Keywords: Procrastination, resource management learning strategies, psychological inflexibility procrastination, effort regulation, study time, psychological (in)flexibility, higher education

Poor academic performance is known to be linked to the tendency to procrastinate. The latter is thought to reflect deficits in effort regulation and study time management (i.e., learningrelated resources), but some recent results have suggested that it could stem from psychological inflexibility. The main objective of the present study was thus to ascertain whether effort regulation, study time management, and psychological (in)flexibility predict variations in French students' academic procrastination. The results of an online survey among 259 first-year humanities and social sciences students revealed that 42.4% of the variance in procrastination was predicted by effort regulation and study time management, and 4% by psychological inflexibility. There was also a negative relationship between academic performance and procrastination. These results are discussed in terms of the usefulness of implementing interventions at the start of university to promote academic success and student wellbeing.

Poor Learning-Related Resource Management and Psychological Inflexibility as Predictors of Procrastination in First-Year University Students In modern society, the life domains of work and studies are characterized by a tendency to procrastinate. Procrastination refers to the unnecessary and intentional postponing of activities, even though it may have negative consequences for the individual [START_REF] Steel | The nature of procrastination: A meta-analytic and theoretical review of quintessential self-regulatory failure[END_REF]. It is linked to poor mental health [START_REF] Stead | I'll go to therapy, eventually": Procrastination, stress and mental health[END_REF], high levels of stress [START_REF] Shaked | Exploring academic procrastination: Perceptions, selfregulation, and consequences[END_REF], feelings of shame [START_REF] Fee | Procrastination: A means of avoiding shame or guilt?[END_REF], guilt [START_REF] Blunt | Project systems of procrastinators: A personal projectanalytic and action control perspective[END_REF], depression and anxiety [START_REF] Ferrari | Compulsive procrastination: Some self-reported characteristics[END_REF], poor quality of life and low satisfaction with life [START_REF] Rozental | Psychometric evaluation of the Swedish version of the Pure Procrastination Scale, the Irrational Procrastination Scale, and the Susceptibility to Temptation Scale in a clinical population[END_REF]. In academic settings, procrastination has negative effects on academic performances [START_REF] Martinie | Predictors of procrastination in first-year university students: Role of achievement goals and learning strategies[END_REF], including lower assignment grades, course grades and grade point averages [START_REF] Corkin | Effects of positive college classroom motivational environments on procrastination and achievement[END_REF], and a higher course dropout rate [START_REF] Balkis | Academic procrastination, academic life satisfaction and academic achievement: The mediation role of rational beliefs about studying[END_REF]. It is also positively correlated with academic burnout [START_REF] Abdi Zarrin | The role of emotional regulation and academic burnout in the prediction of academic procrastination in students[END_REF][START_REF] Hall | Self-efficacy, procrastination, and burnout in postsecondary faculty: An international longitudinal analysis[END_REF]. In sum, academic procrastination is problematic for both the academic performance and wellbeing of students. It is important to identify its predictors, in order to design interventions that can effectively reduce it. University teachers and administrators can limit but not eliminate procrastination among students by acting on situational factors. For instance, changes in examination rules have been shown to reduce study delay [START_REF] Schmidt | Changes in examination practices reduce procrastination in university students[END_REF]. The present study therefore investigated how far academic procrastination among French first-year university students is predicted by effort regulation and study time management, as well as by students' internal state. We focused specifically on this population because for multiple reasons (e.g., change in personal interests, wrong academic choice, health crisis, etc.) including procrastination, more than half of all French students fail to complete their first year (Ministère de l'Enseignement Supérieur et de la Recherche, 2020). This population therefore appears to be particularly at risk and to have fewer resources to cope with academic situations. The purpose of the present study was to inform educators about educational practices that can reduce academic procrastination.

Learning-Related Resource Management and Procrastination

In academic settings, students need to have knowledge and skills, but they also need to use personal resources such as effort regulation. The latter refers to persistence in the face of difficulty or tedium, and relates to the ability to control one's attention and effort in situations where distractions may be welcome. In other words, it reflects a commitment to pursue goals even when the individual is distracted and the task is difficult. This resource is key to learning, in that it determines the continued use of learning strategies. Authors have observed that the less students regulate their effort, the more they procrastinate [START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF][START_REF] Martinie | Predictors of procrastination in first-year university students: Role of achievement goals and learning strategies[END_REF][START_REF] Rakes | The impact of online graduate student's motivation and self-regulation on academic procrastination[END_REF][START_REF] Wolters | Investigating grit and its relations with college students' self-regulated learning and academic achievement[END_REF][START_REF] Ziegler | The development of academic procrastination in first-year secondary education students: The link with metacognitive self-regulation, self-efficacy, and effort regulation[END_REF]. This negative relationship therefore suggests that procrastination is linked to a deficit of effort regulation.

Procrastinators are characterized by poor effort self-regulation, but also by poor study time management. As time is a resource that cannot be changed, individuals must choose how to use it. Compared with nonprocrastinators, chronic procrastinators spend less time preparing for activities that may lead to success and more time on projects that may lead to failure, and underestimate the time needed to perform a task [START_REF] Lay | Trait procrastination and the perception of person-task characteristics[END_REF][START_REF] Mccown | An experimental study of some hypothesized behaviors and personality variables of college student procrastination[END_REF].

Moreover, many students find themselves with insufficient time to prepare for their exams because they have not planned their working time in advance, and therefore fail to achieve the grade needed to pass [START_REF] Asikainen | Understanding the variation in bioscience students' conceptions of learning in the 21st century[END_REF]. Authors have observed that the less students regulate their study time, the more they procrastinate [START_REF] Aribas | The relationship of time management and academic procrastination: A case of university students[END_REF][START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF][START_REF] Martín-Antón | Psychometric properties of the Academic Procrastination Scale in Spanish university students[END_REF][START_REF] Zhao | The relationship between self-control and procrastination based on the self-regulation theory perspective: The moderated mediation model[END_REF]. This negative relationship suggests that procrastination is related to poor study time management skills. However, not all studies have observed a significant relationship [START_REF] Ackerman | My instructor made me do it: Task characteristics of procrastination[END_REF][START_REF] Pychyl | Procrastination and the planning fallacy: An examination of the study habits of university students[END_REF]. For example, [START_REF] Pychyl | Procrastination and the planning fallacy: An examination of the study habits of university students[END_REF] found that individuals with high versus low procrastination scores did not differ on the number of errors in study time estimates. This result does not support idea that procrastination is a study time management problem and that procrastinators therefore have problems with their study time estimations. The relationship between study time management and procrastination therefore requires further clarification.

Experiential Avoidance and Procrastination

Procrastination should not be regarded solely as the outcome of deficits in effort regulation and study time management, as it may also be a strategy for avoiding negative affect [START_REF] Sirois | Procrastination and the priority of short-term mood regulation: Consequences for future self[END_REF]2016;[START_REF] Tice | Emotional distress regulation takes precedence over impulse control: If you feel bad, do it![END_REF]. Procrastination most often occurs when the task to be performed is viewed as aversive (e.g., [START_REF] Blunt | Task aversiveness and procrastination: A multidimensional approach to task aversiveness across stages of personal projects[END_REF][START_REF] Martín-Antón | Psychometric properties of the Academic Procrastination Scale in Spanish university students[END_REF][START_REF] Shaked | Exploring academic procrastination: Perceptions, selfregulation, and consequences[END_REF][START_REF] Steel | The nature of procrastination: A meta-analytic and theoretical review of quintessential self-regulatory failure[END_REF] or lacking an immediate reward [START_REF] Schouwenburg | Study motivation under social temptation: Effects of trait procrastination[END_REF]. Students often have a negative perception of academic work, viewing it as boring, unpleasant and anxiety-provoking [START_REF] Kaftan | A motivational perspective on academic procrastination: Goal focus affects how students perceive activities while procrastinating[END_REF]. Aversive tasks generate negative affect (anxiety and worry), so one way of avoiding further negative affect associated with an aversive task is to not perform that task and to engage in another more pleasant one instead. The results of three studies are consistent with this view. First, in a daily diary study, [START_REF] Pollack | Prior day negative affect influences current day procrastination: A lagged daily diary analysis[END_REF] observed that negative affect predicted next-day procrastination, whereas procrastination did not predict next-day negative affect. Results therefore showed that negative affect precedes procrastination. Second, [START_REF] Forstervold | Students' time management and procrastination in the wake of the pandemic[END_REF] reported that students' perceived stress increased procrastination. Third, [START_REF] Tice | Emotional distress regulation takes precedence over impulse control: If you feel bad, do it![END_REF] found that if participants with induced bad mood believed they could change their mood by engaging in a pleasant distractor task instead of practicing for a multiplication test (i.e., aversive task), they procrastinated. This effect was not observed if participants were given to believe that their bad mood could not be changed. Results therefore suggested that participants procrastinated in order to go from a bad mood to a good mood, and their procrastination served an emotion regulation purpose.

