Guillaume R Coudevylle

Maria Popa-Roch

´phane Ste

Olivier Sinnapah

Nicolas Hue

Robin

Impact of Tropical Climate on Selective Attention and Affect

Keywords: environment, concentration, inattention, impulsivity, affective states, PANAS

Heat has an impact on several aspects of human cognition but the effects of the tropical climate (i.e., hot and wet) have rarely been explored. The purpose of this study was to determine whether selective attention and affect are negatively impacted by the tropical climate. The study followed a within-participants design: participants responded to an affective scale (PANAS) and performed an attention task (d2 Test) in two experimental climate conditions (tropical vs. neutral) with a one-week interval between sessions. The results indicated that they had lower positive affect and selective attention in the tropical climate than in the neutral climate. However, there was no significant difference in the effect on negative affect between conditions. The impact of tropical climate on affects and selective attention is discussed.

The tropical climate (TC), where temperatures can exceed 31 ˚C and relative humidity (rH) exceeds 70%, can be considered an environmental stressor [START_REF] Robin | Effects of tropical climate on mental rotation: The role of imagery ability[END_REF]. Thus far, research has been consistent regarding the negative impact of TC on aerobic exercise [START_REF] Hue | The challenge of performing aerobic exercise in tropical environments: Applied knowledge and perspectives[END_REF], and generally all individuals undergo a period of adjustment to very hot and humid conditions [START_REF] Salati | Precipitation and water recycling in tropical rain forests with special reference to the Amazon basin[END_REF]. Accordingly, [START_REF] Kosonen | Assessment of productivity loss in airconditioned buildings using PMV index[END_REF] suggested that the thermal environment (including heat and humidity) is one of the most important environmental factors affecting human performance. However, research has not clearly determined whether performance decline is due to the direct impact of the climate (the physical and psychological aspects being affected independently) or whether this negative impact is mediated by a decrease in aspects of an individual's cognition (e.g., selective attention) and/or affect. The aim of the current study focused only on the direct impact of the climate on the psychological aspects and explored specifically whether affect and attention are impacted by exposure to TC.

Impact of Tropical Climate on Selective Attention

Although the literature shows that the separate effects of heat and humidity have been investigated, little is known about the combined impact of these two variables on cognitive performance. [START_REF] Pepler | Warmth and performance: An investigation in the tropics[END_REF] observed changes in the performance of a tracking task conducted in high and low rH (respectively, 80 and 20%), with each humidity condition crossed with four temperatures (i.e., 22, 26, 29, and 34 ˚C). The author observed a significant performance decline between 22 and 26 ˚C for the 80% rH condition, whereas a significant decline was observed between the higher temperatures of 26 and 29 ˚C for the 20% rH condition, demonstrating that a high level of rH can be particularly deleterious to mental task performance.

In general, the literature has focused more on the effects of heat than on the influence of TC. In addition, the effects of heat exposure on cognitive function are a relatively recent research topic. Thus, although the physiological reactions to heat stress are well understood, the psychological reactions to heat have not been fully explored (see [START_REF] Robin | The influences of tropical climate on imagined walking time[END_REF]. Chase, Karwowski, Benedict, Queseda, and Irwin-Chase (2003) reported poor dual-task performance at 30 and 35˚C and observed that participants were unable to successfully allocate their attention to the tasks, although they managed to maintain accurate performances. [START_REF] Gaoua | Alterations in cognitive performance during passive hyperthermia are task dependent[END_REF] tested the effect of passive heat exposure on working memory and attention. They showed that memory was impaired but not the attentional processes. These two studies addressed the impact of heat on attention and need to be confirmed in other environmentally stressful conditions such as TC. Although the results in such tasks have been inconsistent across studies, it can be expected that the combination of heat and humidity would induce even higher levels of stress that might negatively impact performance in TC. Indeed, [START_REF] Gaoua | Cognitive function in hot environments: A question of methodology[END_REF] suggested that humans have limited cognitive capacities to ensure successful outcomes due to the continuous external stimulations constantly competing for limited conscious access to the vast global workspace (see [START_REF] Baars | How does a serial, integrated and very limited stream of consciousness emerge from a nervous system that is mostly unconscious, distributed, parallel and of enormous capacity[END_REF].

Impact of Tropical Climate on Positive and Negative Affects

Given the characteristics of TC (i.e., high heat and humidity), it is likely that it has an impact on affective states. [START_REF] Watson | Development and validation of brief measures of positive and negative affect: The PANAS scales[END_REF] developed and validated the Positive and Negative Affect Schedule (PANAS), a scale that measures positive affect (PA) and negative affect (NA). The PA scale assesses the extent to which a person feels enthusiastic, active, and alert and the NA scale measures such feelings as anger, disgust, guilt, fear, and nervousness. However, there is no evidence in the literature regarding TC effects on affective state. According to Gaoua, Grantham, Racinais, and Massoui (2012), affective state is modified by exposure to heat alone. They showed that participants reported higher NA in a hot environment (50 ˚C-30% rH) than in neutral conditions (24 ˚C-30% rH) but PA did not vary as a function of the temperature condition. If the heat stress in hot conditions influences the affective state, one could argue that heat and humidity would have a similar impact on affect. Recently, Robin, Sinnapah, Hue, and Coudevylle (under review) examined how a hot (30 ˚C) and humid (70% rH) environment (i.e., TC) influences PA and NA. In their study, TC reduced PA scores relative to the neutral climate (NC) condition. However, this result needs to be confirmed and more evidence is required before conclusions can be drawn about the impact of TC on affective states.

Research has been scant on whether TC has an impact on attention and affect according to sex. [START_REF] Wyon | Spontaneous magnitude estimation of thermal discomfort during changes in the ambient temperature[END_REF] reported that men were less able than women to withstand the negative effects of heat stress. Likewise, [START_REF] Karjalainen | Gender differences in thermal comfort and use of thermostats in everyday thermal environments[END_REF] noted that men preferred cool environments and were more sensitive than women to heat. [START_REF] Lan | Investigation of gender difference in thermal comfort for Chinese people[END_REF] found similar results, observing that men preferred cooler room temperatures than women. Indeed, [START_REF] Robin | Effects of tropical climate on mental rotation: The role of imagery ability[END_REF] recently showed that men had lower cognitive performance (e.g., Vandenberg mental rotation test) in TC than in NC. Thus, it can be expected that the impact of TC on attention and affect will be greater for men than for women.

Overview

Little is currently known about the impact of TC on attention and affect. The purpose of this study was therefore to determine whether TC has an influence on selective attention and positive and negative affect. We hypothesized that the combination of heat and high humidity would be a stressful factor impacting these two psychological aspects of an individual's functioning. We expected that the selective attention score would be lower in TC than in NC and that positive affect would be lower and negative affect higher in TC compared with NC. Last, we expected that the negative impact of TC would be greater for men than for women.

Method

Participants

Thirty-two physical education students (27 men M age 5 19.8 years, SD 5 1.44; and 5 women M age 5 21.2 years, SD 5 1.48) volunteered for this study and gave written consent before participating. All were living in the West Indies, which has a TC (i.e., hot and wet) characterized by consistently high monthly temperatures, often exceeding 18 ˚C throughout the year, and rainfall that exceeds evapotranspiration for at least 270 days per year [START_REF] Hue | The challenge of performing aerobic exercise in tropical environments: Applied knowledge and perspectives[END_REF].

Experimental Conditions

The experiment was conducted during a regularly scheduled class. Each participant took part to two experimental conditions: Tropical and Neutral. They were randomly assigned by lottery to one of the experimental conditions first, and the two sessions were then conducted at a oneweek interval. The tropical and neutral climates (respectively, TC and NC) were strictly monitored to standardize the experimental conditions. In the TC condition, the ambient temperature was 30.8 ˚C¡1.2 and the humidity was 70.1% ¡2.1 ; this environment was maintained by electric heating and kettle systems. In the NC condition, the ambient temperature was 22.5˚C ¡2.8 and the humidity was 47.6% ¡2.1 ; this environment was obtained using an air conditioning system. The ambient light was artificial and strictly the same (fluorescent light 4000 K) in both conditions. Participants rested in a seated position for 30 min before starting the cognitive assessments.

Measures Selective Attention: d2 Test

Attention capacity was assessed using the d2 Test of Attention (d2T), which was developed to measure sustained attention and concentration under the stress imposed by a time limit for completion [START_REF] Brickenkamp | The d2 Test of Attention[END_REF].

The d2T is a paper-and-pencil letter-cancellation test comprising 14 lines, each containing 47 randomly mixed letters (''p'' and ''d''), for a total of 658 letters. The letters ''p'' and ''d'' appear with one or two dashes above or below each letter. The test participant has to carefully check whether each letter ''d'' has two dashes either above or below it, at a rate of 20 s per line. The complete duration of the test is 4 min 40 s.

The test allows the calculation of several different attention parameters. In the present study, we used four d2Tbased measures of attention as dependent variables. First, we measured processing speed, which was calculated as the sum of the number of characters processed before the final cancellation on each trial (i.e., the total number of items processed). Then, we measured inattention, which was operationalized as the sum of the number of target symbols not canceled (i.e., errors of omission). Next, we measured impulsivity, which corresponded to the sum of the number of non-target symbols canceled (i.e., errors of commission). Last, we measured concentration performance, which was the difference between the total number of correctly canceled items and the number of errors of omission and commission (for a similar procedure, see [START_REF] Vanhelst | Physical activity is associated with attention capacity in adolescents[END_REF].

