J Zuo

C Cadet

Z Li

C Bérenguer

R Outbib

Post-prognostics decision-making strategy to manage the economic lifetime of a two-stack PEMFC system

Keywords: fuel cell stacks, durability, post-prognostics decision-making, decision criterion

The durability challenge of fuel cell technology is one of the main obstacles that hinder its wide commercialization. This work proposes a post-prognostics decision-making strategy to manage the economic lifetime of a two-stack PEMFC system. The influence of fuel cell deterioration and fuel consumption on fuel cell economic lifetime is studied. These two items are combined to construct the mixed decision criterion. Both the constant and dynamic type of system load demand are tested by the decision-making strategy. In simulation work, the hydrogen consumption/system-lifetime ratio and system lifetime are proposed as two evaluation criteria to assess the efficiency of the proposed decisionmaking strategy. Both the constant load and dynamic load simulation results show that when more weight is put on mitigating fuel cell deterioration, a bigger hydrogen consumption/system-lifetime ratio is achieved. For the system lifetime, an obvious increasing trend is observed when putting more weight on mitigating fuel cell deterioration. The proposed strategy can adequately handle the two items in the mixed criterion, generating the optimal load repartition for two-stack fuel cell systems. Therefore, a better economic lifetime can be achieved for the two stacks fuel cell system.

INTRODUCTION

Proton exchange membrane fuel cell (PEMFC) can achieve higher energy conversion efficiency and zero onboard emission compared to the traditional power unit. Therefore, it is well recognized as one of the most promising power units in industrial applications [START_REF] Zheng | Prolonging fuel cell stack lifetime based on pontryagin's minimum principle in fuel cell hybrid vehicles and its economic influence evaluation[END_REF]. However, despite the promising application advantages, the PEMFC technology still suffers from limited durability and high application cost [START_REF] Zhao | A review of polymer electrolyte membrane fuel cell durability for vehicular applications: Degradation modes and experimental techniques[END_REF]. Durability is especially challenging. The US Department Of Energy (DOE) durability targets for stationary and transportation fuel cell applications are 40,000 and 5,000 hours respectively [START_REF] Garland | Doe fuel cell program: Durability technical targets and testing protocols[END_REF]. This target is hardly achieved in present fuel cell technology.

Several approaches have been proposed to deal with the durability challenge of PEMFC. Prognostics and Health Management (PHM) [START_REF] Jouin | Phm of proton-exchange membrane fuel cellsa review[END_REF][START_REF] Zhang | Phm-oriented degradation indicators for batteries and fuel cells[END_REF][START_REF] Lechartier | Proton exchange membrane fuel cell behavioral model suitable for prognostics[END_REF] is one of the most effective methods that can help to extend the lifetime of PEMFC. PHM approach includes multiple application layers including data acquisition, data processing, diagnostic and prognostics. For the prognostics layer, model-based, data-driven and hybrid approaches are the three major types of developed methods. For model-based and hybrid prognostics, Jouin et al. [START_REF] Jouin | Prognostics of proton exchange membrane fuel cell stack in a particle filtering framework including characterization disturbances and voltage recovery[END_REF] used a particle filtering (PF) algorithm based on a polarization model, developed for this purpose. The more recently developed data-driven approaches do not rely directly on analytical deterioration model and they aim at taking benefit from the rapid growth in the dataset of fuel cell. For instance, Ma et al. [START_REF] Ma | Data-driven proton exchange membrane fuel cell degradation predication through deep learning method[END_REF] applied deep neural networks to form a prognostic model. However, the problem of how to use the results of the prognostic layer to further improve the fuel cell durability is rarely addressed in literature.

Energy management strategy (EMS) is a recently emerged approach for the decision-making process to improve fuel cell durability. K. Ettihir et al. [START_REF] Ettihir | Optimizationbased energy management strategy for a fuel cell/battery hybrid power system[END_REF] proposed an adaptive optimal power splitting EMS for PEMFC/battery hybrid system to improve its durability. More recently, several researchers have dealt with the prognostics related decisionmaking for multi-stack fuel cell systems. Yue et al. [START_REF] Yue | Health-conscious energy management for fuel cell hybrid electric vehicles based on prognostics-enabled decision-making[END_REF] brought forward a health-conscious EMS based on the prognostics-enable decision-making in the framework of PHM, thus linking management strategy and PHM. Herr et al. [START_REF] Herr | Decision process to manage useful life of multi-stacks fuel cell systems under service constraint[END_REF] proposed a decision process to manage the useful life of a multi-stacks fuel cell system. When considering the operation of fuel cell systems, fuel operation, besides the lifetime, should also be taken into consideration. Chen et al. [START_REF] Chen | Lifetime prediction and the economic lifetime of proton exchange membrane fuel cells[END_REF] proposed the concept of economic lifetime. Fletcher et al. [START_REF] Fletcher | An energy management strategy to concurrently optimise fuel consumption & pem fuel cell lifetime in a hybrid vehicle[END_REF] put forward an EMS based on dynamic programming to minimize the fuel consumption and the system deterioration for a hybrid vehicle. From the above research works, two aspects need to be further developed: 1) it is necessary to integrate PHM into EMS as a crucial post-prognostics decision-making action, and 2) the system consumption should be considered together with the system durability performance.

In this work, a mixed criterion for post-prognostics decision-making is proposed for a two-stack fuel cell system. The objective of this work is to improve the system lifetime by considering the fuel cell deterioration as well as the fuel consumption. A Gamma process-based load dependent stochastic deterioration model is proposed to describe the deterioration of fuel cell stack. In the prognostic part, a conditional decision probability is calculated to collect the end of life (EoL) information of the system based on an analytical method. Then a fuel consumption rate function is introduced to calculate the system fuel consumption, combing the fuel consumption and the deterioration information, forming a mixed decision criterion. Finally, the decision-making phase outputs the optimal load repartition for the stacks according to the defined criterion.

Fig. 1 Schematic of the post-prognostics decision-making strategy 2 PROBLEM STATEMENT

The objective of the decision-making strategy is to find the optimal load split for a two-stack fuel cell system during the operation period, in order to maximize the system service lifetime while taking into account the fuel consumption.