When they choose a distractor activity, procrastinators may not realize that their task avoidance will subsequently increase their negative affect [START_REF] Tice | Longitudinal study of procrastination, performance, stress, and health: The costs and benefits of dawdling[END_REF]. And yet procrastination has been shown to be linked to negative emotions such as despair, anxiety, guilt, shame, regret, and stress [START_REF] Blunt | Project systems of procrastinators: A personal projectanalytic and action control perspective[END_REF][START_REF] Zeenath | Exploring academic procrastination among undergraduates[END_REF]. Furthermore, procrastination is correlated with negative self-assessment [START_REF] Flett | Procrastination, negative selfevaluation, and stress in depression and anxiety: A review and preliminary model[END_REF][START_REF] Flett | Procrastination automatic thoughts as a personality construct: An analysis of the Procrastinatory Cognitions Inventory[END_REF], in the form of self-deprecating thoughts after delaying tasks [START_REF] Mccown | Content analyses of the beliefs of academic procrastinators[END_REF], and self-blame related to past procrastination [START_REF] Stainton | Trait procrastinators and behavior/traitspecific cognitions[END_REF]. These negative self-assessments fuel procrastinators' stress [START_REF] Flett | Procrastination automatic thoughts as a personality construct: An analysis of the Procrastinatory Cognitions Inventory[END_REF][START_REF] Sirois | Out of sight, out of time? A meta-analytic investigation of procrastination and time perspective[END_REF]. A metaanalysis carried out by [START_REF] Sirois | Less adaptive or more maladaptive? A meta-analytic investigation of procrastination and coping[END_REF] showed that procrastination is linked to greater use of maladaptive coping strategies (i.e., immediately relieving negative emotions related to a stressor without addressing the source of the stress), and less use of adaptive coping strategies (i.e., addressing the cause of a problem or learning to deal with the negative emotions associated with that problem).

Procrastinating to avoid aversive feelings is consistent with experiential avoidance, but also with the notion of psychological (in)flexibility that subtends acceptance and commitment therapy [START_REF] Hayes | Acceptance and Commitment Therapy as a unified model of behavior change[END_REF]. Psychological flexibility is defined as the "ability to contact the present moment more fully as a conscious human being, and to either change or persist when doing so serves valued ends" (Hayes et al., 2006, p. 5). It is characterized by six key psychological processes: (a) acceptance (i.e., ability to face situations rather than trying to avoid them), (b) cognitive defusion (i.e., ability to consider thoughts as mental events rather than reality), (c) self as context (i.e., ability to stand back from one's internal experience and observe it from another perspective: the person does not identify with the content of an internal experience, but with the context in which this experience emerges), (d) committed action (i.e., engagement in actions that are coherent with the person's values), (e) clear values (i.e., identification of own meaningful values), and (f) present moment awareness (i.e., ability to focus attention on the present moment, with openness and curiosity rather than being constantly distracted or oriented by one's thoughts or judgments).

In contrast to psychological flexibility, psychological inflexibility is defined as the "rigid dominance of psychological reactions, over chosen values and contingencies, in guiding action" (Bond et al., 2011, p. 678). It too is characterized by six key psychological processes: (a) experiential avoidance (i.e., avoiding potentially meaningful situations because of the fear of experiencing negative emotions), (b) cognitive fusion (i.e., identifying with one's thoughts), (c) self-as-content (i.e., identifying with the content of one's internal experiences), (d) inaction (i.e., lack of engagement with activities related to one's values), (e) lack of contact with values (i.e., lack of clarity of one's values), and (f) lack of contact with the present moment [START_REF] Hayes | Acceptance and Commitment Therapy as a unified model of behavior change[END_REF].

Previous research has highlighted positive correlations between inflexibility and depression, anxiety, stress, and psychological distress [START_REF] Bond | Preliminary psychometric properties of the Acceptance and Action Questionnaire -II: A revised measure of psychological inflexibility and experiential avoidance[END_REF], while some studies [START_REF] Eisenbeck | From psychological distress to academic procrastination: Exploring the role of psychological inflexibility[END_REF][START_REF] Gagnon | Committed action: An initial study on its association to procrastination in academic settings[END_REF][START_REF] Glick | A preliminary investigation of the role of psychological inflexibility in academic procrastination[END_REF][START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF] have highlighted a negative link between experiential avoidance and procrastination, suggesting that procrastination results from psychological inflexibility. More specifically, [START_REF] Eisenbeck | From psychological distress to academic procrastination: Exploring the role of psychological inflexibility[END_REF] showed that experiential avoidance mediates the relationship between general psychological distress and academic procrastination. For their part, [START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF] found that the variance in procrastination in Finnish students was predicted positively by experiential avoidance, and negatively by time and effort management. To our knowledge, however, their study had at least two limitations. First, participants were preselected, insofar as they had to have problems with their studies and were therefore probably more motivated to improve their learning-related resource management. Second, only one of the 12 (in)flexibility processes was measured.

Purpose of the Present Study

The present study explored whether the tendency to procrastinate is predicted by psychological (in)flexibility processes, effort regulation, and study time management in an unselected French student population. Given that procrastination is considered to be a deficit in learning-related resource management, and may also result from psychological inflexibility, we expected it to be predicted negatively by effort regulation, study time management and psychological flexibility, and positively by psychological inflexibility. We also expected procrastination to be negatively correlated with academic performance. This study was not preregistered, but our data are publicly available on the Open Science Framework and can be access at: https://osf.io/cb8r9/?view_only=b9dd4f139033460ca2de4db50cf5b1a0

Method

The study was preregistered on the institution's register of data processing activities. It was carried out in accordance with the Declaration of Helsinki, and complied with the General Data Protection Regulation and the French Data Protection Act. All participants gave their written informed consent.

Sample and Procedure

During the seventh week of the first semester of the 2021-2022 academic year, French students (N = 580), enrolled in a humanities and social sciences bachelor program were invited by the authors to complete an online survey comprising scales measuring academic procrastination, effort regulation, study time management, and psychological (in)flexibility.

The order of the scales was randomized across participants. A total of 276 students completed the survey. Of these, four outliers were identified (z score > 3), and their data were excluded from the analyses. We had to exclude the data of a further 13 students, as they did not sit the university's exams. The final sample was therefore composed of 259 students (239 female and 20 male) (Mage = 18.64 years, SDage = 1.312).

Measures Academic procrastination.

We used an 11-item validated French version [START_REF] Osiurak | Déterminants de la procrastination académique: Motivation autodéterminée, estime de soi et degré de maximation[END_REF] of [START_REF] Solomon | Academic procrastination: Frequency and cognitive-behavioral correlates[END_REF]'s Academic Procrastination Scale. The internal consistency of this scale was satisfactory (α = .73). All items were rated on a 7-point Likert scale ranging from 1 (not at all true for me) to 7 (totally true for me). A mean procrastination score was calculated.

Resource management.

In the absence of a validated tool in French, 12 items (time and study environment: eight items; effort regulation: four items) from the Motivated Strategies for Learning Questionnaire [START_REF] Duncan | A manual for the Motivated Strategies for Learning Questionnaire (MSLQ)[END_REF] were translated by the authors. An English-French bilingual then translated the French version back into English. Any discrepancies between the original items and the back-translation were discussed by the authors and the back-translator until a satisfactory solution was found. The internal consistency of the two subscales was satisfactory (α = .67 for study time management; α = .64 for effort regulation). A mean score was calculated for each type of resource management.

Psychological (in)flexibility

Psychological (in)flexibility was measured with the French version of the Multidimensional Psychological Flexibility Inventory short form [START_REF] Grégoire | Validation of the English and French versions of the Multidimensional Psychological Flexibility Inventory short form (MPFI-24)[END_REF].

Internal consistency was satisfactory for both the flexibility (α = .85) and the inflexibility (α = .83) subscales. We calculated mean scores for flexibility and inflexibility.

Academic performance

Participants' mean grade for the first semester of the first year of the bachelor's degree program in psychology was used as an indicator of their academic performance. This mean grade was calculated from the grades for five teaching units. Each teaching unit was assessed by a written exam. Grades range from 0 to 20 in the French system.

Results

Data Processing

We conducted two types of analyses. First, correlation analyses were performed between factors (Bravais-Pearson r). Second, we conducted two-step hierarchical regression analyses in which procrastination scores were regressed on the two types of resource management in the first step. Psychological flexibility and inflexibility were added in the follow-up step. Analyses were performed using jamovi software version 2.2.5 (The jamovi Project, 2021).

Descriptive Statistics

The variables in the study are described in Table 1. All variables had a satisfactory level of normality, with skewness values below two and kurtosis values below four [START_REF] Kline | Principles and practice of structural equation modeling[END_REF] reported by [START_REF] Kane | The generality of working memory capacity: A latent-variable approach to verbal and visuospatial memory span and reasoning[END_REF].

Insert Table 1 about here

Correlations Between Procrastination, Psychological Flexibility and Inflexibility,

Learning-Related Resource Management, and Performance

The results of correlation analyses are set out in Table 2. They revealed clear relations between factors. Procrastination was negatively correlated with study time management, effort regulation, psychological flexibility, and performance (all ps < .001). It was also positively correlated with psychological inflexibility (p < .001). Effort regulation was linked positively with study time management, flexibility, and performance (all ps < .001), and negatively with inflexibility (p < .01). Study time management was correlated positively with psychological flexibility and performance (both ps < .001), and negatively with psychological inflexibility (p < .01). Finally, psychological inflexibility was negatively correlated with psychological flexibility (p < .001).

