exemplify the proposed monitoring system, and how it works. In figure 2, there are 3 traffic flows which pass the backbone of the mesh network formed of nodes A, B, C and D. The collector is a station attached to node C. Therefore, node C is the better positioned probe, only 1 hop away from it. Node B is 2 hops away, and nodes A and D, 3 hops. After having exchanged routing table entries, and their distance from the collector, the probes (A, B, C and D) are now capable of deciding what flows to monitor. For instance, for flow F 1I nodes A, B and C all share the knowledge that the entry point is A, the exit point is C, and B is the intermediary node, and also know the distance of each of them from the collector. Since C is the closest, probes A and B decide, locally, not to monitor F 1I . Node D, does not have any decision to take for flow F 1I , since this flow does not pass any of its interfaces. The same applies to flow F I3 , which is monitored by C, and F 23 , monitored by B. In the figure, the collector has been chosen to be attached to node C. Therefore node C is privileged to monitor all flows originating or destined from/to the Internet. This might mean a gain in the probe selection cost which is due to the fact that most flows are destined or originated to/from the Internet ( [START_REF] Jun | The nominal capacity of wireless mesh networks[END_REF] states that practically all traffic is to/from a gateway). It also means that node C can become a bottleneck, slowing down normal traffic in the network. The problem would be agravated for a large topology; in that case, it would be best to chose it's location away from the gateway.

The monitored flows, are exported by each of the probes to the collector for storing/analyzing. The export paths from the probes to the collector are common with the traffic paths in the backbone of network. In figure 2, for router A to export monitored flow information to the collector, the multi-hop path via B and C is used, in competition with regular traffic. Since bandwidth provisioning can be an issue, it is important to coordinate the probes' monitoring decisions such that a flow is not counted multiple times on its path through the backbone and that the path from the probe that monitors the flow to the collector is the most convenient. In the provided example, all flows are monitored only on B and C.

Probe organization

Probe organization in a flow monitoring context means organizing the nodes such that monitoring decisions can be made within the confines of the cost associated with the number of monitoring nodes per flow defined in section 1. Probe organization in a non-reliable environment also means adapting the monitoring system to changes in the topology, which, although not so frequent as in mobile ad-hoc networks, still occur due mainly to node start/shut-down/failure or to unstable links (ie. caused by interference or obstruction).

For the nodes to share a global view of the routing tables of all the backbone nodes, a distribution system is needed, to help disseminate the routing table entries. Normal flooding (see OSPF [START_REF] Moy | OSPF Version 2. Request for Comments 2328[END_REF]) is not optimal, especially where we send routing table information, which for a proactive routing protocol in-cludes all backbone nodes as destinations. We suppose a mesh architecture, where clients are hidden behind NATs installed on each router (see the Roofnet project [START_REF]b mesh network implementation[END_REF]); routing tables on backbone nodes therefore only have entries for all mesh routers and a default entry for the gateway. The number of records to be sent per node is roughly equal to the number of nodes in the backbone. Data organization optimizations can be made to reduce this number, by including lists like: {N ext hop i : Destination N ode ID List} in a flooded message sent by node k, instead of (Destination N ode i : N ext hop) pairs, where Destination N ode ID List represents a list of all the nodes in the backbone that can be reached from k via a N ext hop i node. In addition, k's neighbors need not be sent explicitly, if they are already a next-hop for some destination in the network.

In order to reduce the number of control messages we use the concept of Multi Point Relay (MPR) employed by the OLSR routing protocol to flood topology control messages. The MPR Set selection scheme is that of OLSR (Hello messages are used). For flooding the network with routing entries, routing control message (RC) are sent by every node and broadcast via the MPRs, containing the entire routing table of the sender.

Whenever a node joins, or fails, the topology changes. The system self-adapts, in the sense that the one and two-hops neighbor lists stored inside each node change their components and the MPR Set is recomputed. Similarly, a new global routing table vision on each node is recomputed with the change of routing tables.

Configurable parameters allow the tuning of the granularity of topology changes as seen by the monitoring system. The Hello P eriod is the duration of time between two successive Hello messages sent by a node. The RC P eriod is the duration of time between two successive RC messages. We determine the optimum values for these parameters through simulation with the NS-2 simulator (see section 6).

Flow monitoring decision

A node floods the network with Route-control (RC) messages (UDP messages with the TTL set to 255) via its computed MPR set. An RC message contains a node's routing table records and a sequence number specific to the sending node. The sequence number is used by the node to rule out the possibility of outdated information.