Presently, as the deterioration mechanism of PEMFC is not yet fully known, it is difficult to express the deterioration in a single formula. The research work developed by Jouin et al. [START_REF] Jouin | Degradations analysis and aging modeling for health assessment and prognostics of pemfc[END_REF]. shows that the deterioration of PEMFC can be directly related to the load applied to the system, using an empirical polarization equation [START_REF] Kim | Modeling of proton exchange membrane fuel cell performance with an empirical equation[END_REF]: 𝑉 = 𝑛(𝑬 𝟎 -𝑹𝐼 -𝑨𝑙𝑛(𝐼) -𝒎 𝟏 exp (𝒎 𝟐 𝐼))

(1) the detailed physical meanings of parameters in equation ( 1) are summarized and listed in Table 1. Voltage term related to the reversible potential of PEMFC In equation [START_REF] Zheng | Prolonging fuel cell stack lifetime based on pontryagin's minimum principle in fuel cell hybrid vehicles and its economic influence evaluation[END_REF], all fitting parameters that may vary with aging are written with bold letters. According to the research work of Bressel et al. [START_REF] Bressel | Remaining useful life prediction and uncertainty quantification of proton exchange membrane fuel cell under variable load[END_REF], the overall resistance (R) increases as the stack ages, which indicates that the overall resistance can be used as a PEMFC health indicator.

The hydrogen consumption rate of one PEMFC stack is denoted as 𝑐 𝐻2 (𝑡) , which is defined as a function of the current load 𝐼(𝑡) . The global hydrogen consumption of a PEMFC stack during time period 𝑡 0 is calculated by: 𝐶 𝐻2 = ∫ 𝑐 𝐻2 (𝐼)𝑑𝑡 𝑡 0 0

(2) The proposed post-prognostic decision-making strategy is developed to make a trade-off between the PEMFC deterioration and the fuel consumption. Therefore, two separate but complementary objectives to be minimized are defined:

1. PEMFC deterioration level in order to extend the system lifetime as long as possible. 2. Hydrogen consumption to lower the operation cost while satisfying the operation of the system demand. These criteria are contradictory. Decreasing the deterioration will lead to keep the current in an intermediate, whereas minimizing H 2 consumption will lead to minimize the current. Thus, a suitable current for each fuel cell stack has to be defined by the post-prognostic decision.

The general problem schematic is shown in Fig. 1. Firstly, the two-stack fuel cell system is operating for a certain period. Then, lifetime and consumed hydrogen fuel are calculated for both fuel cells. At last, the decision block is used to integrate the input information from previous blocks, and outputs the optimal load repartition for the two fuel cell stacks. The following assumptions are made on the two-stacks fuel cell system:

1. Two fuel cell stacks are connected in parallel 2. The two fuel cell stacks in the networking system are identical. The output power of each stack ranges from the minimal output power 𝑃 𝑚𝑖𝑛 to maximum output power 𝑃 𝑚𝑎𝑥 .

The value of fuel cell overall resistance ( 𝑅 ) is

considered to be measured in this work. The resistance values can be further estimated thanks to an observer.

POST-PROGNOSTICS DECISION-MAKING

This section focuses on the methodology of the proposed post-prognostics decision-making strategy. More details on the load-dependent deterioration modeling and the associated decision probability are presented in [START_REF] Zuo | Post-prognostics decision making for a two-stacks fuel cell system based on a load-dependent deterioration model[END_REF].

Gamma process deterioration modeling

From equation (1), the overall resistance of PEMFC stack is chosen as the health indicator and is further modeled by a Gamma process to study system deterioration [START_REF] Jouin | Degradations analysis and aging modeling for health assessment and prognostics of pemfc[END_REF][START_REF] Zhang | Proton exchange membrane fuel cell remaining useful life prognostics considering degradation recovery phenomena[END_REF][START_REF] Zhang | An ensemble of models for integrating dependent sources of information for the prognosis of the remaining useful life of proton exchange membrane fuel cells[END_REF]. The overall resistance is assumed to be measurable in realtime. A Gamma process is a stochastic process with independent, positive increments that obeys a Gamma distribution Γ(𝛼, 𝛽) . There are two key parameters in a standard Gamma process, the shape parameter 𝛼 and scale parameter 𝛽. Thanks to the stochastic characteristics, various deterioration behaviors can be simulated by using different 𝛼 and 𝛽 values. When given the 𝛼 and 𝛽 values, the deterioration rate r is calculated by: 𝑟 = 𝛼 • 𝛽 (3) In this work, the deterioration level of the indicator 𝑅 is modeled by a Gamma process. The scale parameter 𝛽 is defined as a constant in this work. The shape parameter 𝛼 is further modeled as a function of the demand load 𝐼, i.e. the deterioration rate is also a load-dependent function 𝑟(𝐼).

PEMFC load-dependent deterioration and fuel consumption modeling

To describe the deterioration rate function 𝑟(𝐼) , the parabola function shown in Fig. 2 is used. This is developed based on the deterioration characteristics of PEMFC [START_REF] Zuo | Post-prognostics decision making for a two-stacks fuel cell system based on a load-dependent deterioration model[END_REF]. As shown in Fig. 2, from minimal operation load 𝐼 𝑚𝑖𝑛 to nominal load 𝐼 𝑛𝑜𝑚 and maximum load 𝐼 𝑚𝑎𝑥 , the corresponding average deterioration rate follows an order of 𝑟 1 > 𝑟 3 > 𝑟 2 .

Furthermore, according to Dépature C. et al. [START_REF] Dépature | Ieee vts motor vehicles challenge 2017-energy management of a fuel cell/battery vehicle[END_REF], the hydrogen fuel consumption rate can be described as a linear function. However, in practical applications, when considering the global stack system, the energy consumption auxiliary components including the air compressor, the cooling pump, and the electronic control device should also be taken into consideration with electrical energy consumption [START_REF] Chen | Lifetime prediction and the economic lifetime of proton exchange membrane fuel cells[END_REF]. This work introduced a quadratic equation to describe the global system fuel consumption considering the characteristics of fuel cell auxiliary components, as shown in Fig. 2.

Fig. 2 Deterioration and hydrogen consumption rates

Fig. 3 Dynamic system load demand

Mixed criterion for a two-stack PEMFC system

A multi-step decision-making strategy is designed for the decision-making phase. The same decision criterion is applied to each decision step at each decision time 𝑘 * 𝜏, where 𝑘 is the decision step, and 𝜏 is the time length of a single decision step.