Insert Table 2 about here

Predictors of Procrastination

We checked the multicollinearity between our variables. The variance inflation factor, an index of statistical multicollinearity between factors, was below five, suggesting an absence of multicollinearity between our variables (see Table 1). Results of the first step of the regression analyses revealed that resource management accounted for 42.4% (adjusted R 2 ) of the variance in procrastination, F(2, 256) = 95.9, p < .001. Procrastination was significantly and negatively predicted by study time management, t(256) = -6.39, p < .001, and effort regulation, t(256) = -6.55, p < .001. The results of the second step revealed that 46.7% of the variance was accounted for when flexibility and inflexibility variables were added, ΔF(2, 254) = 11.3, p < .001. In this model, only inflexibility significantly and positively predicted procrastination, t(254) = 4.26, p < .001.

Discussion

The present study investigated whether learning-related resource management and psychological (in)flexibility predict procrastination in French first-year university students, with a view to using these factors as levers for improving student learning. Drawing on previous studies, we expected the two types of resource management (effort regulation and study time management) to negatively predict procrastination, and psychological inflexibility to positively predict it. Finally, we expected procrastination to be negatively correlated with academic performance.

Correlation analyses revealed that study time management, effort regulation, psychological flexibility, and performance were negatively correlated with procrastination, while the latter was positively correlated with psychological inflexibility. These results are consistent with those reported in the literature (e.g., [START_REF] Aribas | The relationship of time management and academic procrastination: A case of university students[END_REF][START_REF] Eisenbeck | From psychological distress to academic procrastination: Exploring the role of psychological inflexibility[END_REF][START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF][START_REF] Martinie | Predictors of procrastination in first-year university students: Role of achievement goals and learning strategies[END_REF]. Effort regulation and study time management were positively correlated with academic performance, again consistent with literature findings [START_REF] Baker | Metacognition, comprehension monitoring, and the adult reader[END_REF][START_REF] Haarala-Muhonen | Factors affecting the study pace of first-year law students: In search of study counselling tools[END_REF][START_REF] Häfner | Decreasing students' stress through time management training: An intervention study[END_REF]. Finally, effort regulation and study time management were correlated positively with flexibility, and negatively with inflexibility. Concerning our model, 42% of the variance in procrastination was negatively predicted by effort regulation and study time management, and 4% was positively predicted by inflexibility. Taken together, these results reinforce the idea that procrastination reflects impaired study time management [START_REF] Burka | Mind games procrastinators play[END_REF][START_REF] Glick | An investigation of the efficacy of acceptance-based behavioral therapy for academic procrastination[END_REF] and effort regulation [START_REF] Rakes | The impact of online graduate student's motivation and self-regulation on academic procrastination[END_REF][START_REF] Wolters | Investigating grit and its relations with college students' self-regulated learning and academic achievement[END_REF], as well as psychological inflexibility. A recent study in a similar population found that 24% of the variance in procrastination was negatively predicted by effort regulation [START_REF] Martinie | Predictors of procrastination in first-year university students: Role of achievement goals and learning strategies[END_REF]. By adding study time management and psychological inflexibility, our model had greater power to predict procrastination variance.

The idea that procrastination is a deficit of flexibility has only recently emerged in the literature. The results of several studies [START_REF] Eisenbeck | From psychological distress to academic procrastination: Exploring the role of psychological inflexibility[END_REF][START_REF] Gagnon | Committed action: An initial study on its association to procrastination in academic settings[END_REF][START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF] point to a negative link between psychological flexibility and procrastination.

Nevertheless, these studies had various methodological limitations. Psychological (in)flexibility was measured by means of the Acceptance and Action Questionnaire-II [START_REF] Bond | Preliminary psychometric properties of the Acceptance and Action Questionnaire -II: A revised measure of psychological inflexibility and experiential avoidance[END_REF], which was developed to measure experiential avoidance. This tool has been criticized (see [START_REF] Grégoire | Validation of the English and French versions of the Multidimensional Psychological Flexibility Inventory short form (MPFI-24)[END_REF] because it has too few items to capture all the dimensions of psychological (in)flexibility [START_REF] Gámez | Development of a measure of experiential avoidance: The Multidimensional Experiential Avoidance Questionnaire[END_REF]. Furthermore, given that it taps into experiential avoidance, it should be regarded as a measure of psychological inflexibility rather than psychological flexibility. Other authors have also pointed out that this questionnaire does not distinguish between the tendency to use experiential avoidance strategies and the consequences of such strategies, such as more negative emotions [START_REF] Wolgast | What does the Acceptance and Action Questionnaire (AAQ-II) really measure?[END_REF]. This is problematic, as it may lead researchers to overestimate the strength of the relationship between experiential avoidance and mental health problems. One added value of our study was the use of the Multidimensional Psychological Flexibility Inventory to probe the 12 dimensions of psychological flexibility and inflexibility, with no other items measuring other constructs or experiences [START_REF] Rolffs | Disentangling components of flexibility via the Hexaflex model: Development and validation of the Multidimensional Psychological Flexibility Inventory (MPFI)[END_REF].

The strong contributions of effort regulation and study time management to the prediction of procrastination are hardly surprising. Today's students grew up in the age of computers, and these have become an integral part of their lifestyle [START_REF] Haverila | Cell phone usage and broad feature preferences: A study among Finnish undergraduate students[END_REF].

Smartphones and computers can create any number of distractions, and encourage students to focus their attention on short-term distractions (e.g., surfing on the Internet) rather than on academic work, thereby impairing effort regulation and reducing the amount of time spent learning. Moreover, procrastination can be conceptualized as a form of disconnect between the individual's present and future selves. More specifically, procrastinators try to regulate their present mood by avoiding unpleasant, challenging or boring tasks that generate negative emotions (for a review, see [START_REF] Sirois | Procrastination and the priority of short-term mood regulation: Consequences for future self[END_REF]. They prefer immediacy, which prevents them from giving priority to the future. Hence, Sirois (2014)'s meta-analysis showed that procrastination is negatively and moderately associated with the future time perspective, and positively related to the present time perspective.

Our results converge with those reported by [START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF]. These authors found that the tendency to procrastinate among students with study problems was predicted negatively by time and effort management, and positively by experiential avoidance. One added value of our study is that by capturing all the psychological (in)flexibility processes, we were able to show that effort regulation, study time management, and psychological inflexibility are all predictors of academic procrastination in an unselected sample of students.

Our study was therefore not just a simple replication of [START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF]'s study, but an important extension, as it is vital to take all the psychological (in)flexibility processes in account when designing interventions to combat procrastination by helping students deal constructively with their internal states.

Limitations and Future Directions

The present study had four main limitations. First, the correlation analyses did not allow us to reach any conclusions about the causal nature of the relationships we observed. Second, the internal consistency of the resource management scales was below the satisfactory level of .70, reducing the reliability of our results. Third, our results specifically concerned French humanities and social sciences students, who tend to procrastinate more than students in other subject areas [START_REF] Nordby | Do procrastination-friendly environments make students delay unnecessarily[END_REF]. It is worth mentioning that in this subject area, students are mainly female, and 92% of our sample was female. Given that gender-related differences have been observed [START_REF] Ozer | Gender orientation and academic procrastination: Exploring Turkish high school students[END_REF], our results cannot be generalized to male students. [START_REF] Ozer | Gender orientation and academic procrastination: Exploring Turkish high school students[END_REF] observed that among Turkish students, male students reported more frequent academic procrastination than female students. Furthermore, female and male students gave different reasons for their procrastination. Female students cited fear of failure and laziness, while male students cited risk taking and rejection of control.

Moreover, we only included first-year students, meaning that different conclusions might be reached for students in other years. Fourth, our model did not take self-efficacy into consideration. This personal resource refers to individuals' ability to act in a way that enables them to achieve their goals (Bandura, 1997;[START_REF] Heo | Influences of depression, self-efficacy, and resource management on learning engagement in blended learning during COVID-19[END_REF][START_REF] Üner | Study effort, learning strategies and test anxiety when striving for language competence: The role of utility value, self-efficacy, and reasons for learning English[END_REF]. Self-efficacy can play an important role in dealing with stress and increasing mental wellbeing (e.g., [START_REF] Llorens | Does a positive gain spiral of resources efficacy beliefs and engagement exist?[END_REF][START_REF] Salanova | Self-efficacy specificity and burnout among information technology workers: An extension of the Job Demands-Control model[END_REF], and has been found to be linked to procrastination [START_REF] Liu | Academic self-efficacy and postgraduate procrastination: A moderated mediation model[END_REF]Svartdal et al., 2022), effort regulation and study time management [START_REF] Rakes | The impact of online graduate student's motivation and self-regulation on academic procrastination[END_REF][START_REF] Wolters | Investigating grit and its relations with college students' self-regulated learning and academic achievement[END_REF], and psychological flexibility [START_REF] Hailikari | Understanding procrastination: A case of a study skills course[END_REF]. In future research, it would be useful to bring these factors together to explain procrastination by taking each psychological (in)flexibility process into account.

