Dr Mohammad Sultan Al-Akhali

Ali Sadeq

Al-Maweri

Hashem M Al-Shamiri
email: hashem_alshamiri@yahoo.com

Esam Halboub

Khaled Alhadad

The association between periodontitis and non-alcoholic fatty liver disease: a systematic review

Keywords: Periodontitis, NAFLD, association, systematic review

Objectives: Animal and human studies have shown associations between periodontitis and non-alcoholic fatty liver disease (NAFLD). The purpose of the present systematic review was to assess the association between periodontitis and NAFLD Methods: This systematic review aimed to answer the focused question "Is there any association between periodontitis and NAFLD?" PubMed/ Medline, Scopus, Embase, and Web of Science databases were thoroughly searched to identify all relevant studies using the following keywords: periodontal disease, periodontitis, periodontal health, gingival health, periodontal pathogens, NAFLD, and hepatic steatosis.

Due to marked heterogeneity and inconsistency in the included studies, no statistical analyses were conducted.

Results: A total of 12 studies comprising 53,384 patients were included in the present systematic review.

The sample size in the included studies ranged from 52 to 24,470 patients. All studies except one found significant associations between clinical and/or microbial periodontal parameters and NAFLD.

Conclusions:

The available evidence suggests that periodontitis may be a risk factor for development and progression of NAFLD. However, due to limited number of prospective cohort studies included in this review, and the substantial heterogeneity among the included studies, further well-designed prospective cohort studies are highly warranted.

Introduction:

Periodontitis is a chronic disease of the tooth supporting structures characterized by immunological breakdown of bone and soft tissues in response to the bacterial dental plaque [START_REF] Meyle | Molecular aspects of the pathogenesis of periodontitis[END_REF]. It is one of the most common chronic diseases with an estimated rank of being the 6 th most prevalent long-term disease worldwide (GBD 2015 Disease and Injury Incidence and Prevalence [START_REF] Collaborators | Global, regional, and national incidence, prevalence, and years lived with disability for 310 diseases and injuries, 1990-2015: a systematic analysis for the Global Burden of Disease Study[END_REF]. When untreated or unsuccessfully treated, it ultimately leads to progressive loss of attachment, tooth mobility and, in severe cases, tooth loss with associated cost burden (Kassebaum et al., 2014a, Kassebaum et al., 2014b[START_REF] Listl | Global Economic Impact of Dental Diseases[END_REF]. Worthy to mention that its health impact crosses the boundaries of oral cavity as it has been reported to be associated with many systemic diseases [START_REF] Linden | Periodontitis and systemic diseases: a record of discussions of working group 4 of the Joint EFP/AAP Workshop on Periodontitis and Systemic Diseases[END_REF] including, but not limited to, cardiovascular diseases [START_REF] Beukers | Periodontitis is an independent risk indicator for atherosclerotic cardiovascular diseases among 60 174 participants in a large dental school in the Netherlands[END_REF], chronic obstructive pulmonary disease [START_REF] Chung | Associations Between Periodontitis and Chronic Obstructive Pulmonary Disease: The 2010 to 2012 Korean National Health and Nutrition Examination Survey[END_REF], pneumonia [START_REF] Gomes-Filho | Influence of periodontitis in the development of nosocomial pneumonia: a case control study[END_REF], chronic kidney disease [START_REF] Deschamps-Lenhardt | Association between periodontitis and chronic kidney disease: Systematic review and meta-analysis[END_REF], rheumatoid arthritis [START_REF] Potempa | The case for periodontitis in the pathogenesis of rheumatoid arthritis[END_REF], cognitive impairment and dementia [START_REF] Sanchez-Lara | Is periodontitis a risk factor for cognitive impairment and dementia? A case-control study[END_REF], metabolic syndrome [START_REF] Kumar | Association of chronic periodontitis with metabolic syndrome: A cross-sectional study[END_REF] and cancer [START_REF] Corbella | Is periodontitis a risk indicator for cancer? A meta-analysis[END_REF]. Otherwise, periodontal therapy results in glycemic control in diabetic patients [START_REF] Hasuike | Systematic review and assessment of systematic reviews examining the effect of periodontal treatment on glycemic control in patients with diabetes[END_REF][START_REF] Li | Effect of non-surgical periodontal treatment on glycemic control of patients with diabetes: a meta-analysis of randomized controlled trials[END_REF]. Such an argument supports the concept of "periodontal medicine" which emphasizes on the importance of bidirectional oral-systemic link [START_REF] Gulati | Essentials of periodontal medicine in preventive medicine[END_REF][START_REF] Kumar | From focal sepsis to periodontal medicine: a century of exploring the role of the oral microbiome in systemic disease[END_REF]. Recently, many animal and human studies have linked periodontitis with non-alcoholic fatty liver disease (NAFLD) (Akinkugbe et al., 2017b[START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF][START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF]. NAFLD is a condition in which ≥5-10% of hepatocytes reveal macroscopic steatosis upon light microscope examination in the absence of other etiologic risk factors of liver disease. It represents a spectrum of liver disease ranging from simple steatosis (non-alcoholic fatty liver) to non-alcoholic steatohepatitis [START_REF] Neuschwander-Tetri | Nonalcoholic steatohepatitis: summary of an AASLD Single Topic Conference[END_REF]. The former shows a non-progressive clinical course, while the latter is a more serious form of NAFLD. Being associated with features of the metabolic syndrome and has a similar global prevalence of 25% [START_REF] Araujo | Global epidemiology of non-alcoholic fatty liver disease/non-alcoholic steatohepatitis: What we need in the future[END_REF], NAFLD predisposes individuals to type 2 diabetes and cardiovascular diseases [START_REF] Anstee | Progression of NAFLD to diabetes mellitus, cardiovascular disease or cirrhosis[END_REF]. Additionally, untreated NAFLD predispose to liver cirrhosis and hepatocellular carcinoma and the subsequent need for liver transplantation [START_REF] Ponziani | Hepatocellular Carcinoma is Associated with Gut Microbiota Profile and Inflammation in Non-Alcoholic Fatty Liver Disease[END_REF]. Many risk factors for the development of NAFLD have been reported including obesity, diabetes, insulin resistance, oxidative stress and inflammation [1][2][3][4][5][6]from Yoneda]. Recent studies, as noted above, have also reported a link between periodontitis and NAFLD.

Periodontitis and NAFLD are assumed to share (associated with) many common risk factors: namely obesity, diabetes, and insulin resistance. Owing to its subclinical and persistent nature, periodontitis is considered a mild chronic disease with systemic effects. In addition to the elevated blood levels of many inflammatory markers in periodontitis patients [START_REF] Loos | Elevation of systemic markers related to cardiovascular diseases in the peripheral blood of periodontitis patients[END_REF], which have been accused to predispose to many systemic manifestations including NAFLD , periopathogenes injection to mice have been reported to be a risk factor for NAFLD [START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF]. A number of epidemiological and clinical studies have recently evaluated the association between periodontitis and NAFLD (Included studies). However, the published studies on potential association between periodontitis and NAFLD are scarce, of different designs and with conflicting results. Therefore, the present systematic review was conducted to answer the following question: Is periodontitis a risk factor for non-alcoholic fatty liver disease?

Materials and methods:

Focused question:

This systematic review was conducted following the Preferred Reporting Items for Systematic Review and Meta-Analysis (PRISMA) guidelines [START_REF] Moher | Preferred reporting items for systematic review and meta-analysis protocols (PRISMA-P) 2015 statement[END_REF]. The PECO research question that guided this review was: Is chronic periodontitis a risk factor NAFLD?

Eligibility criteria:

The eligibility criteria were as follows: clinical observational studies, cross-sectional studies, case-control, cohort studies, case series or case reports that assessed the relationship between periodontitis and NAFLD in humans. Exclusion criteria were in-vitro studies, pure animal studies, review papers, monographs, conference papers, letters to editors, unpublished data, and studies published in a language other than English.

Literature search:

A literature search was conducted through PubMed/Medline, Scopus, Embase, and Web of Science (ISI) databases to identify all relevant articles published in English from date of inception up to May 30, 2018, using the following keywords in different combination: periodontal disease; periodontitis; periodontal health; gingival health; periodontal pathogens; non-alcoholic fatty liver diseases (NAFLD); and hepatic steatosis.

Titles and abstracts of retrieved articles were screened by two authors (SA and MA) and irrelevant studies were excluded. Full texts of articles obtained from the previous step were read and evaluated independently by the two authors for inclusion. Moreover, the reference lists of included articles were manually searched for additional studies.

Our initial aim was to conduct meta-analysis but due to marked heterogeneity and inconsistency of data among the included studies, no statistical analysis was performed.

Assessment of quality:

Critical appraisal of the included studies was performed by two independent authors. Criteria for assessing the quality of studies were adapted from the Strengthening the Reporting of Observational studies in Epidemiology Statement (STROBE). Seven criteria were considered the most important ones in context of this review and were included in the checklist [START_REF] Vandenbroucke | Strengthening the Reporting of Observational Studies in Epidemiology (STROBE): explanation and elaboration[END_REF].

The STROBE checklist comprised the following: whether the study design is clearly stated, study participants are fully described, sample size is justified, variables are clearly defined, potential confounders are addressed, outcomes are accurately measured and appropriate statistical analysis tests were used. Each criterion was given a response of either "Yes" or "NO". Each study could have a maximum score of 7. After the scores were summed, the methodological quality was graded as low (0-3), acceptable (4-5), and high (6-7).