At each decision time, the deterioration level (i.e. the value of 𝑅) is monitored and based on this measured level 𝑅 𝑜𝑏𝑠 , a conditional probability to reach a deterioration level threshold 𝐹𝑇 𝑑 at the end of the next period is computed for the different possible load repartitions. The calculation approach used here is an exhaustive search. The load repartition decision is eventually made by considering a criterion including the conditional failure probabilities for each stack. The conditional decision probability 𝑃 𝑑 at decision step 𝑘 (𝑃 𝑑 (𝑘) ) is calculated using the following equation:

where 𝐹𝑇 𝑑 (𝑘) is the decision threshold at decision step 𝑘. R(𝑘𝜏) is the simulated deterioration level at decision step 𝑘. 𝑅 (𝑘-1) 𝑜𝑏𝑠 represents the measured deterioration level at step (𝑘 -1).

The consumption of the system under the different possible repartition is also evaluated for each of the stacks. The mixed criterion is constructed to consider both deterioration and fuel consumption, and is formulated as follows:

𝐶𝑟𝑖𝑡𝑒𝑟𝑖𝑜𝑛 = max{𝑤 • 𝑁𝑜𝑟𝑚{(1 -𝑃 𝑑1 ) • (1 -𝑃 𝑑2 )} + (1 -𝑤) • 𝑁𝑜𝑟𝑚{1 (𝐹𝑢𝑒𝑙 1 + 𝐹𝑢𝑒𝑙 2 ) ⁄ }} ( 5
)
where 𝑤 is the weight between fuel cell stack deterioration and system fuel consumption. 𝑁𝑜𝑟𝑚 represents the normalization calculation to make sure that the two items in equation ( 5) have the same order of magnitude. 𝑃 𝑑1 and 𝑃 𝑑2 represent the conditional failure probability of fuel cell stack 1 and stack 2. 𝐹𝑢𝑒𝑙 1 and 𝐹𝑢𝑒𝑙 2 represents the total fuel consumption of two PEMFC stacks.

Decision-making process

The decision is eventually made based on the joint criterion and the load repartition decision is determined as the one that maximizes the joint criterion defined in equation [START_REF] Zhang | Phm-oriented degradation indicators for batteries and fuel cells[END_REF]. A trade-off between the fuel cell deterioration and system fuel consumption is achieved by assigning different weights, 𝑤.

At each decision step 𝑘, an exhaustive search approach is used in search of the optimal load repartition. Whenever the system failure probability and global system hydrogen consumption is evaluated based on the Gamma process simulation, the maximum value of the mixed criterion defined as equation ( 5) is applied to choose the optimal load repartition for each stack. The available control variable is decided as follows: for a given system load demand 𝐼 𝑙𝑜𝑎𝑑 , and 𝐼 1 for FC1(ranges from 𝐼 𝑚𝑖𝑛 to 𝐼 𝑚𝑎𝑥 with a fixed increment step), the operation load for FC2 is decided by the following:

{ 𝐼 𝑙𝑜𝑎𝑑 -𝐼 1 ≤ 𝐼 2 𝐼 𝑚𝑖𝑛 ≤ 𝐼 2 ≤ 𝐼 𝑚𝑎𝑥 (6)
Both constant and dynamic system load demands are studied. For constant system load demand, 𝐼 𝑙𝑜𝑎𝑑 is defined as 𝐼 𝑙𝑜𝑎𝑑2 during the whole operation period as shown in Fig. 3. And the dynamic system demand is shown in Fig. 3 and its value varies among 𝐼 𝑙𝑜𝑎𝑑1 , 𝐼 𝑙𝑜𝑎𝑑2 and 𝐼 𝑙𝑜𝑎𝑑3 .

Performance evaluation

To estimate the performance of the proposed decisionmaking strategy, a performance estimation index is the system lifetime.

Another performance estimation index is the system level hydrogen consumption rate defined by the following equation is chosen as the first estimate index:

𝐻 2 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑆𝑦𝑠𝑡𝑒𝑚 -𝑙𝑖𝑓𝑒𝑡𝑖𝑚𝑒 ⁄ (7) where the 𝐻 2 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 represents the total hydrogen consumption during the service period, and system lifetime equals to EoL. The calculation method of the system lifetime is the same as introduced in [START_REF] Zuo | Post-prognostics decision making for a two-stacks fuel cell system based on a load-dependent deterioration model[END_REF].

The corresponding results of these two indexes are summarized and discussed in the following.

RESULTS AND DISCUSSION

Basic value settings

The initial resistance value R 0 is set as 0.1798 Ω𝑐𝑚 2 . The values of 𝐼 𝑚𝑖𝑛 , 𝐼 𝑛𝑜𝑚 and 𝐼 𝑚𝑎𝑥 are set as 2, 8, and 10 𝐴𝑐𝑚 -2 for each fuel cell stack. The deterioration rate values in Fig. 3 are calculated using the same method introduced in [START_REF] Zuo | Post-prognostics decision making for a two-stacks fuel cell system based on a load-dependent deterioration model[END_REF]. A failure threshold (𝐹𝑇) is defined for PEMFC stack to calculate the EoL value and the 𝐹𝑇 is chosen as 3. Three key values of the dynamic load demand shown in Fig. 3 are set as 12.4, 13, and 15 𝐴𝑐𝑚 -2 , for 𝐼 𝑙𝑜𝑎𝑑1 , 𝐼 𝑙𝑜𝑎𝑑2 and 𝐼 𝑙𝑜𝑎𝑑3 respectively. The constant load demand in this work is set as 13 𝐴𝑐𝑚 -2 . The scale parameter of Gamma process 𝛽 equals to 0.36. For the Gamma process simulation, the total simulation length is 2700 hours. The single decision step time length 𝜏 is 150 hours.

Post-prognostics decision-making results

To study the influence of different weights of the criteria as introduced in equation ( 5), 100 simulations runs were performed, and four different 𝑤 values were used. Each simulation is compared with the results developed under constant load (𝐼 1 = 𝐼 2 = 𝐼 𝑙𝑜𝑎𝑑 2 ⁄ ). For constant system load demand simulations, the results on the overall system level hydrogen consumption rate/System-lifetime ratio are shown in Fig. 4. Globally, the hydrogen consumption rate/systemlifetime ratio decreases with 𝑤 value. By contrast, the results without decision show rough variations between the different simulation times. This phenomenon shows that the proposed decision-making strategy can decrease the failure risk of the system, achieving a steadier operation.