The RC P eriod is the duration of time between two successive RC messages sent by a node. After a node A receives an RC message from a node B, it updates the routing table record information for B , and keeps this information for a RC Store P eriod time duration. If it doesn't receive a RC message from B during this time it will consider B is lost and will discard all routing information from it. This does not affect the accuracy of the monitoring system, since if B is lost, or there is no path from B to A, there cannot exist a flow that passes both A and B.

Data representation and manipulation

The content of an RC message encloses a list of all (N ext hop, Destination List) pairs contained in the routing table of the RC message sender, and the metrics it advertises (distance from the collector, degree of connectivity and up link quality). A receiving node stores this information in a tuple (N ode ID, {N ext Hop N ode 1 , Dst List 1 }, ...{N ext Hop N ode n , Dst List n }). Dst List i represents the list of destination nodes reachable from N ode ID via N ext Hop N ode i .

The procedure for determining the path of a flow passing through a node's interface is shown in algorithm 1. In line 5, the function search next node() returns the next-hop node for the current node towards the destination. Flow end points corresponding to mesh clients are mapped to the routers they connect to. If the flow has one of its end points outside the wireless mesh network (ie, other networks), then either the entry or exit points of the flow are mapped to the gateway. Since the current node holds routing information for all nodes in the backbone, there is exactly one entry in its table that yields the pair (destination: next hop), no matter what node in the backbone the destination is.

Once the complete path of a flow is obtained, the node compares its metrics to those of the nodes on the paths. The distance from the collector is the number of hops a message sent from the node to the collector has to travel, and it is computed by periodically using traceroute to the collector. The connectivity degree is the number of one-hop neighbors of a node. This can be easily deduced from the size of its one-hop neighbor list. In case both collector distance and connectivity degree are identical for two nodes, an up link quality metric, which estimates the quality of the first hop link towards the collector from the node is used to choose between the nodes. As soon as a node finds that there is another node on the flow's path with better metrics, it gives up comparing, and does not monitor the flow.

In this section, we evaluate the effectiveness of our approach. We first describe simulation settings and parameters and then show the results of the evaluation.

Simulation setup and scenario

We simulated a mesh network using the NS-2 simulator [START_REF]The Network Simulator[END_REF]. For the MAC layer we used the 802.11b with RTS/CTS (Request to Send/Clear to Send). The RT-S/CTS mechanism is enabled to effectively realize multi-hopping in the backbone. For the routing protocol, we chose OLSR, and use the UM-OLSR patch [START_REF]UM-OLSR patch for ns-2[END_REF] for ns-2. We conducted our experiments on a variety of network topologies with sizes of between 20 and 50 nodes, and then for scalability reasons, 100 nodes. Nodes in all topologies are fixed, to simulate the static routers in the wireless mesh network backbone.

The traffic pattern is modeled such that 80% of the traffic is directed to or coming from the gateway nodes ( [START_REF] Jun | The nominal capacity of wireless mesh networks[END_REF] states that practically all traffic is to/from a gateway). TCP flows are used to emulate user traffic, with two-way TCP agents randomly disposed over the backbone nodes, and application-level data sources (traffic generator) attached to the agents. We use two-way TCP because it implements SYN/FIN connection establishment/teardown.

Accuracy

The accuracy measures how close to the proposed goal the monitoring system is, in terms of the number of monitored flows, and the number of monitoring probes per flow.

We first measured the accuracy for a stable mesh network (topology did not change during one experiment and the routing tables on all of the nodes were stable). We conducted 5 experiments for a network size of 20 nodes, and another 5 for a network size of 50 nodes. As expected, all flows were counted exactly once. The average number of probes for different topologies that monitored flows was 43% of all probes. The minimum was 34% and the maximum 66% (the values are averaged over the two network sizes).

Next, we compute the accuracy for an instable mesh network topology. We simulated this by adding 5 nodes, one node every 5 minutes to the low-density network (size=20). The total number of TCP flows sent in this experiment was 3000. The experiment lasted 30 minutes. We repeated it 5 times for different topologies (same network size). The first node was added after 5 minutes. The average non-monitored flow percentage was of 5.3%. This is explicable because of the chosen policy: not to monitor flows, when the full flow path can not be reconstructed. We have intentionally set OLSR in our simulator to be more reactive to topology changes than the monitoring overlay (T C Interval is set to 5 seconds and RC P eriod to 7 seconds). Instability occurs immediately after the change in topology, and before all the nodes have a knowledge of the new routing information. Figure 3 shows an example of a scenario where this situation occurs. Node D joins, the links A-D and D-E are active, and the routing protocol chooses to route packets from A to F, through D and E, instead of through B and C. Flows from A to E were monitored by node C (as it is the closest to the collector) before the change in topology. As it takes some time to propagate all routing information to all nodes (the RC P eriod is set to 7 seconds), A, D, E and F only know the next hop for the flow A-E, but not the whole path, therefore they do not monitor any flow that passes through D.