Practical Implications and Conclusions

Taken together, our results suggest that in order to overcome procrastination, students must at the very least manage their effort, study time, and internal state. To facilitate effort regulation, it is important for students to avoid ego depletion, by alternating tasks of varying levels of difficulty, taking breaks during learning sessions, and reducing the number of shortterm distractions available on their phones and computers. Moreover, given that effort regulation is correlated with self-efficacy [START_REF] Heo | Influences of depression, self-efficacy, and resource management on learning engagement in blended learning during COVID-19[END_REF][START_REF] Üner | Study effort, learning strategies and test anxiety when striving for language competence: The role of utility value, self-efficacy, and reasons for learning English[END_REF], one way of promoting effort regulation would be to enhance students' self-efficacy through, say, positive feedback from family, peers, and teachers [START_REF] Schunk | Self-efficacy and classroom learning[END_REF][START_REF] Usher | Sources of middle school students' self-efficacy in mathematics: A qualitative investigation[END_REF]. Furthermore, given that procrastination is negatively associated with the future time perspective [START_REF] Sirois | Out of sight, out of time? A meta-analytic investigation of procrastination and time perspective[END_REF], it would be interesting to raise students' awareness of the importance of planning their study time and organizing their study environment to avoid procrastination. This would enable them to be more active in their learning, by reducing the strategy of working at the last minute under time pressure. The objective here would be to facilitate the conditions for continuous work, which is a source of academic success and wellbeing. Finally, regarding psychological (in)flexibility, interventions specifically designed to develop greater psychological flexibility among students have been tested in acceptance and commitment therapy workshops. The Korsa program was devised to prevent and reduce mental health problems in the student population [START_REF] Grégoire | The use of acceptance and commitment therapy to promote mental health and school engagement in university students: A multisite randomized controlled trial[END_REF]. Although the original aim was not to reduce procrastination, given that the latter is known to be related to mental health issues (e.g., [START_REF] Haycock | Procrastination in college students: The role of self-efficacy and anxiety[END_REF], this type of program might also be useful for reducing procrastination in students. One of the studies on the impact of the Korsa program highlighted the relationship between psychological flexibility, stress and wellbeing, and results showed that the association between psychological flexibility and wellbeing became stronger over time [START_REF] Grégoire | Ecological momentary assessment of stress, well-being and psychological flexibility among college and university students during Acceptance and Commitment Therapy[END_REF]. Future studies involving the Korsa program could therefore measure its potential benefits in terms of reducing procrastination.

In conclusion, helping first-year students to improve their effort regulation and study time management, and to cope constructively with their internal states seems a relevant means of reducing their procrastination. These three factors alone accounted for 46.4% of the variance in procrastination, underlining their influence and the need for targeted interventions in this area.

Table 1

 1 Descriptive Statistics for Measurement ScalesProCra Effort-reg Time Flexi Inflexi Perf Note. Likert scale ranging from 1 (not at all true for me) to 7 (totally true for me). ProCra: procrastination; Effort-reg: effort regulation; Time: study time management; Flexi: psychological flexibility; Inflexi: psychological inflexibility; Perf: performance; Skewness: perfect symmetry or asymmetry equal to 0; Kurtosis: normal kurtosis equal to 0; VIF: variance inflation factor.

	n	259	259	259	259	259	259
	M	3.98	4.75	4.92	4.58	4.18	10.9
	SD	0.99	1.12	0.84	0.95	1.11	2.52
	Skewness	-0.01	-0.32	-0.07 -0.11	-0.00	-0.01
	Kurtosis	-0.80	-0.31	-0.02 -0.19	-0.56	-0.57
	VIF		1.58	1.47	1.22	1.11

Table 2 Correlations for Study Variables

 2 Note. ProCra: procrastination; Effort-reg: effort regulation; Time: study time management; Flexi: psychological flexibility; Inflexi: psychological inflexibility; Perf: performance. * p < .05. ** p < .01. *** p < .001.

		1	2	3	4	5	6
	1. ProCra	-
	2. Effort-reg	-0.581 ***	-
	3. Time	-0.577 ***	0.563 ***	-
	4. Flexi.	-0.293 ***	0.346 ***	0.246 ***	-
	5. Inflexi	0.335 ***	-0.189 **	-0.146 *	-0.304 ***	-
	6. Perf	-0.209 ***	0.331 ***	0.242 ***	0.041	-0.042	-

Table 3

 3 Model Coefficients -ProcrastinationNote. Effort-reg: effort regulation; Time: study time management; Flexi: psychological flexibility; Inflexi: psychological inflexibility. R 2 = .42 (p < .001) for Step 1; Δ R 2 = .04 (p < .001) for Step 2.

	Predictor	B	B SE	t	p	b
	Step 1
	Effort-reg	-0.33	0.05	-6.55	< .001	-0.37
	Time	-0.42	0.06	-6.39	< .001	-0.36
	Step 2.
	Effort-reg	-0.29	0.05	-5.83	< .001	-0.33
	Time	-0.41	0.06	-6.36	< .001	-0.35
	Inflexi	0.19	0.04	4.26	< .001	0.21
	Flexi	-0.02	0.05	-0.54	0.58	-0.02

Procrastination, resource management learning strategies, psychological inflexibility

Declarations

Conflict of interest:

The authors declare that they have no conflict of interest in relation to this article.


domain: math sde

text:
Guerlain Lambert
email: guerlain.lambert@master.ec-lyon.fr

Céline Helbert
email: celine.helbert@ec-lyon.fr

Claire Lauvernet
email: claire.lauvernet@inrae.fr

Claire Lauvernet Deep Gaussian

Deep Gaussian Process Metamodeling with Mixed Variables for Vegetative Filter Strip Design in Agricultural Context

come   L'archive ouverte pluridisciplinaire

Previous work has employed a metamodelling approach to simplify the use of BUVARD and aid users in designing buffer strips adapted to specific contexts [START_REF] Lauvernet | Metamodeling methods that incorporate qualitative variables for improved design of vegetative filter strips[END_REF][START_REF] Lauvernet | Metamodeling approaches to help designing vegetative filter strips and improve the water quality[END_REF]. Bayesian and non-Bayesian surrogates, such as Polynomial Chaos Expansions (PCE) and Mixed-kriging, have been compared, addressing the mixed variables issue for GP and PCE, but not for Deep Gaussian Processes (DGP). In this work, we explore new surrogate methods for BUVARD with quantitative and qualitative variables: Deep Gaussian Processes (DGP) [START_REF] Damianou | Deep Gaussian processes[END_REF] and Deep Sigma Point Processes (DSPP) [START_REF] Jankowiak | Deep Sigma Point processes[END_REF], which are capable of handling mixed variables. These methods leverage their deep structure to efficiently handle zero value plateaus, given the non-stationary and parsimonious structure of BUVARD data.

Figure 1: Depiction of the model chain integrated in BUVARD and its meta modelling as a function of only a few input variables, quantitative (black) and qualitative (green).

We examine various approaches for incorporating categorical variables into the DGP and DSPP models, including processing data by pairs of modalities, converting modalities to quantitative values, and using appropriate kernel combinations. Our findings contribute to the development of more accessible and effective buffer zone management tools for the agricultural community, addressing the challenges posed by mixed variables and the complex model chain in BUVARD, while also tackling non-stationarity and zero value plateaus. Furthermore, the metamodel serves as a relevant first guess for pollution reduction device design and provides an effective uncertainty tool for visualizing the impact of parameter uncertainty on filter efficiency in risk analysis and management.


Guilhem P Baeza

Julian Oberdisse

Angel Alegria

Kay Saalwaechter

Marc Couty

Anne-Caroline Genix
email: acgenix@univ-montp2.fr

Kay Saalwächter

Depercolation of aggregates upon polymer grafting in simplified industrial nanocomposites studied with dielectric spectroscopy

INTRODUCTION

Nanocomposites made of nanometric filler particles embedded in a viscoelastic polymer matrix may have outstanding macroscopic properties, among which one can cite mechanical, electrical, or optical ones. [START_REF] Jancar | Current issues in research on structure-property relationships in polymer nanocomposites[END_REF][START_REF] Heinrich | Reinforcement of elastomers[END_REF][START_REF] Mark | Science and Technology of Rubber[END_REF][START_REF] Schmidt | Properties of polymer-nanoparticle composites[END_REF] This broad range makes such systems suitable candidates for applications in very different fields, from car tires to, e.g., biological tissue engineering [START_REF] Armentano | Biodegradable polymer matrix nanocomposites for tissue engineering: A review[END_REF].

Microscopic understanding is necessary if one wishes to tune macroscopic properties of interest, like for instance the optimization of (non-)linear dynamic mechanical properties of car tire material in order to reduce rolling resistance, and thus fuel consumption and CO 2 production. The basis of macroscopic properties is the microstructure, i.e. the dispersion state of the nanoparticles (NPs), which may be individually dispersed, organized in small aggregates, or form sample-spanning networks of percolated particles or aggregates. [START_REF] Oberdisse | Aggregation of colloidal nanoparticles in polymer matrices[END_REF][START_REF] Jouault | Well-Dispersed Fractal Aggregates as Filler in Polymer-Silica Nanocomposites: Long-Range Effects in Rheology[END_REF][START_REF] Capuano | Universal Features of the Melt Elasticity of Interacting Polymer Nanocomposites[END_REF][START_REF] Aranguren | Effect of reinforcing fillers on the rheology of polymer melts[END_REF][START_REF] Choi | Influence of filler type and content on properties of styrene-butadiene rubber (SBR) compound reinforced with carbon black or silica[END_REF] The microstructure sets the frame for the dynamical properties, which themselves control many macroscopic properties, like, e.g., the time-dependent reaction of samples to mechanical stress which is often based on structural reorganization [START_REF] Payne | Low strain dynamic properties of filled rubbers[END_REF][START_REF] Montes | Particles in model filled rubber: Dispersion and mechanical properties[END_REF].