Table 2 further summarizes the hydrogen consumption rate results. The results of the comparison experiment have the lowest rate value, 40.88 𝑔𝑠 -1 . And the hydrogen consumptions rate results for 𝑤 = 0.3, 𝑤 = 0.5, 𝑤 = 0.7, and 𝑤 = 0.9 are 41.04, 41.5, 47.67, and 58.16 𝑔𝑠 -1 respectively. This result is logical. The higher value of 𝑤 means that, based on equation ( 6), the trade-off rules tend to make a decision that consumes more fuel to mitigate the deterioration. Otherwise, decreasing the weights 𝑤 means that less fuel consumption is appreciated. During the operation, the decision strategy tends to increase the fuel consumption but the lifetime can be considerably improved.

For the system lifetime distribution histograms are plotted in Fig. 5 and Fig. 6. Fig. 5 a) shows the system lifetime results developed without a decision-making strategy. 68% of the system lifetime in Fig. 5 are distributed between 1200 and 1800 hours. Whereas, when applying the proposed decisionmaking strategy, the system lifetime is shifting to a bigger distribution interval. 65% percent of the system lifetime is distributed between 1400 and 2800 hours for 𝑤 = 0.3. For 𝑤 = 0.7 and 0.9, the system lifetime distributed between 1400 and 2800 hours is 86% and 97% respectively. The median and mean system lifetime statistic values shown in Table 2 also prove that the system can achieve a longer lifetime by using bigger 𝑤. Therefore, in order to achieve a maximum economic lifetime for fuel cell systems, the practitioners can choose a proper 𝑤 for the mixed criterion, then apply the decisionmaking strategy to deal with the trade-off between the fuel cell stack deterioration and system fuel consumption, achieving the desired economic lifetime.

For the simulation results derived from dynamic system load demand simulation, the general conclusion is similar to the constant load demand. From Table 2, the hydrogen consumption value is slightly larger than with constant load demand. The value for 𝑤 = 0.7 and 𝑤 = 0.9 are 52.38 and 58.62 𝑔𝑠 -1 respectively, much greater than the value for 𝑤 = 0.3 and 0.5. The system lifetime distribution is exhibited in Fig. 7. Compared with the results in Fig. 5 b), 81% of the system lifetime is distributed between 1400 and 2800 hours for 𝑤 = 0.3 . And for 𝑤 = 0.5 , 𝑤 = 0.7 , and 𝑤 = 0.9 , the lifetime distributed between 1400 and 2800 hours is 89%, 93%, and 98% respectively. Therefore, for the dynamic load demand, the proposed decision-making strategy still works well. The median and mean system lifetime statistic results in Table 2 give the same conclusions.

CONCLUSIONS

A load-dependent deterioration model is developed to model the deterioration of a multi-stack fuel cell system. To manage its economic system life, a joint decision-making criterion which includes both fuel cell deterioration and fuel cell consumption is proposed to formulate the post-prognostics decision-making strategy. For the constant type system load demand, a larger hydrogen consumption / system-lifetime ratio (𝑤) is achieved when applies more weights on mitigating fuel cell deterioration. 68% of the system life developed without decision-making strategy is distributed between 1200 and 1800 hours. When applying the decision-making strategy, the system's life is improved. For 𝑤 = 0.7 and 0.9, the percent of system life distributed between 1400 and 2800 hours reach to 86% and 97%. The dynamic system load demand simulation results prove the same trend. The percent of system life distributed between 1400 and 2800 hours for 𝑤 = 0.3, 0.5, 0.7, and 0.9 are 81%, 89%, 93, and 98% respectively. The proposed decision-making strategy can help to make a compromise between the fuel cell system deterioration and system fuel consumption, and thus greatly improve the system life.

  = 𝑃 (𝑇 𝐹𝑇 𝑑 (𝑘) ≤ 𝑘𝜏) = 𝑃(𝑅(𝑘𝜏) > 𝐹𝑇 𝑑 (𝑘) ) = 𝛤(𝛼𝑡,(𝐹𝑇 (𝑘) 𝑑 -𝑅 (𝑘-1) 𝑜𝑏𝑠 )•𝛽) 𝛤(𝛼𝑡)

Fig. 4

 4 Fig. 4 Hydrogen consumption/System-lifetime ratio (constant load demand).

Fig. 5

 5 Fig. 5 System lifetime distribution histogram -without decision-making strategy. a) constant load demand, b) dynamic load demand.

Fig. 6

 6 Fig. 6 System lifetime distribution histogram -Constant load demand. a) w = 0.3, b) w = 0.5, c) w = 0.7, d) w = 0.9.

Fig. 7

 7 Fig. 7 System lifetime distribution histogram -Dynamic load demand. a) w = 0.3, b) w = 0.5, c) w = 0.7, d) w = 0.9.

Table 1

 1 Physical meaning of the parameters in equation[START_REF] Zheng | Prolonging fuel cell stack lifetime based on pontryagin's minimum principle in fuel cell hybrid vehicles and its economic influence evaluation[END_REF]

	Parameters	Physical meaning
	n	Number of cells in the stack
	𝐼	Operating current load
	R	The overall resistance of PEMFC
	𝑚 1	Constant related to the mass
	𝑚 2	transport overpotential
	𝐸 0

Table 2

 2 System hydrogen consumption and system life results

		Mean hydrogen consumption rate (g s -1 )	Median system life (h)	Mean system life (h)
	Without
	decision-	40.88	1411.5	1387.9
	making
	W = 0.3
	Constant	41.04	1540.5	1524.2
	load


Kennedy Nexon

Benjamin Dollet
email: benjamin.dollet@univ-grenoble-alpes.fr

Chagua Encarnación

Philippe Marmottant

K N Chagua Encarnación

Pervaporation

published or not. The documents may come

Introduction

Global warming is an increasing thread for the survival of forests [START_REF] Choat | Triggers of tree mortality under drought[END_REF][START_REF] Brodribb | Hanging by a thread? Forests and drought[END_REF]. Evapotranspiration at the surface of the leaves, the mechanism whereby sap cir-culates in the xylem from the roots to the leaves, puts the sap in a state of negative pressure [START_REF] Stroock | The physicochemical hydrodynamics of vascular plants[END_REF][START_REF] Jensen | Sap flow and sugar transport in plants[END_REF], all the more that the ambient atmospheric conditions are dry and warm. Cavitation bubbles may then appear in the sap network, and subsequently invade it, which can stop the sap circulation. It is therefore of primary importance to understand this invasion mechanism in leaves, known as air seeding among the plant physiologists.

Like the vascular network of animals, leaf venation forms complex networks with different levels of hierarchy, reconnections, and different sections. The channel section tends to decrease in size after each bifurcation, when a channel divides into several channels: the section becomes smaller and smaller until the terminal veins. It was shown the section follows Murray's law as in animal networks [START_REF] Mcculloh | Water transport in plants obeys Murray's law[END_REF]: the sum of conduit diameters raised to power 3 is conserved.