Accuracy vs Management Overhead

The management overhead measures the number of coordination packets sent by the probes. The only packets sent by the monitoring overlay are Hello and RC messages. We consider 20 and 50 node mesh networks. We try different values for the Hello and RC message periods. In all experiments, randomly, nodes fail for a duration of maximum 1 minute (only one node at a time) and then come back alive. This way we simulate more realistically periodic link failure or quality variations. The experiments are all 5 minutes long. The table 1 lists the average values of the accuracy and number of control messages over a 5 minute period for network sizes of 20 and 50 nodes. The results show that the monitoring system is pretty accurate for a value of RC P eriod of 10, only missing around 13% of the flows. If the interest is to obtain most of the traffic, decreasing the values of Hello P eriod and RC P eriod gives a good monitoring accuracy, but it triples the number of packets sent per node. Further increasing the two parameters yields an accuracy which may no longer be acceptable (62% of flows counted).

We next see how increasing the number of nodes in the backbone affects the control plane.

Scalability

The amount of control traffic and message size were measured as the network size increased. We performed a set of experiments, each one 5 minutes long, where we varied the number of nodes from 20 to 50 (in increments of 5) and then 100. Figure 4 shows the effect of increasing the number of nodes in the topology to the number of control packets conveyed in the network and their size for a (Hello P eriod, RC P eriod) pair of (2,5) seconds.

Both the number and the size of packets vary linearly with the number of nodes. While the number of control messages can be reduced provided the accuracy of monitoring is lowered, the RC message size depends on the routing table size of the nodes; with a pro-active routing protocol, each node holds entries for all nodes. Therefore, RC messages, even though optimized to include a node only once (see section 4) still holds at least all backbone nodes' identities.

Conclusions and Future Work

In this article we have proposed a novel flow monitoring system for Wireless Mesh Networks with the goal of counting all of the flows passing through the backbone on only one mesh node, aiming to reduce export overhead. We build a monitoring overlay, in which a probe is able to decide if it monitors a flow that passes through its interfaces. We have described a protocol for probe organization that aims to reach the proposed goal. The protocol is based on a simple and minimal overhead routing table information flooding by all the probes. By sharing a global routing table view, the probes can trace the paths of the flows within the backbone, and therefore know what other nodes compete for monitoring the flows it sees. The metrics that make the difference in the decision process are chosen to minimize the flow record export: the distance from the collector, the connectivity degree and the up link quality.

We have evaluated the monitoring system to see its accuracy and the impact of the introduced management overhead. We determined the values of the intervals for sending control messages (Hello and RC messages) for a random number of topologies with 20 and 50 nodes, that optimize the number of management packets while keeping the accuracy of the monitoring system within acceptable limits.

In the future we plan to implement the proposed monitoring system on a small scale 802.11 mesh testbed, that we have set up in the lab. Secondly, since the monitoring system is based on the metrics advertised by the participating nodes, we investigate the situation where one or more nodes maliciously send faulty metrics, to either avoid monitoring (to save processing and communication bandwidth) or to monitor all traffic flows with the intention of eavesdropping, for instance. Another track of action is to find means to dynamically compute the values of the control message intervals, to adapt to the topology dynamics of the network. Recommended values (like the ones computed by us) are too general to fit the specificity of a community wireless mesh network (which depends a lot on conditions like geography or sources of interference).