Results

:

Search Results

The PRISMA flowchart for identification of relevant studies is presented in figure1. The first electronic and manual literature search retrieved a total of 137 studies collected from various databases (Embase = 23, Web of Science = 28, PubMed = 65, and Scopus = 21). After removal the duplicate studies, 57 studies were screened through their titles and abstracts, of which 34 studies were found irrelevant and thus excluded. Only 23 publications were eligible for full text evaluation. Of these, 12 studies [START_REF] Ahmad | Association of periodontal status with liver abnormalities and metabolic syndrome[END_REF][START_REF] Furusho | Dental infection of Porphyromonas gingivalis exacerbates high fat diet-induced steatohepatitis in mice[END_REF][START_REF] Furuta | Relationship between periodontitis and hepatic abnormalities in young adults[END_REF][START_REF] Hyvarinen | Chlamydial and periodontal pathogens induce hepatic inflammation and fatty acid imbalance in apolipoprotein E-deficient mice[END_REF][START_REF] Kuraji | Porphyromonas gingivalis induced periodontitis exacerbates progression of non-alcoholic steatohepatitis in rats[END_REF][START_REF] Lins | Oral health profile of cirrhotic patients awaiting liver transplantation in the Brazilian Northeast[END_REF][START_REF] Nagao | Association of Periodontal Diseases and Liver Fibrosis in Patients With HCV and/or HBV infection[END_REF][START_REF] Saito | Relationship between periodontitis and hepatic condition in Japanese women[END_REF][START_REF] Takamura | Porphyromonas gingivalis attenuates the insulin-induced phosphorylation and translocation of forkhead box protein O1 in human hepatocytes[END_REF][START_REF] Tomofuji | Effects of improvement in periodontal inflammation by toothbrushing on serum lipopolysaccharide concentration and liver injury in rats[END_REF][START_REF] Vasconcelos | Decrease of Pericytes is Associated With Liver Disease Caused by Ligature-Induced Periodontitis in Rats[END_REF][START_REF] Widita | Relationship between clinical periodontal parameters and changes in liver enzymes levels over an 8-year period in an elderly Japanese population[END_REF] have been excluded for some reasons( Table1). One publications (Alazawi et al) reported on two independent studies, both of which were eligible for inclusions. Eventually, 11 publications (Akinkugbe et al., 2017a[START_REF] Akinkugbe | Periodontitis and prevalence of elevated aminotransferases in the hispanic community health study/study of latinos[END_REF], Akinkugbe et al., 2017b[START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF][START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF][START_REF] Morita | Serum γ-glutamyltransferase level is associated with periodontal disease independent of drinking habits in Japanese adults[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF] comprising 12 studies were eligible for inclusion in this systematic review and processed for data extraction.

General characteristics:

Out of the 137 retrieved studies, 12 studies obtained from nine publications involving 53,384 patients were included in the present systematic review (Akinkugbe et al., 2017a[START_REF] Akinkugbe | Periodontitis and prevalence of elevated aminotransferases in the hispanic community health study/study of latinos[END_REF], Akinkugbe et al., 2017b[START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF][START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF][START_REF] Morita | Serum γ-glutamyltransferase level is associated with periodontal disease independent of drinking habits in Japanese adults[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF]. With respect to study design, nine studies were crosssectional (Akinkugbe et al., 2017a[START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF][START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF][START_REF] Morita | Serum γ-glutamyltransferase level is associated with periodontal disease independent of drinking habits in Japanese adults[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF], one prospective cohort (Akinkugbe et al., 2017b), one retrospective observational [START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF] and one was a case report [START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF]. Six studies were conducted in Japan [START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF][START_REF] Morita | Serum γ-glutamyltransferase level is associated with periodontal disease independent of drinking habits in Japanese adults[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF], two in the USA [START_REF] Akinkugbe | Periodontitis and prevalence of elevated aminotransferases in the hispanic community health study/study of latinos[END_REF][START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF], two in Germany (Akinkugbe et al., 2017a, Akinkugbe et al., 2017b), one in China (mmmmmm) and one in England [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF]. The sample size in these studies ranged from 52 to 24,470 patients. The mean age of the participants was reported in 9 studies (refrence), which ranged from 39.2 to 51.5 years.

All studies except one [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF] reported gender of the participants (Table 1).

NAFLD parameters:

Regarding diagnosis and assessment progression NAFL condition, biochemical data were used in all included studies (all studies ), computed tomography and/or Ultrasound was used in 10 studies (Akinkugbe et al., 2017a, Akinkugbe et al., 2017b[START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF][START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF], and biopsy findings were used in only 4 studies [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF].

Periodontal parameters: Seven studies (Akinkugbe et al., 2017a[START_REF] Akinkugbe | Periodontitis and prevalence of elevated aminotransferases in the hispanic community health study/study of latinos[END_REF], Akinkugbe et al., 2017b[START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF] assessed clinical periodontal parameters that included one or more of the following: clinical attachment level (CAL), periodontal pockets (PD), bleeding upon probing, or community periodontal index (CPI). One study assessed both clinical and microbial parameters [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF]. The remaining four studies [START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF][START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF][START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF][START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF] evaluated microbial parameters: bacterial loads including prophyromonas gingivalis (P.g)using PCR [START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF]; IgG antibody titers to three periopathic bacteria including Aggregatibacter actinomycetemcomitans (Aa), Fusobacterium nucleatum (Fn), Porphyromonas gingivalis [START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF]; IgG antibody titers to P.g [START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF]; and isolation of P.g [START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF]) (Table 2).

Main outcomes:

All included studies except one [START_REF] Akinkugbe | Periodontitis and prevalence of elevated aminotransferases in the hispanic community health study/study of latinos[END_REF] revealed a significant association between clinical and/or microbial periodontal parameters and NAFLD progression.

Akinkugbe et al 2017 (Akinkugbe et al., 2017b) in their large population-based prospective cohort study investigating the epidemiological association of periodontitis and the incidence of NAFLD among 2,623participants from Pomerania, Germany found a significant association between periodontitis and NAFLD. The findings showed a significantly higher incidence of NAFLD in subjects with moderate and severe CAL periodontitis compared to periodontally healthy participants even after adjusting for potential confounders. The authors observed a dose-response relationship in the incidence of NAFLD in relation to severity of CAL periodontitis. Similar results were seen in PD periodontitis but with no dose-response relationship. The investigators concluded that periodontitis might be an independent risk factor for NAFLD. In another cross-sectional study by Akinkugbe et al 2017 (Akinkugbe et al., 2017a) evaluating the role of Genetic Markers of Inflammation (serum C-reactive protein (CRP) in modifying the Relationship between periodontitis and NAFLD, the findings revealed a significant positive association between periodontitis and the prevalence odds of NAFLD, and such relationship was modified by serum CRP levels.

In one cross-sectional study (reference) investigating the relationship between periodontitis and NAFLD among Japanase adults found that periodontitis was significantly associated with NAFLD. The authors concluded that PD > 4mm could be a risk factor to NAFLD.

Another cross-sectional study (Reference) in China (n=24,470) found that number of missing teeth was significantly associated with presence of NAFLD.

Furthermore, ALazawi [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF] and colleagues conducted two separate studies to ascertain the association between periodontitis and NAFLD: one population-based survey [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF] that based on data from NHANES III participants (n=8,172); and another clinical hospital-based study involving biopsy-proven NAFLD patients [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF]. In the epidemiological based study, authors found a significant correlation between periodontitis (CAL and PD) and NAFLD even after adjustment for potential confounders. Moreover, the authors found some significant correlation between hepatic steatosis and serum antibodies to certain oral pathogens especially S. noxia and S. oralis.

Similar association was replicated in the hospital-based study, with stronger association was noted in patients with advanced liver fibrosis. Additionally, Morita et al 2014 [START_REF] Morita | Serum γ-glutamyltransferase level is associated with periodontal disease independent of drinking habits in Japanese adults[END_REF] observed that ALT and GGT levels were higher in patients with periodontal pockets depth (PPD ≥ 4 mm) compared to healthy controls. [START_REF] Alazawi | Periodontitis is with significant hepatic fibrosis in patients with non-alcoholic fatty liver disease[END_REF] .

However, in a large-scale cross-sectional study conducted by Akinkugbe et al 2018 [START_REF] Akinkugbe | Periodontitis and prevalence of elevated aminotransferases in the hispanic community health study/study of latinos[END_REF], among Hispanic community (n=11,914), the authors did not find any significant association between periodontitis and NAFLD after controlling for potential confounders.

In the four pure microbial studies, investigators found a significant association between periopathogens and NAFLD. Yoneda et al [START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF] investigated the frequency of P.g. in 150 biopsy-proven NAFLD patients and 60 healthy controls using PCR, and found a significant higher frequency of P.g in NAFLD patients than healthy controls (46.7% vs. 21.7%, odds ratio: 3.16). The authors also noted that the non-surgical periodontal treatment carried out on 10 NAFLD patients for 3 months has improved the liver function parameters. The authors concluded that P.g is a risk factor for progression of NAFLD [START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF]. Similarly, Nakahara et al [START_REF] Nakahara | Involvement of Porphyromonas gingivalis in the progression of nonalcoholic fatty liver disease[END_REF] reported a significant correlation between the progression of NAFLD and serum antibodies titre against P.g ().Komazaki et al [START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF] also reported a significant correlation between the progression of NAFLD and serum antibodies titre against Aa and Fn [START_REF] Komazaki | Periodontal pathogenic bacteria, Aggregatibacter actinomycetemcomitans affect non-alcoholic fatty liver disease by altering gut microbiota and glucose metabolism[END_REF].