During evapotranspiration, water is transported from the veins to the stomata at the surface of the leaves, by diffusion through the mesophyll tissue which surround xylem sap in leaves. This transport is also called pervaporation in other contexts, such as chemical engineering; see e.g. [START_REF] Eijkel | Osmosis and pervaporation in polyimide submicron microfluidic channel structures[END_REF][START_REF] Ziane | Drying with no concentration gradient in large microfluidic droplets[END_REF][START_REF] Ziemecka | Hydrogen peroxide concentration by pervaporation of a ternary liquid solution in microfluidics[END_REF], and [START_REF] Bacchin | Microfluidic evaporation, pervaporation and osmosis: from passive pumping to solute concentration, under review in[END_REF] for a full review. To make a biomimetic system mimicking water transport through the mesophyll tissue, we use channels made of polydimethylsiloxane (PDMS), following the method by [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF]. This material is permeable to water vapour, like the mesophyll tissues. Among the different microfluidic or nanofluidic systems which have been used for synthetic trees or leaves, soft PDMS was preferred over the stiff hydrogels [START_REF] Wheeler | The transpiration of water at negative pressures in a synthetic tree[END_REF][START_REF] Vincent | Birth and growth of cavitation bubbles within water under tension confined in a simple synthetic tree[END_REF] or nanoporous silicon [START_REF] Vincent | The fast dynamics of cavitation bubbles within water confined in elastic solids[END_REF], which can withstand negative pressure. Here we do not focus on negative pressure and cavitation, but on latter stages when air invades the network during drying, at pressures close to atmospheric pressure. PDMS has then the advantage that it reproduces better the thinness and flexibility of real leaves.

From a physical standpoint, to understand pervaporation-induced air invasion dynamics in biomimetic networks, we study the simplest systems, increasing the complexity by steps. We started by single channels [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF], then elementary networks [START_REF] Dollet | Drying by pervaporation in elementary channel networks[END_REF], but still at constant section. In the current study, we release this restriction, and focus on the effect of the variation of section in single channels and simple branched networks.

Materials and methods

All our channels are molded in a thin PDMS layer glued on a glass plate, as follows. The glass slide was exposed for 30 s to a plasma, then the PDMS imprint was manually deposited on the plasma-activated glass surface, taking care to avoid folds and blisters of the flexible PDMS sheet. In all this study, the height h of the channel and the thickness H of the PDMS layer (Fig. 1b) are kept as much constant as possible, within the uncertainties of the fabrication process: h = 42 ± 3 µm and H = 88 ± 2 µm (see later Tab. 1 for the values of h and H measured for separate designs). Hence, to study variations of the cross-section, we focus on variations of the channel width w (Fig. 1a). We study first single channels of variable width, then simple networks of channels Fig. 1 (a) Top view sketch of a channel of varying width w(x), where x is the streamwise coordinate along its centreline. The channel is filled with water over a length L along its centreline. The origin x = 0 is located at the closed end of the channel, and the water occupies the region 0 ≤ x ≤ L; the meniscus separating the water-filled part of the channel and its air-filled part is thus located at x = L. (b) Transverse side view of the channel of width w(x) and height h between a glass slide and a PDMS sheet of thickness H. of different widths. However, for the sake of simplicity, in the latter case, each portion of network between consecutive nodes is of constant width.

The experiments are run as follows. The channels are manually cut with a scalpel at the entrance point (the cuts are clearly visible in Fig. 2). The glass slide on which the channels are bond is plunged into a beaker of deionised water under low pressure for one hour, until the channels are completely filled. They are then placed under a constant flux of dry air, and their drying from the entrance points onwards is recorded with a CCD camera, from which the location of the meniscus (or menisci) separating the water-filled part and airfilled part of the channels is measured by image analysis.

First, we study two single channels of continuously variable width between the entrance point and a terminal wall (Fig. 2a). We focus on channels of total length L c = 12 mm with width varying linearly along the channel between 50 µm and 200 µm, either decreasing or increasing from the entrance to the terminal wall. Second, we study two single channels of total length L c = 10 mm with variations of width by steps (Fig. 2b). One channel present five portions (each of length 2 mm) with widths decreasing from 175 µm to 75 µm, and the other has four portions (each of length 2.5 mm) with widths increasing from 75 µm to 150 µm.

We study next simple channel networks, and we revisit the branched networks which we have studied in [START_REF] Dollet | Drying by pervaporation in elementary channel networks[END_REF]. We focus on two configurations. First, we study single-node networks, where an entrance channel splits into two, three, four or five branches (Fig. 2c). In this paper, each of these branches has a different width. Second, we consider a "tree" with consecutive subdivisions of the channels into branches of smaller widths (Fig. 2d). The choice of the geometry of the tree results from a compromise between the presence of certain ingredients of complexity (several levels of branch divisions, branches of different lengths and widths), with yet a simple enough design to allow for simple analysis. 3 Single channels of variable width

Theory

In this Section, we model the drying dynamics by pervaporation of a single channel of varying width w(x), where x is the streamwise coordinate along the centreline of the channel (Fig. 1a). The channel height is denoted h and the PDMS thickness H (Fig. 1b); δ = H -h is therefore the PDMS thickness between the channel top and the outer air. If Q is the volumetric flux of water leaving the water-filled part of the channel, the conservation of mass writes:

-Q = d dt h L(t) 0 w(x)dx = h Lw(x = L), (1)
where the term in brackets is the volume of water inside the channel at time t, L being the length of the water-filled part (henceforth called water length) and | L| the velocity of the meniscus. It is worth noting that at a given flux Q, the larger the width w(x = L) at the location of the meniscus, the slower the meniscus.