Fig. 1 :

 1 Fig. 1: Wireless Mesh Network flow monitoring architecture

Fig. 2 :

 2 Fig. 2: Wireless Mesh Network -flow example

Input: 5 N 7 P

 57 Entry P oint, Exit P oint Output: Flow path: the sequence of Node IDs that can potentially monitor the flow 1 Current N ode = Entry P oint; 2 Destination N ode = Exit P oint; 3 P ath = {Current N ode}; 4 while Current N ode is not Destination N ode do ext N ode = search next node(Current N ode, Destination N ode); 6 Current N ode = N ext N ode; ath = P ath + {Current N ode}; end Algorithm 1: Algorithm for building the path of the flow through the backbone of the wireless mesh network

Fig. 3 :

 3 Fig. 3: Topology Instability Scenario -with route for flow A-F changed from A-B-C-F to A-D-E-F

Fig. 4 :

 4 Fig. 4: (a) the average number of control packets per node per second as node size increases; (b) the average packet size as node size increases;

Table 1 :

 1 Protocol accuracy vs overhead for different pairs of (Hello P eriod, RC P eriod)

	Hello and RC Period Pair Accuracy (%) Overhead (packets/node/sec)
	(2,5)	99.2	10.3
	(2,7)	93.7	9.8
	(3,7)	93.5	6.3
	(5,10)	86.9	2.1
	(5,15)	62	1.5

Acknowledgement

This paper was partially supported by the French Ministry of Research project, AIRNET, under contract ANR-05-RNRT-012-01.


Krishan Bumma

Axel Huerre

Juliette Pierre

Thomas Séon

Early freezing dynamics of an aqueous foam

Early freezing dynamics of an aqueous foam †

Krishan Bumma, * a Axel Huerre, b Juliette Pierre, a and Thomas Séon a Liquid foam freezing is an essential and unavoidable processing to obtain solid foam. We study experimentally the solidification dynamics of a model aqueous foam in contact with a cold substrate. The substrate temperature, the foam bubble radius and the liquid fraction are changed. We show that the freezing dynamics always starts by following a self similar square root of time diffusive dynamics. This early dynamics is then predicted as a function of the control parameters using a 1D diffusion model and by treating our foam as a homogeneous fluid with equivalent thermophysical properties. In particular, we build a new expression for the foam conductivity. Finally, experimental and theoretical results are compared and interpreted. This study paves the way towards the understanding of the complex foam freezing dynamics at longer time, when the freezing is then coupled to water migration in the foam.

The freezing of a porous material initially filled with a liquid is a widely encountered phenomenon. Among its many natural occurrences we can cite the upward displacement of ground surface, called frost heave, caused by the formation of ice lenses within the underlying soil [1][2][3] ; and the frost hardiness of terrestrial plants due to ice formation in specific intercellular and extracellular spaces that induces freezing avoidance within dehydrated living cells [START_REF] Sakai | Frost survival of plants: responses and adaptation to freezing stress[END_REF][START_REF] Beck | [END_REF][6] . From an industrial standpoint, understanding and controlling the freezing of porous materials is essential for cryosurgery and cryo-preserving food, plants, and animals 7 . The quality of frozen products depends on the size of the ice crystals, the degree of cell dehydration, and the amount of damage to the cell walls. All of these factors are determined by the rate of freezing [START_REF] Rahman | Handbook of food preservation[END_REF] .

A solid foam consists of pockets of gas surrounded by solid walls. For example, the sea sponge is an open-cell foam that allows water to penetrate and retains it by capillary action. Since the foam properties are so interesting, many artificial foams have been created, such as polymeric and metallic foams [START_REF] Cantat | Foams: structure and dynamics[END_REF] . Most polymeric foams are created by the formation of gas bubbles in a melt or liquid system through nucleation, growth, and expansion. These foams then solidify through a complex physico-chemical process influenced by many variables controlling the final product. It is typically achieved through thermosetting of molten thermoplastic polymers that harden as the foam expands or cools down [START_REF] Weaire | The physics of foams[END_REF] . In contrast, solid metal foam is produced through a different process. A melt is foamed using a blowing agent that releases gas, and the resulting liquid foam is cooled at the precise moment before the liquid foam collapses [START_REF] Banhart | [END_REF] . This process freezes the unstable foam structure in a solid state, resulting in solid metal foam [START_REF] Banhart | [END_REF] . These particular foams are widely used due to their impressive properties: softness, light weight, yet strength, as well as useful thermal and acoustic insulation, and energy absorption capabilities. In each of these process, the freezing of liquid foam is an essential and unavoidable processing to obtain solid foam. However, foam solidification have been the object of only very few studies 12,13 .

In this paper, we propose to investigate the freezing of an aqueous foam. We begin by describing the experimental setup and then analyze the freezing dynamics of the foam by changing the temperature and bubble radius, as well as the liquid fraction. To predict the freezing dynamics, we treat the foam as a homogeneous fluid with equivalent thermophysical properties and propose a new expression for the foam conductivity. Finally, we compare and interpret the experimental and theoretical results.