Affect Scales

Affect was assessed using the French version of the PANAS [START_REF] Lapierre | E ´valuation de l'affectivite ´positive et ne ´gative en contexte sportif : Traduction francophone du PANAS. Evaluation of positive and negative affectivity in a sports context: French translation of the PANAS[END_REF]. This scale is a 20-item self-report psychometric scale developed to measure the two largely independent constructs [START_REF] Watson | Development and validation of brief measures of positive and negative affect: The PANAS scales[END_REF]. The ten PA items (i.e., active, alert, attentive, determined, enthusiastic, excited, inspired, interested, proud, and strong) are grouped into two sub-dimensions (i.e., attentive and excited) and the ten NA items (i.e., guilty afraid, ashamed, distressed, guilty, hostile, irritated, jittery, nervous, scared, and upset) are grouped into three subdimensions (i.e., scared, nervous, and shameful). Participants rated the PANAS items on a 5-point scale (from 1 ''very slightly'' to 5 ''very much'').

Procedure

At the beginning of each session, the goal was presented: ''The purpose of this experiment is to test your cognitive capacities. Before performing this test of response time, you are going to respond to the items of various questionnaires. All the questionnaires are confidential and anonymous. Your test results will remain anonymous and confidential. For the needs of the study, the room conditions where you will do the tests are standardized.'' Before each session, the participants completed a control self-report questionnaire evaluating their use of medications, other drugs, and alcoholic and non-alcoholic drinks in the last 24 hours. None of them indicated having been thirsty before the sessions. After answering these questions, the participants successively completed subjective measures of environmental perceptions (comfort and thermal sensations), the PANAS, and the d2T.

Analyses

To determine whether the climate condition influenced attention and affect, the dependent variables (i.e., the four d2T-based measures of attention and the PA and NA scores) were submitted to mixed ANOVAs (with repeated measures on the second factor) using sex (female vs. male) as the between-participant factor and climate condition (TC vs. NC) as the within-participant factor. We checked that all variables were normally distributed with the Kolmogorov-Smirnoff test. Alpha was set at 0.05 for all analyses.

Results

Selective Attention: d2T Results

The ANOVAs for processing speed, inattention, and concentration performance revealed the main effect of climate condition [F(1, 30) 5 4.326, p , 0.05; F(1, 30) 5 4.487, p , 0.05; and F(1, 30) 5 4.751, p , 0.05, respectively]. Participants had a lower score in TC than in NC (Table 1). Analysis of the impulsivity scores did not show any main effect of climate (p . 0.05). The ANOVAs for all the variables revealed no significant main effect of sex and no interaction between sex and climate condition (p . 0.05).

PANAS Results

Positive Affect

The ANOVA for PA revealed the main effect of climate condition [F(1, 30) 5 7.391, p , 0.05], and the ANOVA for the attentive score also revealed the main effect of climate condition [F(1, 30) 5 9.157, p , 0.01]. The ANOVA for the excited score revealed a tendency toward significance for the climate condition [F(1, 30) 5 3.651, p 5 0.066]. Participants thus had lower overall PA and attentive scores and showed a tendency toward lower excited scores in TC than in NC (Table 2). The ANOVAs of all the variables revealed no significant main effect of sex and no interaction between sex and climate condition (p . 0.05).

Negative Affect

The analysis of the NA scores showed no main effect of climate or sex and no interaction between these two factors (p . 0.05). The analysis of the three NA sub-dimensions separately (i.e., scared, nervous, shameful) showed no main effect of climate or sex and no interaction between these two factors (p . 0.05).

Discussion

Tropical Climate Impacts Selective Attention

The primary purpose of this study was to explore the impact of the TC on selective attention and positive and negative affect. In line with our predictions, the results showed that the participants had lower selective attention scores in TC than in NC (except for impulsivity). Our results contrast with those reported by [START_REF] Gaoua | Alterations in cognitive performance during passive hyperthermia are task dependent[END_REF], who did not find that attention was impaired by the hot climate. We may therefore assume that the combination of high humidity and high temperature played an important role. The results of our study are in line with those obtained by [START_REF] Chase | A study of computer-based task performance under thermal stress[END_REF], who tested dual-task performance (i.e., shared attention as opposed to selective attention). Those authors suggested that the discomfort induced by the hot climate provoked a cognitive cost and that the participants therefore used more cognitive resources to remain focused on the attention task [START_REF] Chase | A study of computer-based task performance under thermal stress[END_REF][START_REF] Sanders | Towards a model of stress and human performance[END_REF]. This suggests that the individual must cope with competition for cognitive resources between the TC-induced stress and the attention task [START_REF] Delignie `res | Influence de la chaleur humide sur le traitement de l'information et la performance[END_REF]. These results are important because no study to our knowledge has yet shown the deleterious effects of TC on selective attention. Indeed, cognitive psychologists have indicated that humans have limited cognitive capacities to cope with the multiple external stimuli constantly competing for the limited conscious access [START_REF] Gaoua | Cognitive function in hot environments: A question of methodology[END_REF]. It seems that the combination of heat and high rH imposes a cognitive load and therefore negatively impacts cognitive performance.

Tropical Climate Impacts Positive Affect

Consistent with our predictions, TC had a negative impact on PA (i.e., total score and the attentive and excited sub-dimensions). Participants had lower PA in TC than in NC. However, the effects of the two climate conditions on NA (i.e., total score and the sub-dimensions of scared, nervous, and shameful) showed no significant effects. Our results contrast with those of [START_REF] Gaoua | Sensory displeasure reduces complex cognitive performance in the heat[END_REF], who reported no difference in PA between climates but higher NA in a hot environment (50 ˚C) than in neutral condition, but they agree with those of Robin et al. (under review) (i.e., they found an impact of TC on PA but not on NA). It is fairly easy to understand that TC would have a negative influence on the scores for the attentive and excited sub-scales; it also seems reasonable that this climate would not have an effect on the scared, nervous, and shameful scores, particularly in these acclimatized participants. Indeed, in the study of Robin et al. (under review), participants were acclimatized to TC (i.e., born and or living in West Indies for more than 5 years). De Dear and Bragger (1998) have proposed that individuals adapt to the climatic conditions to which they are most exposed. It is therefore highly likely that people acclimatized to thermal and hygrometric stress will maintain a low level of negative affects, remaining calm and serene [START_REF] Watson | Development and validation of brief measures of positive and negative affect: The PANAS scales[END_REF] in an environment in which they live daily (Robin, under review). Finally, it is possible that these sub-scales are not appropriate for detecting the possible negative influence of an environmental climate.

Contrary to our predictions, no sex difference was observed. Although men may be more sensitive than women to the TC while performing certain cognitive tasks (e.g., mental rotation test), they did not differ with regard to attention or affect. The results contrast with previous studies showing that men prefer cool environments and are more sensitive to heat [START_REF] Karjalainen | Gender differences in thermal comfort and use of thermostats in everyday thermal environments[END_REF][START_REF] Lan | Investigation of gender difference in thermal comfort for Chinese people[END_REF][START_REF] Robin | Effects of tropical climate on mental rotation: The role of imagery ability[END_REF][START_REF] Wyon | Spontaneous magnitude estimation of thermal discomfort during changes in the ambient temperature[END_REF]). However, it should be kept in mind that our women's group was smaller than the men's group, and more research is needed to better understand the influences of TC according to sex.

Conclusion

In sum, the present research showed that in a regularly scheduled class (a) participants had lower selective attention scores (i.e., concentration performance) in TC compared with NC, (b) participants had lower positive affect (i.e., attentive and excited) in TC compared with NC, and (c) no sex difference was observed. These results advance our understanding of the psychological impact of TC. It should be underlined that these results were obtained in a sample of acclimatized participants and that the deleterious effects of the climate condition on attention and affect might well be worse in non-acclimatized participants. Further research should explore this hypothesis with the acclimatization characteristic as a factor for comparison between participants. Despite these contributions, some limitations should be mentioned. The main limitation is that in TC, it is not possible to dissociate the impact of heat stress from that of high rH. Additional research could assess the rH effects on selective attention and affect using, for example, climatic chambers.

Table 1

 1 Means and standard deviations for selective attention scores as a function of the climate condition (N 5 32).

	Task	Climate condition
		Tropical M (SD)	Neutral M (SD)
	Processing speed	187.63 (42.87)*	200.84 (46.15)*
	Inattention	112.78 (44.91)*	98.47 (47.12)*
	Impulsivity	2.81 (6.59)	1.91 (2.37)
	Concentration performance	72.03 (89.05)*	100.47 (93.82)*

* Main effect for climate, p , .05.

Table 2

 2 Means and standard deviations for affect scores as a function of climate condition (N 5 32). Main effect for climate, p , 0.05; **main effect for climate, p , 0.01.

	Task a	Climate condition
		Tropical M (SD)	Neutral M (SD)
	PA	21.59 (7.97)*	28.06 (7.66)*
	Attentive	13.50 (4.92)**	17.47 (4.49)**
	Excited	8.09 (3.45)	10.59 (3.57)
	NA	13.97 (4.63)	12.28 (2.54)
	Scared	3.97 (1.37)	3.44 (1.13)
	Nervous	7.94 (3.47)	6.81 (1.73)
	Shameful	2.06 (0.43)	2.03 (0.30)

a PA 5 positive affect; NA 5 negative affect. *

This study was funded by PO-Feder 2015-2018, Re ´gion Guadeloupe, ACTES-APPLI.