Omura et al [START_REF] Omura | Morbidly obese patient with non-alcoholic steatohepatitisrelated cirrhosis who died from sepsis caused by dental infection of Porphyromonas gingivalis: A case report[END_REF] reported a 54 year-old Japanese female with nonalcoholic liver cihrosis who died of sepsis caused by a periodontal pathogen . Computed tomography revealed that the patient had liver cirrhosis with massive ascites, and oral examination revealed severe periodontitis with tendency for bleeding. The patient had no history of drinking habits. The oral flora Parvimonus micra, a periodontal pathogen, was identified in her blood. The autopsy findings revealed nonalcoholic fibrosis of the liver, and on immunohistochemical staining, P.g was detected in the hepatocytes.

Discussion:

Findings from animal and human studies have suggested a relationship between periodontal diseases and NAFLD. However, no attempt has been made to systematically review the available evidence on the association between these two chronic health conditions. Hence, the present systematic review was envisioned to address a focused research question, "Is periodontal disease a risk factor for NAFLD?".

The literature search identified 12 eligible studies that were reviewed and qualitatively analyzed. All of these studies reported on clinical and/or microbiological periodontal parameters in relation to NAFLD.

Interestingly, all studies except one (Akinkugbe et al 2018) found a statistically significant association between periodontal parameters and NAFLD, and further portrayed periodontitis as an independent risk factor for NAFLD progression. Nevertheless, it is pertinent to mention that due to relatively low number of included studies especially cohort studies and the substantial heterogeneity among these studies, the findings of the present review should be interpreted with caution.

Periodontal diseases have already been linked to numerous systemic disorders including diabetes mellitus, hypertension, coronary heart diseases, stroke, and metabolic syndrome, demonstrating the impact of periodontal inflammation at systemic level. The exact mechanism by which periodontitis is linked to NAFLD is still unclear. One possible biological explanation is the low-grade inflammatory nature of periodontitis that elicits systemic inflammation. Low-grade (Chronic) inflammation is pivotal to pathogenesis of obesity-related insulin resistance, a precursor to NAFLD, which in turn contributes to the initiation and progression of NAFLD (Hotamisligil, 2006). The elevated lipopolysachraides and proinflammatory cytokines initiated by P.g, a gram negative periopathogen, have been reported to induce and worsen insulin resistance (Santos Tunes et al 2010) (Panormia study ), a condition involved in the pathogenesis of NAFLD (Alam et al., 2016).

Another possible biological explanation is the alteration in gut microbial composition resulting from the swallowed periodontal pathogens especially P. gingivalis (Arimatsu et al 2014) . The P.g.-induced endotoxins and lipopolysaccharides stimulate hepatocytes and trigger production of cytokines and reactive oxygen species, eventually leading to liver injury and initiation of NAFLD (demmer et al 2016). P.gingivalis, a gram negative bacteria, is the main causative agent for periodontitis. Interestingly, all included microbial studies found a significant association between periodontal pathogens especially P. gingivalis and NAFLD progression (Microbial). Yoneda et al found a significant higher frequency of P.g in NAFLD patients than healthy controls (46.7% vs. 21.7%, odds ratio: 3.16). Moreover, the same authors noted that the infection of type II P. gingivalis on NAFLD mouse model accelerated the NAFLD progression. Another clinical study (Nakhara et al ) also revealed a significant correlation between the progression of NAFLD and serum antibodies titre against P.gingivalis . Also Omura et al., reported isolation of p.g from hepatocytes of a 54 -year-old NAFLD female patient who died of acute sepsis.

Our findings support previous reports that linked P.gingivalis to several systemic diseases such as ….. Periodontal treatments aim to eliminate the inflammation and reduce the bacterial load. Among the included studies, only one study [START_REF] Yoneda | Involvement of a periodontal pathogen, Porphyromonas gingivalis on the pathogenesis of nonalcoholic fatty liver disease[END_REF] investigated the effect of periodontal treatment on NAFLD. The authors observed a significant improvement in liver function parameters (ALT and AST) after successful non-surgical periodontal treatment. These findings suggest that clinical periodontal improvement could have supportive effects in ameliorating severity of NAFLD, and also strengthens the correlation between periodontal diseases and NAFLD.

Biopsy is the gold standard for diagnosis NAFLD. However, ultrasonography is more appropriate for population-based studies (Qaio et al 2018). In our review, although most of the included studies used imaging means and/or biopsy to characterize NAFLD, two studies (Akinkugbe 2018; Morita et al) relied only on biochemical data such as ALT and AST to identify NAFLD. ALT and AST, serum aminotransminase, are non-specific markers of hepatocytes injury and are not always elevated in NAFLD.

Thus, the incidence of outcome (NAFLD) might have not been precisely estimated in the latter two studies, and so the results might have been biased.

Although the available evidence from this review supports the association between periodontitis and NAFLDs, there are several limitations worth considering. One main limitation is the relative small number of studies included, and the great heterogeneity among these studies; therefore no meta-analysis was attempted to quantify the magnitude of the association. Another important limitation is related to the study designs of the included studies, with most of the studies were cross-sectional and only one prospective cohort study, and hence it is difficult to infer causality. Another shortcoming is related to generalization of the results as half of the reported studies were conducted in one country (Japan), and therefore, it is hard to extrapolate the findings to other populations. Further, the marked variability in methods of characterizing both the exposure (periodontal parameters) and the outcome (diagnosis of NAFLD) is an obvious limitation of the present review. However, despite these limitations, this review has several strengths. First, this is the first systematic review to summarize the current evidence on the associations between periodontitis and NAFLD. Second, the study is strengthened by the relative large population size (53,384) and by including both clinical and microbial studies that evaluated the relationship between periodontitis and NAFLD.

In conclusion, the available evidence suggests that periodontitis may be associated with NAFLD.

However, due to limited number of prospective cohorts studies and heterogeneity among these studies, further large-scale prospective cohort studies with long follow up periods targeting different ethnic populations are highly recommended.

Compliance with ethical standards:

Conflict of interests: The authors declare that they have no conflict of interest.

Ethical approval: This is a review article and so no ethical approval is required.

Funding

: We did not receive any external funding Table1: List of excluded studies

	Study	Reason for exclusion
	Hyvärinen et al (2009)	animal study
	Widita et al (2017)	Non-NAFLD
	Ahmad et al (2015)	Non-NAFLD
	Furuta et al (2010)	Non-NAFLD
	Saito et al (2006)	Non-NAFLD
	Nagao et al (2014)	Non-NAFLD
	Takamura et al (2016)	animal study
	Tomofuji et al (2009)	animal study
	Vasconcelos et al(2017) animal study
	Furusho et al (2013)	animal study
	Lins et al (2011)	Non-NAFLD
	Kuraji et al (2016)	animal study

Table 2 : General characteristics of the included studies

 2

	Author and	Study	No	Gender	Mean	NAFLD diagn	Periodontal	Main outcome
	country	design	Participants		age/	Parameters	parameters
					range
	Akinkugbe 2018	Cross-	11,914	M: 45.1%	40.4	-ALT	-CAL & PD	Marginal to null association
	USA	sectional	NAFLD:18.1%	F:54.9%		-AST		between periodontitis and
						-FLI		NAFLD.
	Akinkugbe et al	Prospective	2,623	M: 1,074	46	-Ultasound	-CAL, PD	periodontitis is an
	2017	Cohort		F: 1549		-ALT level		independent risk factor for
	Germany							NAFLD
	Akinkugbe et al	Cross-	2481	M: 1,116	47	-Ultrasound	-CAL, PD	Periodontitis was associated
	2017 Germany	sectional	NAFLD: 654	F: 1365			-CRP	with higher prevalence odds
								of NAFLD and this relation
								was modified by serum CRP
								levels.
	Alazawi et al	population-	8,172	M: 3796	20-74	-Ultrasound	-PPD	NAFLD is associated with
	2017 USA	based survey		F: 4376		-Biochemical	-CAL	periodontitis especially with
							-bleeding	advanced liver fibrosis. Also,
							-Microbial	steatosis was associated with
								serum antibodies of
								periodontal bacteria.
	Alazawi et al	cross-	69 NAFLD	NA	> 18 ys	-Biopsy-proved	PD	periodontitis correlated
	2017	sectional				-Ultrasound		significantly with stage of
	England					-Histological		liver disease,
	Iwasaki 2018	Cross-	1226	M: 772	50	Ultrasound	PD	Significant association
	Japan	sectional		F: 454		Biochemical		between periodontitis and
								NAFLD
	Qiao et al 2018	Cross-	24,470	M: 49.9%	39.2	Ultrasound	missing teeth	missing teeth were associated
	China	sectional		F: 50.1%				with higher presence of
								NAFLD in males.
	Morita et al2014	Cross-	1510	M: 1218	50.4	ALT	CPI	Periodontal disease is
	Japan	sectional		F: 292		GGT		associated with liver function
	Yoneda et al	Comparative	150 NAFLD	M: 64	54.6	-Histological	-PD	Significantly higher
	2012	cross-	60 Controls	F: 86		-Ultrasound	-Microbial	Frequency of P. gingivalis in
		sectional				-ALT, ASP,	-PCR assay	NAFLD/ NASH patients than
						GGT		controls.
	Komazaki et al	Cross-	52 NAFLD	M:27	55	-CT	-Microbial	Aa and Fn antibodies titres
	2018	sectional		F: 25		-Biochemical		were significantly correlated
	Japan					tests		with NAFLD.
	Nakahara et al	Retrospective	200 NAFLD	M: 94	51.5	-Biochemical	-Microbial	A significant correlation
	2018	observational		F: 106		-Histological		between fibrosis progression
	Japan					-CT		& anti-P.g antibody titers
	Omura et al 2015	Case report	1	-	54	Blood analysis	P. gingivalis isolation P. gingivalis was detected in
	Japan					CT		the hepatocytes of NASH
						autopsy		patient
	CRP: C-reactive protein; CPI: Community Periodontal Index

Data extraction:

The following data were extracted by two independent authors using a standard data collection form: authors and year of study; country; study design; age and gender of patients; number of patients; clinical and microbial periodontal parameters, NAFLD parameters, and main outcomes.