The local segmental dynamics in nanocomposites has been studied by several groups in the past [START_REF] Berriot | Evidence for the shift of the glass transition near the particles in silica-filled elastomers[END_REF][START_REF] Papon | Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects[END_REF][START_REF] Roh | Space-Dependent Dynamics in 1,4-Polybutadiene Nanocomposite[END_REF][START_REF] Fragiadakis | Dynamics near the filler surface in natural rubber-silica nanocomposites[END_REF][START_REF] Arrighi | The glass transition and interfacial layer in styrene-butadiene rubber containing silica nanofiller[END_REF][START_REF] Gee | Molecular dynamics studies on the effects of water speciation on interfacial structure and dynamics in silica-filled PDMS composites[END_REF][START_REF] Klüppel | Evaluation of viscoelastic master curves of filled elastomers and applications to fracture mechanics[END_REF][START_REF] Fotiadou | Structure and Dynamics of Hyperbranched Polymer/Layered Silicate Nanocomposites[END_REF][START_REF] Otegui | Influence of Water and Filler Content on the Dielectric Response of Silica-Filled Rubber Compounds[END_REF][START_REF] Boucher | Accelerated physical aging in PMMA/silica nanocomposites[END_REF]. Polymer in contact with NP surfaces may form immobilized, sometimes referred to as "glassy", layers which are mainly detected by NMR [START_REF] Berriot | Evidence for the shift of the glass transition near the particles in silica-filled elastomers[END_REF][START_REF] Papon | Solid particles in an elastomer matrix: impact of colloid dispersion and polymer mobility modification on the mechanical properties[END_REF]. On larger scales, at high temperature (above the glass transition), the occurrence of a Maxwell-Wagner-Sillars (MWS) process has been reported in several composite systems using broadband dielectric spectroscopy (BDS) [START_REF] Otegui | Influence of Water and Filler Content on the Dielectric Response of Silica-Filled Rubber Compounds[END_REF][START_REF] Mijovic | Dynamics in polymer-silicate nanocomposites as studied by dielectric relaxation spectroscopy and dynamic mechanical spectroscopy[END_REF][START_REF] Bohning | Dielectric study of molecular mobility in poly(propylene-graft-maleic anhydride)/clay nanocomposites[END_REF][START_REF] Lee | Accelerated α Relaxation Dynamics in the Exfoliated Nylon-11/Clay Nanocomposite Observed in the Melt and Semicrystalline State by Dielectric Spectroscopy[END_REF]. MWS processes are related to polarization effects due to charge carrier diffusion through the different phases of the nanocomposites. These effects result in the trapping and accumulation of charges at the polymer/silica interface. Charges can diffuse on the surface of individual NPs within aggregates, or on a larger scale on the surface of aggregates [START_REF] Meier | Carbon black networking in elastomers monitored by dynamic mechanical and dielectric spectroscopy[END_REF], processes which have been described theoretically by the interlayer model (ILM) [START_REF] Steeman | AN INTERLAYER MODEL FOR THE COMPLEX DIELECTRIC-CONSTANT OF COMPOSITES[END_REF][START_REF] Steeman | DIELECTRIC MONITORING OF WATER-ABSORPTION IN GLASS-BEAD-FILLED HIGH-DENSITY POLYETHYLENE[END_REF]. The ionic conductivity has also been studied over the whole frequency range [START_REF] Meier | Carbon black networking in elastomers monitored by dynamic mechanical and dielectric spectroscopy[END_REF][START_REF] Fritzsche | Structural dynamics and interfacial properties of fillerreinforced elastomers[END_REF]. In a biphasic material, one may note that the measured conductivity is a global observable, which is related to the polymer, the filler via its surface conductivity, and to the spatial arrangement of the filler which may form conductive pathways of various sizes within the sample.

The dynamical properties of nanocomposites as measured by BDS and NMR have only rarely been compared to macroscopic rheological properties. A correlation has been described in the literature for carbon-black filled elastomers [START_REF] Meier | Carbon black networking in elastomers monitored by dynamic mechanical and dielectric spectroscopy[END_REF][START_REF] Fritzsche | Structural dynamics and interfacial properties of fillerreinforced elastomers[END_REF] and clay nanocomposites [START_REF] Bohning | Dielectric study of molecular mobility in poly(propylene-graft-maleic anhydride)/clay nanocomposites[END_REF]. It is unclear, however, if the reinforcement of mechanical properties is dominated by local dynamical heterogeneities bridging NPs [START_REF] Montes | Influence of the glass transition temperature gradient on the nonlinear viscoelastic behavior in reinforced elastomers[END_REF][START_REF] Mujtaba | Detection of Surface-Immobilized Components and Their Role in Viscoelastic Reinforcement of Rubber-Silica Nanocomposites[END_REF][START_REF] Sotta | The crossover from 2D to 3D percolation: Theory and numerical simulations[END_REF] or large-scale filler structure [START_REF] Oberdisse | Structure and rheological properties of latex-silica nanocomposite films: Stress-strain isotherms[END_REF][START_REF] Oberdisse | Structure of latex-silica nanocomposite films: A small-angle neutron scattering study[END_REF]. As far as we know, it is a novelty to study this question using dielectric spectroscopy of silica nanocomposites with multi-scale structure.

Recently, we have contributed to the understanding of the multi-scale structure of silicastyrene-butadiene (SB) simplified industrial nanocomposites using a combination of smallangle scattering, electron microscopy, and numerical simulations. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Baeza | Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR[END_REF][START_REF] Baeza | Studying twin-samples provides evidence for a unique structure-determining parameter in simplified industrial nanocomposites[END_REF][START_REF] Baeza | Mechanism of aggregate formation in simplified industrial silica styrene-butadiene nanocomposites: effect of chain mass and grafting on rheology and structure[END_REF] Here simplified refers to the absence of any crosslinker and related catalysts. We have built a global model describing structural data where primary silica beads (typical size 17 nm) are organized within small aggregates (typical size 80 nm, aggregation number about 50, giving an internal volume fraction, i.e., compacity, of the order of 35%), which themselves fill large-scale fractal branches (typical width 150 nm). The solid-phase mixing protocol (see SI) applied to industrial silica pellets result in a broad aggregate size distribution of polydispersity 30%. In a first paper focused on dynamics, we have characterized by BDS a new structure-related hightemperature dielectric process termed MWS2, as it occurs in addition to the previously observed MWS process [START_REF] Baeza | A high-temperature dielectric process as a probe of large-scale silica filler structure in simplified industrial nanocomposites[END_REF]. It takes place across polymer between aggregates as found by an analysis of activation energies [START_REF] Baeza | A high-temperature dielectric process as a probe of large-scale silica filler structure in simplified industrial nanocomposites[END_REF]. Under Φ si variation which induces aggregate percolation (network formation), a strong increase of the temperature-independent dielectric strength of MWS2 was found around the mechanical percolation threshold of 12%v.

The key question of this second article on dynamics is to understand the relationship between structure and dynamical processes by now varying principally the amount of graftable SB chains (%D3) in order to control the filler structure [START_REF] Baeza | Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR[END_REF]. Indeed, local dynamics as observed by NMR has been correlated with rheology [START_REF] Mujtaba | Detection of Surface-Immobilized Components and Their Role in Viscoelastic Reinforcement of Rubber-Silica Nanocomposites[END_REF] but the question of the relative importance of large-scale structure with respect to local connections (bridges) through immobilized layers remains open. We therefore focus in the first part of this article on the local scale, and in particular on segmental relaxation investigated by BDS and NMR, as a function of silica content and grafting. Then, the impact of polymer grafting on interfacial dynamics of largescale silica aggregate depercolation (network destruction) is studied. Finally, both local and large-scale dynamical properties are linked to rheology.

EXPERIMENTAL SECTION

2.1 Materials. Details are given in ref [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Baeza | Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR[END_REF]. The polymer matrix is made of styrenebutadiene (SB) random copolymers, with two types of chains of molecular mass 140 kg.mol -1 (PI = 1.07) as determined by size exclusion chromatography. Chains are either linear unmodified SB chains, or bear a single silanol end-function making the chains graftable onto the silica surface by condensation. These functionalized chains are denoted D3, and the matrix composition is referred to by the percentage of D3 chains (%D3). Both types of chains have been purpose-synthesized by Michelin. The silica pellets (Zeosil 1165 MP from Solvay) have a nominal specific surface of 160 m 2 /g, and the size distribution of the nanoparticles obeys a log-normal law with R 0 = 8.55 nm, σ = 27%. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF] Nanocomposite formulation is the same as before [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Baeza | Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR[END_REF] and is detailed in the SI. Real silica volume fractions in nanocomposites have been measured by thermogravimetric analysis (Q500, TA Instruments) and lie in the range Φ si = 8.4%v -21.1%v (i.e., between 21 and 63 phr). The glass-transition temperature was determined using differential scanning calorimetry (Q2000 TMDSC, TA Instruments) in the modulated mode with a heating rate of 3 K/min. For the filled samples, T g shows no significant variation with silica content and %D3 as compared to the pure polymer matrix: T g = -37 ± 0.8°C.

Broadband dielectric spectroscopy (BDS).