Jessica Delavoipière

Bertrand Heurtefeu

Jérémie Teisseire

Antoine Chateauminois

Yvette Tran

Marc Fermigier

Emilie Verneuil

Swelling Dynamics of Surface-Attached Hydrogel Thin Films in Vapor Flows

Keywords: numbers: Hydrogel, coating, swelling, advection diffusion

Introduction

Hydrogel coatings are suitable candidates for applications where both transparency and hydrophilicity are required. Indeed these hydrophilic polymer networks may absorb water by swelling to several times their dry thickness. As such, hydrogel coatings are possibly good candidates for anti-fog applications, as proposed by several teams [1-5] who discussed their efficiency in terms of light transmission while submitted to warm humid air. Indeed, the coating should act as a reservoir for humidity and observation is made [START_REF] Mahmoud | Anti-fogging and anti-frosting behaviors of layer-by-layer assembled cellulose derivative thin film[END_REF][START_REF] Park | Anti-fogging behavior of waterabsorbing polymer films derived from isosorbide-based epoxy resin -ScienceDirect[END_REF] that the higher the film thickness, the more water is absorbed and the later mist appears. Nevertheless, their efficiency to delay mist or fog formation in relation to their kinetics of swelling in humid atmosphere has not been studied yet.

In this paper, we present experimental results and a model allowing for a complete description of the vapor transfer to hydrogel coatings, accounting for the physicalchemical characteristics of the polymer hydrogel. These results are then used to predict the kinetics of swelling of hydrogel coatings.

The generic situation considered here is a coating onto a cold substrate in a vapor flow of higher temperature, for which the transfer of vapor to the coating results from both diffusive processes and the advective flow: it is therefore described by the diffusion-advection equation. If the vapor flow alongside the coating is fast enough, the vapor transverse transfer to the coating by diffusion is confined to a thin layer adjacent to the coating, the diffusion boundary layer. In this case, analytical solutions to the diffusion-advection problem from the literature can be used. Formally, the latter situation corre-sponds to high Péclet numbers Pe, which characterizes the ratio of advective transport by the mean flow to the diffusive transport. To solve the problem, the diffusionadvection equation must be supplemented by a boundary condition at the coating interface with the vapor. To do so, thin films of polymer networks made out of different hydrophilic polymers were prepared and characterized by their equilibrium sorption isotherms in water vapor. We will then determine the relevant boundary condition and show that an approximate analytical solution to the transport problem can be derived in the high Péclet number regime. We will extend our model to the low Péclet numbers thanks to a numerical resolution of the diffusion-advection problem.

As the hygroscopy of the polymer in the film is expected to change the kinetics of swelling, polymer networks with varied affinity for water are chosen. Besides, transfers within the coating may be affected by the glassy or rubbery state of the polymer: indeed, the diffusion of water within glassy polymers can be at least two orders of magnitude slower than in melts, while hygroscopy is also usually lower [START_REF] Dupas | Glass Transition Accelerates the Spreading of Polar Solvents on a Soluble Polymer[END_REF]. Furthermore, some hydrophilic polymers that are glassy in the dry state become rubbery by absorbing water: this plasticization phenomenon may occur at room temperature. Hence, the glassy or melt state of the polymer, as well as the solvent induced glass transition -or plasticization -were tested in their possible effects on the water transfers to the coating. In order to study the effects of hygroscopy and glass transition, coatings made of 3 different hydrophilic polymers were prepared: poly(PEGMA) was chosen as a polymer of low hygroscopy and remaining molten at all water contents; poly(DMA) as a hygroscopic polymer undergoing glass transition and becoming rubbery for water contents above φ g,P DM A ranging between 15% and 20% for temperatures between 0 • C and 25 • C ; poly(NIPAM) as a common thermo-responsive polymer which is hydrophilic at room temperature or below and which, similarly to poly(DMA), is glassy in the dry state at room temperature or below, and becomes rubbery for water content above φ g,P N IP AM ranging between 23% and 30% for temperatures between 0 • C and 25 • C. The transport problem was addressed by working in a cell where the vapor characteristics (controlled laminar flow, temperature and concentration), the geometry and the coating temperature were controlled so that solutions to the advection-diffusion problems could be derived and compared to the measured coating swelling kinetics. Moreover, the experimental conditions were chosen to match those encountered in typical applications, making our study relevant to practical applications, as detailed in the last section of the paper. Within this framework, during the most part of the coating swelling and for coatings thinner than a limit thickness, we were able to show that the advection-diffusive flux in the vapor controls the swelling kinetics while an homogeneous concentration profile is achieved across the coating thickness. In these conditions, we were able to provide a comprehensive description of the vapor absorption kinetics by hydrophilic polymer coatings.

Materials and method

Synthesis and characterization of surface-attached hydrogel coatings

Poly(NIPAM), poly(DMA) and poly(PEGMA) hydrogel coatings were synthetized by crosslinking and grafting preformed polymer chains through thiol-ene click chemistry route as detailed elsewhere [START_REF] Delavoipiere | Poroelastic indentation of mechanically confined hydrogel layers[END_REF][START_REF] Chollet | Multiscale Surface-Attached Hydrogel Thin Films with Tailored Architecture[END_REF][START_REF] Li | Submicrometric Films of Surface-Attached Polymer Network with Temperature-Responsive Properties[END_REF]. The coating is made by spin-coating of the ene-reactive polymer solution on thiol-functionalized silicon wafers with added dithiol crosslinkers. Homogeneous films with wellcontrolled thickness e dry are then obtained.

The thickness of the dry coatings were measured by ellipsometry for thin coatings (e dry < 1.5 µm) and by profilometry for thicker coatings. It typically ranges between 1 µm and 3 µm. The swelling ratio was shown earlier [START_REF] Li | Submicrometric Films of Surface-Attached Polymer Network with Temperature-Responsive Properties[END_REF] to be independent of the coating thickness for thicknesses higher than 200 nm, so that the swelling ratio S was precisely measured from ellipsometry measurements carried out on thin coatings (dry thickness smaller than 1 µm) under dry nitrogen flow and under water: S = ew e dry with e w the coating thickness in water. For poly(DMA) coatings, the swelling ratio S was varied within the range [3 ; 5]. For poly(NIPAM) coatings, S = 7 and for poly(PEGMA) coatings, S = 2.7.

The sorption isotherm of poly(NIPAM), poly(DMA) and poly(PEGMA) films were determined by classical Diffusion Vapor Sorption (Surface Measurement Sys- tems) experiments. Prior to any measurement, the bare substrate is weighed in the DVS chamber, after what the thin film is prepared as detailed above. The sorption isotherm is then measured. Briefly, the coating is exposed to water vapor at a prescribed relative humidity which is increased step by step, and the corresponding water uptake is measured at equilibrium. Within experimental accuracy, no significant dependence of the sorption isotherms with temperature could be detected in a [12;25] • C temperature range. The results are plotted in Fig. ??, where the water volume fraction φ in the coatings at equilibrium is plotted as a function of the water activity a, defined as the ratio of partial vapor pressure p to the saturated water vapor pressure p sat (T ):

a = p p sat (T ) (1)
Water vapor is considered as an ideal gas for which

C sat (T ) = psat(T )M

RT

where C sat (T ) is the saturated water vapor concentration, T the temperature in Kelvin, M = 18 g mol -1 the water molar mass, R = 8.31 J mol -1 K -1 the ideal gas constant so that activity can be identified to relative humidity RH = C Csat(T ) with C the water vapor concentration. In this paper, the saturated water vapor pressure p sat (T ) was calculated from the Rankine empirical formula, derived from Clapeyron model, with the latent heat of vaporization taken independent of temperature:

p sat (T ) = p ref e 13.

Swelling kinetics measurement

The water flux to hydrogel coatings was determined using a homemade setup (Fig. 2) allowing a good control of the transport problem. A closed Plexiglas chamber was built up with length L and width W larger than its height h to avoid side wall effects: h = 1 cm ; W = 4 cm and L = 28 cm. The chamber is equipped with a diverging inlet and converging outlet on opposite sides (see Fig. 2b)), to obtain a laminar flow of humid gas with controlled rate (measured with a flowmeter), and prescribed temperature and humidity (measured with a probe). Experiments with flow rates Q of 0.1 L min -1 and 1 L min -1 corresponding to average velocities U 0 = Q W h of 4.2 10 -3 m s -1 and 4.2 10 -2 m s -1 were carried out. The corresponding Reynolds numbers Re = U0h ν with ν = 1.5 10 -5 m 2 s -1 the kinematic viscosity of gas, are 3 and 30. We carefully paid attention to maintain a laminar flow at the highest Reynolds number by removing the temperature probe. In this case, the temperature is measured after the swelling experiments: it adds an uncertainty of ±1 • C to the temperature in the chamber which is taken into account in the error bars. The humid gas is obtained by bubbling pure nitrogen in deionized water or in saturated salt solutions to adjust the relative humidity RH which was varied between 20% and 90%. The water vapor concentration of the incoming flow C 0 is defined as:

C 0 = RH × C sat (T 0 ) with T 0 its temperature. Typically, C sat (T 0 = 25 • C) = 22.4 10 -3 kg m -3
. The Péclet number P e characterizing the advection-diffusive problem then writes :

P e = U 0 h D (3)
where D = 2.6 10 -5 m 2 s -1 is the water vapor diffusion coefficient in air.