F Meducin

H Zanni

C Noik

G Hamel

B Bresson

Tricalcium silicate (C 3 S) hydration under high pressure at ambient and high temperature (200 °C)

Keywords: Hydration, Temperature, Oil well cement, C 3 S, Pressure

The hydration of a tricalcium silicate paste at ambient temperature and at 200 °C under high pressure (up to 1000 bar) has been studied. Two high pressure cells have been used, one allows in-situ electrical conductivity measurements during hydration under high pressure. The hydration products were characterized by thermal analysis, X-ray diffraction and 29 Si NMR measurements. The pressure has a large kinetic effect on the hydration of a C 3 S paste at room temperature. The pressure was seen to affect drastically the hydration of a C 3 S paste at 200 °C and this study evidences the competition between the different high temperature phases during the hydration.

Introduction

The hydration of a cement paste at high temperature and high pressure is of great importance in determining the durability of oil wells. Many studies have been carried out in hydrothermal conditions or have been carried out on model systems at high temperature [START_REF] Buckle | The hydration of tricalcium and dicalcium silicates in pastes under normal and steam curing conditions[END_REF][START_REF] Odler | Hydration of tricalcium silicate at elevated temperature[END_REF][START_REF] Masse | High temperature hydration of tricalcium silicate, the major component of portland cement: a 29 Si NMR contribution[END_REF][START_REF] Mitsuda | Characterisation of hydrothermally formed C-S-H[END_REF][START_REF] Hong | Phase relations in the CaO-SiO2-H2O system to 200 °C at saturated steam pressure[END_REF][START_REF] Luke | Phase studies of pozzolanic stabilized calcium silicates hydrates at 180 °C[END_REF][START_REF] Colston | An in situ synchrotron energydispersive diffraction study of the hydration of oilwell cement systems under high temperature/autoclave conditions up to 130 °C[END_REF] without additional hydrostatic pressure. And the studies under high pressure concern mainly thermodynamic equilibrium between high temperature and high pressure phases in the ternary H 2 O-CaO-SiO 2 system [START_REF] Harker | Dehydration series in the system CaSiO 3 -SiO 2 -H 2 O[END_REF][START_REF] Pistorius | Thermal decomposition of portlandite and xonotlite to high pressures and high temperatures[END_REF]. However, those results cannot be easily transposed to the hydration of a cement paste at high temperature under high pressure for two reasons. The first is that the behavior of a hydrating cement paste under high pressure and high temperature is very far from the behavior of a system in equilibrium and the second is that the combined effect of the pressure and the temperature produces dramatic changes in the hydration of the paste. This combined effect can be missing in hydrothermal conditions. This paper focuses on the modification induced by high pressure at room and at high temperature (200 °C) on a tricalcium silicate paste hydration (C 3 S in cement chemist's notation: C=CaO, S=SiO 2 , H=H 2 O). The C 3 S is the main component of Portland cement of which oil well cement is composed. The high pressure cells have been specially designed for this study: an original result has been obtained thanks to a high pressure cell which allows in-situ electrical conductivity measurement under high pressure. Under normal conditions of temperature and pressure, C 3 S paste hydration yields to the poorly crystallized and non-stoichiometric C-S-H and Portlandite [START_REF] Taylor | Cement chemistry[END_REF]. At higher temperatures, it has been observed that C-S-H is no longer stable and is replaced by more stable crystallized products [START_REF] Buckle | The hydration of tricalcium and dicalcium silicates in pastes under normal and steam curing conditions[END_REF], the composition and proportions of which depend on the initial conditions (Ca/Si ratio, water/solid ratio). The effect of the high pressure at room temperature has been studied through the measurement of electrical conductivity, an appropriate tool to investigate kinetic processes in these conditions. For the high temperature measurements, single pulse 29 Si NMR experiments, thermal analysis and X-ray diffraction will allow to show the drastic effect of the pressure on the hydration of a C 3 S paste at high temperature.

Experimental

Materials and synthesis

Anhydrous C 3 S was obtained from Ciments Français-Italcementi. Samples for the hydration studies were prepared by mixing C 3 S with deionized and boiled water with a water to solid weight ratio of 0.44. The paste was transferred to a Teflon® container, and placed in the high temperature and high pressure cell. The samples were hydrated for a period extending typically from a few hours to a few days under pressure (Argon gas). Once removed from the cell, further hydration was inhibited by acetone-ether drying before the samples were studied by thermal analysis, X-ray diffraction and 29 Si NMR.

High temperature and high pressure cell description

Two cells were specially designed for high pressure and high temperature hydration. The first cell allowed the in-situ measurement of electrical conductivity during the hydration of the cement at temperatures from ambient to 160 °C and under pressure of 1 to 1000 bar. The body of the cell consisted of a steel cylinder of diameter 65 mm and length 130 mm in which a Teflon® pot containing the hydrating cement paste (typically a few cm 3 ) could be located. A Bridgeman cap-screw-seal system provided a hermetical seal. Two isolated wires (electrodes) were passed through the cap and embedded in the paste during hydration to allow measurement of the electrical conductivity.

Similarly, a thermocouple was also introduced through the cell cap so that the temperature of the paste could also be determined. The cell was connected to a gas compressor providing the required static argon pressure for the hydration experiment. Sample heating was achieved by the addition of a heating collar around the cell connected to a temperature regulator. Conductivity measurement could then be performed with a conductimeter operating with a 16 kHz sampling rate.

A second cell was especially designed to permit longer duration hydration experiments (up to a few days) at high pressure (1000 bar) and high temperature (200 °C). Electrodes were excluded from this design to minimize heat loss and irregularities in the sample environment although a thermocouple was again introduced for temperature control. The system used for pressure and temperature generation was the same as that used with the first cell.

In the operation of both cells, the desired sample conditions were achieved in a three-step process. First the cell was pressurized to half that required. The cell was then heated to the required temperature. Finally, the pressure was raised to the desired value. The time required to reach the target temperature and pressure up to 200 °C and 1000 bar was typically half an hour.

Characterization techniques

Thermal analyses were performed, with a thermobalance TAG24 SETARAM. Samples were submitted to temperatures ranging from 20 °C to 1050 °C with a 10 °C/min ramp rate under ambient pressure during which, the weight loss was recorded. XRD experiments were performed with powdered samples on a Philips PW 1820 device using the Cu Kα radiation. The diffraction patterns were performed in the range 2θ = 10 to 80°w ith a step of 0.03°and 4 s per step. 29 Si NMR experiments were carried out on a Bruker ASX 500 MHz spectrometer operating at 12 T. The spectra were single pulse experiments, with a recycle delay of 60 s. The sample was spun at 5 kHz in a Bruker 7 mm Bruker Magic Angle Spinning probe. The pulse duration was 4 μs.

Results and discussion

3.1. Effect of high pressure at room temperature Fig. 1 shows the conductivity signal of a hydrating C 3 S paste obtained at room temperature under two isostatic pressures (300 bar and 850 bar). The measured conductivity was observed to be dependent upon the conductivity of the solution inside the porous C-S-H and on the texture of this porous material [START_REF] Vernet | Conductimetric test for cement-admixture systems[END_REF][START_REF] Christensen | Influence of silica fume on the early hydration of portland cements using impedance spectroscopy[END_REF]. It is known that the conductivity of the interstitial solution is proportional to the Ca 2+ concentration resulting from the dissolution of C 3 S. Thus, the conductivity versus hydration time curve of a hydrating C 3 S paste (in normal conditions of temperature and pressure) can be roughly divided in two steps. The first step, observed as an increase in the conductivity, corresponds to C 3 S dissolution. The maximum of conductivity is reached just before the precipitation of Portlandite CH when the solution is over-saturated in Ca 2+ ions with respect to this phase. Portlandite precipitation yields to a decrease of conductivity of the hydrating paste, the time constant of this process being very short (∼ several minutes). The second step corresponds to the slow decrease of conductivity, the time constant of this process being several hours. The growth of C-S-H leads to the progressive disconnection of pores. In this second step by comparison to the first one, Ca 2+ concentration in the pores is fixed and equal to an equilibrium concentration between CH and CSH. It is clearly apparent from Fig. 1 that a higher isostatic pressure increases the rate of dissolution of C 3 S. For comparison, the maximum conductivity under the ambient conditions is attained after more than 4 h. The precipitation of CH occurs earlier under high pressure and the porous network closes rapidly. The pressure effect at room temperature is a kinetic effect. This is confirmed by NMR spectroscopy for longer hydration times. 29 Si NMR allows characterisation of poorly crystallized phases C-S-H with a structure based on SiO 4 chains with variable length. 29 Si single pulse NMR spectra were analyzed using the Q n classification [START_REF] Engelhardt | High Resolution 29 Si NMR of Silicates in Zeolites[END_REF], where Q stands for SiO 4 tetrahedron and n refers to the number of tetrahedra linked by oxygen bonds to the previous tetrahedron. In Fig. 2, it can be seen that the hydration of C 3 S under high pressure and room temperature shows that anhydrous C 3 S (Q 0 species between -65 ppm and -75 ppm) disappears more rapidly under higher pressure. Q 2 is a middle chain site (linked to two tetrahedra), Q 1 is an end-chain site and Q 0 is a monomeric site. The peaks located around -80 ppm and -85 ppm correspond to Q 1 and Q 2 entities characteristic of C-S-H [START_REF] Cong | O-17 and Si-29 MAS NMR-study of beta-C2S hydration and the structure of calcium-silicate hydrates[END_REF]. This figure also shows that the same hydrates are formed under pressure as those formed in normal conditions: average short chains composed by three silica tetrahedra (deduced from Q 2 /Q 1 ratio). The presence of C-S-H is also evidenced by the loss of water found below 150 °C during thermal analysis [START_REF] Taylor | Cement chemistry[END_REF] and the presence of a shoulder in the background of the X-ray diffraction pattern at around 2θ = 35°(Kα, Cu).