Experimental setup

The experiment consists in freezing a liquid aqueous foam contained in a tube and placed on a cold substrate, as shown in the schematic of Figure 1(a). The foams are made by mixing a 10 g/L sodium dodecyl sulfate (SDS) solution and perfluorohexane (C 6 F 14 ) saturated air, using a cyclic diphasic flow through the constriction made by two linked syringes 14 . Fluorescent dye is added to help the visualisation. The liquid fraction φ l , defined as the volume percentage of liquid in the foam, is selected a priori by introducing the right amount of soapy solution and gas in the syringes and precisely measured a posteriori by weighing a controlled volume of foam. φ l ranges from 3% to 27% in our experiments. Foams dryer than 3% are not achievable by this method. The liquid fraction is considered unaffected by gravity as the Bond number, Bo = ρgRH/γ with ρ the liquid density, g the gravity, γ the surface tension, R the characteristic bubble radius, and H the foam height, that compares the capillary pressure at the bubble scale and the hydrostatic pressure on the foam height, is smaller than 1 (around 0.1) [START_REF] Cantat | Foams: structure and dynamics[END_REF] .

The bubble size distribution is measured a posteriori, by dispersing the foam at the surface of the soapy solution to obtain a monolayer of bubbles and then measuring their diameters using a microscope 15 . With our 20 mL syringes the bubble size distribution is reproducible and the average bubble size is around R = 25 µm. The typical bubble size distribution is show in the supplementary material †. By using C 6 F 14 saturated air, we ensured that the foams do not undergo coarsening during the time of our experiments. Indeed, the typical time needed to double the average bubble size of such a foam is of a few hours, while the freezing experiment is at most 15 minutes long. The average bubble size measured initially is then not affected by coarsening during the freezing process. However, it can be increased signifi- cantly by waiting at least 3 hours before freezing the foam. This way, a few experiments are done with average bubble size foam of R = 48 µm and 86 µm.

Once produced and characterised, the foam is poured into a cylindrical plastic tube with inner diameter 1.2 cm and height 4.8 cm. We observe that the foam sticks to the walls of the tube. The tube wall is thin enough, down to 100 µm, to limit the vertical heat flux through it. Moreover, the very low thermal conductivity of the air allows us to safely neglect the heat flux in the air and to consider the plastic tube as insulating. The tube filled with the foam is then placed vertically on a copper surface (see Fig. 1 (a)) cooled down using a refrigerated circulator (Julabo CORIO CD-1000F). This enables the substrate to maintain a constant temperature in the range -14 • C ≤ T s ≤-32 • C during the 15 minutes experiments. Temperatures down to -105 • C were also achieved using liquid nitrogen to cool the substrate.

Finally, the freezing foam is observed under ultra-violet (UV) light with a 200 mm macro lens mounted on a Nikon D800 with a 52.5 mm long-extension tube. Typically obtained images are presented in the time sequence of Figure 1 (b), where the liquid foam appears bright green because of the fluorescent dye, and the solid foam appears dark as the fluorescent molecules are largely expelled from the growing ice 16 .

Experimental Results

In this time sequence (Fig. 1 (b)), a slab of foam of 10 mm right above the cold substrate (T s = -30.2 • C) is displayed as a function of time. At t=0 the foam is entirely liquid, but very rapidly the foam solidifies and we observe the dark solid foam growing and decelerating with time. The height of the freezing front, h(t), is then measured and plotted with black dots as a function of time in the inset of Figure 2(a). The qualitative observation is confirmed, the freezing starts quickly and then slows down with time. The main plot of Figure 2(a) presents the same experimental data in logarithmic scale. Two regimes are observed : a first regime where the data aligns along a dashed line of slope 1/2 meaning that the height of the frozen foam grows following the square root of time, and then enters a slower second regime after a few hundred seconds. The dashed line of slope 1/2 fitting the experi-mental diffusive regime can be expressed as h(t) = D exp eff t, where D exp eff is an effective diffusive coefficient determined experimentally. The existence of these two regimes of front propagation, starting with a diffusive regime, is typical of systems where diffusion and convection play a role with different time scales 17,18 . In these experiments, convection in the liquid foam might lead to a water supply at the freezing front, impacting the diffusive growth after a certain time. However, the mechanisms leading to water convection are not clear at all. The appearance of this second regime is then beyond the scope of this study focused on the quantitative description of the first diffusive regime of freezing foam. Therefore, the following panels on Figure 2 present only the results for this early-time regime.