In the bottom wall of the chamber, a Peltier module with dimensions W × W was inserted on which the samples were set. The leading edges of the sample and of the Peltier module were aligned. The Peltier module allows to create an homogeneous temperature field at the sample surface and to regulate the sample temperature to values lower than the ambient temperature, in the range 2 to 25 • C. Samples consist in silicon wafers with dimensions (typically 1 × 2 cm 2 ) smaller than the chamber sizes W and L on which model hydrogel coatings were prepared. Finally, the upper wall of the chamber is a glass window: it allows in a laser beam of wavelength λ = 632 nm, which reflects on the sample with an angle of incidence i = 50 • C. The typical size of the laser spot on the sample is 400 µm. The reflected light is collected by a photodiode. Its intensity is modulated by the light interferences between the reflections on the coating/air and coating/substrate interfaces (Fig. 2 c)) by absorbing water, the coating swells and its thickness variation δe is measured from the photodiode signal (Fig. 2d)) using classical interferometry theory and assuming the gel optical index obeys a linear mixing law with water concentration (see SI for details). Observations from above using a CCD camera and a zoom allow for the detection of mist formation. Prior to any experiment, the sample is cooled down to temperature T sub and dry nitrogen flows in the cell. At initial time, humid nitrogen enters the chamber and the film thickness increase over time is measured, at a distance x = 1 cm from the sample edge.

Experimental results

First, due to the high aspect ratio of the films having a thickness e much smaller than their lateral size, their swelling is one dimensional, and in the direction perpendicular to the plane of the substrate to which the polymer network is grafted. Figure 3 presents the increase in thickness δe over time of a poly(DMA) coating for a given humidity of the incoming advective flow and various temperatures of the substrate T sub for P e = 16 (a) and P e = 1.6 (b). After a short transient regime, the thickness increases linearly over time. In the linear regime, the swelling velocity defined as v = de dt increases with the temperature difference ∆T between incoming air T 0 and substrate T sub , as shown in inset in Fig. 3 a). The coating thickness increment δe can be related to the dry coating thickness e dry , and to the averaged water volume fraction φ. To do so, we assume a homogeneous water fraction along the coating thickness, a hypothesis that will be validated later. This gives: φ = δe e dry +δe . For large enough temperature differences ∆T between the chamber temperature T 0 and the sample one T sub , Fig. 3 shows that the coating always swells to the same maximum thickness e max before mist appearance, as indicated by a dash-dotted line in Fig. 3. We check that this maximum thickness corresponds to the maximum swelling capacity S of the hydrogel independently measured by ellipsometry on thin films. The same results were obtained for the three types of polymers.

Besides, we observe that, for a given polymer, the linear regime for δe(t) starts at the same value denoted δe * , corresponding to the same average volume fraction φ * for all experimental conditions. This result also holds for other humidities of the incoming flow tested. We determine φ * for the three types of polymer: φ * (poly(DM A)) = 40 -45% ; φ * (poly(P EGM A)) = 20 -30% and φ * (poly(N IP AM )) = 40 -45%. For poly(DMA) and poly(NIPAM), φ * is larger than the plasticization water fraction φ g .

Boundary condition at film interface

From these observations, let us first validate the hypothesis of homogeneous water content along the coating thickness by evaluating the order of magnitude of the water concentration gradient along the coating thickness. To do so, let us write the equality of the water mass fluxes in the vapor J vap and across the coating J polymer . The flux in the vapor corresponds to the diffusion across the boundary layer δ and scales as DC 0 /δ. As found in the literature [START_REF] Bird | Lightfoot. Transport Phenomena[END_REF], at high Péclet numbers, the boundary layer scales with the channel height h as h/P e 1/3 . With P e = 16 and h = 1 cm here, we obtain: δ = 4 mm. For a water volume fraction difference accross the film thickness ∆φ, the flux in the polymer scales as ρDp∆φ e with ρ = 10 3 kg m -3 the water density and D p the water diffusion coefficient in the polymer coating: D p ≈ 10 -11 m 2 s -1 for rubbery networks [START_REF] Yoon | Poroelastic swelling kinetics of thin hydrogel layers: comparison of theory and experiment[END_REF][START_REF] Salmon | Humidity-insensitive water evaporation from molecular complex fluids[END_REF]. The equality of the water mass fluxes gives an estimate of the water volume fraction difference accross the film thickness: ∆φ = e δ D ρDp C 0 . At most, we find ∆φ = 0.1 for a swollen film thickness e = 10 µm and typical values C 0 = 20 10 -3 kg m -3 , δ = 4 mm. Consequently, we will make the assumption that the water content in the coating is homogeneous along the thickness of rubbery films (poly(PEGMA) films at all water contents and poly(DMA) and poly(NIPAM) when molten, that is with φ w > φ g ).

For glassy coatings, however, D p typically decreases to 10 -14 m 2 s -1 and the water concentration gradient in the film thickness becomes significant. Consequently, at the beginning of the swelling of poly(DMA) and (poly)NIPAM coatings, diffusion within the coating is limiting. Poly(DMA) and poly(NIPAM) coatings undergo glass transition once a volume fraction of water φ g of order 15% to 30% has been absorbed. This actually occurs early in the experiments, and always before the beginning of the linear regime since φ g < φ * . Meanwhile, in this early regime, the water vapor concentration profile in the chamber is in its transient: given the short duration of this complex transient regime compared to the total swelling time, it will not be further accounted for in this paper.

Let us now focus on the linear regime for the time variations of the coating thickness (δe > δe * ) for which water mass transfers are assumed to be limited by the flux in the vapor. This latter hypothesis is further confirmed by the fact that the measured swelling velocities are independent of the initial coating thickness (Fig. ??insets). Consequently, assumption can be made that the water fraction is homogeneous along the coating thickness holds at all times in the linear regime, and is equal to the average water volume fraction derived from theoretically measured δe. In addition, the evolution of the coating thickness can be ascribed to the water vapor flux J vap through:

J vap = ρ de dt = ρv (4)
The boundary condition in the vapor at the coating surface, namely the water vapor concentration at the surface C sub , remains to be determined. If we report the experimental values of φ * on the sorption isotherms (Fig. ??), we note that it corresponds to activities ranging between 0.9 and 0.95 for all tested polymers. Hence, in the linear regime, the activity ranges between 0.9 and 1 and can therefore be considered as a constant denoted a * . Meanwhile, the balance of water activities at the coating surface, in the polymer and in the vapor, writes:

a(φ) = C sub C sat (T sub ) ( 5
)
Considering a constant activity in the polymer enforces the water vapor concentration at the coating surface C sub to be constant all along the sample. In the following, we make the simple approximation that the water activity a * is equal to 1 in this regime, equivalently C sub ≈ C sat (T sub ). We emphasize that in the narrow range of activity between a * and 1, (a = 1 corresponds to the saturated gel), the water content of the gel varies strongly, as seen from Fig. ??.

Finally, hypothesis can be made that water condensation, if any, does not lead to a temperature increase through latent heat, so that the temperature at the film surface is constant. Also, the temperature dependence of the diffusion coefficient of water in air [START_REF] Marrero | Gaseous diffusion coefficients[END_REF] will be neglected: indeed, a 10 % variation in D is expected for ∆T = 20 o C. Any temperature effect will therefore be disregarded except for that on the saturation concentration of water vapor.

Transport problem

In the following, we analyse the steady state transport from the advective diffusive vapor flow to the hydrogel coating. The vapor volumetric concentration in the chamber is denoted C. The chamber width is larger than the sample width so that the problem reduces to a two dimensional problem: C = C(x, y). As the sample is located in the middle of the chamber with L/2 >> h, the gas flow is well-established above the sample. It can be described with a steady laminar parabolic velocity u = u(y) e x along the longitudinal direction x where:

u(y) = 6U 0 y h (1 -y/h) (6)
U 0 being the average velocity U 0 = Q W h . While the downstream advection renews the vapor concentration to its initial concentration C 0 , the diffusion in the cross-stream drives vapor to the coating where it is absorbed. The steady state concentration of vapor is obtained by solving the advection-diffusion equation:

u • ∇C -D∆C = 0 (7)
with the boundary conditions:

C(y → ∞) = C 0 (8)
C(y = 0) = C sub (9)
In the flow direction, the relative contribution of the advective and diffusive terms to the transport problem is characterized by the Péclet number as defined by the equation [START_REF] Mahmoud | Anti-fogging and anti-frosting behaviors of layer-by-layer assembled cellulose derivative thin film[END_REF].