A broadband high-resolution dielectric spectrometer (Novocontrol Alpha) was used to measure the complex dielectric permittivity ε*(ω) = ε'(ω) -iε''(ω) in the frequency range from f = 10 -2 to 10 7 Hz (ω = 2πf) of samples prepared as before [START_REF] Baeza | A high-temperature dielectric process as a probe of large-scale silica filler structure in simplified industrial nanocomposites[END_REF]. After 20 minutes equilibration at 60°C in the BDS cryostat (under nitrogen atmosphere), isothermal frequency measurements have been performed with temperature stability better than 0.1 K. Two series of nanocomposites have been investigated: samples with matrix composition fixed to 50%D3 and Φ si ranging from 8.4 to 21.1%v, and samples with fixed Φ si (≈ 8.5%v) and %D3 varying from 0 to 100%. For all samples, the temperature sequence starts from 333 K down to 153 K. Afterwards, the high-T range between 333 and 363 K has been explored. BDS fitting. Data treatment was carried out simultaneously on ε' and ε'' using for each temperature the combination of a purely dissipative dc-conductivity term and one Havriliak-Negami (HN) for each dipolar relaxation or MWS process present in the frequency window

ε * ω = ε ∞ + ∆ε 1 + iωτ γ δ -i σ ε ω (1)
where γ j and δ j are two shape parameters ranging between 0 and 1, and describing the symmetric and asymmetric broadening of the j th process, respectively. ∆ε j is the dielectric relaxation strength corresponding to the difference between the unrelaxed and relaxed values of the dielectric constant, and τ HN j is the characteristic relaxation time. In our approach, we systematically reduced the number of free parameters in order to extract reliable information, in particular on dielectric strengths and time scales for overlapping processes. A first fit of the data has been performed with all variables let free in the fit function to identify possible trends with temperature for the HN shape parameters. Then, according to the results, these parameters were forced to follow a linear (possibly constant) behavior (see SI).

Dielectric processes. Processes in the neat SB matrix (50%D3) have been investigated, i.e., the α-relaxation linked to the glass transition and the secondary β-relaxation which has been related to local motions within the butadiene units [START_REF] Cerveny | Dielectric αand β-Relaxations in Uncured Styrene Butadiene Rubber[END_REF]. In nanocomposites, two additional high-temperature dielectric (MWS1 and MWS2) processes at time scales slower than the αrelaxation have been identified [START_REF] Baeza | A high-temperature dielectric process as a probe of large-scale silica filler structure in simplified industrial nanocomposites[END_REF]. The HN parameters for both processes -shape parameters and dielectric strength -were found to hardly vary with T, and have been fixed to their average value in the final fitting procedure. [START_REF] Baeza | A high-temperature dielectric process as a probe of large-scale silica filler structure in simplified industrial nanocomposites[END_REF] We obtain a broad distribution of relaxation times (γ MWS1 = 0.42 and γ MWS2 = 0.55), in agreement with structurally polydisperse systems [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF]. Also, the secondary β-relaxation in pure polymer is masked in nanocomposites by the more intense β'-relaxation, which is also slightly shifted in frequency. A summary plot with all contributions appearing in ε'(ω) and ε"(ω) over a temperature window from 173 K to 323 K, and fit details are given in the SI. The β'-process is well-known in the silica-SB nanocomposites, where it has been assigned to localized fluctuations of water molecules at the silica surface [START_REF] Klüppel | Evaluation of viscoelastic master curves of filled elastomers and applications to fracture mechanics[END_REF][START_REF] Cerveny | Dielectric Study of Hydration Water in Silica Nanoparticles[END_REF]. A complete analysis of its evolution with Φ si and %D3 is given in the SI.

Special care was given to the description of the α-relaxation in the nanocomposites as it was found to overlap with the much more intense MWS1-process. In this case, shape parameters (γ α and δ α ) were imposed to the same values as in the pure matrix (see SI). To account properly for its relaxation strength in the nanocomposite, we have considered the filler effect on the local field according to ref [START_REF] Kremer | Broadband Dielectric Spectroscopy[END_REF], using a simple model for two-phase heterogeneous materials with 2.5 and 5.5 for the dielectric permittivity of SB and silica 36, 37 , respectively.

The characteristic relaxation time was consequently left as the only free fit parameter for this process.

Temperature dependence. For each dielectric process, the relaxation time corresponding to the frequency of maximal loss τ max = (2πf max ) -1 is a convenient parameter which can be read off from the graph ε"(f) or determined from the fit parameters. These times are plotted in Figure 1 for the matrix processes together with all other processes observed in nanocomposites. For all processes except the segmental relaxation, the temperature dependence of the relaxation times in Figure 1 is described by the Arrhenius equation ( )

        T k E exp τ = T τ B 0 (2a)
where E is the activation energy, k B the Boltzmann constant and τ 0 the high-temperature limit associated to vibrational times. In contrast, the temperature dependence of the α-relaxation times display a non-Arrhenius behavior, which can be well described by the Vogel-Fulcher-Tammann equation [START_REF] Kremer | Broadband Dielectric Spectroscopy[END_REF] ( )

        0 0 0 T - T DT exp τ = T τ (2b)
where T 0 is the temperature at which τ would diverge and D a coefficient related to the fragility concept [START_REF] Angell | RELAXATION IN LIQUIDS, POLYMERS AND PLASTIC CRYSTALS -STRONG FRAGILE PATTERNS AND PROBLEMS[END_REF]. Fitting the data in Figure 1 using eq(2a) led to E β = 32 kJ/mol in good agreement with ref [START_REF] Cerveny | Dielectric αand β-Relaxations in Uncured Styrene Butadiene Rubber[END_REF] (E β = 34.3 ± 1.3 kJ/mol independent of the styrene content). For the α-relaxation, we obtained τ 0 = 10 -12.8 s, T 0 = 187 K and D = 8.1 from our data (the corresponding fragility value is m = 76), in good agreement with the work of Ferry [START_REF] Ferry | Viscoelastic properties of polymers[END_REF] and Cerveny et al [START_REF] Cerveny | Dielectric αand β-Relaxations in Uncured Styrene Butadiene Rubber[END_REF] on close SB samples (m = 69 ± 3). The temperature value corresponding to τ α = 100 s gives an estimation of the glass-transition temperature, T g (100 s) = 231.5 K, to be compared with the calorimetric value T g = 236 K. Turning to the silica related processes, they all follow an Arrhenius temperature-dependence with E β' = 56 kJ/mol, E MWS1 = 55 kJ/mol and E MWS2 = 113 kJ/mol, which will also be discussed below.

Low field NMR.

Proton NMR experiments were performed using a Bruker Minispec mq20 with a proton resonance frequency of 20 MHz (0.5 Tesla). Free induction decay (FID) and magic sandwich echo FID (MSE-FID) signals, which are sensitive to molecular motions, were measured between 10 and 70°C in order to extract a solid-like signal fraction in the SB nanocomposites using a recycle delay of 2 s. The MSE-FID allows refocussing the FID signal and thus accesses the whole decay including the very first part, which is not available in simple FID due to the spectrometer dead time. The used methods have been reviewed in ref [START_REF] Saalwächter | [END_REF], and details about the sequences can be found in refs [START_REF] Papon | Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects[END_REF][START_REF] Mauri | Spin-diffusion NMR at low field for the study of multiphase solids[END_REF]. The signal of the nanocomposites can be decomposed in two contributions: a fast decay corresponding to the response of the rigid part and a slow decay associated to the mobile part. The data up to t = 200 µs have then been fitted to [START_REF] Mujtaba | Detection of Surface-Immobilized Components and Their Role in Viscoelastic Reinforcement of Rubber-Silica Nanocomposites[END_REF] ( )

( ) ( ) ( ) β mobil rig 2 rig rig 0 t/τ exp ) a (1 t/τ exp a = M M(t) - - + - (3)
where a rig is the fraction of rigid polymer. In order to avoid (small) systematic errors related to signal loss during the MSE, we used the unfiltered signal (FID) in all cases except at 10°C

where the filtered signal (MSE-FID) led to more stable fits. The value for the short decay time constant τ rig was determined independently from a double-quantum (DQ) filtered signal [START_REF] Mauri | Spin-diffusion NMR at low field for the study of multiphase solids[END_REF],

where the mobile part is almost entirely suppressed. Within error bars, it was found not to vary significantly with Φ si (or %D3) and the temperature: τ rig = (30 ± 3) µs. It has been fixed

to this value in eq 3. It is noted that the main effect of softening with temperature should be an increase of τ rig at constant fraction accompanied by non-trivial changes in the shape of the relaxation (decay) function. [START_REF] Papon | Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects[END_REF] Owing to the small overall fraction, because of which fitting is restricted to a two-component model, the change in relaxation behavior is thus qualitatively reflected in the apparent decrease of the rigid fraction.

2.4 Rheology. The rheological response in the linear regime expressed by the storage modulus, G'(ω), was obtained with a stress-controlled rheometer AR 2000, used in the straincontrolled mode in plate-plate geometry (20 mm diameter). Isothermal frequency sweeps at fixed low deformation level (γ = 0.1%) were performed in the temperature range from 10°C to 80°C allowing to cover a large frequency range from 10 -2 to 10 4 rad/s using the principle of time-temperature superposition for a reference temperature of 50°C.