Combined effect of high pressure and high temperature

It is known that the C-S-H phase ceases to be stable above 120 °C and is replaced by crystallized phases [START_REF] Buckle | The hydration of tricalcium and dicalcium silicates in pastes under normal and steam curing conditions[END_REF][START_REF] Colston | An in situ synchrotron energydispersive diffraction study of the hydration of oilwell cement systems under high temperature/autoclave conditions up to 130 °C[END_REF][START_REF] Bresson | Hydration of tricalcium silicate (C 3 S) at high temperature and high pressure[END_REF]. The hydration of the C 3 S paste has been studied at 200 °C under two pressures over times between 2 days and 16 days. The weight proportion (%) of each phase for different hydration times and for two pressures is displayed in Table 1 (2, 4 and 6 days at 600 bar, and 4 and 16 days at 400 bar). The following phases were identified [START_REF] Hansen | Si-29 chemical shift anisotropies in calcium silicates from high-field Si-29 MAS NMR spectroscopy[END_REF]: Jaffeite Ca 6 (Si 2 O 7 )(OH) 6 (tricalcium silicate hydrate), Hillebrandite Ca 2 (SiO 3 )(OH) 2 (β-dicalcium silicate hydrate), α-C 2 SH Ca 2 (SiO 3 )(OH) 2 (α-dicalcium silicate hydrate) and Portlandite Ca(OH) 2 (CH) according to the literature in the same conditions of temperature. All the phases observed in these experiments were crystallized phases. The combination of thermal analysis and X-ray diffraction allowed quantification of the phases. These results were compared with 29 Si single pulse NMR experiments in the case of the hydration of C 3 S at 600 bar (Fig. 3). The 29 Si NMR experiments performed on these samples are semi-quantitative. Indeed, the α-C 2 SH-Q 0 species seemed to be underestimated by NMR. This could be due to the relaxation time T 1 which can be too low. It is known that the 29 Si relaxation time T 1 of some calcium silicates can be very long (several hundreds of seconds) so that a quantitative 29 Si NMR would require at the same time a large number of scans necessary to give a good signal to noise ratio and a very long delay between two consecutive scans. In our case, the NMR was not used to quantify the phases because the quantification is ensured by other methods. However, the NMR results confirm the relative evolution of each phase. The peaks at -82.9 (Q 1 ) and -86.0 ppm (Q 2 ) are related to Jaffeite and Hillebrandite respectively. The thermal analysis shows the absence of the loss of water before 150 °C confirming the absence of the C-S-H. The role of pressure on the evolution of phases during hydration can be readily observed from the comparison of samples hydrated at 200 °C under 400 bar and 600 bar. After 2 days, the hydration is complete (C 3 S has completely disappeared) but the sample is still evolving through dissolution-precipitation mechanisms. The competition between the three high temperature phases is clearly shown on Fig. 3 and in Table 1. The first observation is that Hillebrandite and Jaffeite amounts always increase with time of hydration.

The second observation is that the α-C 2 SH is mainly formed at the beginning of hydration and is favoured by high pressure.

Discussion

The phases appearing at 200 °C and their evolution are greatly modified by the pressure. After a first step of the hydration (in our case at least after 2 days), the C 3 S is fully hydrated but the sample continues to evolve because the free water (limiting factor of these reactions) is still available to allow the dissolution-precipitation mechanism. It has to be noticed that even if it is not observed in this study, the formation of C-S-H can occur before the formation of the crystallized phases, this preliminary step has been reported in the literature in hydrothermal conditions even at high temperature [START_REF] Buckle | The hydration of tricalcium and dicalcium silicates in pastes under normal and steam curing conditions[END_REF]. In our case, under high pressure, this step occurs too rapidly to be observed in the time scale of these experiments. The amount of Jaffeite is always increasing with hydration time whatever the pressure. It is explained by the fact that it is a stable phase at this temperature and it has the same C/S ratio as in the solution (C/S equal to 3) and as the initial product (C 3 S). This result matches those obtained at a lower temperature [START_REF] Bresson | Hydration of tricalcium silicate (C 3 S) at high temperature and high pressure[END_REF] (160 °C and 120 °C) where it was found that under hydrothermal conditions, Jaffeite was always readily formed for an initial C/S ratio equal to 3. The Hillebrandite also appears along with the Jaffeite and its amount also always increases with hydration time. This is supported by the literature dealing with equilibria at high temperature [START_REF] Hong | Phase relations in the CaO-SiO2-H2O system to 200 °C at saturated steam pressure[END_REF][START_REF] Harker | Dehydration series in the system CaSiO 3 -SiO 2 -H 2 O[END_REF] showing that Hillebrandite is in equilibrium with Jaffeite. It is also known that another metastable phase can appear [START_REF] Buckle | The hydration of tricalcium and dicalcium silicates in pastes under normal and steam curing conditions[END_REF][START_REF] Luke | Phase studies of pozzolanic stabilized calcium silicates hydrates at 180 °C[END_REF][START_REF] Colston | An in situ synchrotron energydispersive diffraction study of the hydration of oilwell cement systems under high temperature/autoclave conditions up to 130 °C[END_REF] when a C 3 S paste is hydrated between 130 °C and 200 °C known as α-C 2 SH. It has been clearly demonstrated [START_REF] Colston | An in situ synchrotron energydispersive diffraction study of the hydration of oilwell cement systems under high temperature/autoclave conditions up to 130 °C[END_REF] that the α-C 2 SH phase appears and develops after 200 min in a hydrating cement paste at 130 °C. Other studies [START_REF] Bresson | Hydration of tricalcium silicate (C 3 S) at high temperature and high pressure[END_REF] showed that the α-C 2 SH phase is present even at 120 °C in a C 3 S hydrating paste with silica flour under high pressure (400 bar). At 160 °C, the pressure highly favours this phase [START_REF] Bresson | Hydration of tricalcium silicate (C 3 S) at high temperature and high pressure[END_REF]. In our case, the results illustrate the competition between the two stable phases (Hillebrandite and Jaffeite) and the metastable phase. It is clearly shown that at the first stage of the hydration, the α-C 2 SH phase is favoured by high pressure. It is progressively replaced by more stable phases with time in accordance with the literature [START_REF] Hong | Phase relations in the CaO-SiO2-H2O system to 200 °C at saturated steam pressure[END_REF]. Indeed, the α-C 2 SH phase dissolves to form a more stable hydrate (Hillebrandite) with the same C/S ratio (equal to 2) or a more stable hydrate as Jaffeite with a higher C/S ratio and with consumption of the Portlandite. It has to be noticed that in this temperature range, the pressure effect is critical in the sense that small changes of the conditions of pressure and temperature can induce great changes in the phase composition of the product after a few days. The unstable system hydrates with a competition between the two stable phases and the metastable phase favoured by the pressure. The results are clearly different from those obtained under hydrothermal conditions.

Conclusion

This study focused on the influence of high pressure on a hydrating tricalcium silicate (C 3 S) paste at room temperature and at 200 °C. At room temperature, the electrical conductivity results (for a short hydration time) as well as the 29 Si NMR results (for a longer hydration time) showed that the effect of the pressure is a kinetic effect. The hydrates formed under high pressure at room temperature are the same as under atmospheric pressure. On the contrary, at high temperature (200 °C), the pressure greatly affects the formation of the crystallized phases. The evolution occurs even after the full hydration of the C 3 S phase. First, it has been shown that the α-C 2 SH is formed at the first stage of hydration. This phase progressively disappears during hydration. It is replaced by a more stable phase with the same C/S ratio equal to 2 or by a more stable phase with a higher C/S ratio (Jaffeite C/S ratio is equal to 3), this reaction being possible with Portlandite consumption. Secondly, the α-C 2 SH phase is favoured by the hydrostatic pressure which can induce drastic changes during the hydration of an instable system at high temperature. These results obtained at 200 °C confirm precedent results obtained at lower temperature (160 °C).

Fig. 1 .

 1 Fig. 1. Comparison of the electrical conductivity obtained during the hydration of a C 3 S paste at room temperature under 300 bar and 850 bar. Both signals are normalized against the maximum conductivity peak.