We first study the regime where P e >> 1. In this regime, we can make two hypotheses. In the flow direction, advection dominates over diffusion so that we neglect longitudinal diffusion. The concentration gradient develops in a boundary layer with a height δ small compared to the chamber height h. Within this depletion zone, the humid gas velocity is then linearized as:

u(y) = 6U 0 h y (10)
Therefore, the advection diffusion problem reduces to:

6U 0 h y ∂C ∂x = D ∂ 2 C ∂y 2 (11)
Introducing the Péclet number, it becomes:

6P e h 2 ∂C ∂x = 1 y ∂ 2 C ∂y 2 (12)
Autosimilar solutions of this classical Leveque problem [START_REF] Bird | Lightfoot. Transport Phenomena[END_REF] can be searched for in the form:

C = C 0 f (ζ) + C sub (13)
with the reduced variable ζ = yP e 1/3

x 1/3 h 2/3 . Equation ( 12) becomes:

-2ζ 2 f (ζ) = f (ζ) (14)
By integrating this differential equation with the boundary conditions ( 8) and ( 9), we obtain:

C(ζ) = C sub + (C 0 -C sub ) ζ 0 e -2 3 t 3 dt ∞ 0 e -2 3 t 3 dt (15)
By noting the constant

A 2 = ∞ l=0 e -2
3 t 3 dt ∼ 1.02, the diffusive vapor flow toward the coating is then given by:

J vap (x) = D ∂C ∂y y=0 = D(C 0 -C sub ) A 2 U 0 hxD 1/3 (16)
J vap (x) = D C 0 -C sub δ(x) (17)
with δ(x) = A 2 hxD U0

1/3 = A 2 xh 2 P e 1/3
the thickness of the depletion layer. The diffusive vapor flow J vap is measured at the laser spot position (x = 1 cm). Our measurement averages the thickness variations over the laser spot size ∆x which is sufficently small compared to x so that the corresponding variation ∆δ of δ can be neglected ( ∆δ δ = 1 3 ∆x x ≈ 10 -2 ). For P e ≈ 1 where the approximation δ << h no longer holds, a solution of the advective diffusive problem is found by numerical resolution [START_REF] Squires | Making it stick: convection, reaction and diffusion in surface-based biosensors[END_REF] of the transport equation for the water vapor ∂C/∂t + u.∇C = D∆C with the finite element code FreeFem++ [START_REF] Hecht | New development in freefem++[END_REF]. The velocity field is prescribed with the parabolic profile (equation ( 6)). The initial condition is a constant concentration in the whole domain. At t > 0, we impose a fixed normalized concentration equal to 1 at the inlet of the flow chamber, a concentration equal to C sub on the part of the bottom wall corresponding to the absorbing gel, and a zero normal flux at all other walls. The concentration field is then iterated in time, until a steady-state situation is reached corresponding to the balance between the advection by the flow and molecular diffusion. At this point, the finite element mesh is refined according to the concentration gradient to resolve accurately the diffusion boundary 1500 nm ; PNIPAM: 1450 nm ; PDMA: 1000 nm (S=4.4), 1200 nm (S=4 and 5), 1500 nm (S=4), 2600 nm (S=4), • 2970 nm (S=3). Dotted lines and full lines represent respectively the analytical solutions (from eq (17)) and the numerical solutions for P e = 1.6 (red), P e = 16 (blue) and P e = 160 (green). layer on top of the adsorbing gel. From the concentration gradient at the bottom wall, we deduce the flux of vapor towards the gel. The concentration maps computed for different Péclet numbers are plotted in Fig. 4. Note that the numerical simulation ignores the limited absorption capability of the film.

Discussion

Taking into account the previously determined boundary condition C sub = C sat (T sub ), Fig. 5 represents the experimental swelling velocities v, or equivalently the vapor flux J vap /ρ through Eq. 4, as a function of ∆C = C 0 -C sat (T sub ) for two series of experiments car-ried out at P e = 16 and P e = 1.6 respectively with different hydrogel films. The difference of concentration ∆C is varied by changing the temperature and/or the relative humidity RH of the humid nitrogen. At each Péclet number, we first find that all data collapse on the same curves within experimental accuracy. No effects of the thickness nor the swelling ratio S is observed. This is consistent with the prediction in Eq. 17. Furthermore, both approximated analytical solutions of the swelling velocity (v anal ) (from equation ( 17)) and numerical solutions (v simul ) expressed by equations ( 7) to ( 9) are plotted for P e = 160, 16 and 1.6. No experiments could be performed at P e = 160, but the simulated and analytical solutions at this Péclet number were compared in order to test the approximated analytical solution. We see indeed that the analytical solution enables to describe the swelling at high Péclet numbers. While it is a very good approximation at P e = 160 (v simul /v anal = 0.98), a 10% difference is obtained for P e = 16. The analytical solution becomes out of range for P e = 1.6 (v simul /v anal = 0.41) where longitudinal diffusion compares with advection. This is visible on the concentration maps in Fig. 4 (Pe=1.62) where the thickness of the boundary layer extends over the flow cell height.

The experimental data with P e = 1.6 enable us to validate our model since the flow has likely less fluctuations at this low Reynolds number Re = 3. The data at P e = 16 (open symbols) are in good agreement with the simulated prediction. The good agreement between the simulation and the experiments whatever the film nature, thickness and swelling ratio confirms that we can describe the swelling of an hydrogel coating cooled down and submitted to a humid air flow thanks to a diffusive-advective equation with the boundary condition: C sub = C sat (T sub ).

These results can be used to predict the efficiency of these hydrogel coatings as anti-fog coatings. By choosing experimental conditions that are relevant to typical applications, our results can be readily transposed to design coatings (thickness, hygroscopy) with prescribed anti-fog capability. The delay to mist formation T can be approximated by T = ρ Jvap ∆e max = ρ Jvap e dry (S -1) with J vap given by Eq.17 and S and e dry are obtained from simple coating characterizations (DVS and profilometry). For vertical plates (mirrors, windows,...) set at a temperature lower than humid atmosphere, our results can also be transposed by replacing the forced advective vapor flow with the thermal advective flow alongside the plate. Indeed, sizes and velocity conditions in these practical situations are such that the Péclet numbers are typically larger than 10. The vapor velocity gradient at the coating interface in Eq. 10 can be taken as the ratio between the advective velocity to the thermal boundary layer thickness. Our work thereby provides an tool for the design of antifog absorbing coating made of hydrophilic polymers.

Supporting Information: Measurement of the hydrogel film thickness by interferometry

The reflecting light resulting from the interferences between the reflections on the coating/air and coating/substrate interfaces depends on the phase shift ϕ between the two reflected beams: with n the index of the hydrated polymer film. During the swelling, this index varies but the associated variation of cos(r) is negligible. We consider cos(r) = cst = cos(r(n av )) with n av ≈ 1.4 the average refractive index. With the linear mixing law, the variation of the phase shift δϕ corresponds to the variation of the film thickness δe so that:

I = I min + I max 2 (1 + Γ cos(ϕ)) (18
δe ≈ λ 4πn w cos(r) δϕ (22)
with n w the water refractive index. Between two consecutive extrema of the intensity (Eq. 18) resulting from the interferences, the phase shift varies from π. Consequently, according to the equation ( 22), the corresponding thickness variation is:

δe min-max = λ 4n w cos(r) = 141 nm (23)

[1] Svenja Grube, Konstantin Siegmann, and Martina Hirayama. A moisture-absorbing and abrasion-resistant transparent coating on polystyrene | SpringerLink. Jour-

  FIG. 1:Sorption isotherms of poly(DMA) (S=5), poly(NIPAM) (S=7) and poly(PEGMA) (S=2.7) thin films at 20 • C. Water volume fraction absorbed by coatings made of the different polymers as a function of the water activity of the vapor at equilibrium, a, defined by Eq.1. The data at a = 1 corresponds to the swelling ratio S measured under water by ellipsometry.

7 - 5120 T ( 2 )

 751202 with p ref = 1 atm the reference pressure. The sorption isotherms in Fig.?? evidence the varied hygroscopy of the polymers used: of the three polymers, poly(DMA) is the most hygroscopic, and poly(PEGMA) the least.

FIG. 2 :

 2 FIG.2: Schematic of the custom-built set-up. a) Side view: at t < 0, the absorbing coating is cooled down to T sub thanks to a Peltier module (not shown) and dried with dry nitrogen. At t = 0, humid nitrogen flow is sent in the Plexiglas chamber: the absorbing hydrogel coating swells and its thickness variation is deduced thanks to an interferometric measurement localized at x = 1 cm. Blue circle: zone zoomed in on c). The camera allows to observe the mist appearance. b) Top view (drawn to scale): the hydrogel coating (dark blue rectangle) is aligned with the leading edge of the Peltier module of dimensions W × W (light blue square). c) Principle of the thickness variation measurement with an interferometric set-up using a laser beam which reflects at the vapor/coating and the coating/substrate interfaces and is next collected by a photodiode. d) Photodiode voltage as a function of time during coating swelling.

6

 6

FIG. 3 :

 3 FIG. 3: Swelling kinetics of a poly(DMA) film submitted to humid N2 for various temperature differences ∆T = T0 -T sub . The dash-dotted line corresponds to the thickness increase for which mist appears. The solid line corresponds to a linear fit in the swelling linear regime. The dashed line marks the onset of the linear regime. a) Q = 1 L min -1 , P e = 16, RH = 88±1%. Dry thickness: e dry = 1.5 µm, swelling ratio S = 4.4. b) Q = 0.1 L min -1 , P e = 1.6, RH = 60 ± 1%. e dry = 1.2 µm, S = 5. Insets: Swelling velocity determined in the linear regime as a function of the temperature difference ∆T for different dry thicknesses: 1000 nm (S=4.4) 1050nm (S=4) 1200 nm (S=4 and 5) 1500 nm (S=4) 2600 nm (S=4) • 2970 nm (S=3)

FIG. 4 :

 4 FIG. 4: Concentration profiles in the vapor normalized by incoming concentration C0 for Pe=162, 16.2, 1.62 from numerical simulations.

6 FIG. 5 :

 65 FIG.5: Experimental swelling velocities measured at x = 1 cm as a function of ∆C = RHCsat(T0) -Csat(T sub ) for experiments at P e = 16 (blue markers) and P e = 1.6 (red markers) with films of poly(PEGMA), PNIPAM, and PDMA of varied thickness and swelling ratio: poly(PEGMA): • 910 nm, 1500 nm ; PNIPAM: 1450 nm ; PDMA: 1000 nm (S=4.4), 1200 nm (S=4 and 5), 1500 nm (S=4), 2600 nm (S=4), • 2970 nm (S=3). Dotted lines and full lines represent respectively the analytical solutions (from eq (17)) and the numerical solutions for P e = 1.6 (red), P e = 16 (blue) and P e = 160 (green).