There are two contributions to the flux Q = Q ℓ + Q g : one by diffusion from the water-PDMS interface (Q ℓ ), and an evaporative flux Q g at the meniscus. We restrict our analysis to a slowly varying width: |dw/dx| ≪ 1. This hypothesis is suitable for slender channels, such as those encountered in leaves. For a slowly varying width, the streamwise dependence of the concentration field of water into PDMS is much weaker than the spanwise dependence, and we may neglect the width variation to estimate the diffusive flux. Hence:

Q ℓ = L 0 q ℓ (w(x))dx, (2)
where q ℓ (w) is the diffusive flux per unit length of a channel of constant width w. Inserting in (1), we get:

h Lw(x = L) = - L 0 q ℓ (w(x))dx -Q g . (3)
In [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF], we have derived the following analytical predictions of the flux q ℓ :

q ℓ = D P csat P (1 -RH)q ℓ , (4)
with D P = 10 -9 m 2 /s the diffusivity of water in PDMS [START_REF] Watson | The behaviour in water in poly(dimethylsiloxane)[END_REF], csat P = M C sat P /ρ the mass fraction of water in PDMS at saturation (with M = 0.018 kg/mol the molar mass of water, ρ = 10 3 kg/m 3 its density, and C sat P = 40 mol/m 3 the saturation concentration of water in PDMS [START_REF] Randall | Permeation-driven flow in poly(dimethylsiloxane) microfluidic devices[END_REF], this gives csat P = 7.2 × 10 -4 ), RH the relative humidity of the outer air (henceforth set to zero, consistently with our experiments) and qℓ a dimensionless shape factor coming from the resolution of a diffusion problem:

qℓ = w δ + ξ, (5)
with ξ another dimensionless shape factor quantifying diffusion between the channel side walls and the outer air; under reasonable geometrical assumptions quantified in [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF] and holding in our experiments, it equals:

ξ = 2 π ln (H + δ)h δ 2 + H δ ln H + δ h .
We also use the prediction of Q g made in [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF] with the local value of the thickness at the meniscus location, simply denoted w in the remainder of this Section. To be rigorous, the assumption |dw/dx| ≪ 1 is not sufficient to make such an approximation, and should be replaced by the more stringent condition |dw/dx| ≪ ε 1/2 where ε = D P /αD a is a small parameter (with α = 0.03 the Henry constant quantifying the water affinity in PDMS [START_REF] Harley | Thermodynamic study in dynamic water vapor sorption in Sylgard-184[END_REF] and D a = 2 × 10 -5 m 2 /s the diffusivity of water vapour in air, hence ε = 2 × 10 -3 ), because the value of Q g depends on the concentration field of water at distances of order ε -1/2 w away from the meniscus [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF]. Under this approximation, the following prediction holds:

Q g = αD a D P hw qℓ . ( 6
)
3.2 Experiments

Channels with linearly variable width

We now test our theory on the two channels with linearly variable width. Fig. 3a shows the time evolution of the water length in these two channels.

It is a decreasing function of time, consistently with the drying process. More interestingly, both curves are convex, but the convexity is larger for the channel with increasing width compared with the one with decreasing width. The meniscus in the first channel is faster at the beginning of the drying process, but slower at the end, and the total drying time is almost similar for both channels.

Qualitatively, this behaviour highlights the influence of the local value of the channel width at the meniscus location in Eq. ( 1): the narrower the channel, the faster the meniscus at given flux Q. Quantitatively, we can apply the model of Sec. 3.1 for a width profile of the form:

w(x) = w 0 + (w 1 -w 0 ) L c -L -x L c , (7)
with L c the channel length, w 0 the width at the channel entrance and w 1 the width at its closed end. Inserting ( 7) in (2) and using ( 4) and ( 5), we get:

Q ℓ = D P csat P w 1 δ + ξ L - w 1 -w 0 2δL c L 2 .
Using the expression (6) of Q g , we get after some algebra: with the dimensionless quantity: L = L/L c , and two physicochemical parameters having the dimension of a diffusivity:

1 -1 - w 0 w 1 L d L dt = - D 1 hw 1 w 1 δ + ξ L + w 1 -w 0 2δ L2 - D 2 L c √ hδ × 1 + w 0 -w 1 w 1 L 1 + w 0 -w 1 w 1 L + δ w 1 ξ , (8)
D 1 = D P csat P , D 2 = αD a D P . (9)
Eq. ( 8) provides an explicit relationship between the instantaneous velocity of the meniscus and its location. Therefore, to compare our data and our model, we plot the velocity of the meniscus as a function of the water length in Fig. 3b, and we fit the curves by ( 8) with D 1 and D 2 as fitting parameters. We anticipate that these two parameters will serve as fitting parameters for each set of experiments to be discussed throughout this study, and we refer to Sec. 5 for a discussion of their values.

Fig. 3 shows that the fit is excellent. Only small discrepancies are visible for largest values of L, which corresponds to cases where the meniscus is close to the open end of the channel and may be influenced by the uncontrolled opening of the channel. Furthermore, if we fit independently the data for each channel, the values of the fitting parameters are in close agreement (see later Tab. 1). This shows the accuracy of our model to explain data in channels of slowly varying width.

Channels with stepwise varying width

We proceed with two channels of stepwise varying length. Fig. 4a shows the time evolution of the water length in these two channels. It is once again a decreasing function of time. However, the novel feature of the curves is the occurrence of finite changes of velocity at each step, which shows more clearly when we plot the velocity of the meniscus as a function of the water length (Fig. 4b). The velocity jump is positive, respectively negative, at the steps where the width decreases, respectively increases. Qualitatively, as discussed in Sec. 3.2.1, this behaviour is consistent with the influence of the local value of the channel width at the meniscus location in [START_REF] Choat | Triggers of tree mortality under drought[END_REF]. We can compare the experimental data with our model, bearing in mind that strictly speaking, the latter does not apply to stepwise variations in width, because it relies on an hypothesis of slowly varying width. We assume that the meniscus is in a portion of width w 0 and at a distance L 0 (t) from the upcoming step, and that the water-filled part of the channel spans N full portions of length L i and constant width w i (for i ranging from 1 to N ). The total water length is then L(t) = L 0 (t) + N i=1 L i , and the total diffusive flux (2) writes:

Q ℓ = q ℓ (w 0 )L 0 + N i=1 q ℓ (w i )L i = D P csat P w 0 δ + ξ L 0 + N i=1 w i δ + ξ L i ,
where (4) and ( 5) have been used. Inserting into (3), we get:

L = - L + L sup τ , (10)
with:

τ = 1 D 1 A , (11)
and:

L sup = D 2 D 1 √ A - N i=1 w i -w 0 w 0 + ξδ L i ,
where A is a channel-dependent geometrical parameter:

A = 1 hw 0 w 0 δ + ξ , (12)
and D 1 and D 2 are given by ( 9). We fit the data of Fig. 4b by Eq. ( 10) with D 1 and D 2 as fitting parameters. Contrary to the channels with linearly variable width, the agreement between the model and the data is not very good; in particular, the model predicts jumps in velocity which are much smaller in amplitude than what is experimentally observed. Possible reasons for this discrepancy are discussed in Sec. 5.2.