Table 1

 1 Weight proportion (%) of the C 3 S hydrated samples at 200 °C under 600 bar for 2, 4 and 6 days and at 200 °C under 400 bar for 4 and 16 days

	Temperature	Pressure	Duration	Portlandite α-	Jaffeite Hillebrandite
	(°C)	(bar)	(days)		C 2 SH
	200	600	2	18	68	13	1
	200	600	4	12	45	40	3
	200	600	6	11	35	46	8
	200	400	4	15	4	36	45
	200	400	16	12	0	39	49

Fig. 3. Comparison of the 29 Si NMR single pulse spectra of a C 3 S paste hydrated at 200 °C under 600 bar for 2, 4 and 6 days.


domain: phys.cond.cm-scm phys spi

text:
Aude Belguise

Sabine Cantournet

François Lequeux

Hélène Montes

Weak nonlinearities in viscoelastic mechanical properties of polymers near their glass transition: local vs. macroscopic laws for stress-induced acceleration of the mechanical response

We focus on the role of dynamical heterogeneities on the weak mechanical nonlinearities of amorphous polymers near and above the glass transition temperature T g by combining experiments and numerical coarse-grained simulations. The acceleration of the macroscopic nonlinear modulus relaxation resulting from the applied stress is measured below yielding. As a result of dynamic disorder, the macroscopic acceleration differs from the local acceleration.

We obtain a good agreement of experimental measurements with simulations computed by using an exponential function of the square stress for the local acceleration. Further, the length scale of dynamical heterogeneities is deduced.

Introduction

Many studies have focused on upscaling the mechanical response of disordered materials as it is encountered in many different systems (granular media [1], foams [2], networks [3], and polymers [4,5]) and situations (elasticity fracture [6] and yield stress [7]). Many nonintuitive behaviours have been reported, particularly in the nonlinear regime. However, change of scale in the nonlinear viscoelasticity of amorphous polymers has not yet been completely understood, particularly near the glass transition, where the dynamical heterogeneities are completely at stake. Glassy polymers can be represented as tiled by domains of few tens of monomers whose collective configuration can be reorganised under thermal agitation [8][9][10]. Each domain has an intrinsic relaxation time that is temperature-dependent. Experiments have shown that the intrinsic relaxation times are randomly distributed over an entire system with a very wide distribution function (more than four decades) [9]. In this frame, numerical simulations showed that mechanical coupling between domains modifies the intrinsic local response of heterogeneities resulting in a complex macroscopic average of the local viscoelastic responses [11][12][13][14][15].

For a polymer glass near its glass transition temperature Tg, relaxation times vary with temperature in the linear regime and with macroscopic stress in the nonlinear regime. In the linear regime, an increase in temperature results in a decrease in the local intrinsic times: the conformational changes are favoured by the thermal energy k B T. At the macroscopic scale, the time-dependent mechanical response of glassy polymers is accelerated. For instance, after a step strain, the macroscopic stress relaxes from its glassy value to its rubber value at shorter times for increasing temperature. However, the form of the macroscopic response does not significantly vary with temperature, i.e., in a good approximation, all intrinsic relaxation times are shifted by the same factor as temperature increases. This feature is known as the timetemperature superposition law. Thus, in the linear regime, the macroscopic stress relaxation measured at two different temperatures T and T ref can be superimposed by applying a shift factor to the time scale which is equal to the factor applied to the local relaxation times. Thus, after an increase in temperature, the macroscopic and local mechanical responses are accelerated similarly.

Experiments have shown that the application of a nonlinear mechanical solicitation (strain or stress) results in an acceleration of the macroscopic response. For instance, after a step strain, the nonlinear stress relaxation of a glassy polymer is faster than that measured in the linear regime [16][17][18][19][20][21][22]. Therefore, nonlinear acceleration might be addressed with certain shift factors, as this approach is generally employed for the temperature effect. However, O'Connell et al. [21][22] have shown that nonlinear effects do not correspond to a simple shift factor applied to the time scale, as is the case for time-temperature superposition. Hence, the nonlinear acceleration of relaxation is challenging.

At the nanometric scale, plastic deformation is attributed to stress-induced molecular rearrangements [23][24][25][26] that occur by crossing energy barriers. Employing photobleaching methods, Lee et al. [27] measured the rotational correlation time as a function of the true stress for N,N0-dipentyl-3,4,9,10-perylenedicarboximide (DPPC) molecules in Polymethylmethacrylate (PMMA) samples near Tg during uniaxial creep deformation.

According to Long et al. [28], these effects can be described by multiplying the intrinsic local relaxation times with a function f of the local stress σ which is exp[-(σ/Y)²], where Y is the critical stress given by , where ξ is the size of the dynamical heterogeneities, k B is Boltzmann constant, T is the temperature, and G G is the glassy shear modulus.

In this study, we measure the macroscopic stress relaxation of cross-linked PMMA samples in the linear and weak nonlinear regimes. The experimental responses are analysed using a finite element approach that was developed in a previous study which mimics dynamical heterogeneities. In the nonlinear regime, intrinsic relaxation times characterising each heterogeneity are assumed to be proportional to the same function of the local stress according to the law proposed by Long et al. [28] To compare the local nonlinear response with the macroscopic response, we define a macroscopic acceleration function F which, is

Materials and methods

We measured the macroscopic relaxation modulus of cross-linked PMMA chains at various temperatures in the glass transition domain.

PMMA samples preparation

PMMA samples were prepared according to the method described by Casas et al. [29]. A 1 mm sheet polymer was obtained using radical polymerisation and reticulation of methylmethacrylate monomers by applying UV irradiation for 8 h. We chose diacrylate butanediol as the cross-linker. To initialise the polymerisation and reticulation, we added a photoinitiator (Irgacure (Ciba, France)) (0.1 wt% monomer). The concentration of the crosslinker was chosen equal to 0.6% per mol of methylmethacrylate monomers such that the average weight between the cross-links Mc was similar to the entanglement distance Me. (Me ≈ 8000 g/mol, corresponding to an entanglement length of approximately 6 nm). Cross-linking allowed us to successively perform step-strain linear and nonlinear tensile relaxation tests on the same sample.

Procedure for stress relaxation measurements

It is very difficult to precisely measure weak deviations to linearity below the yield point.

Hence, the measurements of the linear and nonlinear responses should be conducted on the same sample in the same setup without displacing the sample. In practice, the macroscopic stress relaxations of PMMA samples are measured in the linear and weak nonlinear regimes at different temperatures. We verified that for strain amplitudes below 0.3%, the mechanical responses are linear near Tg. Thus, we apply a step strain of 0.3% for linear measurements.

The nonlinear stress relaxations are measured in the weak nonlinear regime, i.e., below the yield point. Thus, we applied a strain with an amplitude smaller than 2%. To accurately observe the deviation of the nonlinear response with respect to the linear response, we applied a nonlinear condition strain ranging between 0.75% and 2%.

Linear and nonlinear stress relaxations were successively measured on the same sample to compare them with sufficient accuracy. Between each measurement, thermal annealing was achieved by heating the sample at 32 °C above the glass transition temperature. The sample was then cooled to the temperature of the experiment while applying zero stress control. After attaining thermal equilibrium, the relaxation of the modulus was measured. We verified that the results obtained do not vary when an ageing step is added before the measurement.

2. Modelling the mechanical response of a polymer near its glass transition temperature Amorphous polymers near their glass transition temperature are heterogeneous at the nanometric scale. To model nonlinear effects at the macroscopic scale on such a heterogeneous system, we use the 3D version of the model developed by Masurel et al. [START_REF] Belguise | Mechanics and Physics of Solids at Micro-and Nano-Scales[END_REF][START_REF] Masurel | [END_REF][32]. The space is tiled by mechanically coupled domains. The resolution is performed in 3D using the finite element method [START_REF] Belguise | Mechanics and Physics of Solids at Micro-and Nano-Scales[END_REF].

The mechanical response of each domain is given by a Zener system made of two parallel branches. One branch consists of a Maxwell branch that represents the glassy contribution to the stress with an elastic modulus E G and a relaxation time of  i . The second branch has an elastic spring of modulus E R that represents the stress contribution due to the chain entropy.

The glassy and rubber moduli are assumed to be equal for all domains. The compression modulus K is assumed to be independent of time and is set to 2 GPa for the domains. We verified that for values of K larger than 1 GPa, the results of this study are same. Each domain has its own relaxation time τ i which is randomly drawn following a log-normal probability distribution, , where τ 0 is the centre of the lognormal distribution and s is its width. τ 0 is the only time scale in the model. The value of  0 thus defines the position of the glass transition, whereas the width s of the distribution controls the width of the glass transition.

In yielding follows an extended von Mises criterion [33]. Hence, we use the equivalent stress as:

(

with the local deviatoric part of the stress and p its local pressure [START_REF] Belguise | Mechanics and Physics of Solids at Micro-and Nano-Scales[END_REF][START_REF] Masurel | [END_REF]. The coefficient α is set to 0.3 in agreement with experiments [START_REF] Ward | Mechanical Properties of Solid Polymers[END_REF].

In this study, numerical simulations were performed in 3D with systems containing 16 × 16 × 16 cubic domains which are mechanically coupled according to the finite element method.

The resolution is performed using the finite element method with the finite element code Zebulon [35,[START_REF] Ryckelynck | [END_REF]. Each domain was divided into eight quadratic c3d20 cubic elements. To avoid edge effects, periodic boundary conditions were applied. The local stress used in the acceleration function in the nonlinear case is the mean over the domain. The macroscopic stress and strain are the means over the entire system.