  ) with I max et I min the maximum and minimum intensity, and Γ the contrast ratio defined by:Γ = I max -I min I max + I min (19)and ϕ the phase shift:ϕ = π + 4π λ ne cos(r)(20)with e the thickness of the hydrated film and n its refractive index. According to Snell and Descartes' law, the angle of the refracted beam r in the film writes: cos(r(n)) = 1 -


Julien Chopin
email: julien.chopin@ufba.br

Richard Villey

David Yarusso

Etienne Barthel

Costantino Creton

Matteo Ciccotti
email: matteo.ciccotti@espci.fr

Nonlinear Viscoelastic Modeling of Adhesive Failure for Polyacrylate Pressure-Sensitive Adhesives

We investigate experimentally the adherence energy Γ of model polyacrylate Pressure Sensitive Adhesives (PSAs) with combined large strain rheological measurements in uniaxial extension and an instrumented peel test. We develop a nonlinear model for such peel test which captures the dependence of Γ(V ) with peeling rate V revealing the key role played by the extensional rheology. Our model explains in particular why traditional linear viscoelastic approaches correctly predict the slope of Γ(V ) curves for sufficiently elastic PSAs characterized by a simple rate-independent debonding cri-1 terion. However, for more viscoelastic adhesives, we identified a more complex ratedependent debonding criterion yielding a significant modification of the Γ(V ) curves, an effect that has been largely overlooked so far. This investigation opens the way towards the understanding of fibrils debonding, which is the main missing block to predict the adherence of PSAs.

Introduction

It is a very well known result that during the peeling of a typical viscoelastic PSA from a solid substrate, the measured adherence energy Γ (the energy required to peel a unit tape area) is several orders of magnitude above the value given by the thermodynamic work of adhesion between the adhesive and the substrate. Moreover, in the slow steady state regime, Γ is very dependent on the temperature T and peeling rate V . It has long been observed that for viscoelastic adhesives the adherence curves Γ(V, T ) can be collapsed to a single master curve Γ(a T V, T ref ) at a reference temperature T ref by renormalizing the velocity axis by the same shift factor a T that is used for the Time-Temperature Superposition (TTS) of the linear rheological measurements. [START_REF] Ferry | Viscoelastic properties of polymers[END_REF][START_REF] Gent | Adhesion of Viscoelastic Materials to Rigid Substrates[END_REF][START_REF] Kaelble | Theory and Analysis of Peel Adhesion: Rate-Temperature Dependence of Viscoelastic Interlayers[END_REF][START_REF] Andrews | Mechanics of elastomeric adhesion[END_REF][START_REF] Maugis | Subcritical crack growth, surface energy, fracture toughness, stick-slip and embrittlement[END_REF] This has suggested that linear viscoelasticity should be used to quantitatively model the adherence energy Γ(V, T ).

Various linear viscoelastic models have since been developed based on a perturbation of the linear elastic fracture mechanics (LEFM) crack tip singular fields. [START_REF] Mueller | Crack propagation in a linearly viscoelastic strip[END_REF][7][8][9][10][11] These models predict a fracture energy in the separable form Γ(V, T ) = Γ 0 (1 + Φ(a T V )), 12,13 where Γ 0 is an intrinsic adhesion energy and Φ(a T V ) is a factor accounting for the linear viscoelastic losses. However, most attempts to quantitatively check these predictions experimentally for soft adhesives and rubbers have up to now failed. 11,14,15 Even an important protagonist of this domain such as Gent, 14 has pointed out the intrinsic limitation of these theories to describe experimental data since they would predict a process-zone of unphysical subatomic size.

When considering soft PSAs the elastoadhesive lengthscale Γ/E, where large strains are present around the crack tip, 16,17 is comfortably larger than the typical 20 µm thickness of the adhesive layer used for tapes. [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] Since the LEFM approach relies on a clear separability of scales between the crack tip singular field and the bulk sample deformation, it can not be applied to PSAs in most practical situations. Alternative modeling approaches have been developed to account for the strong confinement of the material based on a linear viscoelastic foundation following the pioneering work of Kaelble. [START_REF] Kaelble | Theory and Analysis of Peel Adhesion: Rate-Temperature Dependence of Viscoelastic Interlayers[END_REF][START_REF] Kaelble | Theory and Analysis of Peel Adhesion : and Mechanics Mechanisms[END_REF][START_REF] Kaelble | Theory and Analysis of Peel Adhesion : Bond Stresses and Distributions[END_REF] But the experimentally observed development of a dense array of highly stretched fibrils in the debonding region is at odds with this linear approach. [START_REF] Urahama | Effect of peel load on stringiness phenomena and peel speed of pressuresensitive adhesive tape[END_REF][START_REF] Chiche | Cavity growth in soft adhesives[END_REF][START_REF] Benyahia | The mechanisms of peeling of uncross-linked pressure sensitive adhesives[END_REF][START_REF] Gay | Theory of Tackiness[END_REF] The role played by the fibrillated region in the overall dissipation was pointed out by several polymer rheologists, [START_REF] Gent | Adhesion of Viscoelastic Materials to Rigid Substrates[END_REF][START_REF] Derail | Relationship between Viscoelastic and Peeling Properties of Model Adhesives . Part 1 . Cohesive Fracture[END_REF][START_REF] Yarusso | Quantifying the Relationship Between Peel and Rheology for Pressure Sensitive Adhesives[END_REF] but most experimental investigations were made on non-crosslinked fluid-like adhesives, where failure occurs either cohesively in the fibrils or without fibrillation at all. Such viscoelastic fluids even if able to strain harden in extension are not representative of the lightly crosslinked polymers used for PSA that form an extended fibrillar zone that cleanly debonds from the surface of the substrate. Thus, the respective roles played in the peeling process by the linear, small strain dissipation controlled by the monomer friction coefficient and by the non-linear extensional rheology, where strain hardening due to chain architecture is important, remains elusive. The same conclusion has been reached with other classically used adhesive tests such as the probe-tack test. [START_REF] Deplace | Deformation and adhesion of a periodic softsoft nanocomposite designed with structured polymer colloid particles[END_REF][START_REF] Creton | Large-strain mechanical behavior of model block copolymer adhesives[END_REF] In a previous investigation, [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] we specifically studied a series of model lightly cross-linked polyacrylate adhesives with different behaviors in extensional rheology but nearly identical small strain dissipation at the strain rates relevant for peeling. We showed from steady-state peeling experiments that a more pronounced strain hardening caused an overall decrease of the peeling force and a stronger dependence with peeling rate, which can not be explained by any model based on linear rheology (see Fig. 5(a)). In parallel, a quantitative characterization of the debonding region by optical microscopy revealed that the fibrils always detached cleanly from the substrate after a stretch larger than 500%, which diminished with peeling velocity, confirming that non-linear material properties matter here. Similar experiments were later performed and reported by Barrios. [START_REF] Barrios | Anatomy of the deformation of pressure sensitive adhesives (PSAs) from rigid substrates[END_REF] Moreover, we found that a more pronounced strain stiffening causes the maximum fibril extension before detachment to decrease and to become less sensitive to the peeling rate. [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] In this paper, we provide the first quantitative physically based model connecting nonlinear extensional rheological properties of model crosslinked PSAs with the adherence curves obtained in peel tests. We reconcile the TTS principle observed in the adherence curve and nonlinear deformation in the adhesive. Our results reveal complex debonding criteria, which unambiguously demonstrate the non trivial role played by polymer chain architecture and hence non-linear rheology in determining the Γ(V ) curves.

Experimental

The model PSA were synthesized by free radical polymerization in solution from 85 wt% of 2-ethylhexyl acrylate (EHA), 10 wt% of methyl acrylate (MA) and 5 wt% of acrylic acid (AA). 0.2 wt% or 0.4 wt% of aluminium acetyl acetonate relative to monomer was added after synthesis to provide two levels of cross-linking (labelled A, B) to the adhesive layer-which were coated between two silicone release liners.

The linear viscoelastic characterization of the polymers was carried out with an Anton-Paar rheometer (MCR-301) using a standard parallel plate geometry (Fig. 1(a)). The storage G (ω, T ) and loss G (ω, T ) shear moduli were measured at pulsation ω in the range 2 • 10 -1 to 60 rad/s. Pulsation sweeps were performed for temperatures T in the range -40 to 120 • C.

The maximum applied shear strain is γ = 2 • 10 -2 .

The nonlinear rheological properties are obtained by extensional rheology measurements using the Sentmanat extensional rheometer (SER, see Fig. 1(b)), [START_REF] Sentmanat | Measuring the transient extensional rheology of polyethylene melts using the SER universal testing platform[END_REF] adapted on the same MCR-301 rheometer. A slab of the polymer of width W = 5 mm and thickness e = 600 µm is brought in contact with the cylinders. The stickiness of the materials guarantees a no slip condition at the interface. The cylinders were rotated at a constant angular velocity, resulting in a constant true strain rate ε in the range 10 -2 to 4 s -1 . The temperature was varied in the range 0 to 40 • C. The cross-section area of the deformed slab is given by

A(t) = eW • exp(-εt).
True strain and stress are given by ε(t) = εt and σ(t) = F (t)/A(t), respectively, where t is time and F is the measured force acting on the slab. Nominal stress is σ N = σ/λ, where λ = exp(ε) is the stretch.