Branched channels with different widths

To extend further our study, we now consider branched channels, yet without reconnections, and with different widths. For simplicity, we study only channels where individual branches have constant widths.

Theory

Let us first consider a meniscus in the entrance channel. It is in contact with a water region which spans part of the entrance channel and all branches. All these contribute to the diffusive flux Q ℓ introduced in Sec. 3.1. Hence, Eq. ( 3) is replaced by:

h Lw m = -q ℓ (w(s))ds -Q g , ( 13
)
where L is the total length of the water region, w m is the value of the width at the location of the meniscus, and the integral on the right-hand side spans all the water region, with s a curvilinear coordinate along the centreline of that region.

Let us specify the analysis to the case relevant to the experiments, of N branches of length L i and constant width w i (for i ranging from 1 to N ) extending from the first node. We also consider that the entrance channel has a constant width w 0 , and that the water inside this channel spans a length L 0 (t). The total water length is then L(t) = L 0 (t)+ N i=1 L i , and (13) becomes:

h L0 w 0 = -q ℓ (w 0 )L 0 - N i=1 q ℓ (w i )L i -Q g . ( 14
)
Proceeding in a similar fashion as in Sec. 3.2.2, we obtain that the water length should obey Eq. [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF], where the sum extends over all the branches ahead of the meniscus. The dynamics in intermediate channels also follows [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF] and thus [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF], provided w 0 is taken as the local width at the meniscus location, and the sum runs over all water-filled branches ahead of the meniscus. The dynamics in terminal branches is obtained by suppressing the sum from [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF].

In the current case where all branches have a constant width, the dynamics is therefore of the form L0 = -(L 0 + L sup )/τ , where the characteristic time scale τ depends only on the local width (where the meniscus is located), while the length scale L sup depends on the whole network ahead of the meniscus. We proved in [START_REF] Dollet | Drying by pervaporation in elementary channel networks[END_REF] that this analysis is true as long as all branches are slender (of length much larger than the transverse dimensions) and well separated (of mutual distances much larger than the transverse dimensions), in order to neglect diffusive coupling between channels separated by a small distance [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF].

Experiments

Single-node networks

Fig. 5a, c ande shows the distance travelled by the menisci as a function of time in the single-node networks with respectively two, three and five branches. In the entrance channel, the meniscus velocity increases as the number of branches increases, and there is a velocity jump as the meniscus in the entrance channel splits into several menisci in the branches. This result was already found for single-node networks with uniform width, and can be ascribed to a jump of connected water length when meniscus splitting occurs at the node [START_REF] Dollet | Drying by pervaporation in elementary channel networks[END_REF]. The new feature here is that the dynamics of each branch depends on its width: the smaller the latter, the larger the meniscus velocity and the shorter the drying time. Qualitatively, this is a signature of the fact that the typical drying time τ is an increasing function of the width. This can be seen from Eqs. ( 11) and ( 12), which show that such a dependence on width arises from the interplay between the diffusion between the channel top wall and the outer air [giving the term w/δ in ( 12)], and the diffusion between the channel side walls and the outer air [giving the term ξ in [START_REF] Vincent | Birth and growth of cavitation bubbles within water under tension confined in a simple synthetic tree[END_REF]]. We showed in [START_REF] Dollet | Drying of channels by evaporation through a permeable medium[END_REF] that this interplay is responsible for the faster drying of narrow channels because they are more prone to lose water from their side walls, compared to wider channels.

To test more quantitatively the predictions of Sec. 4.1, we plot the velocity of the meniscus as a function of the water length in the right panel of Fig. 5, and we fit the curves by [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF]. For each single-node network, we impose that D 1 and D 2 be the same for the entrance channel and all the branches. The right panel of Fig. 5 shows that the fit is satisfactory; in particular, the terminal branches are correctly described, also the experimental data show some concavity, which may be ascribed to evaporative coupling effects discussed in [START_REF] Dollet | Drying by pervaporation in elementary channel networks[END_REF]. The entrance channels are also well fitted, with the exception of the five-branch network where the experimental slope is too large.

Tree network

We now turn to the tree network of Fig. 2d. It comprises 27 branches, but some of them share the same dimensions and "order" in the tree hierarchy, and it is found that they then follow a similar dynamics. We gather such branches into 9 different groups, represented by different symbols in Fig. 6a. Fig. 6b shows the distance travelled by the menisci as a function of time in the tree, for all branches. Like for the single-node networks, velocity jumps occur at each node. The new feature here is that at given depth in the tree network, the velocity is an increasing function of the water length ahead of the The lines are fits by [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF], with D 1 and D 2 as fitting parameters. The inset is a zoom on the bottom left part of the main plot.

considered meniscus: the shorter the branches, the slower the dynamics (see inset of Fig. 6b).

Here again, we plot the velocity of the meniscus as a function of the water length (Fig. 6c), and we fit the curves by [START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF]. We find once again a good agreement between the experiments and the data, which suggests that our physical modelling remains adapted to the case of more complex branched networks.

Discussion

Values of the fitting parameters

In each experiment, the two physicochemical parameters D

Limitations of the current model and perspectives

We have compared all our experiments with a model based on the hypothesis of slowly varying width. This enables to consider that water diffusion through PDMS acts mainly in the transverse direction. We have shown that such a model captures well our experimental observations, except the case of channels with stepwise varying width.

Obviously, in such a case, the hypothesis of slowly varying width is violated at the location of the steps. To go beyond and extend the model, the full 3D diffusion of water through PDMS should be solved, at least in the vicinity of the steps. However, this requires time-consuming numerical simulations, beyond the scope of this study.

Moreover, an additional ingredient may become important in the case of steps: when passing an abrupt variation in channel section, the meniscus curvature necessarily varies at given contact angle, to match the geometry of the channel walls. In other contexts, such an effect is important for droplet breakup in microfluidic T-junctions [START_REF] Ody | Transport of wetting liquid plugs in bifurcating microfluidic channels[END_REF] and in soap films and foams in porous media [START_REF] Xu | Effective viscosity of foam in periodically constricted tubes[END_REF][START_REF] Cox | A theory of the effective yield stress of foam in porous media: the motion of a soap film traversing a three-dimensional pore[END_REF][START_REF] Géraud | Lamella division in a foam flowing through a two-dimensional porous medium: A model fragmentation process[END_REF]. This induces variations of capillary pressure, hence of pressure in the liquid, which may deform the channel, an effect hitherto disregarded in the model. This might explain why velocity jumps in Fig. 4 are much more marked in experiments that in the model. Interestingly, in the context of tree physiology, such effects are relevant in the xylem network of trees [START_REF] Tyree | Vulnerability of xylem to cavitation and embolism[END_REF][START_REF] Choat | Pit membrane porosity and water stress-induced cavitation in four co-existing dry rainforest tree species[END_REF],

where xylem vessels are separated by pits of nanometric dimensions where menisci between embolism and liquid can get trapped while undergoing huge variations of capillary pressure. In an upcoming study, we will revisit this effect by mimicking pits in microfluidic experimental micromodels, to clarify the couplings between drying, meniscus dynamics, capillary pressure variations and channel deformability.