Results

Segmental relaxation in nanocomposites

The α-relaxation is detected in the range from 243 and 293 K in nanocomposites with various silica fractions and %D3. The dielectric response at 273 K is shown in Figure 2 where one can also see the MWS1-process at lower frequency. The fit function by means of eq 1 (see SI for the detailed fit contributions) is also included. Note that, for all samples, the description is possible using highly constrained parameters. Namely, the shape and the dielectric strength of the α-process can be fixed to pure matrix parameters, see section 2.2, implying that its spectral shape remains unchanged. The relaxation times obtained in this way are displayed in Figure 3, where they are compared to those of the matrix. Times for the MWS1-process are given in the SI with a description in terms of the interlayer model [START_REF] Steeman | AN INTERLAYER MODEL FOR THE COMPLEX DIELECTRIC-CONSTANT OF COMPOSITES[END_REF][START_REF] Steeman | DIELECTRIC MONITORING OF WATER-ABSORPTION IN GLASS-BEAD-FILLED HIGH-DENSITY POLYETHYLENE[END_REF]. Within error bars, we observe that the segmental dynamics is not modified by the presence of filler up to the highest silica fraction, in agreement with ref [START_REF] Otegui | Influence of Water and Filler Content on the Dielectric Response of Silica-Filled Rubber Compounds[END_REF]. The impact of grafting with %D3 from 0 to 100% is shown in the inset of Figure 3 for Φ si ≈ 8.5%v. There is also no effect of grafting visible in BDS on the segmental dynamics. The investigation of local segmental dynamics in nanocomposites has been the goal of a large number of works in past years [START_REF] Berriot | Evidence for the shift of the glass transition near the particles in silica-filled elastomers[END_REF][START_REF] Papon | Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects[END_REF][START_REF] Roh | Space-Dependent Dynamics in 1,4-Polybutadiene Nanocomposite[END_REF][START_REF] Fragiadakis | Dynamics near the filler surface in natural rubber-silica nanocomposites[END_REF][START_REF] Arrighi | The glass transition and interfacial layer in styrene-butadiene rubber containing silica nanofiller[END_REF][START_REF] Gee | Molecular dynamics studies on the effects of water speciation on interfacial structure and dynamics in silica-filled PDMS composites[END_REF][START_REF] Klüppel | Evaluation of viscoelastic master curves of filled elastomers and applications to fracture mechanics[END_REF][START_REF] Fotiadou | Structure and Dynamics of Hyperbranched Polymer/Layered Silicate Nanocomposites[END_REF][START_REF] Otegui | Influence of Water and Filler Content on the Dielectric Response of Silica-Filled Rubber Compounds[END_REF][START_REF] Boucher | Accelerated physical aging in PMMA/silica nanocomposites[END_REF]. A slowing down of the α-process due to interaction or grafting between the polymer chain and the filler has been reported [START_REF] Papon | Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects[END_REF][START_REF] Fragiadakis | Dynamics near the filler surface in natural rubber-silica nanocomposites[END_REF][START_REF] Arrighi | The glass transition and interfacial layer in styrene-butadiene rubber containing silica nanofiller[END_REF]. In this context, distinct contributions have been detected: one part of the chain segments is not affected by the filler, whereas an interfacial polymer layer with frozen dynamics at nanosecond time scales has been evidenced by solid-state NMR experiments, the so-called "glassy" fraction [START_REF] Berriot | Evidence for the shift of the glass transition near the particles in silica-filled elastomers[END_REF][START_REF] Papon | Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects[END_REF]. Such an immobilized layer with a thickness of 1 to 2 nm at the particle surface close to T g would lead to an increase of the solid-like fraction within the nanocomposite and is thus of utmost importance for filler network percolation [START_REF] Mujtaba | Detection of Surface-Immobilized Components and Their Role in Viscoelastic Reinforcement of Rubber-Silica Nanocomposites[END_REF][START_REF] Merabia | A Microscopic Model for the Reinforcement and the Nonlinear Behavior of Filled Elastomers and Thermoplastic Elastomers (Payne and Mullins Effects)[END_REF][START_REF] Mujtaba | Mechanical Properties and Cross-Link Density of Styrene-Butadiene Model Composites Containing Fillers with Bimodal Particle Size Distribution[END_REF].

We found that the time-scale for the α-relaxation is not modified with Φ si , which means that either there are no immobilized layers, or that BDS is not sensitive enough. To check the existence of an immobilized fraction of organic material (which apart from SB could also comprise the silanization agent octeo) in our nanocomposites, we have performed proton lowfield NMR experiments in the temperature range between 10 and 70°C. The characteristic time for segmental dynamics as seen by NMR stands in the µs range. Thus, the chosen temperatures are just above the "NMR T g ", which is roughly located 40 K above the calorimetric T g [START_REF] Saalwächter | [END_REF].

Based on eq 3, the apparent rigid polymer fraction a rig has been determined for the silica fractions between 8.4 and 21.1%v using a constant short decay time τ rig = 30 µs as obtained from the DQ measurement. An example of data fitting is shown in Figure 4a, and one can see that there is no rigid contribution in the neat matrix. As can be seen in Figure 4b, the apparent amount of rigid polymer increases with Φ si and decreases with temperature as expected due to the softening of the immobilized part. The obtained fractions are rather low (between ≈ 1.5 and 3.5%) as compared with other systems like e.g., silica-polyethylacrylate nanocomposites [START_REF] Papon | Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects[END_REF]. Note that our results are in agreement with a recent work by Mujtaba et al [START_REF] Mujtaba | Detection of Surface-Immobilized Components and Their Role in Viscoelastic Reinforcement of Rubber-Silica Nanocomposites[END_REF] on industrial SB rubber nanocomposites.

We can rationalize our results based on the structure of aggregates filling fractal branches in our system, cf. [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF]. In this paper, we have combined a statistical analysis of TEM pictures with SAXS results and numerical simulations, in order to quantitatively describe the spacefilling properties of silica NPs inside aggregates, and the ones of aggregates inside branches.

The main ingredients of structure description in reciprocal space are the NP form factor, the construction of polydisperse aggregates by NPs, the repulsive interactions between aggregates caused by excluded volume and leading to a measurable suppression of the low-q scattering.

Finally, the large-scale fractal network is described by a power-law behavior. This SAXS analysis is combined with structural input given by the fraction of branches in the samples obtained by TEM. Aggregates contain typically some 50 primary beads and together with the aggregate radius one can define the internal silica volume fraction termed compacity, κ. The latter value evolves from 31 to 38% with Φ si in these samples [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF], and one can calculate the immobilized polymer fraction per aggregate ( )

κ 1 κ Φ a Φ 1 a si rig si agg rig - - = (4)
The latter corresponds to the ratio between the volume of the immobilized polymer and the total polymer volume in one aggregate. At 30°C, a rig agg = (7.3±0.7) %. Within error bars, it is not found to vary with Φ si .

A simple estimation of the layer thickness based on the hypothesis of individual bead dispersion gives 0.33 and 0.29 nm for Φ si = 8.4 and 21.1%v, respectively, i.e. the immobilized layer is very thin and its thickness independent of Φ si . This remains also true in presence of small aggregates, like here. Lastly, there was no detectable variation of the immobilized fraction as a function of the quantity of graftable chains (see the insert in Figure 4b at fixed Φ si ≈ 16.7%v), presumably due to the small number of (end-)grafting functions.

The low fraction of immobilized polymer obtained from NMR confirms the BDS results,

where it has been found that the α-relaxation was not affected by the presence of silica in our concentration range. Indeed, the resolution of BDS does not allow the detection of the slowdown of such a small polymer fraction.

Influence of grafting on percolation

Grafting of hydrophobic molecules at low grafting density has been shown to affect neither the water-dependent β' (see SI) nor the segmental dynamics (α-process, see Figure 3). We now concentrate on the impact of grafting on the high-temperature processes MWS1, MWS2, and the conductivity. The imaginary part of the dielectric permittivity is shown in Figure 5 for samples at fixed silica volume fraction and various fractions of graftable chains in the matrix ranging from 0% to 100%D3 at 333 K. One notes that the 0%D3 sample which contains no graftable chains at all is singular, with a high conductivity and a strong MWS2 contribution (see the detailed fit contributions in the figure), whereas the other samples display much lower values of ε''(ω) in the low-frequency range. The different MWS-processes can be identified in the broad maximum in Figure 5.

They are described by means of the HN expression using Φ si -and %D3-independent shape parameters (see section 2.2). Their temperature-independent strengths have been averaged from 293 to 363 K and are plotted as a function of matrix composition %D3, in Figure 6. The data in Figure 6 show that the strength of the MWS1 process at fixed Φ si is essentially unaffected by grafting as observed for the β'-process (see SI). Note that the comparison with the interlayer model [START_REF] Steeman | AN INTERLAYER MODEL FOR THE COMPLEX DIELECTRIC-CONSTANT OF COMPOSITES[END_REF][START_REF] Steeman | DIELECTRIC MONITORING OF WATER-ABSORPTION IN GLASS-BEAD-FILLED HIGH-DENSITY POLYETHYLENE[END_REF] is favorable, as the strength is determined by the quantity of silica at constant humidity. For both the logarithm of conductivity and dielectric strength of MWS2, there is a continuous decrease with %D3 leading to saturation. These quantities have been superimposed using the appropriate scale for each, which allows one to see that the shape of the decrease is correlated. MWS1 is found not to be affected by grafting, whereas MWS2 and the logarithm of the conductivity -which show a similar T-dependence -both display the same decrease. This common behavior will be discussed in parallel with the structural and rheological evolution in the next section. To finish, the temperature dependence of the relaxation time of the high-T processes is given in the SI for different matrix compositions, i.e., different grafting, and activation energies are summarized in Table 2. As previously observed for Φ si variation [START_REF] Baeza | A high-temperature dielectric process as a probe of large-scale silica filler structure in simplified industrial nanocomposites[END_REF], the MWS2 process displays a similar activation energy as the ionic conductivity and the high-temperature α-process (E a = 80 kJ/mol, see dotted line in Figure 3), suggesting again that it is primarily controlled by segmental motion of the polymer matrix via the ionic mobility (diffusion across the matrix).