Finally, instrumented peel tests were performed at a 90 o angle by applying a force F at a peeling rate V (Fig. 1(c)). As detailed elsewhere, [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] the peeling occurred either at imposed load or imposed velocity allowing us to explore a large range of peeling rates from 1 µm/s to 4 m/s. The same model adhesives were coated on a 38µm thick, PET film. The adhesive layer thickness a 0 = 19 µm. All adhesive tapes were carefully bonded to the release side of a commercially available adhesive tape which is itself bonded to a microscope glass slide for small and intermediate peeling rates and long aluminum profiles for higher peeling rates.

This protocol ensures clean peeling and repeatable results. Because the PET backing layer is much stiffer than the adhesive, the energy release rate G in this geometry is given by :

G = F b , (1)
where b = 20 mm is the tape width. In steady state, an energy balance yields G = Γ(V ).

Further, the experimental setup was equipped with a lateral optical microscope allowing to image the structure of the debonding region with a micrometric resolution (Fig. 1(d)).

Thus, the peeling test was intrumented to characterize the structure of the debonding region while measuring macroscopically the adherence curves Γ(V ). Insights of viscoelastic dissipative processes occurring in the debonding region have been obtained from linear and nonlinear rheological measurements in controlled and imposed geometries. In the remaining part of the paper, we will present in turn the results obtained from each experimental setup.

We will finally present our model, developed to predict the macroscopic adherence curves of lightly crosslinked PSA from bulk rheological properties and detailed knowledge of the structure of the debonding region.

Results

In Fig. 2, we present the results obtained from linear rheological measurements. We plot the rescaled storage shear modulus b T • G in Fig. 2 temperature can be captured using the Williams-Landel-Ferry model :

ln a T = -C 1 T -T ref C 2 + T -T ref (2)
G (t = ω -1 ) G 2 (ω) + G 2 (ω).
This can be roughly approximated by a power law G ∼ t -β with β = 0.23 in the relevant range for peel. We can readily see that the level of cross-linking does not significantly affect the linear rheological response of the material nor the shift factors.

Typical nonlinear extensional curves σ N (λ) are shown in Fig. 3(a-c) for independent changes of the crosslinking level, strain rate and temperature. In all cases, after a linear regime at λ ∼ 1, we observe a softening followed by a strain hardening, as expected for PSAs. [START_REF] Deplace | Fine tuning the adhesive properties of a soft nanostructured adhesive with rheological measurements[END_REF] that the variation of σ N with ε and T can be captured by a single prefactor A( ε, T ) allowing to collapse all the curves on a reference master curve, where

A( εref = 1 s -1 , T ref = 296K) = 1.
The stress curves can then be written in a separable form σ N (λ, ε,

T ) = A( ε, T ) • σ ref N (λ),
where σ ref N (λ) only depends on the stretch λ and on the cross-linking level. This separability is less obvious for the polymer A with a lower level of crosslinking, especially for low T and large λ where nonlinear relaxation processes typical of viscoelastic fluids are active. [START_REF] Wagner | A constitutive analysis of uniaxial elongational flow data of low-density polyethylene melt[END_REF] However, it provides a good description of the nonlinear rheology in the domain of stretch experienced by fibrils before debonding in peeling.

In the inset of Fig. 4, the prefactor A for polymers A and B is plotted in a log-log scale as A( ε, T ) depends both on ε and T (inset:raw data), but can be collapsed to A ∼ (a T ε) 0.27 when using the same shift factor a T measured from the linear viscoelastic behavior.

a function of ε for the range of temperatures for which we have performed nonlinear tests. In the main graphic of Fig. 4, we show that the data can be collapsed onto a master curve when plotted against a T ε where a T is the shift factor determined from linear rheology. Remarkably, the linear and nonlinear rheology are thus found to share the same TTS. Moreover, for both polymers, we find that A ∼ (a T ε) 0.27 for ε > 10 -2 s -1 . Deviations are observed at smaller ε but, as it will be shown later, this occurs outside the range of strain rates experienced by the fibrils during our peeling tests. By construction, the same prefactor A can be used to rescale the true stress curves σ(ε), yielding :

σ(ε, ε, T ) = A(a T ε) • σ ref (ε) (3)
which is compatible with a Rivlin-Sawyers type of nonlinear constitutive law. [START_REF] Bird | Dynamics of polymeric liquids[END_REF] Since Eq. 3 is also valid in the small strain regime where we have σ ∼ G(t) • ε(t), we obtain that

A(a T ε) ∼ G(a T ε) ∼ (a T ε) β
where we have used the Cox-Merz rule ( ε ∼ ω ∼ 1/t). [START_REF] Cox | Correlation of dynamic and steady flow viscosities[END_REF] The exponent β = 0.23 measured by linear rheology (cf. Fig. 2(e)) is found to be in reasonable agreement with the 0.27 exponent obtained for nonlinear rheology. This strongly indicates that most of the rate and temperature dependence of both polymers is encapsulated in a single linear rheology function A, in the range relevant for the measured peeling curves. The reduced lateral size decreases with speed as /a 0 ∼ V -p with p = 0.16 (resp. p = 0.10) for polymer A (resp. B). (c) The final strain ε f = ln(λ f ) is described by ε f ∼ V -q where q = 0.05 (resp. q = 0.007) for polymer A (resp. B).

Next, we focus on the characterization of the fibrillated debonding region during steadystate peeling using an instrumented peel test [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF][START_REF] Villey | In-situ measurement of the large strain response of the fibrillar debonding region during the steady peeling of pressure sensitive adhesives[END_REF] at a 90 • angle and T = 23 • C. From images of the fibrillated debonding region, we measured both the lateral size of the fibrillar debonding region and the maximum fibril length a f at debonding (see Fig. 1(d)). In Fig. 5(a), we reproduced the adherence curves as a function of the peeling rate for polymer A and B from data collected in Villey et al. [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] As mentioned in the introduction, polymer B which has a more pronounced strain hardening presents a lower adherence than polymer A but a stronger dependence with the peeling rate. The adherence curves for the two polymers are distinct until reaching a critical velocity V c = 13 mm/s above which they exhibit an identical trend characterized by a negative slope. The region V > V c corresponds to an unsteady peeling caused by a stick-slip instability. 37 However, in the present manuscript, we focus on the steady-state peeling regime with a well-defined peeling rate. In this regime, the normalized lateral size /a 0 shown in Fig. 5(b) is found to decrease as a power law ∼ V -p with an exponent p 0.13 weakly dependent on the crosslinking level. The characteristic timescale for stretching the fibrils t f = /V is in the range 0.01 -20 s a regime where linear rheological measurements show no difference between polymers A and B (see Fig. 2). The maximal fibril stretch λ f = a f /a 0 is plotted in Fig. 5(c). For both polymers, the fibrils are stretched outside the regime of linear viscoelasticity and reach the strain hardening regime before debonding from the substrate (see Fig. 3). Therefore, we expect that material nonlinearities are also

a key ingredient at smaller scale to model the failure mechanism of fibrils. The relation between the level of crosslinking and the maximal fibril stretch is however non trivial. For polymer B, λ f is roughly independent of the peeling rate and λ f ≈ 5. But for polymer A, we observe not only an overall larger λ f but also a clear rate dependency. A fit of the data with a power law yields ε f = ln(λ f ) ∼ V -q with q = 0.05 (resp. q = 0.007) for polymer A (resp.

B). The fact that the effect of the level of crosslinking is more pronounced at smaller peel rate is rather counter intuitive. Indeed, it is generally believed that the models of adhesion based on the assumption of linear viscoelasticity work best at smaller peel rates. This point will be clarified in the discussion of our model. Furthermore, we evaluate the average strain rate εa experienced by a fibril as εa ≈ ε f V / ∝ V 1+m where m = p -q 1. For the measured peeling rates in the range 10 -5 -10 -2 m/s, the equivalent strain rate is in the range of 5 • 10 -2 -10 2 s -1 which is overlapped by the rheological measurements.

We are now in a position to propose a model to predict the adherence energy by combining the nonlinear material behavior and the measured characteristics of the fibrillated debonding region. Since the cavitation of the adhesive locally relaxes the effect of confinement, 38 we initially make the assumption that the large stretch of the fibrils can be treated as uniaxial.

The adherence energy can thus be modeled as:

Γ c (V ) = k • a 0 ε f ( εa) 0 σ(ε , εa )dε (4) 0 0.5 1 1.5 2 2.5
< (MPa) where σ(ε, εa ) is provided by the elongational measurements corresponding to the average strain rate εa and k is an adjustable factor compensating the crude approximation of uniaxial extension of fibrils, while the more complex fibril drawing condition will be investigated in future work. In the present implementation ε f (V ) and (V ) are measured quantities, since a model able to capture their functional dependence with the peeling speed has yet to be developed. The linear dependence of the adherence energy on the adhesive layer thickness has been demonstrated in previous experimental and theoretical studies. [START_REF] Kaelble | Theory and analysis of peel adhesion: adhesive thickness effects[END_REF]39 In Fig 6, we replot the σ(ε, ε) curves for polymer A and B in terms of true stress and strain varying the strain rate at T = 20 • C, unless otherwise stated in caption. The area under the curves is proportional to Γ c /a 0 by the factor k. In Fig 7, we plot the model prediction Γ c as function of peeling rate V according to Eq. 4, along with the steady-state peeling data from our previous work. [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] The two datasets are in good agreement if the value of peel prediction is multiplied by a dimensionless prefactor k = 5 demonstrating that Eq. 4

0 1 2 " 0 0.5 1 1.5 2 2.5 < (MPa) 0 1 0 1 0 1 0 2 " 0 1 0 2 " 4 s !1 0.2 s !1 0.1 s !1 2 s !1 1 s !1 0.05 s !1 0.01 s !1 0.5 s !1 (b) (c) (a)

Polymer B Polymer A

Figure 7: Calculated adherence curves Γ c given by Eq. 4 (large blue and red symbols) superposed with peeling data from Villey et al. [START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] (small grey symbols), already shown in Fig. 5(a). Using a numerical prefactor k = 5 quantitative agreement is obtained. Empty symbols are calculated from measurements taken at 0 • C. captures the clear difference in the adherence energy curves Γ(V ) between polymer A and B both in terms of relative values and slope, while a model based on linear rheology alone cannot explain this difference. The dimensionless prefactor k, can be interpreted as the sign that the fiber drawing process from the bulk adhesive layer is affected by a higher stress triaxiality than uniaxial extension tests, which will be the focus of further investigation.