Conclusions

Our study revealed that drying in networks with channels of variable sections are well described by a generalised version of our model, consisting in the integration of the section-dependent diffusive flux over the length of the channels (Eq. 3). This model fits very well with experiments for linearly varying section, stepwise-decreasing width and division in branches with different sections. The drying dynamics is always smooth, without any arrest, just velocity changes. However, studies on real leaves [START_REF] Ponomarenko | Ultrasonic emissions reveal individual cavitation bubbles in water-stressed wood[END_REF][START_REF] Brodribb | Revealing catastrophic failure of leaf networks under stress[END_REF] showed a stop-and-go dynamics with arrests and sudden jumps. This peculiar dynamics may be the consequence of the fact that conduits are not fully continuous in plants, but are made of collection of separate conduits connected by pits. Future work will focus on the effect of tiny constrictions in drying dynamics.

Fig. 2

 2 Fig. 2 Snapshots of the channel designs studied in this paper. (a) Single channels of linearly varying width. Notice their slight curvature, due to a slight distortion of the flexible PDMS layer when it is bond to glass; this has no noticeable influence on the dynamics. (b) Single channels with widths varying by steps, which are marked by circles outside the channels. (c) Single-node networks with varying number of branches (the results with the network with four branches will not be presented here). All entrance channels have a width 100 µm and a length Le = 3 mm. All branches have a length L b = 4 mm. (d) "Tree" design. Branches have a length of 2, 1 or 0.5 mm, and their width decrease at each bifurcation: 175 µm for the entrance channel, then 150, 125, 100 and 75 µm.

Fig. 3

 3 Fig.3Drying dynamics in channels of width varying linearly between 50 and 200 µm, for a decrease (•) and an increase ( ) of the width over a length of 12 mm. (a) Time evolution of the water length, and (b) velocity of the meniscus as a function of the water length. In (b), the plain curves are fits by Eq. (8) with D 1 and D 2 as fitting parameters. In (a), the plain curves result from numerical integration of the fitting curves of (b). The dotted curves show the case of a channel of constant width 125 µm.

Fig. 4

 4 Fig.4Drying dynamics in channels of stepwise varying width. (a) Time evolution of the water length. Here, we use all available data points, which are close enough to constitute a seemingly continuous curve. The top curve corresponds to a decreasing width, and the bottom one to an increasing width. Successive portions of constant width are alternatively figured in plain and dotted lines. (b) Velocity of the meniscus as a function of the water length in channel of stepwise increasing (triangles) or decreasing (circles) width. The lines are fits by Eq. (10) with D 1 and D 2 as fitting parameters.

Fig. 5

 5 Fig. 5 Drying dynamics in single-node networks. From bottom to top, networks with two, three and five branches. Left column: temporal evolution of the distance travelled by the menisci from the entrance point. Right column: velocity as a function of the water length. Insets in panels (d) and (f) are zooms on the terminal branches. Symbols: entrance channel (•), branches of width 75 (▽), 100 (△), 125 (⊲), 150 (⊳) and 175 µm ( ). The lines are fits by (10), with D 1 and D 2 as fitting parameters.

Fig. 6

 6 Fig. 6 Drying dynamics in a tree. (a) Symbols used for the different branches. (b) Temporal evolution of the distance travelled by the menisci from the entrance point. Right column: velocity as a function of the water length. The inset is a zoom on the terminal branches, to help distinguishing their drying dynamics. (c) Velocity as a function of the water length.The lines are fits by[START_REF] Noblin | Proc. Natl. Acad. Sci[END_REF], with D 1 and D 2 as fitting parameters. The inset is a zoom on the bottom left part of the main plot.

Table 1

 1 1 = D P C sat P and D 2 = √ αD a D P have been used as fitting parameters. Their values are re-Values of the fitting parameters D 1 and D 2 , and of the dimensions h and H, for all experiments. ported in Tab. 1. They show some dispersion, especially for the channels with stepwise varying width, for which it has been shown that the model is inaccurate. The other experiments yield D 1 = 0.4 µm 2 /s and D 2 = 2 × 10 4 µm 2 /s.Taking values from the literature: D P = 10 -9 m 2 /s[START_REF] Watson | The behaviour in water in poly(dimethylsiloxane)[END_REF], D a = 2 × 10 -5 m 2 /s, α = 0.03[START_REF] Harley | Thermodynamic study in dynamic water vapor sorption in Sylgard-184[END_REF] and csat P = M C sat P /ρ = 7.2 × 10 -4 (using in particular C sat P = 40 mol/m -3[START_REF] Randall | Permeation-driven flow in poly(dimethylsiloxane) microfluidic devices[END_REF]), we compute D 1 = 0.7 µm 2 /s and D 2 = 2 × 10 4 µm 2 /s. Bearing in mind that the values of the physicochemical parameters D P , α and C sat P are known with relatively poor precision, we can conclude that the values of our fitting parameters are in good agreement with the reported values of D P , α and C sat P .

	Experiment	Figure	D 1	D 2	h	H
			(µm 2 /s)	(×10 4 µm 2 /s)	(µm)	(µm)
	linearly decreasing width	3	0.410	2.53	39.3	89.3
	linearly increasing width		0.394	2.42
	stepwise decreasing width	4	0.184	2.83	45.6	84.5
	stepwise increasing width		0.540	0.79
	2-branch network	5a,b	0.389	1.83
	3-branch network	5c,d	0.395	1.62	42.2	87.4
	5-branch network	5e,f	0.437	1.32
	tree	6	0.265	1.57	41.4	89.2

Acknowledgements We thank Aurélien Caumont, Danièle Centanni and Ludovic Keiser for experimental help. Funding by ANR, under the grant PHYSAP ANR619-CE30-0010-02, is gratefully acknowledged.

Conflict of interest

The authors declare that they have no conflict of interest.