In Figure 2(b), the solidification dynamics in the first regime for a foam, with the same liquid fraction φ l ∼ 16.5% and mean bubble size R = 25µm, is plotted for three different substrate temperature: T s = -39.9 • C, -69.8 • C and -96.0 • C. We notice that the three curves have the same shape on a log-log scale with a freezing dynamics following a square root of time. The curves are ordered by temperature, with colder substrates causing faster solidification of a given volume of foam: the effective diffusive coefficient depends on the temperature and, as expected, is larger for colder substrates. Figure 2(c) presents the solidification dynamics in logarithmic scale for foams with three liquid fractions φ l = 12.7, 18.8, and 26.9% but same mean bubble size R = 25 µm and same substrate temperature T s = -20 • C. The experiments always follow the square-root dynamics but are shifted: the effective diffusive coefficient clearly depends on the liquid fraction. From these curves it seems that the wetter the foam (high liquid fractions, darker symbols) the faster the freezing dynamics. Finally, Figure 2(d) shows foams freezing with different bubble size distributions †, with mean bubble sizes: R = 23, 48 and 86 µm, all other parameters being equal: φ l ∼ 10% and T s = -30 • C. At early times, we recover the square root of time regime and all the curves are superimposed. Therefore, it seems that the bubble radius does not influence the solidification dynamics during the first regime.

In the following sections of the paper, we focus on building a physical model that predicts the effective diffusion coefficient

D th

eff of a one dimensional foam freezing and compare it with our experimental measurements. As observed experimentally in Figure 2, the diffusion coefficient has to vary with the substrate temperature T s and the liquid fraction in the foam φ l but not with the average bubble size R.

Predicting the freezing dynamics

Stefan model

The freezing front propagation dynamics for a one-dimensional liquid system is known as the Stefan problem [START_REF] Rubinstein | The Stefan Problem[END_REF] . The growth of the solid layer follows the classical diffusive dynamics: h(t) = √ D eff t, where D eff is an effective diffusion coefficient that depends on the temperatures at the boundaries and the thermal properties of the two phases 20 . When a layer of ice rests between a semi-infinite solid substrate and semi-infinite water, a variant of the Stefan problem exists [21][22][START_REF] Kant | Proceedings of the National Academy of Sciences[END_REF] . To find it one consider the 1-D heat equations in the substrate, in the ice and in the water, one write the continuity of temperature and heat fluxes at the ice-substrate interface, and the discontinuity of heat fluxes due to latent heat at the ice-water interface. This way one obtain an implicit relationship between D eff , the substrate temperature T s , and the thermal parameters of the water, the ice, and the substrate. This equation can be solved numerically to obtain D eff . It was computed for the experimental configuration of Figure 1 and the resulting dynamics for the ice front propagation is plotted as a solid line in Figure 2(a). This prediction of freezing dynamics for water lies above the fit of the experimental freezing dynamics for foam in the diffusive regime (dashed line): the freezing front propagates much faster in water than in our foam. Experimental freezing of the soapy water shows no difference with freezing of distilled water, so the difference observed between the freezing of a foam and of water lies in the physical characteristics of the foam. Figure 2(b) confirms this difference as D exp eff appears to be dependent on the foam liquid fraction φ l . Therefore, to predict the freezing dynamics of a foam, the geometrical configuration (randomly oriented Plateau borders, vertices, soap films) and the physical properties of the phases (liquid and encapsulated gas) have to be taken into account.

Foam conductivity

We will consider the foam as a homogeneous fluid with effective thermo-physical properties and in particular, an effective thermal conductivity. This treatment is classical and has been used in many other examples of heat conduction through heterogeneous media [START_REF] Boomsma | [END_REF][START_REF] Kaviany | Principles of heat transfer in porous media[END_REF] . The problem of heat conduction in heterogeneous materials is mathematically analogous to the problem of electrical conductivity in such materials [START_REF] Pietrak | [END_REF] . Maxwell was the first to give analytical expressions for the effective conductivity of an heterogeneous medium in his famous work on electricity and magnetism [START_REF] Maxwell | A treatise on electricity and magnetism[END_REF] . He considered the problem of dilute dispersion of spherical inclusions, namely the very-wet limit. In the case of air bubbles in water the expression for the effective conductivity reduces to 2φ l λ l /(3φ l ), with λ l the conductivity of water. Naturally, the other limit is a very-dry foam, where bubbles are tightly compressed in a small volume of liquid and form a network of randomly oriented Plateau borders at which three soap films meet. In this limit, the effective conductivity of the foam is given by Lemlich's expression [START_REF] Lemlich | [END_REF] : φ l λ l /3.