The direct experimental access to the model parameters provides interesting insights on the physical mechanisms at the origin of the adherence curves in peeling. When focusing on the fibril debonding data in Fig. 6(a), we clearly see that the debonding criterion cannot be simply expressed in terms of a critical stress or strain. [START_REF] Yarusso | Quantifying the Relationship Between Peel and Rheology for Pressure Sensitive Adhesives[END_REF]40 However, for the more crosslinked polymer B, the debonding criterion is closer to a critical strain condition. In Fig. 6(c), we compare the stress curves for polymers A and B at specific values of ε. We observe that although the stress level for B is always higher than for A, the overall dissipation, hence Γ c , is larger for A due to the larger values of ε f . Moreover, the observation that the two polymers present a different rate dependence of ε f with ε explains why at higher ε the peeling energy Γ c for both polymers tends to get closer. Therefore, we demonstrate that both the increase in the stress level with ε, and the decrease of ε f are crucial to determine the general trend of Γ c (V ), making it difficult to derive a priori predictions unless we identify a sound fibril debonding criterion.

We can now discuss the link between our model and previous linear models. Taking advantages of the separability between strain rate dependence and strain dependence observed for our polymers (see Eq. 3), Eq. 4 can be further simplified as :

Γ c (V ) ∝ A(a T εa ) • ε f ( εa) 0 σ ref (ε)dε (5)
which more clearly reveals two contributions to Γ c (V ) : 1-a rate-dependent linear viscoelastic factor, and 2-a nonlinear factor whose rate-dependence originates from the debonding criterion. If we first consider polymer B alone, for which ε f is essentially rate independent, the dependence on rate of the adherence curve Γ c (V ) ∝ V n is completely determined by linear rheology and the power law exponent of n = 0.30 is quite close to the measured values of β ∼ 0.23 to 0.27. This result can explain why several authors have previously observed the TTS superposition for the adherence curves and the correlations between the n and β exponents. However, when considering the behaviour of polymer A, which has a less elastic character, the maximum fibril stretch at debonding ε f becomes significantly dependent on the peel rate and the competition between the increase in stress and the decrease in maximum stretch affects the Γ c (V ) curve in a deeply nonlinear manner. It is worth noting that even in the low speed regime, where LEFM based approaches are generally considered to be more appropriate, extremely soft materials, such as polymer A, are shown to be stretched increasingly outside their linear viscoelastic regime, thus precluding the use of linear models.

Conclusion

We have shown that the peel force of model soft viscoelastic PSAs can be quantitatively modeled for a range of strain rates with a single adjustable dimensionless prefactor based 14 on experimentally measured values of the maximum stretch in the fibrils at the detachment point. For more crosslinked adhesives the stress can be separated into the product of a strain dependent term that characterizes the strain softening and hardening behaviour of the adhesive, and a strain rate dependent term, which characterizes the molecular friction and is dependent on the glass transition temperature of the adhesive. However, for weakly crosslinked adhesives the two factors are no longer easily separable and the functional form of the strain hardening depends on the strain rate especially for very large strains.

A detailed modeling of both the relevant stress triaxiality during the fibril drawing process and of the fibril debonding criterion are certainly the next step, but we believe that the present model provides a clear understanding of the ingredients needed to understand the adherence energy in the peeling of very soft and dissipative materials.

(

Figure 1 :

 1 Figure 1: Schematics of experimental setups. (a) The linear shear storage G (ω) and loss G (ω) moduli of the adhesives are measured by dynamical mechanical analysis in a parallel plate geometry at imposed pulsation ω. (b) Nonlinear rheological properties are measured with the Sentmanat extensional rheometer which measures the applied force on an adhesive slab under uniaxial stretching condition at constant strain rate. (c) Instrumented peeling tests are performed with the same adhesives coated on a PET backing which is peeled by a vertical force F and peeling rate V . (d) Sketch of the fibrillar debonding region where is the lateral size and a f the maximal fibril size before detachment from the substrate. A commercial adhesive tape is bonded to a glass or aluminum substrate to ensure clean peeling and repeatable results.

  Fig. 2(b) as a function of the rescaled pulsation a T •ω at a reference temperature T ref = 296 K. The horizontal shift factor a T and vertical shift factor b T are shown in Fig. 2(c) and (d), respectively. Over the range of temperatures and rates used in the peel tests and extensional rheology, b T ≈ 1 within 10% error. The dependence of the horizontal shift factor with

  Fig. 2(e), we calculate the relaxation shear modulus G(t) at temperature T ref using the Cox-Merz rule, 27 yielding G (t = ω -1 )G 2 (ω) + G 2 (ω). This can be roughly approximated

Fig. 3 (Figure 3 :

 33 Figure3: (a-c) Nonlinear extensional curves representing the nominal stress σ N versus stretch λ, varying (a) concentration of crosslinker level, (b) strain rate ε, and (c) temperature T . All the curves show a softening regime followed by a strain hardening. Higher concentrations of crosslinker induce stiffening at large λ but makes no observable difference at small λ. (d) All the rheological curves collapse on a master curve when rescaled by a single paremeter A( ε, T ) independent of λ.

Figure 4 :

 4 Figure 4: The rescaling factor A for both polymers is obtained from a fitting procedure over the entire range of strain rates for various temperatures in the range 0 to 40 • C(colorbar).A( ε, T ) depends both on ε and T (inset:raw data), but can be collapsed to A ∼ (a T ε) 0.27 when using the same shift factor a T measured from the linear viscoelastic behavior.

Figure 5 :

 5 Figure 5: (a) Measured adherence energy Γ(V ) for polymer A and B. Above a characteristic peeling rate V c , a stick-slip instability develops for both polymers. Data from Villey et al.[START_REF] Villey | Rate dependent elastic hysteresis during the peeling of pressure sensitive adhesives[END_REF] (b) The reduced lateral size decreases with speed as /a 0 ∼ V -p with p = 0.16 (resp. p = 0.10) for polymer A (resp. B). (c) The final strain ε f = ln(λ f ) is described by ε f ∼ V -q where q = 0.05 (resp. q = 0.007) for polymer A (resp. B).

Figure 6 :

 6 Figure 6: Evolution of the dissipated energy (proportional to the shaded areas) with a T • ε (shifted at a common temperature T = 20 • C), for (a) polymer A and (b) polymer B. Upper bounds for the integral ε f = ln λ f are obtained from data given in Fig. 5(c). Curves with a star label are measured at 0 • C and rescaled. (c) Comparison of the stress-strain curves and debonding criteria for polymer A and B for the same ε at T = 20 • C.

  7) Schapery, R. A theory of crack initiation and growth in viscoelastic media. Part I : Theoretical Development. Int. J. Fracture 1975, 11, 141-159 (8) de Gennes, P.-G. Fracture of a Weakly Crosslinked Adhesive. CR. Acad. Sci. II 1988, 307, 1949-1953 (9) Saulnier, F.; Ondarçuhu, T.; Aradian, A.; Raphaël, E. Adhesion between a viscoelastic material and a solid surface. Macromolecules 2004, 37, 1067-1075 (10) Persson, B.; Brener, E. Crack propagation in viscoelastic solids. Phys. Rev. E 2005, 71, 036123 (11) Barthel, E.; Frétigny, C. Adhesive contact of elastomers: effective adhesion energy and creep function. J. Phys. D Appl. Phys. 2009, 42, 195302 (12) Gent, A. N.; Schultz, J. Effect of Wetting Liquids on the Strength of Adhesion of Viscoelastic Material. J. Adhesion 1972, 3, 281-294 (13) Maugis, D.; Barquins, M. Adhesion of Viscoelastic Materials to Rigid Substrates. In: Allen KW, editor. Adhesion 12. London: Elsevier Applied Science 1988, 310, 205 (14) Gent, A. N. Adhesion and strength of viscoelastic solids. Is there a relationship between adhesion and bulk properties? Langmuir 1996, 12, 4492-4496 (15) Cristiano, A.; Marcellan, A.; Keestra, B. J.; Steeman, P.; Creton, C. Fracture of model polyurethane elastomeric networks. J. Polym. Sci. B 2011, 49, 355-367 (16) Hui, C.-Y.; Jagota, A.; Bennison, S. J.; Londono, J. D. Crack blunting and the strength of soft elastic solids. Proc. Roy. Soc. A-Math. Phy. 2003, 459, 1489-1516 (17) Creton, C.; Ciccotti, M. Fracture and adhesion of soft materials: a review. Rep. Prog.

Physics 2016, 79, 046601