Discussion of relationship between structure, rheology, and MWS processes.

The large scale structure as observed by TEM [START_REF] Baeza | Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR[END_REF] shows the transition from a heterogeneous distribution of silica with large space-spanning branches filled with aggregates at 0%D3 to a much more homogeneous distribution of clusters of aggregates at 100%D3. This transition with grafting can be described as the depercolation of aggregates, and our dielectric measurements confirm this in a rather unique manner. Indeed, the BDS response in Figure 6 shows that the second interfacial polarization process MWS2 is strongly related to these global structural features. Also, all dielectric processes are associated with a broad distribution of relaxation times, in agreement with structurally polydisperse systems [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF].

The depercolation of the large-scale filler structure with increasing polymer grafting has direct consequences on the rheological response of the samples to shear. One of the characteristics of the G'(ω) curves is the plateau modulus G, which can also be determined for the unfilled matrix (G 0 ), allowing to define the reinforcement factor as the ratio of moduli G/G 0 . In Figure 7, we compare the evolution of this reinforcement factor with silica volume fraction of two matrix compositions, 0%D3 and 50%D3 grafting. The reinforcement was determined at a common frequency of 60 Hz in raw data taken from refs [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Bouty | Nanofiller Structure and Reinforcement in Model Silica/Rubber Composites: A Quantitative Correlation Driven by Interfacial Agents[END_REF]. Both sets show a weak increase up to some critical percolation volume fraction, and a much stronger one above. However, in the case of the non-functional SB (0%D3), the critical percolation occurs at lower Φ si . This illustrates thus the more percolated behavior of the 0%D3 at a given volume fraction. For a quantitative comparison, we have fitted the following function [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF] to the data ( )

agg 0 0 G 1 2.5 Φ + G G 1 b c f agg agg c agg agg c agg G   Φ -Φ = + Θ Φ -Φ     -Φ   (5)
where Φ agg is the volume fraction of aggregates within the branches and defined as Φ si /(κΦ fract ). Note that average values of the fraction of branches, Φ fract , have been determined by electron microscopy before [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Baeza | Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR[END_REF]. The fit allows us to determine the relative strength of the network G f /G 0 = 90 (0%D3) and 50 (50%D3). The critical silica volume fractions are indicated by arrows in the graph. We obtain 7.5±0.5% and 11.5±0.5% for 0%D3 and 50%D3, respectively. This confirms the result of the visual inspection: with a shift of approximately 4% in volume fraction, this macroscopic technique illustrates that grafting moves the system away from percolation. Such a result agrees nicely with literature results. Indeed, Kumar et al.

[51] recently reviewed the different situations arising as a function of the mass and grafting density of chains with respect to matrix chains. Depending on the wetting ability of the polymer brush by the (short or long) polymer matrix chains, different regimes are evidenced leading to very different dispersion of the filler particles in the matrix. In our particular case (same mass of grafted and matrix polymer, high reduced grafting density), grafting induces good dispersion, thus shifting the system further away from the percolation threshold as observed in our work. Incidentally, one may also comment on the NMR results in the light of the reinforcement factors shown in Figure 7. It has been shown recently by Mujtaba et al [START_REF] Mujtaba | Detection of Surface-Immobilized Components and Their Role in Viscoelastic Reinforcement of Rubber-Silica Nanocomposites[END_REF] that reinforcement may be coupled with the fraction of immobilized ("glassy") polymer layers on the surface of the NPs. The interpretation is that even thin layers convey connections between the NPs.

These results were based on samples of very similar structure. In the present study, the fraction of (equally thin) immobilized layers was found to be independent of the grafting fraction, which in itself is a surprising result but may be due to the low grafting densities. On the other hand, the corresponding rheological data shown in Figure 7 correlate with the strong structural and dynamical evolution as seen by BDS. It may thus be concluded that the structural contribution -and not the quantity of immobilized polymer -dominates the rheological response in our samples.

CONCLUSION

The dielectric properties of silica-filled nanocomposites have been studied by BDS. The insensitivity of the α-relaxation to the addition of silica was confirmed by an independent NMR-experiment aiming at the determination of the fraction of an immobilized polymer layer. The equivalent layer thickness was found to be of the order of a few Angstroms, and the "glassy" polymer fraction in an aggregate below 10%. The latter value was determined using the results of a multi-scale structural model based on aggregates of some tens of primary nanoparticles filling large-scale branches and developed by us recently [START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF]. This calculation thus accounts for the heterogeneous distribution of silica in the matrix, and allows the estimation of the hard fraction of aggregates given by both silica (via aggregate compacity)

and immobilized polymer to some 40%. This last value suggests that aggregates can be considered as rigid as compared to the surrounding viscoelastic polymer matrix.

Finally, a second Maxwell-Wagner-Sillars process (MWS2) is shown to evolve strongly with grafting at fixed volume fraction (8.5%v). Knowledge of the microstructure and linear rheology of these nanocomposites enables us to correlate these evolutions of dynamics with the structure. Unlike MWS1 but similar to rheology, the MWS2 process is sensitive to depercolation upon polymer grafting. This establishes that dielectric spectroscopy can be used as a simultaneous probe for large-scale structural reorganization in complex silica-polymer nanocomposites and local dynamics. In combination with NMR, it shows that structural contributions to the rheological response dominate those of immobilized polymer in our samples.

Figure 1 :

 1 Figure 1: Complete relaxation map for the neat SB (black empty symbols) and a nanocomposite filled with 8.4%v of silica (plain symbols with different colors for the different processes investigated, see the dielectric spectra and other fit parameters in SI). The solid lines through data points are fits to the VFT equation for the α-relaxation and to the Arrhenius equation for the others.

Figure 2 :

 2 Figure 2: Dielectric loss spectra at T = 273 K for nanocomposites with a) different silica volume fractions (Φ si = 8.4, 12.7, 16.8 and 21.1%v from bottom to top, 50%D3), and b) different amounts of graftable chains (%D3 = 0, 25, 50, 75 and 100, from top to bottom, Φ si ≈ 8.5%v). The matrix signal is shown as black squares. Solid lines are fits to the experimental data by means of eq 1.

Figure 3 :

 3 Figure 3: Temperature dependence of the relaxation times for the α-process in nanocomposites with Φ si = 8.4 -21.1%v and 50%D3. Dotted line is a linear fit of the data above 283 K, see text for details. Inset: same data for different %D3 at fixed Φ si ≈ 8.5%v.

Figure 4 :

 4 Figure 4: a) FID data at 30°C for the matrix and a nanocomposite with Φ si = 21.1%v and 50%D3. The corresponding non-quantitative DQ data used to determine the apparent rigid-fraction time constant, τ rig , are shown in the lower part. Lines are fits using eq 3. b) NMR-determined fraction of immobilized polymer as a function of temperature for the SB nanocomposites with Φ si = 8.4 -21.1%v and 50%D3. Inset: rigid fraction versus %D3 at 50°C and fixed Φ si = (16.7±0.1) %v.

Figure 5 :

 5 Figure 5: Dielectric loss spectra at T = 333 K for nanocomposites with different amounts of graftable chains at fixed Φ si ≈ 8.5%v (%D3 = 0, 25, 50, 75 and 100, from top to bottom). Solid lines are fits to the experimental data by means of eq 1. The fit contributions are included for the sample with 0%D3: the dashed-dotted line stands for the MWS1 contribution, the dashed line for MWS2, and the dotted line for the dc-conductivity.

Figure 6 :

 6 Figure 6: DC-conductivity at 393 K (diamonds) and average value of the dielectric strengths versus T for MWS1 (circles) and MWS2 (squares) as a function of the fraction of graftable chains (Φ si ≈ 8.5%v). Lines are a guide to the eye.

Figure 7 :

 7 Figure 7: Comparison of rheological response at 60 Hz in terms of the reinforcement factor G/G 0 as a function of silica volume fraction, for two different grafting matrix compositions, 0%D3 (circles) and 50%D3 (squares). Solid lines are fits with eq 5. The purely hydrodynamic reinforcement, 1+2.5Φ agg , is also indicated (dotted lines). Reinforcement values are obtained from data in refs[START_REF] Baeza | Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM[END_REF][START_REF] Bouty | Nanofiller Structure and Reinforcement in Model Silica/Rubber Composites: A Quantitative Correlation Driven by Interfacial Agents[END_REF].

Table 2 :

 2 Activation energy for the dielectric processes MWS1 and MWS2, and the dc-conductivity in nanocomposites with fixed Φ si ≈ 8.5%v and different fractions of %D3.

	%D3	E MWS1 (kJ/mol)	E MWS2 (kJ/mol)	E σ σ σ σ (kJ/mol)
	0	58.5	63	53
	25	56	121	90
	50	55	120	88
	75	53	108	88
	100	52	107	85

Acknowledgements: We are thankful for a "Chercheur d'Avenir" grant by the Languedoc-Roussillon region (J.O.) and Ph.D. funding "CIFRE" by Michelin (G.P.B.). The authors acknowledge financial support from the European Commission under the Seventh Framework Program by means of the grant agreement for the Integrated Infrastructure Initiative N. 262348 European Soft Matter Infrastructure (ESMI). We are indebted to C. Dupas (L2C) and A. Mujtaba (Halle) for experimental support.