We computed the relaxation of the system undergoing uniaxial elongation and applying either linear or nonlinear conditions.

Experimental results

The modulus relaxation is measured in the linear regime at different temperatures by applying a step strain. The linear relaxation function of the modulus, called E L , is determined by applying the time-temperature superposition principle. The data measured at temperature T are superimposed with the data measured at the reference temperature T ref by multiplying the time scale by a factor a T/Tref . Figure 1 presents the master curve we obtained which corresponds to the linear relaxation function E L . Here, the reference temperature was chosen to be 108 °C. Consequently, we analyse the nonlinear response with respect to the linear response by comparing the nonlinear modulus relaxation to the master curve built in the linear regime at the same temperature. A comparison is thus performed for each temperature. The time scale accessible through the direct measurement of stress relaxation is significantly shorter than that related to the linear relaxation function E L which shows the complete relaxation of the modulus from its glassy value to its rubber value. Thus, the comparison of nonlinear and linear experimental data can be performed only on a limited (restricted) time range, and thus, a limited range of modulus values for each measurement temperature.

Experiments via numerical approach

We employ our numerical model to describe the linear and nonlinear responses measured at different temperatures on our PMMA samples.

Linear regime

First, we consider the experimental master curve E L built in the linear regime. The values of the glassy and rubber moduli, E G and E R , of each heterogeneity are equal to the macroscopic moduli measured on our samples, i.e., E G = 1.2 GPa and E R = 1.5 MPa. Further, for these simulations, K = 2 GPa. Consequently, the value of the Poisson coefficient  is equal to 0.4999 in the rubber state and to 0.40 in the glassy state; these values are in agreement with the experimental values of  reported for PMMA, that are ranged between 0.34 and 0.4 [START_REF] Livermore | Material Properties Database[END_REF].

The fitting of the master curve using the numerical model is performed by adjusting the values of the width s and centre  0 of the intrinsic relaxation time distribution. At the reference temperature of 108 °C, we obtained a good agreement between the simulations and experiment when s = 4,83 and  0 = 60 s with E G = 1.2 GPa, E R = 1.5 MPa, and K = 2 GPa.

By employing the values of the five parameters determined from data measured in the linear regime, we now model the nonlinear response measured on the same PMMA samples.

Nonlinear regime

We now compare the predictions of our model by applying nonlinear conditions to experimental results measured on the PMMA samples. We recall that according to our numerical approach, the intrinsic relaxation time of each heterogeneity is multiplied in the nonlinear regime by a local stress acceleration function that we assume to be equal to the function that was theoretically predicted by Long et al., i.e.:

(

where the local equivalent stress is given by equation 1.

Computations were performed using the values of E R , E G , s, and  0 obtained from the fitting of the linear response measured on our PMMA samples. The critical stress Y is thus a parameter that must be adjusted.

Using Y = 5.75 MPa, we obtained a good agreement between the simulations and nonlinear response measured at the reference temperature of 108 °C and for a strain step of 1% amplitude. The simulation and experimental data are compared in figure 2.

If Y is assumed to be independent of temperature, the model predicts a time-temperature superposition for the nonlinear response for a given strain amplitude. Because no timetemperature superposition in the nonlinear regime is experimentally observed, the value of Y must be varied with temperature. PMMA samples (see figure 2). The corresponding values of the size of heterogeneities computed following the relation suggested by Long et al. [28] are presented in the inset.

We analysed the distribution of local strain during nonlinear relaxation. In such heterogeneous systems, the strain is locally distributed with domains undergoing a larger local strain than the other. However, we observed that the local strain is always smaller than 4% for a 1% macroscopic strain applied. In the strain range studied in this study, the local stress is not modified by the limit extensibility of polymer chains; an extension of 4% at the length scale of heterogeneity (i.e., a few nanometres) is significantly smaller than the maximal extension at this scale, which is given by whereR max = bN is the contour length of a polymer chain,  is its meansquare end-to-end distance, b is the Kühn length, and N is the number of Kühn segments of a chain segment included in heterogeneity [START_REF] Rubinstein | Polymer Physics[END_REF]. For a size of heterogeneity of 3 nm, applying b = 1.7 nm for PMMA chains, the approximate value of N = (3/1.7) 2 = 3.5, leading to a maximum value of approximately  max = 76% for PMMA chains. The extension undergone by the chain segment is thus negligible at the scale of heterogeneities. The local response we assume in our model is thus valid in the weak nonlinear regime.

Our model provides a satisfactory description of macroscopic response in the linear and weak nonlinear regimes. The model provides also information on the local stress field and the local relaxation times distribution during the nonlinear macroscopic stress relaxation.

We analyse the distribution of local relaxation times that are defined as during the stress relaxation for nonlinear conditions. The evolution of the local relaxation times predicted by our numerical model for nonlinear stress relaxation agrees with observations reported by Lee et al [19] for nonlinear creep experiments on PMMA samples: they not only observed a decrease of the local mean relaxation time under stress, previously discussed by Long et al [28], but also a narrowing of the local time distribution followed by its broadening after the flow onset.

Our numerical approach shows that mechanical percolation drives the nonlinear macroscopic stress relaxation in our PMMA samples. We emphasize the main difference between stepstrain and creep experiments is that, in the first case, the mechanical percolation threshold is crossed keeping a small macroscopic strain i.e. smaller than the yield strain. In contrast in creep experiments, the strain continuously increases. Having this effect in mind, we only performed and discussed step strain experiments, in order to remain in the small strain regime, and to avoid effects due to limit extensibility. Now that we have shown that our model is able to describe the feature observed by Lee et al [19] at a microscopic scale, we turn to the comparison of behaviours of the macroscopic stress.

Comparison of macroscopic and local stress relaxations

As shown in figure 2, the nonlinear relaxation is accelerated when compared with the linear relaxation. To analyse the effect of disorder on the nonlinear response, we first define a macroscopic acceleration function F which we will compare with the local function f that is applied at the scale of heterogeneities.

Definition of the macroscopic acceleration function

We use the macroscopic linear and nonlinear response of a system to define a macroscopic acceleration function F such that, for a single Zener system, it is equal to the local stress acceleration function f. For a step strain, the relaxation of the modulus in the linear regime is given in 1D by (3) and in the nonlinear regime by: (4)

In equations 3 and 4, and are the relaxation moduli (i.e., the ratio of stress over strain), for linear and nonlinear behaviours, respectively, and ε 0 is the deformation. Further, the equations driving the modulus relaxation are written in 1D. They can also be written in 3D as

shown in Annex 1. The 1D and 3D relaxation equations result in the same equations regarding the direction of the deformation (uniaxial elongation in this study).

From equations 3 and 4, we deduce that (5) where the two time derivatives of the L.H.S. of equation 3 are considered not at the same time but for the same values of E L z and E NL z that are equal to E.

We extend equation 5 to the macroscopic system, and thus, define the macroscopic acceleration function F as the ratio of the derivatives of linear (E L ) and nonlinear relaxations (E NL ) measured at the same state of relaxation:

with . The function F depends on the relaxation state which is characterised by the value of the modulus and strain that has been applied.

According to the definition we choose for F, for a single Zener system, the macroscopic function F depends only on the stress undergone by the Zener system that is equal to E NL  0.

In the next section, we show that because of disorder, the macroscopic acceleration function F deviates from the local function f. Further, we analyse the shape of the macroscopic acceleration function F resulting from measurements performed on our PMMA samples.

Shape of the macroscopic acceleration function F

Experimental results

Applying equation 5, the experimental macroscopic acceleration function of our PMMA samples is determined from the linear master curve and nonlinear response measured at the same temperature T. Figure 6 presents the macroscopic acceleration functions obtained at each temperature for a step strain of 1% amplitude. We observed that the macroscopic function of our samples, F, depends on temperature. We represent the macroscopic function F by plotting -ln[F(E,ε 0 )] as a function of the macroscopic stress E 0 with log scales. Here, we obtained a linear curve with a slope of approximately 0.9 +/-0.1. The value of the slope does not significantly vary with temperature in the range probed by experiments.

Numerical predictions

We compute the macroscopic acceleration function F for a disordered system and compare it with the acceleration function f of a Zener system. Figure 7 presents the curves obtained by plotting -ln(F) and -ln(f) as a function of E/E 0 on a log scale for Y= 5 MPa with E 0 = E G + E R .

Here, F and f exhibit very different behaviours.

First, we consider the case of a single Zener system. Here, -ln(f)=(E 0 /Y) 2 results in a straight line with a slope of 2 on a log scale.

In the heterogeneous case, we identify a range of modulus values over which the quantity ln(F(E, 0 )) varies linearly with E/E 0 in log scale. In this modulus range, the slope is equal to 0.83. This means that owing to dynamical heterogeneities, the macroscopic and local acceleration functions are different.

However, the local and macroscopic acceleration would have been equal if the local acceleration function was assumed to depend on the mean stress rather than the local stress.

The difference between the macroscopic and local acceleration functions occurs because the stress field is disordered. However, at short times, immediately after the strain step where E/E 0 is close to 1, the stress field is homogeneous and f and F collapse. With increasing time, a disordered stress field occurs owing to the distribution of intrinsic times. This stress field results in heterogeneous acceleration. The macroscopic acceleration deviates from the local acceleration.