The foams considered here have intermediate liquid fractions and thus their conductivities lie between these two limits. We propose to build an expression that accounts for the foam structure evolution between the very-dry and very-wet limits. In a very-dry foam, φ l → 0, the water is mainly along the length of the Plateau borders and not in the vertices (where Plateau borders meet in fours), forming mostly 1D structures. Then when φ l increases to reach the very-wet limit, φ l → φ lc = 0.33, the foam becomes a suspension of spheres and Plateau borders no longer exist, the water is in what would be the vertices: 3D liquid structures between 4 bubbles. Note that the films are considered to contain a negligible amount of water, not contributing significantly to the conductivity. In the aim of expressing the conductivity of the liquid matrix of the foam, we propose to weigh the Lemlich's expression by the proportion of liquid in Plateau borders and to add Maxwell's expression weighed by its complement, the proportion of vertices. Lacking an exact expression for the proportion of Plateau borders relative to vertices, we will take 1φ l /φ lc as the proportion of Plateau borders. This function of φ l goes to 1 when the foam is dry and the liquid is within the straight segments of the Plateau borders, and goes to 0 at φ l = φ lc , when the bubbles are at close packing and surrounded only by vertices. The proposed expression of the conductivity of the liquid matrix is thus:

λ matrix = 1 3 φ l λ l (1 -φ l /φ lc ) + 2 3 -φ l φ l λ l φ l /φ lc (1)
This new theoretical expression of the conductivity of the foam can be compared to experimental data on electric conductivities. Figure 3(a) shows measurements of the electrical conductivity of foams for liquid fractions between 1 and 30%. These data are realised or extracted from the literature by Feitosa et al. [START_REF] Cantat | Foams: structure and dynamics[END_REF]29 . The dashed line represents a one-parameter best-fit to the data, proposed by the authors, that describes all the data well and, in particular, works better than the previous ones [START_REF] Weaire | The physics of foams[END_REF][START_REF] Lemlich | Industrial & Engineering Chemistry Process Design and Development[END_REF] for the intermediate liquid fractions. This convenient fit is a rational function formed by the ratio of second-order polynomials. The model proposed here (Eq. ( 1)) is plotted as a black solid line and shows an excellent agreement with the experimental values. Therefore this new model works well, has no adjustable parameter and is built using physical arguments.

The net heat transfer in a foam is the superposition of the conduction and the radiation considered separately [START_REF] Hilyard | Low density cellular plastics: Physical basis of behaviour[END_REF] . The thermal conductivity of a foam can thus be expressed as the sum of these different contributions: the conduction through the foam matrix (solid or liquid) which is analogous to the electrical conduction and therefore given by the Eq. ( 1), and the two contributions that take place inside the bubbles, namely the conduction through air and the radiation. For the thermal conductivity of the gas phase, we will simply weight the conductivity of air λ air by the amount of air, 1φ l , neglecting the contribution of the films. As we use relatively wet foams (> 5%) with small bubbles (∼ 100 µm), radiation can be neglected [START_REF] Kaviany | Principles of heat transfer in porous media[END_REF][START_REF] Hilyard | Low density cellular plastics: Physical basis of behaviour[END_REF] .

Finally, we end up with the following expression for the thermal conductivity of foam :

λ l,s foam = (1 -φ l )λ air + 1 3 φ l λ l,s (1 -φ l /φ lc ) + 2 3 -φ l φ l λ l,s φ l /φ lc (2)
The superscripts l and s stand for liquid and solid, here water and ice. In this expression, we observe that neither the average bubble size distribution, nor the interfacial properties play a role.

Related Stefan problem

The Stefan problem can now be solved as described previously, considering the foam as an effective medium, with conductivity λ l,s foam defined by Equation ( 2). We assume in the following that the foam solidifies keeping the same porosity 1φ l , and only the conductivities of the water is changed to the one of ice after foam freezes.

The Stefan condition describing phase change in our case is then written as:

φ l ρ s L dh dt = λ s foam ∂ T ∂ z (h -) -λ l foam ∂ T ∂ z (h + ) (3)
with L the latent heat of solidification of the water solution, and ρ s the density of ice. eff . This plot shows the good agreement between our model and the solidification behaviour of the foam during the first hundred seconds of the experiment that defines the diffusion regime of foam solidification.