In practice, the range of modulus values over which the quantity ln(F(E, 0 )) varies linearly with E/E 0 in log scale corresponds to the experimental windows for which mechanical measurements are sufficiently accurate to experimentally determine the acceleration function.

Thus, we focus on this modulus range, where the form of the macroscopic acceleration function is given by . Figure 10 presents the experimental master curves similarly obtained for 1% and 1.5% deformation. Similar slopes were observed for the two deformation amplitudes, as was the case in our simulations. As shown in figure 10, the macroscopic acceleration functions measured for a reference temperature of 108 °C and 1% and 1.5% strain can be described using Y = 5.75 MPa. In the frame of our model, the vertical shift observed between the F functions measured at different temperatures can be attributed to the temperature dependence of the critical stress Y.

According to Long et al. [28], the parameter Y of the local acceleration function depends on the temperature and size of dynamical heterogeneities. According to the relation proposed by and considering G G = 427 MPa and K = 2 GPa, we found that the size of heterogeneity (T) decreases from 6.5 nm to 4.5 nm for a temperature increase of approximately 15 K in the temperature range probed by our experiments, as shown in the inset of figure 3.

Conclusion

In conclusion, by combining experiments and a numerical approach accounting for dynamical heterogeneities, we show that experimental relaxation measurements preformed in the weak nonlinear regime before yielding can be described in our model by applying the same local acceleration law for all the dynamical heterogeneities. The microscopic time distribution under strain are in agreement with experimental observations of Lee et al [19]. The macroscopic responses predicted by our model using the expression of Long et al. [28] for the local acceleration function are in good agreement with the mechanical data. These results lead us to estimate the characteristic length scale of dynamical heterogeneities, which decreases with temperature and is close to 5 nm. Finally, we show that -under nonlinear solicitationstress-induced accelerations of the mechanical relaxation are strikingly different at the local and macroscopic scales.

Annex 1:

In 3D, the differential equation of a Zener element for a linear case is where and are the time dependent deviatoric stress and strain tensors, respectively, and are the fourth-order glassy and rubber isotropic elasticity tensors, respectively, and τ 0 is the relaxation time.

In the case of a relaxation test, this leads to the following equation:

where is the fourth-order isotropic modulus tensor such that

In the nonlinear case, this is where is fourth-order isotropic nonlinear modulus tensor and is an acceleration function that depends on an equivalent stress.

Regarding the term corresponding to uniaxial elongation direction, this leads to the 1D equations:

for linear condition and for nonlinear condition,

Figure 1 :

 1 Figure 1: Linear relaxation master curve (empty markers) and nonlinear relaxation measured at 108 °C (filled circles). The black line corresponds to simulation predicted by our model for linear condition by applying the following values: K = 2 GPa, E R = 1.5 MPa, E G = 1.200 GPa, and ln( 0 ) = 4.1 s = 4.83

Figure 2

 2 Figure 2 presents the modulus relaxation measured in the nonlinear regime at different temperatures as a function of the variable , where are the time-temperature shift factors determined in the linear regime. As shown in figure 2, data measured in the nonlinear regime at different temperatures do not collapse into a master curve.

Figure 2 :

 2 Figure 2: Nonlinear relaxation measured at different temperatures by applying a deformation of 1% are plotted as a function of the shifted time a T/Tref t where a T/Tref are the time -temperature equivalence shift factors measured in the linear regime at the reference temperature of 108°C. The master curve measured in the linear regime is added. Experimental data are compared to numerical simulation predicted by our model in the linear and nonlinear regimes by applying: E R = 1.5 MPa, E G = 1.200 GPa, s = 4.83, ln( 0 ) = 4.1. In the nonlinear regime, a good description of experiments is obtained adjusting the value of the critical stress Y involved in the local acceleration function f (see equation 2). For T = 96 °C, Y= 4 MPa, for T = 108 °C, Y = 5.75 MPa, and for T = 112 °C, Y = 7 MPa.

Figure 2

 2 Figure 2 compares the experiments performed for 1% strain at different temperatures to the numerical curves predicted by our model in the nonlinear regime assuming a temperature dependence of Y. We adjust the value of Y to obtain the best description of the experimental result. We found values for Y varying from 4 to 7 MPa for temperatures ranging between 96 °C and 112 °C, as shown in figure 3.

Figure 3 :

 3 Figure 3: Parameter Y as a function of temperature inferred from the fitting by our model of the nonlinear responses measured at different temperatures applying 1% step strain on

Figure 4

 4 presents the distribution functions of the napierian logarithm of the relaxation times at different steps of the nonlinear relaxation, computed with Y = 6 MPa and = 0.01. The corresponding macroscopic stress relaxation is shown in figure 5-a. We can identify several regimes.

Figure 4 :

 4 Figure 4: distribution function of the napierian logarithm of local relaxation times at different steps of the macroscopic stress relaxation. Computation were performed applying Y= 6 MPa,   =0.01. The width and the center time of the intrinsic

Figure 5 :

 5 Figure 5: a) macroscopic stress relaxation as a function of macroscopic time. Computation were performed for   =0.01 and for varying Y values. The inverse of the width of the corresponding local relaxation time distributions 1/s is shown as a function of macroscopic time. b) the geometric mean time of the corresponding local relaxation time distributions divided by the geometric mean time value of the intrinsic time distribution  0 is plotted as a function of the macroscopic stress 

First, we consider

  a single Zener system with a Maxwell branch representing the glassy contribution to the stress with an elastic modulus E G z and relaxation time  0 that is in parallel with an elastic spring of modulus E R z , which represents the stress contribution owing to the chain entropy. In our approach, we model the nonlinear behaviour of the polymer by multiplying the relaxation time  0 with an acceleration function f of the stress σ. According to Long et al., we assume an acceleration function of the form exp[-(σ/Y) 2 ].

Figure 6 :

 6 Figure 6: The quantity -ln[F(E, 0 )] deduced from experiments performed at different temperatures applying a deformation  0 of 1% vs. macroscopic stress E 0 in log-log scale

Figure 7 :

 7 Figure 7: The quantity -ln[F(E,   ] as a function of the modulus normalised by E 0 , the modulus at t = 0 in log-log scale. Data in filled blue squares are computed assuming a 3D heterogeneous system applying a local acceleration function f given by equation 2 with Y=5MPa

Figure 8 :

 8 Figure 8: (a) -ln(F(E,ε 0 )) as a function of E for different strains in log scales. (b) vertical shift factors 1/g(ε 0 ) are applied to the data. The inset presents the variation of g(ε 0 ) as a function of the strain amplitude in log scales

Figure 9 : 5 . 3 .

 953 Figure 9: a) The quantity -ln(F(E,ε 0 )) predicted by our model for varying values of the critical stress Y are plotted as a function of the modulus value E divided by the modulus value in the glassy state E 0 in log-log scale. Computations were performed by applying: E R = 1.5 MPa, E G = 1.2 GPa, s = 4.83, ln( 0 ) = 4.1 and  = 1%. Y value was varied from 3 MPa to 12 MPa. b) numerical curves collapse if a shift factor b is applied to the quantity -ln(F(E,ε 0 )). The inset presents the variation of b as a function of Y in log-log scale. The line corresponds of a power law function with an exponent equal to 1.87.

Figure 10 :

 10 Figure 10: The quantity -ln[F(E, 0 )] deduced from experiments vs. macroscopic stress E 0 in log-log scale. The master curves are obtained at T ref = 108 °C for strains of 1% and 1.5% applying a multiplicative shift factor  T/Tref . to -ln[F(E, 0 )]. Continuous lines are simulation results computed using Y = 5.75 MPa and f() = exp[-(σ/Y) 2 ] with K = 2 GPA, E R = 1.5 MPa, E G = 1.200 GPa, s = 4.83, and ln( 0 ) = 4.1.

  local acceleration function f(σ)=exp[-(σ/Y) n ] with values of n different from 2, the curves -ln[F(E,ε 0 )] as a function of E with log scales, exhibit a scaling regime in a limited range of relaxation modulus, i.e., . The exponent m depends on the power n of the local acceleration function. It further depends on the width s of the relaxation time distribution. In the limit case s = 0 which corresponds to a homogeneous case, f = F and thus, m = n. The variations of m with n in the local acceleration function are plotted in figure 11 for various widths of the relaxation time distribution s. For the width of relaxation time distribution s > 2.3, the slope m does not significantly depend on s.

Figure 11 :

 11 Figure 11: (a) The modulus relaxation E(t, 0 ) as a function of time computed assuming s = 4.83, ln( 0 ) = 4.1, E G = 1.2 GPa, E R = 1.5 MPa, and a local stress dependence of the local acceleration function that follows f() = 1 and f( eq ) = exp[-(σ eq /Y) n ] with Y = 5 MPa for linear and nonlinear conditions, respectively. Computation were performed for n = 2 and n = 1. (b) The quantity -ln[F(E, 0 ] as a function of the modulus normalised by E 0 in log-log scale. The function F computed assuming a local stress field disorder is compared to the local acceleration function f for different values of n and  0 = 0.01. (c) The variation of the

  this study, nonlinear effects are modelled in the framework of the theory proposed by Long et al. For nonlinear simulations, the intrinsic relaxation time of domains is multiplied by the stress acceleration function predicted by Long et al. at the scale of heterogeneity and is equal

to . In the Long et al. theory, the stress is scalar. In this study, we extended the relation proposed by Long et al. to 3D tensor stress. According to the expression of , intrinsic times are significantly accelerated for  > Y. The 3D criteria for polymer