The inset of Figure 3(b) presents the variation of the effective diffusion coefficient with liquid fraction. Two series of experiments at T s = -20 ± 1 • and -30 ± 1 • C are plotted for liquid fractions varying between 3 and 28%. The experimental measurements are plotted with dots and theoretical prediction with two thick lines corresponding to the two substrate temperatures. As the experiments are realized at temperatures that can be slightly different in the given range, the prediction is given in the same range represented by the thickness of the two lines. As expected, the comparison between experimental measurements and theoretical prediction is satisfying. For liquid fraction higher than φ l ∼ 7 -8%, the wetter the foam, the faster it freezes. This is what was observed in Figure 2(c), or in Figure 2(a) where the freezing is observed faster in a liquid column than in a foam, and is explained by a better conduction of heat in ice than in water.

However, counter-intuitively we predict that the curve is nonmonotonic and dry foams (φ l < 5%) freeze faster than wetter ones. Indeed, the thermal conductivity of the foam does not tend to zero when the liquid fraction tends to zero. Consequently, the heat exchanged through the gas phase is small but significant compared to the very small amount of water to freeze. This nonmonotonic variation of the effective diffusion coefficient with liquid fraction is predicted theoretically and confirmed experimentally.

Conclusion and perspective

In this study, it is shown that the freezing of an aqueous foam starts by following a self-similar square root of time-diffusive dynamics. A new model for the foam conductivity, physical, taking into account the foam structure, and without adjustable parameters is proposed. A good agreement is found between experimental measurements and theoretical prediction for the front propagation dynamics, highlighting the key role of the liquid fraction in the freezing dynamics. In the experiment, we show that after about a hundred seconds the freezing foam leaves its diffusive regime and the freezing slows down. In the model, it is assumed that the foam porosity (1φ l ) stays constant when it freezes. Most probably, the freezing dynamics loses its diffusive character when this strong assumption is broken and the porosity starts to change while the foam freezes. The understanding of mechanisms at the origin of this modification of the foam structure when it solidifies and the characterisation and prediction of the evolution of the foam porosity constitute interesting leads for future research.

Fig. 1 a

 1 Fig. 1 a) Schematic of the experiment b) Time sequence of a foam freezing experiment with liquid fraction φ l = 13% and substrate temperature T s = -30.2 • C. The liquid foam appears green due to the fluorescein only fluorescing in the liquid phase. The frozen layer, appearing purple under UV light, undergoes fast growth in the first seconds before slowing down.

Fig. 2

 2 Fig. 2 (a) Height of the freezing front for T s = -30.2 • C, R = 25µm, and φ l = 13%. The grey zone represents the uncertainty of the automated measurements, estimated at 15 pixels on the original images, it is about 0.2 mm. The solid line is the result of the 1D model for pure water, and the dashed line is a 1/2 power-law fit of the experimental data. (b) h(t) measured for φ l ∼ 16.5% and R = 25µm for temperatures T s = -39.9 • C, -69.8 • C and -96.0 • C. The solid line is the 1/2 slope. (c) h(t) measured for T s = -20 ± 1 • C and R= 25µm for φ l = 12.7%, 18.9% and 26.9%. (d) h(t) measured for T s = -30 ± 1 • C and φ l = 10 ± 1% for 3 different radii R= 23µm, 48µm, and 86µm.

Fig. 3 (

 3 Fig.3(a)Electrical conductivity of the foam relative to the conductivity of the liquid, as a function of the liquid fraction (experimental data from29 ). The dashed line is a fit of the data proposed in29 . The solid line shows our prediction based on Eq. (1). (b) Experimental measurement of D eff , plotted against the predicted value of D eff for experiments with radii between 18µm and 30µm for temperatures ranging from -14 • C to -105 • C. The measurement of this effective diffusion coefficient is obtained by fitting a square root to the first 100 s of our experimental measurement h(t). Inset: Experimental measurement of D eff for experiments done at -20 and -30°C for varying liquid fractions. The blue and red regions represent the predicted value of D eff at -20 ± 1 • C and -30 ± 1 • C.

  The resolution of this Stefan problem for given values of liquid foam and substrate temperatures gives the prediction of the freezing front propagation dynamics in the foam h(t) = D th eff t (see Supplementary Materials for details † ). The experimental value of D exp eff is plotted as a function of the theoretical prediction D th eff in Figure 3(b) for substrate temperatures ranging from -14 • C to -105 • C, liquid fractions between 3 and 27% and mean bubble radii between 18 and 30 µm. The solid line represents D

	exp eff =D th

Conflicts of interest

There are no conflicts to declare.